diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzaabf" "b/data_all_eng_slimpj/shuffled/split2/finalzzaabf" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzaabf" @@ -0,0 +1,5 @@ +{"text":"\\section*{Dedication}\n\nThis thesis is dedicated to my parents, Sonia and Oscar, and my brother John Paul.\nTheir love and support are everything to me, and this milestone is as much a celebration\nof accomplishments as it is a testament to the strength of family.\n\\clearpage\n\\acknowledgements\n\nThese past five years have been an incredible privilege for, if nothing else,\nthe extraordinary individuals with whom I met and worked.\nDay in and day out, my group members challenge, encourage, and support\nme, and I am thankful for their companionship throughout this journey:\nDavid Hicks,\nDemet Usanmaz,\nEric Gossett,\nPinku Nath,\nMarco Esters,\nRico Friedrich,\nPranab Sarker,\nDenise Ford,\nPauline Colinet,\nCamilo Calderon,\nCarlo De Santo,\nGeena Gomez,\nHarvey Shi,\nAllison Stelling,\nYoav Lederer,\nLuis Agapito,\nand\nManuela Damian.\n\nI am fortunate to have so many mentors and guides who, in various\ncapacities, have opened my eyes to new fields and fueled my scientific curiosity:\nCormac Toher,\nFrisco Rose,\nOlexandr Isayev,\nKesong Yang,\nStefano Sanvito,\nAmir Natan,\nMichael Mehl,\nPatrick McGuire,\nJes\\'{u}s Carrete,\nNatalio Mingo,\nMatthias Scheffler,\nClaudia Draxl,\nValentin Stanev,\nIchiro Takeuchi,\nand\nOhad Levy.\n\nI am incredibly appreciative of my professors, teachers,\nand advisors from Cornell University and Bloomfield High School\nwho helped me construct the vision in which this milestone is achievable:\nSara Xayarath Hern\\'{a}ndez,\nJoel Brock,\nErnest Fontes,\nKenneth Card,\nDaniel Di Domenico,\nMarian Connolly,\nBrian Miller,\nLou Cappello,\nand\nManuela Gonnella.\n\nAbove all, I am especially grateful to my PhD Advisor, Stefano Curtarolo,\nfor the opportunity to discover my passion in this field.\nI have been blessed with many in my life who believe in me, but few as\nemphatically as Stefano.\n``\\textit{Non ducor, duco}''.\n\nFinally, I acknowledge support from the National Science Foundation Graduate Research Fellowship under Grant No. DGF1106401.\n\n\\clearpage\n\\tableofcontents\n\\listoffigures\n\\listoftables\n\\chapters\n\\chapter{Introduction}\n\n\\begin{center}\n``\\textit{Nihil est in intellectu quod non sit prius in sensu}''\\footnote{``Nothing is in the intellect that was not first in the senses.''} \\\\\n--- Thomas Aquinas's \\textit{Quaestiones Disputatae de Veritate}, \\\\ quaestio 2, articulus 3, argumentum 19.\n\\end{center}\n\nMaterials discovery drives technological innovation, spanning\nthe stones and simple metals that forged the first tools to the semiconductors that power today's computers.\nHistorically, these advancements follow from intuition and\nserendipity~\\cite{curtarolo:art94,curtarolo:art124,MGI,Norman_RPP_2016,Eberhart_NMat_2004}.\nAs such, major breakthroughs --- which are few and far between --- are seldom predictable.\nFortunately, ``\\textit{big data}'' is powering a paradigm shift:\nmaterials informatics.\nIntegration of data-centric approaches in an otherwise \\textit{a posteriori} field promises to\nbridge the widening gap between observation and understanding,\naccelerating the pace of technology.\nMore importantly, data-driven modeling --- offering predictions grounded in empirical evidence ---\nmay finally break with tradition, enabling control over discovery and\nachieving rational materials design.\n\nWielding data to accelerate innovation is not a new idea,\nsince it constitutes standard practice in biology~\\cite{Reichhardt_Nature_1999,Luscombe_MIM_2001}\nand chemistry~\\cite{Brown_Chemoinformatics_1998,Gasteiger_Chemoinformatics_2003}.\nYet its adoption in materials science has been slow, as it was first introduced in the early 2000's~\\cite{curtarolo:art13}.\nThis delay can be attributed to the ongoing development of standard \\nobreak\\mbox{\\it ab-initio}\\\npackages~\\cite{kresse_vasp,VASP4_2,vasp_cms1996,vasp_prb1996,quantum_espresso_2009,gonze:abinit,Blum_CPC2009_AIM},\nparticularly to better address calculation of the exchange correlation energy~\\cite{PBE,Perdew_SCAN_PRL_2015}.\nNevertheless, the impact of density functional theory ({\\small DFT}) on computational materials science cannot be understated~\\cite{nmatHT},\noffering a reasonable compromise between cost and accuracy~\\cite{Haas_PRB_2009}.\nThe success of these implementations has stimulated the rapid development\nof automated frameworks and corresponding data repositories,\nincluding {\\small AFLOW}\\ (\\underline{A}utomatic \\underline{Flow} for Materials Discovery)~\\cite{aflowPAPER,curtarolo:art110,curtarolo:art85,curtarolo:art63,aflowBZ,curtarolo:art57,curtarolo:art53,curtarolo:art49,monsterPGM,aflowANRL,aflowPI},\nNovel Materials Discovery Laboratory~\\cite{nomad},\nMaterials Project~\\cite{APL_Mater_Jain2013},\nOpen Quantum Materials Database~\\cite{Saal_JOM_2013},\nComputational Materials Repository~\\cite{cmr_repository},\nand Automated Interactive Infrastructure and Database for Computational Science~\\cite{Pizzi_AiiDA_2016}.\nThese house an abundance of materials data.\nFor instance, the {\\small AFLOW}\\ framework, described in Section~\\ref{sec:aflow_chp},\nhas characterized more than 2 million compounds, each by about\n100 different properties accessible via the {\\sf \\AFLOW.org}\\ online database~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}.\nInvestigations employing this data have not only led to advancements in modeling\nelectronics~\\cite{nmatTI,curtarolo:art94,curtarolo:art124,ceder:nature_1998},\nthermoelectrics~\\cite{curtarolo:art96,curtarolo:art114,curtarolo:art115,curtarolo:art119,curtarolo:art120,curtarolo:art125,curtarolo:art129},\nsuperalloys~\\cite{curtarolo:art113},\nand metallic glasses~\\cite{curtarolo:art112},\nbut also to the synthesis of two new magnets --- the first\ndiscovered by computational approaches~\\cite{curtarolo:art109}.\n\nFurther advancements are contingent on continued development and expansion of these materials repositories.\nNew entries are generated both by\n\\textbf{i.} calculating the properties of previously observed compounds\nfrom sources such as the Inorganic Crystal Structure Database~\\cite{ICSD} ({\\small ICSD}),\nand\n\\textbf{ii.} decorating structure prototypes~\\cite{aflowANRL,curtarolo:art130}.\nConsidering all possible crystals of different arrangements and decorations~\\cite{curtarolo:art124,Walsh_NChem_2015},\nthe analysis of existing structures --- a small subset --- is a critical first-step in determining fruitful directions for exploration.\nFor example, Section~\\ref{sec:art130} presents a general overview of the structure types appearing in an important\nclass of the solid compounds, \\nobreak\\mbox{\\it i.e.}, binary and ternary compounds of the 6A column oxides, sulfides, and selenides.\nIt contains an in-depth statistical analysis of these compounds, including the prevalence of various structure types,\ntheir symmetry properties, compositions, stoichiometries and unit cell sizes.\nResults reveal that these compound families include preferred stoichiometries and structure\ntypes that may reflect both their specific chemistry and research bias in the\navailable empirical data.\nDetection of non-overlapping gaps and missing stoichiometries in such\npopulations will guide subsequent studies: structures are avoided in the event that they are chemically\nunfavorable, or targeted to complement existing measurements.\n\nWith materials of interest identified, accurate computation of their properties demands\na set of reliable calculation parameters\/thresholds~\\cite{curtarolo:art104}.\nThese inputs need to be understood by researchers, and should be reported by the originators\nto ensure reproducibility and enable collaborative database expansion.\nAs described in Section~\\ref{sec:art104}, the {\\small AFLOW}\\ Standard defines these parameters\nfor high-throughput electronic structure calculations of crystals --- the basis for all {\\small AFLOW}\\ characterizations.\nStandard values are established for reciprocal space grid density,\nplane wave basis set kinetic energy cut-off, exchange-correlation\nfunctionals, pseudopotentials, {\\small DFT}$+U$ parameters, and convergence criteria.\n\nExploration of more complex properties~\\cite{curtarolo:art96,curtarolo:art115} and materials~\\cite{curtarolo:art110,curtarolo:art112}\ntypically warrants advanced (and expensive) characterization techniques~\\cite{Hedin_GW_1965,GW,ScUJ,Malashevich_GW_TiO_PRB2014,Patrick_GW_TiO2_JPCM2012}.\nFortunately, state-of-the-art workflows~\\cite{curtarolo:art96,curtarolo:art110,curtarolo:art115} and\ncareful descriptor development~\\cite{curtarolo:art112} have\nenabled experimentally-validated modeling within a {\\small DFT}\\ framework.\nFor instance, a thorough description of thermomechanical properties\nrequires difficult and time-consuming experiments.\nThis limits the availability of data:\none of the main obstacles for\nthe development of effective accelerated materials design strategies.\nSection~\\ref{sec:art115} introduces an automated, integrated workflow with robust error-correction\nwithin the {\\small AFLOW}\\ framework {that combines} the newly devised\n``Automatic Elasticity Library'' with the previously implemented {\\small GIBBS}\\ method~\\cite{curtarolo:art96}.\nThe former extracts the mechanical properties from several automatic self-consistent stress-strain calculations,\nwhile the latter employs those mechanical properties to evaluate the thermodynamics within the Debye model.\n{The} thermomechanical {workflow} is benchmarked against a set of\n74 experimentally characterized systems to pinpoint a\nrobust computational methodology for the evaluation of bulk and shear moduli,\nPoisson ratios, Debye temperatures, Gr{\\\"u}neisen parameters, and thermal conductivities of a wide variety of materials.\nThe effect of different choices of equations of state {and exchange-correlation functionals}\nis examined and the optimum combination of properties for the\nLeibfried-Schl{\\\"o}mann prediction of thermal conductivity is identified,\nleading to improved agreement with experimental results compared to the {\\small GIBBS}-only approach.\nThe {\\small AEL}-{\\small AGL}\\ framework has been applied to the {\\sf \\AFLOW.org}\\ data repositories to compute the thermomechanical properties\nof over 5,000 unique materials.\n\nSimilar to thermomechanical characterizations,\ndescriptions of thermodynamic stability and\nstructural\/chemical disorder are also resolved through an analysis of aggregate sets of \\nobreak\\mbox{\\it ab-initio}\\ calculations.\n\\textit{A priori} prediction of phase stability\nrequires\nknowledge of all energetically-competing structures at formation conditions.\nLarge materials repositories\noffer a path to prediction through the construction of\n\\nobreak\\mbox{\\it ab-initio}\\ phase diagrams, \\nobreak\\mbox{\\it i.e.}, the convex hull\nat a given temperature\/pressure.\nHowever, limited access to relevant data and software infrastructure has\nrendered thermodynamic characterizations largely peripheral,\ndespite their continued success in dictating synthesizability.\nIn Section~\\ref{sec:art146}, a new module is presented for autonomous thermodynamic stability analysis\nimplemented within {\\small AFLOW}.\nPowered by the {\\small AFLUX}\\ Search-{\\small API}, {\\small \\AFLOWHULLtitle}\\ leverages data of more than\n2 million compounds characterized in the {\\sf \\AFLOW.org}\\ repository,\nand can be employed locally from any {\\small UNIX}-like computer.\nThis module integrates a range of functionality:\nthe identification of stable phases and equivalent structures, phase coexistence,\nmeasures for robust stability, and determination of decomposition reactions.\nAs a proof-of-concept, thermodynamic characterizations have been performed\nfor more than 1,300 binary and ternary systems, enabling the identification of several\ncandidate phases for synthesis based on their relative stability criterion --- including\n17\\ promising $C15_{b}$-type structures and two half-Heuslers.\nIn addition to a full report included herein, an interactive online web application\nhas been developed, showcasing the results of the analysis, and is\nlocated at {\\sf aflow.org\/aflow-chull}.\n\nThe convex hull construction has fueled the generation of\nnovel descriptors for glass forming ability~\\cite{curtarolo:art112} and,\nmore generally, modeling structurally disordered systems.\nStatistical methods are employed to address chemically disordered\nstructures, where system-wide properties are resolved through an analysis of\nrepresentative ordered supercells~\\cite{curtarolo:art109}.\nIncorporating the effects of disorder is a necessary, albeit difficult, step in materials modeling.\nNot only is disorder intrinsic to all materials,\nbut it also offers a route to enhanced and even otherwise inaccessible functionality,\nas demonstrated by its ubiquity in technological applications.\nProminent examples include glasses~\\cite{kelton1991crystal,kelton2010nucleation,kelton1998new},\nsuperalloys~\\cite{Donachie_ASM_2002},\nfuel cells~\\cite{Xie_ACatB_2015},\nhigh-temperature superconductors~\\cite{Bednorz_ZPBCM_1986,Maeno_Nature_1994},\nand low thermal conductivity thermoelectrics~\\cite{Winter_JACerS_2007}.\n\nPredicting material properties of chemically disordered systems remains a\nformidable challenge in rational materials design.\nA proper analysis of such systems by means of a supercell approach requires\nconsideration of all possible superstructures, which can be a time-consuming process.\nOn the contrary, the use of quasirandom-approximants, while\ncomputational effective, implicitly bias the analysis toward disordered states with the lowest site correlations.\nIn Section~\\ref{sec:art110}, a novel framework is proposed\nto investigate stoichiometrically driven trends of disordered systems\n(\\nobreak\\mbox{\\it i.e.}, having partial occupation and\/or disorder in the atomic sites).\nAt the heart of the approach is the identification and analysis of unique supercells of a\nvirtually equivalent stoichiometry to the disordered material.\nBoltzmann statistics are employed to resolve system-wide properties at a high-throughput level.\nTo maximize efficiency and accessibility, this method has been integrated within {\\small AFLOW}.\nAs proof of concept, the approach is applied to three systems of interest,\na zinc chalcogenide (ZnS$_{1-x}$Se$_x$),\na wide-gap oxide semiconductor (Mg$_{x}$Zn$_{1-x}$O),\nand an iron alloy (Fe$_{1-x}$Cu$_{x}$)\nat various stoichiometries.\nThese systems exhibit properties that are highly tunable as a function of composition,\ncharacterized by optical bowing and linear ferromagnetic behavior.\nNot only are these qualities\npredicted, but additional insight into underlying physical mechanisms is revealed.\n\nThe aforementioned frameworks --- offering characterizations of thermomechanical\nand thermodynamic properties, as well as resolving features of disordered systems ---\nhave both benefited from and stimulated the development of the {\\sf \\AFLOW.org}\\ repository.\nThe combination of plentiful and diverse materials data~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}\nand its programmatic accessibility~\\cite{aflowAPI,aflux} also\njustify the application of data-mining techniques.\nThese methods can\nresolve subtle trends and correlations among materials and their\nproperties~\\cite{curtarolo:art94,Ghiringhelli_PRL_2015,curtarolo:art124,curtarolo:art129,curtarolo:art135},\nas well as motivate the formulation of novel property descriptors~\\cite{curtarolo:art112,curtarolo:art139}.\nIn fact, materials data generated by automated frameworks are\nconducive to such approaches,\nwhere strict standardizations of calculation parameters~\\cite{curtarolo:art104} not only ensure\nreproducibility, but also a minimum accuracy threshold.\nErrors from approximations or choice in parameters can therefore be treated as systematic,\nwhich are easily identified and rectified by \\underline{m}achine \\underline{l}earning ({\\small ML}) algorithms.\nModels have been generated for predicting electronic~\\cite{curtarolo:art124},\nthermomechanical~\\cite{curtarolo:art124,deJong_SR_2016} and vibrational~\\cite{curtarolo:art120,curtarolo:art129} properties,\nas well as the thermodynamic stability of both ordered~\\cite{Ghiringhelli_PRL_2015}\nand disordered~\\cite{Ward_ML_GFA_NPGCompMat_2016} phases.\nIn Section~\\ref{sec:art124}, data from the {\\small AFLOW}\\ repository for \\nobreak\\mbox{\\it ab-initio}\\ calculations\nis combined with Quantitative Materials Structure-Property Relationship models to predict important properties:\nmetal\/insulator classification, band gap energy, bulk\/shear moduli, Debye temperature, and heat capacities.\nThe prediction's accuracy compares well with the quality of the training data for virtually any\nstoichiometric inorganic crystalline material, reciprocating the available thermomechanical experimental data.\nThe universality of the approach is attributed to the construction of the descriptors: Property-Labeled Materials Fragments.\nThe representations require only minimal structural input allowing straightforward implementations of simple heuristic design rules.\n\n{\\small ML}\\ approaches are expected to become indispensable in two specific scenarios, prediction\nof complex properties and screening of large sets of materials.\nFor example, feature-importance analyses have informed on the interactions that elicit high-temperature\nsuperconductivity~\\cite{curtarolo:art94,curtarolo:art137}, an\nelusive phenomenon in which the driving mechanisms are still contested.\nSuperconductivity has been the focus of enormous research efforts since its discovery more than a century ago.\nYet, some features\nremain poorly understood; mainly the connection\nbetween superconductivity and chemical\/structural properties of materials.\nTo bridge the gap, several machine learning schemes are developed in Sections~\\ref{sec:art094} and \\ref{sec:art137}\nto model the critical temperatures $\\left(T_{\\mathrm{c}}\\right)$ of\nknown superconductors available via the SuperCon database.\nAs expected, these analyses suggest distinct mechanisms are responsible for driving superconductivity in\ndifferent classes of materials.\nHowever, they also hint at very complex physical interactions.\nFortunately, {\\small ML}\\ algorithms like random forests~\\cite{randomforests} are\ncapable of extracting very complicated functional relationships.\nIn the case of predicting $T_{\\mathrm{c}}$, these ``black-box'' models are\nquite valuable as\nfew alternative practical modeling schemes exist.\n\nIn Section~\\ref{sec:art094}, novel analytical approaches are introduced based on structural and electronic materials fingerprints\nand applied to predict the $T_{\\mathrm{c}}$\nof known superconductors.\nThe framework is employed to \\textbf{i.} query large databases of materials using similarity concepts,\n\\textbf{ii.} map the connectivity of materials space (\\nobreak\\mbox{\\it i.e.}, as materials cartograms)\nfor rapidly identifying regions with unique organizations\/properties,\nand \\textbf{iii.} develop predictive Quantitative Materials Structure-Property Relationship models for guiding materials design.\nThe materials fingerprinting and cartography approaches are\neffective computational tools to analyze,\nvisualize, model, and design new materials.\n\nSuperconductors are revisited in a much more in-depth study presented in Section~\\ref{sec:art137},\nleveraging the full set of the $12,000+$ materials in the SuperCon database.\nMaterials are first divided into two classes based on their $T_{\\mathrm{c}}$ values,\nabove and below $10$~K,\nand a classification model predicting this label is trained.\nThe model uses coarse-grained features based only on the chemical compositions.\nIt shows strong predictive power, with out-of-sample accuracy of about $92\\%$.\nSeparate regression models\nare developed to predict the values of $T_{\\mathrm{c}}$ for cuprate, iron-based, and low-$T_{c}$ compounds.\nThese models also demonstrate good performance,\nwith learned predictors offering\ninsights into the mechanisms behind superconductivity in different families of materials.\nTo improve the accuracy and interpretability of these models,\nnew features are incorporated using materials data\nfrom the {\\sf \\AFLOW.org}\\ repository.\nTo find potential new superconductors, the classification and regression models are combined into a single integrated pipeline\nand employed to search the entire Inorganic Crystallographic Structure Database ({\\small ICSD}).\nMore than 30 non-cuprate and non-iron-based oxides are selected as candidate materials.\n\nBeyond superconductors, {\\small ML}\\ models are created to predict properties of thermoelectrics (Section~\\ref{sec:art120})\nand permanent magnets (Section~\\ref{sec:art109}).\nThermoelectric materials generate an electric voltage when subjected to a temperature gradient, or\nconversely create\na temperature gradient when a voltage is applied~\\cite{snyder_complex_2008, nolas_thermoelectrics:_2001}.\nWith no moving parts and their resulting scalability, thermoelectrics\nhave potential applications in power generation for spacecraft,\nenergy recovery from waste heat in automotive and industrial facilities~\\cite{bell_cooling_2008, disalvo99},\nand spot cooling for nanoelectronics using the Peltier cooling effect~\\cite{bell_cooling_2008, disalvo99}.\nHowever, most of the available thermoelectric materials have low efficiency, only converting a few percent\nof the available thermal energy into electricity.\nTherefore, a major goal of thermoelectrics research is to develop new materials that have\nhigher thermoelectric efficiency as determined by a figure of merit~\\cite{snyder_complex_2008, nolas_thermoelectrics:_2001}.\nThe metric is dependent on quantities such as the Seebeck coefficient and electrical\/thermal conductivities.\nOne promising path to optimizing the figure of method is to\nminimize the lattice thermal conductivity.\n\nIn Section~\\ref{sec:art120}, the thermal conductivity $\\left(\\kappa\\right)$ is analyzed for\nsemiconducting oxides and fluorides with cubic perovskite structures.\nUsing finite-temperature phonon calculations and {\\small ML}\\ methods,\nthe mechanical stability of about 400 structures is resolved at 0~K, 300~K, and 1000~K.\nOf these, 92 compounds are determined to be mechanically stable at high temperatures\n--- including 36 not mentioned in the literature so far --- for which $\\kappa$ is calculated.\nSeveral trends are revealed, including\n\\textbf{i.} $\\kappa$ generally being smaller in fluorides than in oxides,\nlargely due to the lower ionic charge,\nand \\textbf{ii.} $\\kappa$ decreasing more\nslowly than the usual $T^{-1}$ behavior for most cubic perovskites.\nAnalyses expose the simple structural descriptors that correlate with $|\\kappa|$.\nThis set is also screened for materials exhibiting negative thermal expansion.\nThe study highlights a general strategy coupling force constants calculations with an iterative {\\small ML}\\ scheme\nto accelerate the discovery of mechanically stable compounds at high temperatures.\n\nThe role of {\\small ML}\\ models in predicting magnetic properties is of particular significance,\nas their \\textit{a priori} predictions were validated with the discovery of two new magnets.\nMagnetic materials underpin modern technologies, ranging from data storage to energy conversion and contactless sensing.\nHowever, the development of a new high-performance magnet is a long and often unpredictable process, and only\nabout two dozen feature in mainstream applications.\nIn Section~\\ref{sec:art109}, a systematic pathway is described to the discovery of novel\nmagnetic materials.\nBased on an extensive electronic structure library of Heusler alloys containing 236,115 compounds,\nalloys displaying magnetic order are selected, and it is determined\nwhether they can be fabricated at thermodynamic equilibrium.\nSpecifically, a full stability analysis is carried out for intermetallic Heusler alloys made only of transition metals.\nAmong the possible 36,540 candidates, 248 are found to be thermodynamically stable but only 20 are magnetic.\nThe magnetic ordering temperature, $T_\\mathrm{C}$, has then been estimated by a regression\ncalibrated on the experimental $T_\\mathrm{C}$ of about 60 known compounds.\nAs a final validation, the synthesis is attempted for a few of the predicted compounds,\nand two new magnets are produced.\nOne, Co$_2$MnTi, displays a remarkably high $T_\\mathrm{C}$ in perfect agreement with\nthe predictions, while the other, Mn$_2$PtPd, is an antiferromagnet.\nThis work paves the way for large-scale design of novel magnetic materials at unprecedented speed.\n\nOverall, data-driven approaches have extended materials modeling capabilities within a {\\small DFT}\\ framework.\nDescriptors for thermodynamic stability and formation\/features of disordered materials\nare accessible through analyses of ensembles of ordered structures,\nstimulating the development of large materials repositories.\nTo match the growth of these databases, insight-extraction must also be automated.\n{\\small ML}\\ methods are employed to reveal structure-property relationships and expose similarities among materials.\nUltimately, the power in {\\small ML}\\ lies in the speed of its predictions, which out-paces\n{\\small DFT}\\ calculations by orders of magnitude~\\cite{Isayev_ChemSci_2017}.\nEfforts to explore the full materials space through brute-force {\\small DFT}\\\ncalculations are impractical;\nstudies conservatively enumerate the size\nof possible hypothetical structures to be as large as 10$^{100}$~\\cite{Walsh_NChem_2015}.\nGiven that the number of currently characterized materials pales in comparison\nto the true potential diversity, methods --- like those presented here ---\nto filter\/screen the most interesting candidate materials\nwill play an integral role in future materials discovery workflows.\n\\clearpage\n\\chapter{The Automatic Flow Framework for Materials Discovery}\nMaterials informatics requires large repositories of materials data to identify trends in and correlations between materials properties,\nas well as for training machine learning models.\nSuch patterns lead to the formulation of descriptors that guide rational materials design.\nGenerating large databases of computational materials properties requires robust, integrated, automated frameworks~\\cite{nmatHT}.\nBuilt-in error correction and standardized parameter sets enable the production and analysis of data without direct intervention from human researchers.\nCurrent examples of such frameworks include\n{\\small AFLOW}\\ (\\underline{A}utomatic \\underline{{\\small FLOW}})~\\cite{aflowPAPER, aflowBZ, aflowlibPAPER, aflowAPI, curtarolo:art104, aflowlib.org, aflow_fleet_chapter, aflowPI, paoflow},\nMaterials Project~\\cite{materialsproject.org, APL_Mater_Jain2013, CMS_Ong2012b, Mathew_Atomate_CMS_2017},\n{\\small OQMD}\\ (\\underline{O}pen \\underline{Q}uantum \\underline{M}aterials \\underline{D}atabase)~\\cite{Saal_JOM_2013, Kirklin_AdEM_2013, Kirklin_ActaMat_2016},\nthe Computational Materials Repository~\\cite{cmr_repository} and its associated scripting interface {\\small ASE} (\\underline{A}tomic \\underline{S}imulation \\underline{E}nvironment)~\\cite{ase},\n{\\small AiiDA}\\ (\\underline{A}utomated \\underline{I}nteractive \\underline{I}nfrastructure and \\underline{Da}tabase for Computational Science)~\\cite{aiida.net, Pizzi_AiiDA_2016, Mounet_AiiDA2D_NNano_2018},\nand the Open Materials Database at \\verb|httk.openmaterialsdb.se| with its associated \\underline{H}igh-\\underline{T}hroughput \\underline{T}ool\\underline{k}it ({\\small HTTK}).\nOther computational materials science resources include the aggregated repository maintained by the \\underline{No}vel \\underline{Ma}terials \\underline{D}iscovery ({NOMAD}) Laboratory~\\cite{nomad},\nthe Materials Mine database available at \\verb|www.materials-mine.com|,\nand the \\underline{T}heoretical \\underline{C}rystallography \\underline{O}pen \\underline{D}atabase ({\\small TCOD})~\\cite{Merkys_TCOD_2017}.\nFor this data to be consumable by automated machine learning algorithms,\nit must be organized in programmatically accessible repositories~\\cite{aflowlibPAPER, aflowAPI, aflowlib.org, materialsproject.org, APL_Mater_Jain2013, Saal_JOM_2013, nomad}.\nThese frameworks also contain modules that combine and analyze data from various calculations to predict complex thermomechanical phenomena, such as lattice thermal conductivity and mechanical stability.\n\nComputational strategies have already had success in predicting materials for\napplications including photovoltaics~\\cite{YuZunger2012_PRL},\nwater-splitters~\\cite{CastelliJacobsen2012_EnEnvSci},\ncarbon capture and gas storage~\\cite{LinSmit2012_NMAT_carbon_capture, Alapati_JPCC_2012},\nnuclear detection and scintillators~\\cite{Derenzo:2011io, Ortiz09, aflowSCINT, curtarolo:art46},\ntopological insulators~\\cite{nmatTI, Lin_NatMat_HalfHeuslers_2010},\npiezoelectrics~\\cite{Armiento_PRB_2011, Vanderbilt_Piezoelectrics_PRL2012},\nthermoelectric materials~\\cite{curtarolo:art68, madsen2006, aflowKAPPA, curtarolo:art85},\ncatalysis~\\cite{Norskov09},\nand battery cathode materials~\\cite{Hautier-JMC2011, Hautier-ChemMater2011, Mueller-ChemMater2011}.\nMore recently, computational materials data has been combined with machine learning approaches\nto predict electronic and thermomechanical properties~\\cite{curtarolo:art124, deJong_SR_2016},\nand to identify superconducting materials~\\cite{curtarolo:art94}.\nDescriptors are also being constructed to describe the formation of disordered\nmaterials, and have recently been used to predict the glass forming\nability of binary alloy systems~\\cite{curtarolo:art112}.\nThese successes demonstrate that\naccelerated materials design can be achieved by combining structured data sets generated\nusing autonomous computational methods with intelligently formulated descriptors and machine learning.\n\n\\section{Automated computational materials \\texorpdfstring{design \\\\ frameworks}{design frameworks}}\n\\label{sec:aflow_chp}\n\nRapid generation of materials data relies on automated frameworks such as\n{\\small AFLOW}~\\cite{aflowPAPER, aflowBZ, aflowlibPAPER, aflowAPI, curtarolo:art104},\nMaterials Project's \\verb|pymatgen|~\\cite{CMS_Ong2012b} and \\verb|atomate|~\\cite{Mathew_Atomate_CMS_2017},\n{\\small OQMD}~\\cite{Saal_JOM_2013, Kirklin_AdEM_2013, Kirklin_ActaMat_2016},\n{\\small ASE}~\\cite{ase}, and {\\small AiiDA}~\\cite{Pizzi_AiiDA_2016}.\nThe general automated workflow is illustrated in Figure~\\ref{fig:aflow_chp:materials_design_workflow}.\nThese frameworks begin by creating the input files required by the electronic structure\ncodes that perform the quantum-mechanics level calculations, where the initial geometry is\ngenerated by decorating structural prototypes (Figure~\\ref{fig:aflow_chp:materials_design_workflow}(a, b)).\nThey execute and monitor these calculations, reading any error messages written to the\noutput files and diagnosing calculation failures.\nDepending on the nature of the errors, these frameworks are equipped with a catalog of prescribed solutions ---\nenabling them to adjust the appropriate parameters and restart the calculations (Figure~\\ref{fig:aflow_chp:materials_design_workflow}(c)).\nAt the end of a successful calculation, the frameworks parse the output files to extract the relevant materials data\nsuch as total energy, electronic band gap, and relaxed cell volume.\nFinally, the calculated properties are organized and formatted for entry into machine-accessible, searchable and sortable databases.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig001}\n\\mycaption[Computational materials data generation workflow.]\n{({\\bf a}) Crystallographic prototypes are extracted from databases such as the\n{\\small ICSD}\\ or the NRL crystal structure library, or generated by enumeration algorithms.\nThe illustrated examples are for the rocksalt, zincblende, wurtzite, Heusler, anti-Heusler and\nhalf-Heusler structures.\n({\\bf b}) New candidate materials are generated by decorating the atomic sites with different elements.\n({\\bf c}) Automated {\\small DFT}\\ calculations are used to optimize the geometric structure and calculate energetic, electronic,\nthermal, and elastic properties.\nCalculations are monitored to detect errors.\nThe input parameters are adjusted to compensate for the problem and the calculation is re-run.\nResults are formatted and added to an online data repository to facilitate programmatic access.\n({\\bf d}) Calculated data is used to plot the convex hull phase diagrams for each alloy system to identify stable compounds.}\n\\label{fig:aflow_chp:materials_design_workflow}\n\\end{figure}\n\nIn addition to running and managing the quantum-mechanics level calculations, the frameworks also\nmaintain a broad selection of post-processing libraries for extracting additional properties,\nsuch as calculating x-ray diffraction (XRD) spectra from relaxed atomic coordinates, and the\nformation enthalpies for the convex hull analysis to identify stable compounds (Figure~\\ref{fig:aflow_chp:materials_design_workflow}(d)).\nResults from calculations of distorted structures can be combined to calculate\nthermal and elastic properties~\\cite{aflowPAPER, curtarolo:art96, curtarolo:art100, curtarolo:art115},\nand results from different compositions and structural phases can be amalgamated to generate thermodynamic phase diagrams.\n\n\\subsection{Generating and using databases for materials discovery}\n\nA major aim of high-throughput computational materials science is to identify new, thermodynamically stable compounds.\nThis requires the generation of new materials structures, which have not been previously reported in the literature,\nto populate the databases. The accuracy of analyses involving sets of structures, such as that used to determine thermodynamic stability,\nis contingent on sufficient exploration of the full range of possibilities. Therefore, autonomous materials design frameworks\nsuch as {\\small AFLOW}\\ use crystallographic prototypes to generate new materials entries consistently and reproducibly.\n\nCrystallographic prototypes are the basic building blocks used to generate the wide range of materials entries involved in\ncomputational materials discovery.\nThese prototypes are based on \\textbf{i.} structures commonly observed in nature~\\cite{ICSD, navy_crystal_prototypes, aflowANRL},\nsuch as the rocksalt, zincblende, wurtzite or Heusler structures illustrated in Figure~\\ref{fig:aflow_chp:materials_design_workflow}(b),\nas well as \\textbf{ii.} hypothetical structures, such as those enumerated by the methods described in References~\\onlinecite{enum1, enum2}.\nThe {\\small AFLOW}\\ Library of Crystallographic Prototypes~\\cite{aflowANRL} is also available online at \\url{aflow.org\/CrystalDatabase\/}, where\nusers can choose from hundreds of crystal prototypes with adjustable parameters, and which can be decorated to generate new input\nstructures for materials science calculations.\n\nNew materials are then generated by decorating the various atomic sites in the crystallographic prototype with different elements.\nThese decorated prototypes serve as the structural input for \\nobreak\\mbox{\\it ab-initio}\\ calculations.\nA full relaxation of the geometries and energy determination follows, from which phase diagrams for stability analyses can be constructed.\nThe resulting materials data are then stored in an online data repository for future consideration.\n\nThe phase diagram of a given alloy system can be approximated by considering the low-temperature limit in\nwhich the behavior of the system is dictated by the ground state~\\cite{monster, monsterPGM}.\nIn compositional space, the lower-half convex hull defines the minimum energy surface and the\nground-state configurations of the system.\nAll non-ground-state stoichiometries are unstable, with the decomposition described by the\nhull facet directly below it.\nIn the case of a binary system, the facet is a tie-line as illustrated in Figure~\\ref{fig:aflow_chp:convex_hulls}(a).\nThe energy gained from this decomposition is geometrically represented by the (vertical-)distance of the\ncompound from the facet and quantifies the excitation energy involved in forming this compound.\nWhile the minimum energy surface changes at finite temperature (favoring disordered structures),\nthe $T=0$~K excitation energy serves as a reasonable descriptor for relative thermodynamic\nstability~\\cite{curtarolo:art113}.\nThis analysis generates valuable information such as ground-state structures,\nexcitation energies, and phase coexistence for storage in the\nonline data repository.\nThis stability data can be visualized and displayed by online modules,\nsuch as those developed by {\\small AFLOW}~\\cite{curtarolo:art113}, the Materials Project~\\cite{Ong_ChemMat_2008},\nand the {\\small OQMD}~\\cite{Akbarzadeh2007, Kirklin_AdEM_2013}.\nAn example visualization from {\\small AFLOW}\\ is shown in Figure~\\ref{fig:aflow_chp:convex_hulls}(b).\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.8\\linewidth]{fig002}\n\\mycaption[Convex hull phase diagrams for multicomponent alloys systems.]\n{({\\bf a}) Schematic illustrating construction of convex hull for a general\nbinary alloy system $A_{x}B_{1-x}$. Ground state structures are depicted as red points, with the minimum energy\nsurface outlined with blue lines. The minimum energy surface is formed by\nconnecting the lowest energy structures with tie lines which form a convex hull.\nUnstable structures are shown in green, with the decomposition reaction indicated\nby orange arrows, and the decomposition energy indicated in purple.\n({\\bf b}) Example ternary convex hulls as generated by {\\small AFLOW}.}\n\\label{fig:aflow_chp:convex_hulls}\n\\end{figure}\n\nConvex hull phase diagrams have been used to discover new thermodynamically\nstable compounds in a wide range of alloy systems, including hafnium~\\cite{curtarolo:art49, curtarolo:art51},\nrhodium~\\cite{curtarolo:art53}, rhenium~\\cite{curtarolo:art63}, ruthenium~\\cite{curtarolo:art67}, and technetium ~\\cite{curtarolo:art70}\nwith various transition metals, as well as the Co-Pt system~\\cite{curtarolo:art66}. Magnesium alloy systems such as the lightweight\nLi-Mg system~\\cite{curtarolo:art55} and 34 other Mg-based systems~\\cite{curtarolo:art54} have also been investigated.\nThis approach has also been used to calculate the solubility of elements in titanium alloys~\\cite{curtarolo:art47}, to study the effect of hydrogen\non phase separation in iron-vanadium~\\cite{curtarolo:art74}, and to find new superhard tungsten nitride compounds~\\cite{curtarolo:art90}.\nThe data has been employed to generate structure maps for hcp metals~\\cite{curtarolo:art57},\nas well as to search for new stable compounds with the Pt$_8$Ti phase~\\cite{curtarolo:art56},\nand with the $L1_1$ and $L1_3$ crystal structures~\\cite{curtarolo:art71}.\nNote that even if a structure does not lie on the ground state convex hull, this does not rule out its existence.\nIt may be synthesizable under specific temperature and pressure conditions, and then be metastable under ambient\nconditions.\n\n\\subsection{Standardized protocols for automated data generation}\n\nStandard calculation protocols and parameters sets~\\cite{curtarolo:art104} are essential to\nthe identification of trends and correlations among materials properties.\nThe workhorse method for calculating quantum-mechanically resolved materials properties\nis \\underline{d}ensity \\underline{f}unctional \\underline{t}heory ({\\small DFT}).\n{\\small DFT}\\ is based on the Hohenberg-Kohn theorem~\\cite{Hohenberg_PR_1964}, which proves that for a ground state system,\nthe potential energy is a unique functional of the density: $V (\\mathbf{r}) = V(\\rho(\\mathbf{r}))$.\nThis allows for the charge density $\\rho(\\mathbf{r})$ to be used as the central variable for the calculations\nrather than the many-body wave function $\\Psi(\\mathbf{r}_{1}, \\mathbf{r}_{2}, ..., \\mathbf{r}_{N})$,\ndramatically reducing the number of degrees of freedom in the calculation.\n\nThe Kohn-Sham equations~\\cite{DFT} map the $n$ coupled equations for the system of $n$ interacting particles\nonto a system of $n$ independent equations for $n$ non-interacting particles:\n\\begin{equation}\n\\label{eq:aflow_chp:kohnshameqns}\n\\left[ -\\frac{\\hbar^2}{2m} \\nabla^2 + V_s (\\mathbf{r}) \\right] \\phi_i (\\mathbf{r}) = \\varepsilon_i \\phi_i(\\mathbf{r}),\n\\end{equation}\nwhere $\\phi_i(\\mathbf{r})$ are the non-interacting Kohn-Sham eigenfunctions and $\\varepsilon_i$ are their eigenenergies.\n$V_s (\\mathbf{r})$ is the Kohn-Sham potential:\n\\begin{equation}\n\\label{eq:aflow_chp:kohnshampotential}\n V_s (\\mathbf{r}) = V(\\mathbf{r}) + \\int e^2 \\frac{\\rho_s (\\mathbf{r}^{\\prime})}{|\\mathbf{r} - \\mathbf{r}^{\\prime}|}\n d^3 \\mathbf{r}^{\\prime} + V_{\\substack{\\scalebox{0.6}{XC}}}\\left[\\rho_s(\\mathbf{r})\\right],\n\\end{equation}\nwhere $V(\\mathbf{r})$ is the external potential\n(which includes influences of the nuclei, applied fields, and the core electrons when pseudopotentials are used),\nthe second term is the direct Coulomb potential, and $V_{\\substack{\\scalebox{0.6}{XC}}}\\left[\\rho_s(\\mathbf{r})\\right]$ is the exchange-correlation term.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig003}\n\\mycaption{Standardized paths in reciprocal space for calculation of the electronic band\nstructures for the 25 different lattice types~\\cite{aflowBZ}.}\n\\label{fig:aflow_chp:band_structure_paths}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig004}\n\\mycaption[Side-by-side visualization of the crystal structure and Brillouin Zone using Jmol~\\cite{Jmol_Hanson,Jmol}.]\n{(\\textbf{a}) The structure highlighted is Ag$_{3}$KS$_{2}$ ({\\small ICSD}\\ \\#73581): \\url{http:\/\/aflow.org\/material.php?id=Ag6K2S4_ICSD_73581}.\n(\\textbf{b}) The {\\small AFLOW}\\ Standard path of high-symmetry \\textbf{k}-points is illustrated in the Brillouin Zone~\\cite{aflowBZ}.}\n\\label{fig:aflow_fleet:jmol_bz}\n\\end{figure}\n\nThe mapping onto a system of $n$ non-interacting particles comes at the cost of introducing the\nexchange-correlation potential $V_{\\substack{\\scalebox{0.6}{XC}}}\\left[\\rho_s(\\mathbf{r})\\right]$, the exact form of which is unknown and must be approximated.\nThe simplest approximation is the \\underline{l}ocal \\underline{d}ensity \\underline{a}pproximation ({\\small LDA})~\\cite{Perdew_prb_1981},\nin which the magnitude of the exchange-correlation energy at a particular point in space is\nassumed to be proportional to the magnitude of the density at that point in space.\nDespite its simplicity, {\\small LDA}\\ produces realistic results for atomic structure, elastic and vibrational properties\nfor a wide range of systems. However, it tends to overestimate the binding energies of materials, even\nputting crystal bulk phases in the wrong energetic order~\\cite{Zupan_LDAperformance_PRB_1998}.\nBeyond {\\small LDA}\\ is the \\underline{G}eneralized \\underline{G}radient \\underline{A}pproximation ({\\small GGA}), in which the exchange correlation term\nis a functional of the charge density and its gradient at each point in space.\nThere are several forms of {\\small GGA}\\, including those developed by Perdew, Burke and Ernzerhof ({\\small PBE}~\\cite{PBE}), or by Lee, Yang and Parr ({\\small LYP}~\\cite{LYP_1988}).\nA more recent development is the meta-{\\small GGA}\\ \\underline{S}trongly \\underline{C}onstrained and \\underline{A}ppropriately \\underline{N}ormed ({\\small SCAN})\nfunctional~\\cite{Perdew_SCAN_PRL_2015}, which satisfies all 17 known exact constraints on meta-{\\small GGA}\\ functionals.\n\nThe major limitations of {\\small LDA}\\ and {\\small GGA}\\ include their inability to adequately describe systems with strongly correlated or localized electrons,\ndue to the local and semilocal nature of the functionals.\nTreatments include the Hubbard $U$ corrections~\\cite{LiechDFTU, Dudarev_dftu}, self-interaction corrections~\\cite{Perdew_prb_1981}\nand hybrid functionals such as Becke's 3-parameter modification of {\\small LYP} ({\\small B3LYP}~\\cite{B3LYP_1993}), and that of Heyd, Scuseria and Ernzerhof ({\\small Heyd2003}~\\cite{Heyd2003}).\n\nWithin the context of \\nobreak\\mbox{\\it ab-initio}\\ structure prediction calculations, {\\small GGA}-{\\small PBE}\\ is the usual standard since it tends to produce\naccurate geometries and lattice constants~\\cite{monster}.\nFor accounting for strong correlation effects, the {\\small DFT}$+U$ method~\\cite{LiechDFTU, Dudarev_dftu}\nis often favored in large-scale automated database generation due to its low computational overhead.\nHowever, the traditional {\\small DFT}$+U$ procedure requires the addition of an empirical factor to the potential~\\cite{LiechDFTU, Dudarev_dftu}.\nRecently, methods have been implemented to calculate the $U$ parameter self-consistently from first-principles, such as the ACBN0 functional~\\cite{curtarolo:art93}.\n\n{\\small DFT}\\ also suffers from an inadequate description of excited\/unoccupied states, as the theory\nis fundamentally based on the ground state.\nExtensions for describing excited states include time-dependent {\\small DFT}\\ ({\\small TDDFT})~\\cite{Hedin_GW_1965} and the GW correction~\\cite{GW}.\nHowever, these methods are typically much more expensive than standard {\\small DFT}, and are not generally considered for large scale database generation.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig005}\n\\mycaption[Example band structure and density of states images automatically generated and\nserved through the {\\sf \\AFLOW.org}\\ data repository.]\n{The structure highlighted is AlCo$_{2}$Fe ({\\small ICSD}\\ \\#57607): \\url{http:\/\/aflow.org\/material.php?id=Al1Co2Fe1_ICSD_57607}.\nThe results of the spin-polarized calculation are differentiated by: color on the band structure plot\n(black\/red for majority\/minority spin), and sign on the density of states plot (positive\/negative for majority\/minority spin).\nThe band structure is calculated following the {\\small AFLOW}\\ Standard path of high-symmetry \\textbf{k}-points~\\cite{aflowBZ}.}\n\\label{fig:aflow_fleet:bs_plot}\n\\end{figure}\n\nAt the technical implementation level, there are\nmany {\\small DFT}\\ software packages available, including\n{\\small VASP}~\\cite{kresse_vasp, vasp_prb1996, vasp_cms1996, kresse_vasp_paw},\n\\QUANTUMESPRESSO~\\cite{qe, Giannozzi:2017io}, {\\small ABINIT}~\\cite{gonze:abinit, abinit_2009},\n{\\small FHI-AIMS}~\\cite{Blum_CPC2009_AIM}, {\\small SIESTA}~\\cite{Soler2002SIESTA} and {\\small GAUSSIAN}~\\cite{Gaussian_2009}.\nThese codes are generally distinguished by the choice of basis set.\nThere are two principle types of basis sets: plane waves, which take the form $\\psi (\\mathbf{r}) = \\sum e^{i \\mathbf{k}\\cdot\\mathbf{r}}$,\nand local orbitals, formed by a sum over functions $\\phi_a (\\mathbf{r})$ localized at particular points in space, such as\ngaussians or numerical atomic orbitals~\\cite{Hehre_self_consistent_molecular_orbit_JCP1969}.\nPlane wave based packages include {\\small VASP}, \\QUANTUMESPRESSO\\ and {\\small ABINIT}, and are generally better suited to periodic systems such as bulk inorganic materials.\nLocal orbital based packages include {\\small FHI-AIMS}, {\\small SIESTA}\\ and {\\small GAUSSIAN}, and are generally better suited to non-periodic systems such as organic molecules.\nIn the field of automated computational materials science, plane wave codes such as {\\small VASP}\\ are generally preferred:\nit is straightforward to automatically and systematically generate well-converged basis sets\nsince there is only a single parameter to adjust, namely the cut-off energy determining the number\nof plane waves in the basis set.\nLocal orbital basis sets tend to have far more independently adjustable degrees of freedom,\nsuch as the number of basis orbitals per atomic orbital as well as their respective cut-off radii,\nmaking the automated generation of reliable basis sets more difficult.\nTherefore, a typical standardized protocol for automated materials science calculations~\\cite{curtarolo:art104} relies on\nthe {\\small VASP}\\ software package with a basis set cut-off energy higher than that recommended by the {\\small VASP}\\ potential files,\nin combination with the {\\small PBE}\\ formulation of {\\small GGA}.\n\nFinally, it is necessary to automate the generation of the \\textbf{k}-point grid and pathways in\nreciprocal space used for the calculation of forces, energies and the electronic band structure.\nIn general, {\\small DFT}\\ codes use standardized methods such as the Monkhorst-Pack scheme~\\cite{MonkhorstPack} to generate reciprocal lattice \\textbf{k}-point grids,\nalthough optimized grids have been calculated for different lattice types and are available online~\\cite{Wisesa_Kgrids_PRB_2016}.\nOptimizing \\textbf{k}-point grid density is a computationally expensive process that is difficult to automate,\nso instead standardized grid densities based on the concept of\n``\\underline{$k$}-\\underline{p}oints \\underline{p}er \\underline{r}eciprocal \\underline{a}tom'' ({\\small KPPRA}) are used.\nThe {\\small KPPRA}\\ value is chosen to be sufficiently large to ensure convergence for all systems.\nTypical recommended values used for {\\small KPPRA}\\ range from 6,000 to 10,000~\\cite{curtarolo:art104},\nso that a material with two atoms in the calculation cell will have a \\textbf{k}-point mesh of at least 3,000 to 5,000 points.\nStandardized directions in reciprocal space have also been defined for the calculation of the\nband structure as illustrated in Figure~\\ref{fig:aflow_chp:band_structure_paths}~\\cite{aflowBZ} and Figure~\\ref{fig:aflow_fleet:jmol_bz}.\nThese paths are optimized to include all of the high-symmetry points of the lattice.\nA standard band structure plot as generated by {\\small AFLOW}\\ is illustrated in Figure~\\ref{fig:aflow_fleet:bs_plot}.\n\n\\subsection{Integrated calculation of materials properties}\n\\label{subsec:aflow_chp:thermomechanical}\n\nAutomated frameworks such as {\\small AFLOW}\\ combine the computational analysis of properties including symmetry, electronic structure,\nelasticity, and thermal behavior into integrated workflows.\nCrystal symmetry information is used to find the primitive cell to reduce the size of {\\small DFT}\\ calculations,\nto determine the appropriate paths in reciprocal space for electronic band structure calculations (see Figure~\\ref{fig:aflow_chp:band_structure_paths}~\\cite{aflowBZ}),\nand to determine the set of inequivalent distortions for phonon and elasticity calculations.\nThermal and elastic properties of materials are important for predicting the thermodynamic and mechanical stability\nof structural phases~\\cite{Greaves_Poisson_NMat_2011, Poirier_Earth_Interior_2000, Mouhat_Elastic_PRB_2014, curtarolo:art106}\nand assessing their importance for a variety of applications.\nElastic properties such as the shear and bulk moduli are important for predicting the hardness\nof materials~\\cite{Chen_hardness_Intermetallics_2011, Teter_Hardness_MRS_1998},\nand thus their resistance to wear and distortion.\nElasticity tensors can be used to predict the properties of composite\nmaterials~\\cite{Hashin_Multiphase_JMPS_1963, Zohdi_Polycrystalline_IJNME_2001}.\nThey are also important in geophysics for modeling the propagation of seismic waves\nin order to investigate the mineral composition of geological\nformations~\\cite{Poirier_Earth_Interior_2000, Anderson_Elastic_RGP_1968, Karki_Elastic_RGP_2001}.\nThe lattice thermal conductivity $\\left(\\kappa_{\\substack{\\scalebox{0.6}{L}}}\\right)$ is a crucial\ndesign parameter in a wide range of important\ntechnologies, such as the development of new thermoelectric\nmaterials~\\cite{zebarjadi_perspectives_2012,aflowKAPPA,Garrity_thermoelectrics_PRB_2016},\nheat sink materials for thermal management in electronic devices~\\cite{Yeh_2002},\nand rewritable phase-change memories~\\cite{Wright_tnano_2011}.\nHigh thermal conductivity materials, which typically have a zincblende or diamond-like structure, are essential\nin microelectronic and nanoelectronic devices for achieving\nefficient heat removal~\\cite{Watari_MRS_2001}, and have\nbeen intensively studied for the past few decades~\\cite{Slack_1987}.\nLow thermal conductivity materials constitute\nthe basis of a new generation of thermoelectric materials and thermal\nbarrier coatings~\\cite{Snyder_jmatchem_2011}.\n\nThe calculation of thermal and elastic properties offer an excellent example of the power of\nintegrated computational materials design frameworks.\nWith a single input file, these frameworks can automatically set-up and run calculations of\ndifferent distorted cells, and combine the resulting energies and\nforces to calculate thermal and mechanical properties.\n\n\\subsubsection{Autonomous symmetry analysis}\n\nCritical to any analysis of crystals is the accurate determination of the symmetry profile.\nFor example, symmetry serves to\n\\textbf{i.} validate the forms of the elastic constants\nand compliance tensors, where the crystal symmetry dictates equivalence or absence\nof specific tensor elements~\\cite{nye_symmetry, curtarolo:art100, Mouhat_Elastic_PRB_2014}, and\n\\textbf{ii.} reduce the number of \\nobreak\\mbox{\\it ab-initio}\\ calculations needed for phonon\ncalculations, where, in the case of the finite-displacement method, equivalent\natoms and distortion directions are identified through factor group and site symmetry\nanalyses~\\cite{Maradudin1971}.\n\nAutonomous workflows for elasticity and vibrational characterizations\ntherefore require a correspondingly robust symmetry analysis.\nUnfortunately, standard symmetry packages~\\cite{stokes_findsym,Stokes_FROZSL_Ferroelectrics_1995,platon_2003,spglib},\ncatering to different objectives, depend on tolerance-tuning to\novercome numerical instabilities and atypical data --- emanating from\nfinite temperature measurements and uncertainty in experimentally reported observations.\nThese tolerances are responsible for validating mappings and identifying isometries,\nsuch as the $n$-fold operator depicted in Figure~\\ref{fig:aflow_chp:sym}(a).\nSome standard packages define separate tolerances for space, angle~\\cite{spglib},\nand even operation type~\\cite{stokes_findsym,Stokes_FROZSL_Ferroelectrics_1995,platon_2003}\n(\\nobreak\\mbox{\\it e.g.}, rotation \\nobreak\\mbox{\\it vs.}\\ inversion).\nEach parameter introduces a factorial expansion of unique inputs, which can result in\ndistinct symmetry profiles as illustrated in Figure~\\ref{fig:aflow_chp:sym}(b).\nBy varying the spatial tolerance $\\epsilon$, four different space groups can be observed\nfor AgBr ({\\small ICSD}\\ \\#56551\\footnote{{h}ttp:\/\/www.aflow.org\/material.php?id=56551}), if one is found at all.\nGaps in the range, where no consistent symmetry profile can be resolved, are\nparticularly problematic in automated frameworks, triggering critical failures in subsequent analyses.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig006}\n\\mycaption[Challenges in autonomous symmetry analysis.]\n{(\\textbf{a}) An illustration of a general $n$-fold symmetry operation.\n(\\textbf{b}) Possible space group determinations with mapping tolerance $\\epsilon$ for AgBr ({\\small ICSD}\\ \\#56551).\n(\\textbf{c}) Warping of mapping tolerance sphere with a transformation from cartesian to fractional basis.}\n\\label{fig:aflow_chp:sym}\n\\end{figure}\n\nCell shape can also complicate mapping determinations.\nAnisotropies in the cell, such as skewness of lattice vectors, translate\nto distortions of fractional and reciprocal spaces.\nA uniform tolerance sphere in cartesian space, inside which points are considered mapped,\ngenerally warps to a sheared spheroid, as depicted in Figure~\\ref{fig:aflow_chp:sym}(c).\nHence, distances in these spaces are direction-dependent, compromising the integrity\nof rapid minimum-image determinations~\\cite{hloucha_minimumimage_1998} and generally warranting\nprohibitively expensive algorithms~\\cite{curtarolo:art135}.\nSuch failures can result in incommensurate symmetry profiles, where the real space\nlattice profile (\\nobreak\\mbox{\\it e.g.}, bcc) does not match that of the reciprocal space (fcc).\n\nThe new {\\small AFLOW-SYM}\\ module~\\cite{curtarolo:art135} within {\\small AFLOW}\\ offers careful treatment of tolerances, with extensive\nvalidation schemes, to mitigate the aforementioned challenges.\nAlthough a user-defined tolerance input is still available, {\\small AFLOW}\\ defaults to one of two pre-defined\ntolerances, namely \\texttt{tight} (standard) and \\texttt{loose}.\nShould any discrepancies occur, these defaults are the starting values of a large tolerance scan,\nas shown in Figure~\\ref{fig:aflow_chp:sym}(b).\nA number of validation schemes have been incorporated to catch such discrepancies.\nThese checks are consistent with crystallographic group theory principles, validating operation\ntypes and cardinalities~\\cite{tables_crystallography}.\nFrom considerations of different extreme cell shapes, a heuristic threshold has been defined\nto classify scenarios where mapping failures are likely to occur --- based on skewness and mapping tolerance.\nWhen benchmarked against standard packages for over 54,000 structures in the Inorganic Crystal Structure Database,\n{\\small AFLOW-SYM}\\ consistently resolves\nthe symmetry characterization most compatible with experimental observations~\\cite{curtarolo:art135}.\n\nAlong with accuracy, {\\small AFLOW-SYM}\\ delivers a wealth of symmetry properties and representations\nto satisfy injection into any analysis or workflow.\nThe full set of operators --- including that of the point-, factor-, crystallographic point-, space groups,\nand site symmetries --- are provided in matrix, axis-angle, matrix generator, and quaternion representations in\nboth cartesian and fractional coordinates.\nA span of characterizations, organized by degree of symmetry-breaking, are available, including\nthose of the lattice, superlattice, crystal, and crystal-spin.\nSpace group and Wyckoff positions are also resolved.\nThe full dataset is made available in both plain-text and {\\small JSON}\\ formats.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig007}\n\\mycaption[Calculation of thermomechanical properties.]\n{({\\bf a}) {\\small AEL}\\ applies a set of independent normal and shear strains to the crystal structure to obtain\nthe elastic constants.\n({\\bf b}) {\\small AGL}\\ applies a set of isotropic strains to the unit cell to obtain\nenergy \\nobreak\\mbox{\\it vs.}\\ volume data, which is fitted by a polynomial in order to\ncalculate the bulk modulus as a function of volume, $B_{\\mathrm S} (V)$.\n$B_{\\mathrm S} (V)$ is then used to calculate the Debye temperature as a function of volume and thus\nthe vibrational free energy as a function of temperature.\nThe Gibbs free energy as a function of volume is then minimized for each pressure and temperature\npoint to obtain the equilibrium volume and other thermomechanical properties.\n({\\bf c}) {\\small APL}\\ obtains the harmonic \\underline{i}nteratomic \\underline{f}orce \\underline{c}onstants ({\\small IFC}{}s) from supercell calculations where inequivalent\natoms are displaced in inequivalent directions, and then the changes in the forces on the other atoms are calculated.\nThe {\\small IFC}{}s are then used to construct the dynamical matrix, which is diagonalized to obtain the phonon eigenmodes.\n{\\small AAPL}\\ calculates three-phonon scattering effects by performing supercell calculations where pairs of inequivalent atoms are displaced in inequivalent directions, and\n the changes in the forces on the other atoms in the supercell are calculated to obtain the third-order anharmonic {\\small IFC}{}s.}\n\\label{fig:aflow_chp:thermomechanical}\n\\end{figure}\n\n\\subsubsection{Harmonic phonons}\n\nThermal properties can be obtained by directly\ncalculating the phonon dispersion from the dynamical matrix of {\\small IFC}{}s.\nThe approach is implemented within the {\\small \\underline{A}FLOW} \\underline{P}honon \\underline{L}ibrary\n({\\small APL})~\\cite{aflowPAPER}.\nThe {\\small IFC}{}s are determined from a set of supercell calculations in which the atoms are\ndisplaced from their equilibrium positions~\\cite{Maradudin1971} as shown in Figure~\\ref{fig:aflow_chp:thermomechanical}(c).\n\nThe {\\small IFC}{}s derive from a Taylor expansion of the potential energy, $V$, of the crystal\nabout the atoms' equilibrium positions:\n\\begin{multline}\n V=\\left.V\\right|_{\\mathbf{r}(i,t)=0,\\forall i}+\n \\sum_{i,\\alpha}\\left.\\frac{\\partial V}{\\partial r(i,t)^{\\alpha}}\\right|_{\\mathbf{r}(i,t)=0,\\forall i} r(i,t)^{\\alpha} \\\\+\n \\frac{1}{2}\\sum_{\\substack{i,\\alpha,\\\\ j,\\beta}}\\left.\\frac{\\partial^2V}\n{\\partial r(i,t)^{\\alpha}\\partial r(j,t)^{\\beta}}\\right|_{\\mathbf{r}(i,t)=0,\\forall i}\n r(i,t)^{\\alpha}r(j,t)^{\\beta}+\n\\ldots,\n\\label{eq:aflow_chp:PE_harmonic}\n\\end{multline}\nwhere $r(i,t)^{\\alpha}$ is the $\\alpha$-cartesian component ($\\alpha=x,y,z$) of the time-dependent atomic displacement\n$\\mathbf{r}(t)$ of the $i^{\\mathrm{th}}$ atom about its equilibrium position,\n$\\left.V\\right|_{\\mathbf{r}(i,t)=0,\\forall i}$ is the potential energy of the crystal in its equilibrium configuration,\n$\\left.\\partial V\/\\partial r(i,t)^{\\alpha}\\right|_{\\mathbf{r}(i,t)=0,\\forall i}$\nis the negative of the force acting in the $\\alpha$ direction on atom $i$ in the equilibrium configuration\n(zero by definition), and\n$\\left.\\partial^2V\/\\partial r(i,t)^{\\alpha}\\partial r(j,t)^{\\beta}\\right|_{\\mathbf{r}(i,t)=0,\\forall i}$\nconstitute the {\\small IFC}{}s $\\phi(i,j)_{\\alpha,\\beta}$.\nTo first approximation, $\\phi(i,j)_{\\alpha,\\beta}$ is the negative of the force exerted\nin the $\\alpha$ direction on atom $i$ when atom\n$j$ is displaced in the $\\beta$ direction with all other atoms maintaining their equilibrium positions,\nas shown in Figure~\\ref{fig:aflow_chp:thermomechanical}(c).\nAll higher order terms are neglected in the harmonic approximation.\n\nCorrespondingly, the equations of motion of the lattice are as follows:\n\\begin{equation}\n M(i)\\ddot{r}(i,t)^{\\alpha}=\n-\\sum_{j,\\beta}\\phi(i,j)_{\\alpha,\\beta}\nr(j,t)^{\\beta}\\quad\\forall i,\\alpha,\n\\label{eq:aflow_chp:eom_harmonic}\n\\end{equation}\nand can be solved by a plane wave solution of the form\n\\begin{equation}\nr(i,t)^{\\alpha}=\\frac{v(i)^{\\alpha}}{\\sqrt{M(i)}} e^{\\mathrm{i}\\left(\\mathbf{q}\\cdot\\mathbf{R}_{l} - \\omega t \\right)},\n\\end{equation}\nwhere $v(i)^{\\alpha}$ form the phonon eigenvectors (polarization vector),\n$M(i)$ is the mass of the $i^{\\mathrm{th}}$ atom,\n$\\mathbf{q}$ is the wave vector,\n$\\mathbf{R}_{l}$ is the position of lattice point $l$,\nand $\\omega$ form the phonon eigenvalues (frequencies).\nThe approach is nearly identical to that taken for electrons in a periodic potential (Bloch waves)~\\cite{ashcroft_mermin}.\nPlugging this solution into the equations of motion (Equation~\\ref{eq:aflow_chp:eom_harmonic}) yields the following set of linear equations:\n\\begin{equation}\n\\omega^{2}v(i)^{\\alpha}=\n \\sum_{j,\\beta}D_{i,j}^{\\alpha,\\beta}(\\mathbf{q})\n v(j)^{\\beta}\\quad\\forall i,\\alpha,\n\\end{equation}\nwhere the dynamical matrix $D_{i,j}^{\\alpha,\\beta}(\\mathbf{q})$ is defined as\n\\begin{equation}\nD_{i,j}^{\\alpha,\\beta}(\\mathbf{q})=\n \\sum_{l}\\frac{\\phi(i,j)_{\\alpha,\\beta}}{\\sqrt{M(i)M(j)}} e^{-\\mathrm{i}\\mathbf{q}\\cdot\\left(\\mathbf{R}_{l}-\\mathbf{R}_{0}\\right)}.\n\\end{equation}\nThe problem can be equivalently represented by a standard eigenvalue equation:\n\\begin{equation}\n\\omega^{2}\n\\begin{bmatrix}\n \\\\\n\\mathbf{v} \\\\\n ~\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n & & \\\\\n & \\mathbf{D}(\\mathbf{q}) & \\\\\n & &\n\\end{bmatrix}\n\\begin{bmatrix}\n \\\\\n\\mathbf{v} \\\\\n ~\n\\end{bmatrix},\n\\label{eq:aflow_chp:dyn_eigen}\n\\end{equation}\nwhere\nthe dynamical matrix and phonon eigenvectors have dimensions $\\left(3 n_{\\mathrm{a}} \\times 3 n_{\\mathrm{a}}\\right)$\nand $\\left(3 n_{\\mathrm{a}} \\times 1 \\right)$, respectively, and $n_{\\mathrm{a}}$ is the number of atoms in the cell.\nHence, Equation~\\ref{eq:aflow_chp:dyn_eigen} has $\\lambda=3 n_{\\mathrm{a}}$ solutions\/modes referred to as branches.\nIn practice, Equation~\\ref{eq:aflow_chp:dyn_eigen} is solved for discrete sets of $\\mathbf{q}$-points to compute\nthe phonon density of states (grid over all possible $\\mathbf{q}$) and dispersion\n(along the high-symmetry paths of the lattice~\\cite{aflowBZ}).\nThus, the phonon eigenvalues and eigenvectors are appropriately denoted $\\omega_{\\lambda}(\\mathbf{q})$ and\n$\\mathbf{v}_{\\lambda}(\\mathbf{q})$, respectively.\n\nSimilar to the electronic Hamiltonian, the dynamical matrix is Hermitian, \\nobreak\\mbox{\\it i.e.},\n$\\mathbf{D}(\\mathbf{q})=\\mathbf{D}^{*}(\\mathbf{q})$.\nThus $\\omega_{\\lambda}^{2}(\\mathbf{q})$ must also be real, so $\\omega_{\\lambda}(\\mathbf{q})$ can either be real or purely imaginary.\nHowever, a purely imaginary frequency corresponds to vibrational motion of the lattice that increases exponentially in time.\nTherefore, imaginary frequencies, or those corresponding to soft modes, indicate the structure is dynamically unstable.\nIn the case of a symmetric, high-temperature phase, soft modes suggest there exists a lower symmetry structure\nstable at $T=0$~K.\nTemperature effects on phonon frequencies can be modeled with\n\\begin{equation}\n\\widetilde{\\omega}_{\\lambda}^{2}(\\mathbf{q},T)=\\omega_{\\lambda}^{2}(\\mathbf{q},T=0)+\\eta T^2,\n\\end{equation}\nwhere $\\eta$ is positive in general.\nThe two structures, the symmetric and the stable, differ by the distortion\ncorresponding to this ``frozen'' (non-vibrating) mode.\nUpon heating, the temperature term increases until the frequency reaches zero, and a phase transition occurs from\nthe stable structure to the symmetric~\\cite{Dove_LatDynam_1993}.\n\nIn practice, soft modes~\\cite{Parlinski_Phonon_Software} may indicate:\n\\textbf{i.} the structure is dynamically unstable at $T$,\n\\textbf{ii.} the symmetry of the structure is lower than that considered, perhaps due to magnetism,\n\\textbf{iii.} strong electronic correlations, or\n\\textbf{iv.} long range interactions play a significant role, and a larger supercell should be considered.\n\nWith the phonon density of states computed, the following thermal properties can be calculated:\nthe internal vibrational energy\n\\begin{equation}\n U_{\\substack{\\scalebox{0.6}{vib}}}(\\mathbf{x},T)=\\int_{0}^{\\infty} \\left( \\frac{1}{2} + \\frac{1}{e^{\\left(\\beta \\hbar \\omega\\right)}-1} \\right) \\hbar \\omega g(\\mathbf{x};\\omega) d\\omega,\n\\end{equation}\nthe vibrational component of the free energy $F_{\\substack{\\scalebox{0.6}{vib}}}(\\mathbf{x}; T)$\n\\begin{equation}\nF_{\\substack{\\scalebox{0.6}{vib}}}(\\mathbf{x}; T) \\!=\\!\\! \\int_0^{\\infty} \\!\\!\\left[\\frac{\\hbar \\omega}{2} \\!+\\!\n\\frac{1}{\\beta} \\ \\mathrm{log}\\!\\left(1\\!-\\!{\\mathrm e}^{- \\beta \\hbar \\omega }\\right)\\!\\right]\\!g(\\mathbf{x}; \\omega) d\\omega,\n\\label{eq:aflow_chp:fvib}\n\\end{equation}\nthe vibration entropy\n\\begin{equation}\nS_{\\substack{\\scalebox{0.6}{vib}}}(\\mathbf{x},T)=\\frac{U_{\\substack{\\scalebox{0.6}{vib}}}(\\mathbf{x},T)-F_{\\substack{\\scalebox{0.6}{vib}}}(\\mathbf{x}; T)}{T},\n\\end{equation}\nand the isochoric specific heat\n\\begin{equation}\nC_{{\\substack{\\scalebox{0.6}{V}}}, {\\substack{\\scalebox{0.6}{vib}}}}(\\mathbf{x},T)=\\int_{0}^{\\infty} \\frac{ k_{\\substack{\\scalebox{0.6}{B}}} \\left(\\beta \\hbar \\omega \\right)^2 g(\\mathbf{x};\\omega)}{\\left(1-e^{-\\left(\\beta \\hbar \\omega\\right)}\\right) \\left(e^{\\left(\\beta \\hbar \\omega\\right)}-1\\right) } d\\omega.\n\\end{equation}\n\n\\subsubsection{Quasi-harmonic phonons}\n\nThe harmonic approximation does not describe phonon-phonon scattering, and so cannot be used to\ncalculate properties such as thermal conductivity or thermal expansion.\nTo obtain these properties, either the quasi-harmonic approximation can be used,\nor a full calculation of the higher order anharmonic {\\small IFC}{}s can be performed.\nThe quasi-harmonic approximation is the less computationally demanding of these two methods,\nand compares harmonic calculations of phonon properties at different volumes to predict anharmonic properties.\nThe different volume calculations can be in the form of harmonic phonon calculations as described\nabove~\\cite{curtarolo:art114, curtarolo:art119},\nor simple static primitive cell calculations~\\cite{Blanco_CPC_GIBBS_2004, curtarolo:art96}.\nThe \\underline{Q}uasi-\\underline{H}armonic \\underline{A}pproximation\nis implemented within {\\small APL}\\ and referred to as {\\small QHA-APL}~\\cite{curtarolo:art96}.\nIn the case of the quasi-harmonic phonon calculations, the anharmonicity of the system is described by\nthe mode-resolved Gr{\\\"u}neisen parameters, which are given by the change in the phonon frequencies as a function of volume\n\\begin{equation}\n\\label{eq:aflow_chp:gamma_micro}\n\\gamma_{\\lambda}(\\mathbf{q}) = - \\frac{V}{\\omega_{\\lambda}(\\mathbf{q})} \\frac{\\partial \\omega_{\\lambda}(\\mathbf{q})}{\\partial V},\n\\end{equation}\nwhere $\\gamma_{\\lambda}(\\mathbf{q})$ is the parameter for the wave vector $\\mathbf{q}$ and the $\\lambda^{\\rm{th}}$ mode of the phonon dispersion.\nThe average of the $\\gamma_{\\lambda}(\\mathbf{q})$ values, weighted by the specific heat capacity of each mode $C_{{\\substack{\\scalebox{0.6}{V}}},\\lambda}(\\mathbf{q})$, gives the average\nGr{\\\"u}neisen parameter:\n\\begin{equation}\n\\label{eq:aflow_chp:gamma_ave}\n\\gamma = \\frac{\\sum_{\\lambda,\\mathbf{q}} \\gamma_{\\lambda}(\\mathbf{q}) C_{{\\substack{\\scalebox{0.6}{V}}},\\lambda}(\\mathbf{q})}{C_{\\substack{\\scalebox{0.6}{V}}}}.\n\\end{equation}\nThe specific heat capacity, Debye temperature and Gr{\\\"u}neisen parameter can then be combined to\ncalculate other properties such as the specific heat capacity at constant pressure $C_{\\rm p}$,\nthe thermal coefficient of expansion $\\alpha$, and the lattice thermal\nconductivity $\\kappa_{\\substack{\\scalebox{0.6}{L}}}$~\\cite{curtarolo:art119}, using similar expressions to those described in Section~\\ref{sec:art115}.\n\n\\subsubsection{Anharmonic phonons}\n\nThe full calculation of the anharmonic {\\small IFC}{}s requires performing supercell calculations in which pairs of\ninequivalent atoms are displaced in all pairs of\ninequivalent directions~\\cite{Broido2007, Wu_PRB_2012, ward_ab_2009, ward_intrinsic_2010, Zhang_JACS_2012, Li_PRB_2012, Lindsay_PRL_2013, Lindsay_PRB_2013, Li_ShengBTE_CPC_2014, curtarolo:art125}\nas illustrated in Figure~\\ref{fig:aflow_chp:thermomechanical}(c).\nThe third order anharmonic {\\small IFC}{}s can then be obtained by calculating the change in the forces on\nall of the other atoms due to these displacements.\nThis method has been implemented in the form of a fully\nautomated integrated workflow in the {\\small AFLOW}\\ framework,\nwhere it is referred to as the {\\small \\underline{A}FLOW} \\underline{A}nharmonic\n\\underline{P}honon \\underline{L}ibrary ({\\small AAPL})~\\cite{curtarolo:art125}.\nThis approach can provide very accurate results for the lattice thermal conductivity when combined\nwith accurate electronic structure methods\n\\cite{curtarolo:art125},\nbut quickly becomes very expensive for systems with multiple\ninequivalent atoms or low symmetry.\nTherefore, simpler methods such as the quasi-harmonic Debye model\ntend to be used for initial rapid screening~\\cite{curtarolo:art96, curtarolo:art115}, while\nthe more accurate and expensive methods are used for characterizing systems\nthat are promising candidates for specific engineering applications.\n\n\\subsection{Online data repositories}\n\nRendering the massive quantities of data generated using automated \\nobreak\\mbox{\\it ab-initio}\\ frameworks available\nfor other researchers requires going beyond the conventional methods\nfor the dissemination of scientific results in the form of journal articles.\nInstead, this data is typically made available in online data repositories, which can usually be accessed both\nmanually via interactive web portals, and programmatically via an \\underline{a}pplication \\underline{p}rogramming \\underline{i}nterface ({\\small API}).\n\n\\subsubsection{Computational materials data web portals}\n\nMost computational data repositories include an interactive web portal front end that enables manual data access.\nThese web portals usually include online applications to facilitate data retrieval and analysis.\nThe front page of the {\\small AFLOW}\\ data repository is displayed in Figure~\\ref{fig:aflow_chp:aflow_web_apps}(a).\nThe main features include a search bar where information such as\n{\\small ICSD}\\ reference number, {\\small \\underline{A}FLOW} \\underline{u}nique \\underline{id}entifier ({\\small AUID}) or the chemical formula can be entered\nin order to retrieve specific materials entries.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig008}\n\\mycaption[{\\small AFLOW}\\ web applications.]\n{({\\bf a}) Front page of the {\\small AFLOW}\\ online data repository, highlighting the link to\n({\\bf b}) the {\\small AFLOW}\\ advanced search application, which facilitates complex search\nqueries including filtering by chemical composition and materials properties and\n({\\bf c}) the {\\small AFLOW}\\ interactive convex hull generator, showing the 3D hull for the Pt-Sc-Zn ternary alloy system.}\n\\label{fig:aflow_chp:aflow_web_apps}\n\\end{figure}\n\nBelow are buttons linking to several different online applications such as the advanced search functionality,\nconvex hull phase diagram generators, machine learning applications~\\cite{curtarolo:art124, curtarolo:art129, curtarolo:art136} and {\\small AFLOW}-online data analysis tools.\nThe link to the advanced search application is highlighted by the orange square, and the application page is shown in Figure~\\ref{fig:aflow_chp:aflow_web_apps}(b).\nThe advanced search application allows users to search for materials that contain (or exclude) specific elements or groups of elements,\nand also to filter and sort the results by properties such as electronic band structure energy gap (under the ``Electronics'' properties filter group)\nand bulk modulus (under the ``Mechanical'' properties filter group).\nThis allows users to identify candidate materials with suitable materials properties for specific applications.\n\nAnother example online application available on the {\\small AFLOW}\\ web portal is the convex hull phase diagram generator.\nThis application can be accessed by clicking on the button highlighted by the orange square in\nFigure~\\ref{fig:aflow_chp:aflow_web_apps}(a), which will bring up a periodic table allowing users to\nselect two or three elements for which they want to generate a convex hull.\nThe application will then access the formation enthalpies and stoichiometries of the materials entries in the\nrelevant alloy systems, and use this data to generate a two or three dimensional convex hull phase diagram\nas depicted in Figure~\\ref{fig:aflow_chp:aflow_web_apps}(c).\nThis application is fully interactive, allowing users to adjust the energy axis scale,\nrotate the diagram to view from different directions, and select specific points to obtain more information on the\ncorresponding entries.\n\n\\subsubsection{Programmatically accessible online repositories of computed materials properties}\n\nIn order to use materials data in machine learning algorithms, it should be stored in a structured online database\nand made programmatically accessible via a \\underline{re}presentational \\underline{s}tate \\underline{t}ransfer {\\small API}\\ ({\\small REST-API}).\nExamples of online repositories of materials data include {\\small AFLOW}~\\cite{aflowlibPAPER, aflowAPI},\nMaterials Project~\\cite{materialsproject.org}, and {\\small OQMD}~\\cite{Saal_JOM_2013}.\nThere are also repositories that aggregate results from multiple sources such as\n{NOMAD}~\\cite{nomad} and Citrine~\\cite{citrine_database}.\n\n{\\small REST-API}{}s facilitate programmatic access to data repositories.\nTypical databases such as {\\small AFLOW}\\ are organized in layers,\nwith the top layer corresponding to a project or catalog (\\nobreak\\mbox{\\it e.g.}, binary alloys),\nthe next layer corresponding to data sets (\\nobreak\\mbox{\\it e.g.}, all of the entries for a particular alloy system),\nand then the bottom layer corresponding to specific materials entries, as illustrated in Figure~\\ref{fig:aflow_chp:aflow_restapi_layers_aurl}(a).\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig009}\n\\mycaption[{\\small AFLOW}\\ {\\small REST-API}\\ structure.]\n{\\small (\\textbf{a}) The {\\small AFLOW}\\ database is organized as a multilayered system.\n(\\textbf{b}) Example of an {\\small AURL}\\ which enables direct programmatic access to specific materials entry properties in the {\\small AFLOW}\\ database.}\n\\label{fig:aflow_chp:aflow_restapi_layers_aurl}\n\\end{figure}\n\nIn the case of the {\\small AFLOW}\\ database, there are currently four different ``projects'', namely the\n``ICSD'', ``LIB1'', ``LIB2'' and ``LIB3'' projects; along with three\nmore under construction: ``LIB4'', ``LIB5'' and ``LIB6''.\nThe ``ICSD'' project contains calculated data for previously observed compounds~\\cite{ICSD},\nwhereas the other three projects contain calculated data for single elements, binary alloys,\nand ternary alloys respectively, and are constructed by decorating prototype\nstructures with combinations of different elements.\nWithin ``LIB2'' and ``LIB3'', there are many different data sets, each corresponding to a specific\nbinary or ternary alloy system.\nEach entry in the set corresponds to a specific prototype structure and stoichiometry.\nThe materials properties values for each of these entries are encoded via keywords,\nand the data can be accessed via {\\small URL}{}s constructed from the different layer names and the appropriate keywords.\nIn the case of the {\\small AFLOW}\\ database, the location of each layer and entry is\nidentified by an {\\small \\underline{A}FLOW} \\underline{u}niform \\underline{r}esource \\underline{l}ocator ({\\small AURL})~\\cite{aflowAPI},\nwhich can be converted to a {\\small URL}\\ providing the absolute path to a particular layer, entry or property.\nThe {\\small AURL}\\ takes the form \\url{server:AFLOWDATA\/project\/set\/entry\/?keywords},\nfor example \\url{aflowlib.duke.edu:AFLOWDATA\/LIB2_RAW\/Cu_pvV_sv\/15\/?energy_atom},\nwhere \\url{aflowlib.duke.edu} is the web address of the physical server where the data is located,\n\\url{LIB2_RAW} is the binary alloy project layer, \\url{Cu_pvV_sv} is\nthe set containing the binary alloy system Cu-V, \\url{15} is a specific entry with the composition\nCu$_3$V in a tetragonal lattice, and \\url{energy_atom} is the keyword corresponding\nto the property of energy per atom in units of eV, as shown in Figure~\\ref{fig:aflow_chp:aflow_restapi_layers_aurl}(b).\nEach {\\small AURL}\\ can be converted to a web {\\small URL}\\ by changing the ``\\url{:}'' after the server name to a ``\\url{\/}'',\nso that the {\\small AURL}\\ in Figure~\\ref{fig:aflow_chp:aflow_restapi_layers_aurl}(b) would become the\n{\\small URL}\\ \\url{aflowlib.duke.edu\/AFLOWDATA\/LIB2_RAW\/Cu_pvV_sv\/15\/?energy_atom}.\nThis {\\small URL}, if queried via a web browser or using a UNIX utility such as \\texttt{wget},\nreturns the energy per atom in eV for entry \\url{15} of the Cu-V binary alloy system.\n\nIn addition to the {\\small AURL}, each entry in the {\\small AFLOW}\\ database is also associated with an\n{\\small AUID}~\\cite{aflowAPI},\nwhich is a unique hexadecimal (base 16) number constructed from a checksum of the {\\small AFLOW}\\ output file for that entry.\nSince the {\\small AUID}\\ for a particular entry can always be reconstructed by applying the checksum\nprocedure to the output file, it serves as a permanent, unique specifier for each calculation,\nirrespective of the current physical location of where the data are stored.\nThis enables the retrieval of the results for a particular calculation from different servers,\nallowing for the construction of a truly distributed database that is robust\nagainst the failure or relocation of the physical hardware. Actual database versions can be\nidentified from the version of {\\small AFLOW}\\ used to parse the calculation output files and\npostprocess the results to generate the database entry. This information can be retrieved using the\nkeyword \\url{aflowlib_version}.\n\nThe search and sort functions of the front-end portals can be combined with the programmatic\ndata access functionality of the {\\small REST-API}\\ through the implementation of a Search-{\\small API}.\nThe {\\small AFLUX}\\ Search-{\\small API}\\ uses the {\\small LUX}\\ language to enable the embedding of logical\noperators within {\\small URL}\\ query strings~\\cite{aflux}.\nFor example, the energy per atom of every entry in the {\\small AFLOW}\\ repository containing the element Cu or V,\nbut not the element Ti, with an electronic band gap between 2 and 5~eV, can be retrieved using the command:\n\\url{aflowlib.duke.edu\/search\/API\/species((Cu:V),(!Ti)),Egap(2*,*5),energy_atom}.\nIn this {\\small AFLUX}\\ search query, the comma ``\\verb|,|'' represents the logical {\\small AND} operation, the colon ``\\verb|:|'' the logical {\\small OR} operation,\nthe exclamation mark ``\\verb|!|'' the logical {\\small NOT} operation, and the asterisk ``\\verb|*|'' is the ``loose'' operation that defines a range of values to search within.\nNote that by default {\\small AFLUX}\\ returns only the first 64 entries matching the search query.\nThe number and set of entries can be controlled by appending the \\verb|paging| directive to the end of the search query as follows:\n\\url{aflowlib.duke.edu\/search\/API\/species((Cu:V),(!Ti)),Egap(2*,*5),energy_atom,paging(0)},\nwhere calling the \\verb|paging| directive with the argument ``0'' instructs {\\small AFLUX}\\ to return all of the matching entries\n(note that this could potentially be a large amount of data, depending on the search query).\nThe {\\small AFLUX}\\ Search-{\\small API}\\ allows users to construct and retrieve customized data sets, which they can feed into materials\ninformatics machine learning packages to identify trends and correlations for use in rational materials design.\n\nThe use of {\\small API}{}s to provide programmatic access is being extended beyond materials data retrieval,\nto enable the remote use of pre-trained machine learning algorithms.\nThe {\\small AFLOW-ML}\\ {\\small API}~\\cite{curtarolo:art136} facilitates access to the two machine learning models\nthat are also available online at \\url{aflow.org\/aflow-ml}~\\cite{curtarolo:art124, curtarolo:art129}.\nThe {\\small API}\\ allows users to submit structural data for the material of interest using a utility such as \\verb|cURL|,\nand then returns the results of the model's predictions in {\\small JSON}\\ format.\nThe programmatic access to machine learning predictions enables the incorporation of machine learning into\nmaterials design workflows, allowing for rapid pre-screening to automatically select\npromising candidates for further investigation.\n\n\\clearpage\n\\section{The Structure and Composition Statistics of 6A Binary and Ternary Crystalline Materials}\n\\label{sec:art130}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art130}.\n\n\\subsection{Introduction}\nThe creation of novel materials with optimal properties for diverse applications requires a fundamental\nunderstanding of the factors that govern the formation of crystalline\nsolids from various mixtures of elements.\nCompounds of the non-metallic elements of column 6A, oxygen, sulfur and selenium, are of particular interest.\nThey serve in a large variety of applications\nin diverse fields of technology, \\nobreak\\mbox{\\it e.g.}, chemistry, catalysis, optics,\ngas sensors, electronics, thermoelectrics, piezoelectrics,\ntopological insulators, spintronics and more~\\cite{eranna2004oxide,fortunato2012oxide,tsipis2008electrode,jiang1998new,panda2009review,shi2017,lorenz20162016,ruhle2012all}.\nGiven the very large number of possibilities, many of the alloy systems of these elements have not\nbeen fully investigated, some of them even not at all.\n\nIn recent years, high-throughput computational techniques based on \\nobreak\\mbox{\\it ab-initio}\\ calculations\nhave emerged as a potential route to bridge these experimental gaps and\ngain understanding of the governing principles of compound formation~\\cite{nmatHT}.\nThis led to the creation of large databases of computational materials\ndata~\\cite{aflowPAPER,CMS_Ong2012b}.\nYet, these computational approaches are practically limited by the number and size of structures\nthat can be thoroughly analyzed, and fundamental issues that limit\nthe applicability of standard semi-local {\\small DFT}\\ for non-metallic compounds.\nThe sought-after governing principles are thus still largely unknown.\n\nNevertheless, the considerable body of experimental data that is already available,\nalthough by no means complete, is a useful basis for large-scale data analysis.\nThis experimental data is usually presented in\ncompendiums that lack statistical analysis.\nPresenting this data in a structured manner may be conducive for gaining insights\ninto the essential factors that determine structure formation, and may help to provide\nmaterial scientists with the necessary foundation for rational\nmaterials design.\n\nAnalyses recently carried out for the intermetallic binaries~\\cite{dshemuchadse2014some}\nand ternaries~\\cite{dshemuchadse2015more} have uncovered interesting Bravais lattices distributions and an unexpected large prevalence of unique structure types.\nHere we extend the analysis and discuss trends, as well as special phenomena, across\nbinary and ternary compounds of the 6A non-metals.\nThis analysis reveals the following\ninteresting observations:\n\\begin{itemize}[leftmargin=*]\n\n \\item Considerable overlap exists between the sulfides and selenides:\n about a third of the total number of structure types are shared among\n both compound families.\n In contrast, the overlap between the oxides and the other two families is rather small.\n\n \\item The prevalence of different compound stoichiometries in the sulfide\n\tand selenide families is very similar to each other\n\tbut different from that of the oxides. Some stoichiometries\n\tare abundant in the oxides but are {\\it almost\n absent} in the sulfides or selenides, and vice versa.\n\n \\item The number of ternary oxide stoichiometries, $A_{x}B_{y}$O$_{z}$, decreases when the product of\n binary oxide stoichiometries, of participating elements, increases. This behavior can be explained by general thermodynamic arguments and is discussed in the text.\n\n \\item Overall, oxide compounds tend to have richer oxygen content than the sulfur and selenium content in their corresponding compounds.\n\n \\item Across all three compound families, most structure types are represented\n by only one compound.\n\n \\item High symmetry lattices, \\nobreak\\mbox{\\it e.g.}, the orthorhombic face centered,\n orthorhombic body centered and cubic lattices\n\t are relatively rare among these compounds.\n This reflects the spatial arrangement of the compound forming orbitals of the 6A non-metals,\n whose chemistry does not favor these structures.\n\n\\end{itemize}\n\nIn the analysis presented here, we adopt the ordering of the elements by Mendeleev numbers as\ndefined by Pettifor~\\cite{pettifor:1984,pettifor:1986},\nand complement it by investigating the crystallographic properties of\nthe experimentally reported compounds.\nPettifor maps constructed for these compound families exhibit similar separation between different structure types as the\nclassical Pettifor maps for binary structure types~\\cite{pettifor:1984,pettifor:1986}.\nFor some stoichiometries, the structure types show similar patterns in\nthe maps of the three compound families, suggesting\nthat similar atoms tend to form these stoichiometries with all three elements.\nSuch similarity of patterns is more common between\nthe sulfides and selenides than between either of them and the oxides.\n\nThese findings suggest a few possible guiding principles for directed searches of new compounds.\nElement substitution could be used to examine favorable candidates within the\nimperfect overlaps of the structure distributions, especially between the sulfides and selenides.\nMoreover, the missing stoichiometries and structure symmetries mean that data-driven approaches, \\nobreak\\mbox{\\it e.g.},\nmachine learning, must use training sets not limited to one compound family, even in studies directed at that specific set of compounds.\nThis hurdle may be avoided by augmenting the known structures with those of the other families.\nIn addition, identified gaps in the Mendeleev maps suggest potential new compounds,\nboth within each family or by correlations of similar structure maps across the different families.\n\n\\subsection{Data methodology}\n\n\\begin{table}[tp]\\centering\n\\mycaption{Data extraction numerical summary.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\n & compounds & unique compounds & structure types \\\\\n\\hline\ntotal & 88,373 & 50,294& 13,324 \\\\\n\\hline\nunary & 1752 & 499 & 197\\\\\n\\hline\nbinary & 27,487 & 10,122 & 1,962 \\\\\nbinary oxides & 3,256 & 844 & 538 \\\\\nbinary sulfides & 1,685 & 495 & 270 \\\\\nbinary selenides & 1,050 & 332 & 168 \\\\\n\\hline\nternary & 37,907 & 23,398& 4,409\\\\\nternary oxides & 10,350 & 5,435 & 2,079 \\\\\nternary sulfides & 3,190 & 2,041 & 784 \\\\\nternary selenides & 1,786 & 1,256 & 521\\\\\n\\hline\nquaternary & 15,138 & 11,050 & 3,855 \\\\\n5 atoms & 4,638 & 3,899 & 2,053 \\\\\n6 atoms & 1,219 & 1,101 & 682 \\\\\n7 atoms & 212\t& 201 &\t154 \\\\\n8 atoms & 20 & 20 & 12\\\\\n\\end{tabular}\n\\label{tab:art130:ICSD_DATA}\n\\end{table}\n\nThe {\\small ICSD}~\\cite{ICSD_database} includes approximately 169,800 entries (as of August 2016).\nFor this study we exclude all entries with partial or random occupation and those that do not have full structure data.\nThe remaining set of structures has been filtered using the {\\small AFLOW}\\ software~\\cite{curtarolo:art49,curtarolo:art53,curtarolo:art57,aflowBZ,curtarolo:art63,aflowPAPER,curtarolo:art85,curtarolo:art110,monsterPGM,aflowANRL,aflowPI},\nwhich uses an error checking protocol to ensure the integrity of each entry.\n{\\small AFLOW}\\ generates each structure by appropriately propagating the Wyckoff positions of the specified spacegroup.\nThose structures that produce inconsistencies, \\nobreak\\mbox{\\it e.g.}, overlapping atoms or a different stoichiometry\nthan the structure label are ignored.\nIf atoms are detected to be too close ($\\leq 0.6$\\AA), alternative standard ITC\n(International Table of Crystallography)~\\cite{tables_crystallography} settings of the spacegroup are attempted.\nThese settings define different choices for the cell's unique axes, possibly\ncausing atoms to overlap if not reported correctly.\nOverall, these considerations reduce the full set of {\\small ICSD}\\ entries to a\nmuch smaller set of 88,373 ``true'' compounds.\nThese entries are contained in {\\small AFLOW}\\ Database~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}.\nThey include the results of the {\\small AFLOW}\\ generated full symmetry analysis for each structure, \\nobreak\\mbox{\\it i.e.}, Bravais lattice,\nspace group and point group classifications, and Pearson symbol\n(the method and tolerances used for this analysis follow the {\\small AFLOW}\\ standard~\\cite{curtarolo:art104}).\nFor the analysis presented here we identify all the binary and ternary compounds included in this set,\n27,487 binary entries and 37,907 ternary entries.\nFrom these, we extract all the entries that contain oxygen, sulfur or selenium as one of the components.\nOf the binaries, we find 3,256 oxides, 1,685 sulfides and 1,050 selenides.\n10,530 oxides, 3,190 sulfides and 1,786 selenides are found among the ternaries.\nDuplicate entries representing different experimental reports of the same compound,\n\\nobreak\\mbox{\\it i.e.}, the same elements, stoichiometry, space group and Pearson designation, are then eliminated\nto obtain a list in which every reported compound is represented by its most recent corresponding entry in the {\\small ICSD}.\nThis reduces our list of binaries to 844 oxides, 495 sulfides and 332 selenides, and\nthe list of ternaries to 5,435 oxides, 2,041 sulfides and 1,256\nselenides.\nThese results are summarized in Table~\\ref{tab:art130:ICSD_DATA}.\nThroughout the rest of the study, we will refer to these sets of\nbinary and ternary compounds. We choose not to discuss multi-component structures with four or more elements since their\nrelative scarcity in the database most probably indicates incomplete\nexperimental data rather than fundamental issues of their chemistry.\nIt is also instructive to check the effect of element abundance on the number of compounds.\nThe abundance of oxygen in the earth's crust is $\\sim47\\%$ by weight,\naround 1000 times more than that of sulfur ($\\sim697$~ppm) which is around $5,000$\nmore abundant than selenium ($120$~ppb)~\\cite{wedepohl1995composition}.\nComparison with the number of elements (O\/S\/Se) binary compounds, 844\/495\/332,\nor ternary compounds, 5,435\/2,041\/1,256, makes it clear that while a rough correlation\nexists between the elements' abundance and the number of their known compounds,\nit is by no means a simple proportion.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.6\\linewidth]{fig010}\n\\mycaption[Distributions of the compounds among structure types for binary (inset) and ternary compounds.]\n{Oxides are shown in blue, sulfides in yellow and selenides in green.\nThe binary distributions differ mostly by the length of their single-compound prototypes tails,\nwhile the ternary distribution of the oxides deviates significantly from those of the sulfides and selenides.}\n\\label{fig:art130:prototypes_distribution_curves_log}\n\\end{figure}\n\nIn the next stage, we identify unique structure types.\nStructure types are distinguished by stoichiometry, space group, and Pearson designation, without consideration\nof the specific elemental composition.\nThis implicit definition of structure type is common in the literature~\\cite{Villars2013, PaulingFile},\nand we use it throughout the study as\nproviding a good balance of clarity and simplicity.\nHowever, it should be noted that there are a few rare cases of complex structures where a given\nstructure type under this definition includes a few sub-types (see Figure~\\ref{fig:art130:structure_types_comparison}).\nExamples exist of more complex definitions of structure types, formulated to define similarities\nbetween inorganic crystals structures~\\cite{lima1990nomenclature}.\n\nThe binary structure type lists contain 538 oxides, 270 sulfides and 168 selenides.\nThe ternary lists contain 2,079 oxides, 784 sulfides and 521 selenides.\nThis means that 64\\% of the binary oxides, 55\\% of the sulfides and 51\\% of the selenides are distinct structure types.\nThe corresponding ratios for the ternaries are 38\\% of the oxides, 38\\% of the sulfides and 41\\% of the selenides.\nAll the other entries in the compound lists represent compounds of the same\nstructure types populated by different elements.\nDifferently put, this means that there are on average about 1.6 compounds per structure type in the binary oxides,\n$1.8$ in the binary sulfides and $2$ in the binary selenides.\nAmong the ternaries, the corresponding numbers are 2.6 compounds per structure type in the oxides,\n$2.6$ in the sulfides and $2.4$ in the selenides.\nThese numbers may be compared to the intermetallics, where there are\n20,829 compounds of which 2,166, about 10\\%,\nare unique structure types~\\cite{dshemuchadse2014some}.\nThere are about seven compounds per structure types in the binary intermetallics and about nine in the ternaries.\nThe number for binary intermetallics is considerably larger than for ternary oxides, sulfides or selenides. Together with the higher proportion of unique structure types in the latter, this reflects the limits on\nmaterials chemistry imposed by the presence of one of those 6A elements.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig011}\n\\mycaption[Distributions of structure types among (\\textbf{a}) binary and (\\textbf{b}) ternary stoichiometries.]\n{Oxides are shown in blue, sulfides in yellow and selenides in green.\nThe distributions of the selenides and sulfides are quite similar while those of the\noxides deviate significantly, as detailed in the text.}\n\\label{fig:art130:stoi_hist}\n\\end{figure}\n\nIt should be noted that this structure selection procedure produces lists that partially overlap,\n\\nobreak\\mbox{\\it i.e.}, certain structure types may appear in more than one list,\nsince there might be oxide structure types that are also represented among\nthe sulfide or selenide structures, and vice versa.\n11\\% of the binary oxide structure types also appear in the binary sulfides list and 8\\% are represented\nin the binary selenides list.\n33\\% of the binary sulfide are also represented in the selenides list.\nThe total number of binary oxides, sulfides and selenides structure types is 976, which is reduced by 16\\%,\nto 818 structure types, by removing all overlaps.\nThe corresponding overlap ratios for the ternaries are 10\\% for the oxides and sulfides,\n6\\% for the oxides and selenides and 31\\% for the sulfides and selenides.\nThe total number of entries in the ternary oxides, sulfides, and selenides structure type lists is 3,384,\nwhich is reduced to 2,797 structure types by removing all overlaps, a 17\\% reduction.\nTherefore, the overlaps between these three compound families are similar for the binaries and ternaries.\nIn both, the overlap between the oxides and the other two families is rather small,\nwhereas the overlap between the sulfides and selenides represents about a third of the total number of structure types.\n\nThe sequence of Mendeleev numbers includes 103 elements, from hydrogen to lawrencium\nwith numbers 1-6 assigned to the noble gases, 2-16 to the alkali metals and alkaline earths,\n17-48 to the rare earths and actinides, 49-92 to the metals and metalloids and 93-103 to the non-metals.\nOf these, noble gases are not present in compounds and artificial elements\n(metals heavier than uranium) have very few known compounds.\nWe are thus left with 86 elements, of which the above compounds are composed.\nThat means there are about ten times more binary oxides than\nelement-oxygen combinations, about six times more sulfides than element-sulfur\ncombinations and four times more selenides than element-selenium combinations.\nOxides are much more common than sulfides and selenides.\nThe corresponding numbers for the ternaries are much lower.\nThere are about 1.6 times more ternary oxides than two-element-oxygen ternary possible systems,\nabout 0.6 times less ternary sulfides and about 0.4 times less ternary selenides than the corresponding two-element combinations.\nThe ternaries are relatively quite rare, more so as we progress from oxides to sulfides and then to selenides.\nA similar analysis of the intermetallic binaries in Reference~\\onlinecite{dshemuchadse2014some} shows that of the 20,829 intermetallics,\n277 are unaries (about three times more than possible metal elements), 6,441 are binaries\n(about two times more than possible metal binary systems),\nand 13,026 are ternaries (6.5 times less than possible metal ternary systems).\nThis means that unary metal structures are less common among the metallic\nelements than the oxide, sulfide and binary selenide compounds among their corresponding binary systems.\nThis seems to reflect simply the larger space of stoichiometries available to binaries over unaries.\nHowever, on the contrary, the intermetallic binary compounds are more common among the metallic binary\nsystems than the oxide, sulfide and ternary selenide compounds among their corresponding ternary systems.\nThis discrepancy again reflects either the chemical constraints imposed by the presence of a 6A non-metal on the\nformation of a stable ternary structure, or simply gaps in the\nexperimental data since many ternary systems have not been thoroughly investigated.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.55\\linewidth]{fig012}\n\\mycaption[Composition distributions of binary (\\textbf{a}) oxide, (\\textbf{b}) sulfide, and (\\textbf{c}) selenide stoichiometries.]\n{The count indicates the number of different stoichiometries that include the respective element.\nThe colors go from no stoichiometries (white) to the maximal number of stoichiometries\n(dark blue) which is different for each element, 19\/8\/9 for O\/S\/Se.\nIslands of high prevalence appear for the 4B and 5B transition metals and\nthe heavy alkalis in all three compound families.\nAdditional, smaller islands appear in the sulfides and selenides for the 8 and\n1B transition metals and the 3A and 5A semi-metals.}\n\\label{fig:art130:stoi_periodic_oxide}\n\\end{figure}\n\n\\subsection{Results and discussion}\n\\boldsection{Structure types.}\nThe distribution of the binary and ternary compounds among the corresponding structure types is shown\nin Figure~\\ref{fig:art130:prototypes_distribution_curves_log}.\nDetailed data for the most common structure types is presented in Tables~\\ref{tab:art130:oxide_binary_data}-\\ref{tab:art130:selenide_ternary_data}.\n\nAbout 84\\% of the binary oxide structure types represent a single\ncompound, characterizing the tail end of the binary oxide distribution.\nThey include about 53\\% of the binary oxide compounds.\nThe most common structure type represents 29 compounds,\n3.4\\% of the oxide compounds list.\nAmong the binary sulfides, 76\\% of the structure types represent a single compound.\nThey include 41\\% of the binary sulfide compounds.\nThe most common structure type represents 32 compounds, 6.5\\% of the\nsulfide compounds list.\nAmong the binary selenides, 76\\% of the structure types represent a single compound.\nThey include 39\\% of the binary selenide compounds.\nThe most common structure type represents 31 compounds, 9.3\\% of the selenide compounds list.\n\nIn all three binary lists the most common structure type is rock salt (NaCl).\nThe binary oxide structure type distribution has a much longer tail than the sulfides and selenides,\n\\nobreak\\mbox{\\it i.e.}, more oxide compounds have unique structure types.\nThe most common structure type in these three distributions represents\na similar number of compounds but a smaller proportion of the corresponding compounds in the oxides.\nThe middle regions of the distributions are very similar\n(inset Figure~\\ref{fig:art130:prototypes_distribution_curves_log}).\nThis means that the much larger number of binary oxide compounds, compared to the sulfides and selenides,\nis expressed at the margin of the distribution, in the long tail of unique compounds.\n\nThis discrepancy between the three binary distributions is much less\napparent among the ternary compounds.\n64\\% of the ternary oxide structure types represent a single compound.\nThey include 24\\% of the ternary oxide compounds.\nThe two most common structure types, pyrochlore and perovskite, represent 116 and 115 compounds,\nrespectively, about 2\\% each of the entire compounds list.\nAmong the ternary sulfides, 70\\% of the structure types represent a single compound.\nThey include 34\\% of the ternary sulfide compounds.\nThe most common structure type, delafossite, represents 65 compounds,\n4\\% of the entire compounds list.\nAmong the ternary selenides, 62\\% of the structure types represent a single compound.\nThey include 26\\% of the ternary selenide compounds.\nThe most common structure type, again delafossite, represents 51 compounds, 4\\% of the ternary sulfides.\n\nIn contrast to the binaries, the larger count of ternary oxides, compared to the sulfides and selenides,\nis expressed by a thicker middle region of the structure type distribution,\nwhereas the margins have a similar weight in the distributions of the three compound families.\n\n\\boldsection{Binary stoichiometries.}\nThe structure types stoichiometry distribution for the binary oxide, sulfide and selenide compounds is shown in\nFigure~\\ref{fig:art130:stoi_hist}(a).\nWe define the binaries as $A_xB_y$, where $B$ is O, S or Se, and the number of structure types is shown as a function of $y\/(y+x)$.\nA very clear peak is found for the oxides at the stoichiometry 1:2,\n$A$O$_2$, while both the sulfides and selenides have a major peak at 1:1, $A$S and $A$Se, respectively.\n\nFor $y\/(y+x)<0.5$, there are more gaps in the plot (missing stoichiometries) for the oxides compared\nto the sulfides and selenides, while for $y\/(y+x)>0.6$ there are more gaps in the sulfides and selenides,\nthis behavior is shown in detail in Tables~\\ref{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries}-\\ref{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries_3}.\nAn important practical conclusion is that augmenting the binary oxide structure types with\nthose of sulfides and selenides will produce a more extensive coverage of possible stoichiometries.\n\nAnother interesting property is the number of stoichiometries\nfor each of the elements in the periodic table.\nThe prevalence of binary oxide stoichiometries per element is shown in Figure~\\ref{fig:art130:stoi_periodic_oxide}(a).\nA few interesting trends are evident --- the first row of transition metals shows a peak near vanadium (19 stoichiometries)\nand titanium (14 stoichiometries).\nHafnium, which is in the same column of titanium has only a single stoichiometry --- HfO$_{2}$.\nBoth the beginning and end of the $d$-elements exhibit a small amount of stoichiometries --- scandium\nwith only one and zinc with only two.\nThe two most abundant elements, silicon and oxygen, form only a single stoichiometry in the\n{\\small ICSD}\\ --- SiO$_{2}$, with 185 {\\it different} structure types.\nAnother interesting trend is evident for the alkali metals, where rubidium and cesium have more\nstoichiometries --- perhaps related to the participation of $d$-electrons in the chemical bonds.\n\nFigures~\\ref{fig:art130:stoi_periodic_oxide}(b) and (c)\nshow the binary stoichiometries prevalence per element for sulfur and selenium respectively.\nSimilar trends are exhibited --- there are two ``islands'' of large number of stoichiometries\nin the transition metals: one around vanadium and titanium and the other near nickel and copper.\nEvidently, prime candidates for new compounds should be searched among structures in the vicinity of\nthese high density islands, especially for elements that exhibit a considerably higher density in one family.\n\n\\boldsection{Ternary stoichiometries.}\nSimilar to the binaries, the ternary stoichiometries are designated\n$A_xB_yC_z$, where $C$ is O, S or Se.\nThe distributions of the ternaries are, as might be expected, more complex,\nwith maxima at $z\/(x+y+z)=0.6$ for the oxides, $z\/(y+x+z)=0.55$ for the sulfides and $z\/(y+x+z)=0.5$ for the selenides.\nThe major peaks still appear at integer and half integer values, but with more minor peaks at intermediate values.\nThis behavior is shown in Figure~\\ref{fig:art130:stoi_hist}(b).\nThe ternary selenide and sulfides distributions are again nearly identical, and there are\nalmost no compounds with ratios larger than $0.75$ in the oxides or larger than $0.66$ in the sulfides and selenides.\nHowever, there are few sulfide and selenide compounds around 0.8 and 0.85 but no oxides.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Ternary stoichiometry data: $A_xB_yC_z$.]{``$C$-rich'' refers to stoichiometries where $z>x+y$.}\n\\vspace{3mm}\n\\begin{tabular}{ l|r|r|r }\n& oxygen & sulfur & selenium \\\\\n\\hline\nNumber of stoichiometries & 585 & 282 & 206 \\\\\n$C$-rich stoichiometries ratio & 0.85 & 0.67 & 0.66 \\\\\n$C$-rich compound ratio & 0.92 & 0.77 & 0.73 \\\\\n\\end{tabular}\n\\label{tab:art130:ternary_stoi}\n\\end{table}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig013}\n\\mycaption[Prevalence of stoichiometries among ternary compounds.]\n{Panels include\n(\\textbf{a}) oxide,\n(\\textbf{b}) sulfide,\n(\\textbf{c}) selenide compounds,\nand, for reference,\n(\\textbf{d}) all the possible stoichiometries with up to 12 atoms\nof each component per unit cell.\nIn each figure, the smaller circles are normalized to the biggest one, which denotes the highest prevalence, \\nobreak\\mbox{\\it i.e.},\n718 for oxides, 242 for sulfides, and 145 for selenides,\nin addition a heat map color scheme is used where blue means low prevalence and red means the highest prevalence for each element.\nThe $x$ and $y$ axes denote the atomic fractions in the ternaries $A_xB_yC_z$, where $C$ is O, S or Se, respectively.\n$A$ and $B$ are ordered by Mendeleev number where $M_A>M_B$.}\n\\label{fig:art130:triangles}\n\\end{figure}\n\nAnother perspective of ternary stoichiometries is demonstrated in Figure~\\ref{fig:art130:triangles}\nwhich shows the abundance of the most common stoichiometries.\nThe biggest circle in each diagram denotes the prevalence of the most common stoichiometry\n(number of unique compounds for this stoichiometry),\nwhich is 718 ($x=1$, $y=1$, $z=3$) for oxides, 242 ($x=1$, $y=1$, $z=2$) for sulfides, and 145 ($x=1$, $y=1$, $z=2$) for selenides.\nThe smaller circles in each plot are normalized to the corresponding highest prevalence.\n\nThese diagrams highlight the similarities as well as important differences between the three families of compounds.\nIn all three cases, the most common stoichiometries appear on the symmetry axis of the diagram, \\nobreak\\mbox{\\it i.e.},\nat equal concentrations of the $A$ and $B$ components, or very close to it.\nFor the oxides, they are concentrated near 0.5-0.6 fraction of oxygen, representing the\n$A_1B_1$O$_2$ and $A_1B_1$O$_3$ stoichiometries, respectively,\nand form a very dense cluster with many similar reported stoichiometries of lower prevalence.\nOutside this cluster, the occurrence of reported compositions drops sharply, and other regions\nof the diagram are very sparsely populated, in particular near the vertices of the $B$ and O components.\n\nThe sulfide and selenide diagrams also exhibit prominent clusters on the $AB$ symmetry axes,\nbut they appear at a lower S or Se concentration of about 0.5, \\nobreak\\mbox{\\it i.e.}, $A_1B_1C_2$ stoichiometry.\nThey are considerably more spread out and include a significant contribution at the $ABC$ stoichiometry.\nIn both sulfides and selenides, an additional minor cluster appears closer to the $A$ vertex (Figure~\\ref{fig:art130:triangles}).\nA few members of this cluster are ternary oxides, reflecting the high electronegativity and high Mendeleev number (101) of oxygen.\nThe $B$ and $C$ vertex regions are still sparsely populated, but less so than in the oxides case.\nOverall, the sulfide and selenide diagrams are very similar to each other and different from that of the oxides.\nThey are more spread out, less $AB$ symmetric than the oxide diagram and less tilted towards rich $C$-component concentration.\nThis discrepancy may reflect some uniqueness of oxygen chemistry compared to sulfur and selenium,\nor rather simply reflect the oxygen rich environment in which naturally formed compounds are created in the atmosphere.\nThe number of stoichiometries and the differences in the $C$-component concentration are summarized in Table~\\ref{tab:art130:ternary_stoi}.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Distribution of the oxide, sulfide and selenide compounds and structure types among the 14 Bravais lattices.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\n& \\multicolumn{3}{R{3cm}|}{binary compounds}\n& \\multicolumn{3}{R{3cm}|}{binary structure types}\n& \\multicolumn{3}{R{3cm}|}{binary compounds per structure type}\n& \\multicolumn{3}{R{3cm}|}{ternary compounds}\n& \\multicolumn{3}{R{3cm}|}{ternary structure types}\n& \\multicolumn{3}{R{3cm}}{ternary compounds per structure type}\\\\ \\hline\n& O & S & Se & O & S & Se & O & S & Se & O & S & Se & O & S & Se & O & S & Se \\\\ \\hline\naP & 51 & 13 & 5 & 39 & 12 & 5 & 1.3 & 1.1 & 1 & 378 & 79 & 60 & 219 & 56 & 39 & 1.7 & 1.4 & 1.5 \\\\\nmP & 82 & 54 & 31 & 62 & 36 & 20 & 1.3 & 1.5 & 1.6 & 918 & 318 & 198 & 363 & 166 & 109 & 2.5 & 1.9 & 1.8 \\\\\nmS & 88 & 31 & 22 & 58 & 21 & 15 & 1.5 & 1.5 & 1.5 & 672 & 251 & 170 & 292 & 117 & 77 & 2.3 & 2.1 & 2.2 \\\\\noP & 123 & 82 & 48 & 81 & 37 & 30 & 1.5 & 2.2 & 1.6 & 950 & 481 & 266 & 373 & 139 & 105 & 2.5 & 3.5 & 2.5 \\\\\noS & 39 & 24 & 11 & 36 & 19 & 9 & 1.1 & 1.3 & 1.2 & 334 & 84 & 60 & 133 & 40 & 25 & 2.5 & 2.1 & 2.4 \\\\\noF & 11 & 7 & 11 & 10 & 6 & 4 & 1.1 & 1.2 & 2.8 & 51 & 32 & 23 & 28 & 14 & 8 & 1.8 & 2.3 & 2.9 \\\\\noI & 22 & 5 & 2 & 20 & 4 & 2 & 1.1 & 1.25 & 1 & 89 & 36 & 27 & 39 & 15 & 12 & 2.3 & 2.4 & 2.25 \\\\\ntI & 41 & 20 & 10 & 31 & 17 & 8 & 1.3 & 1.2 & 1.25 & 418 & 80 & 72 & 101 & 34 & 23 & 4.1 & 2.4 & 3.1 \\\\\ntP & 78 & 27 & 28 & 48 & 13 & 16 & 1.6 & 2.1 & 1.75 & 239 & 73 & 52 & 107 & 39 & 26 & 2.2 & 1.9 & 2.0 \\\\\nhP & 94 & 87 & 66 & 62 & 50 & 32 & 1.5 & 1.7 & 2.1 & 435 & 224 & 103 & 198 & 75 & 41 & 2.2 & 3.0 & 2.5 \\\\\nhR & 40 & 44 & 20 & 30 & 33 & 15 & 1.3 & 1.3 & 1.3 & 420 & 230 & 133 & 123 & 49 & 33 & 3.4 & 4.7 & 4.0 \\\\\ncP & 42 & 22 & 20 & 21 & 6 & 4 & 2.0 & 3.7 & 5.0 & 187 & 58 & 43 & 45 & 18 & 13 & 4.2 & 3.2 & 3.3 \\\\\ncF & 75 & 65 & 48 & 19 & 10 & 6 & 3.9 & 6.5 & 8.0 & 251 & 80 & 43 & 27 & 17 & 7 & 9.3 & 4.7 & 3.9 \\\\\ncI & 58 & 14 & 10 & 21 & 6 & 2 & 2.8 & 2.3 & 5.0 & 92 & 15 & 6 & 30 & 5 & 3 & 3.1 & 3.0 & 2.0 \\\\\n\\end{tabular} }\n\\label{table:bravais_lattice_distribution}\n\\end{table}\n\nAnother interesting observation is that while some stoichiometries are abundant in the oxides\nthey are almost absent in the sulfides or the selenides. For example,\nthere are 299 compounds with the $A_2B_2$O$_7$ stoichiometry (ignoring\norder between $M_A$ and $M_B$), but only two $A_2B_2$S$_7$ compounds\nand no $A_2B_2$Se$_7$ compounds. Also, there are 71 $A_1B_3$O$_9$\ncompounds but no $A_1B_3$S$_9$ and $A_1B_3$Se$_9$ compounds. On the\nother hand, there are no $A_4B_{11}X_{22}$ oxides, but 20 sulfides and 8 selenides.\nIf we require that $M_A>M_B$, there are no oxides of the $A_3B_2X_2$\nstoichiometry, but 25 sulfides and 7 selenides.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.55\\linewidth]{fig014}\n\\mycaption[Composition distributions of ternary (\\textbf{a}) oxide, (\\textbf{b}) sulfide, and (\\textbf{c}) selenide stoichiometries.]\n{The count indicates the number of different stoichiometries that include the respective element.\nThe colors go from no stoichiometries (white) to the maximal number of stoichiometries (dark blue)\nwhich is different for each element, 96\/59\/51 for O\/S\/Se.\nHigh prevalence appears for the alkali metals in all three compound families.\nAn additional island in the transition metals is much more pronounced in the oxides.\nThe sulfides and selenides distributions are nearly identical, and show high prevalence of oxygen containing ternaries.}\n\\label{fig:art130:tern_stoi_periodic_oxide}\n\\end{figure}\n\nAgain, an important conclusion is that there are many missing stoichiometries,\nFigure~\\ref{fig:art130:triangles}(d) shows all the possible stoichiometries for $A_xB_yC_z$ for $x,y,z \\le 12$,\nclearly showing rich concentration in the middle, which is not the case for oxides, and also to a\nlesser degree to sulfides and selenides.\n\nWe can repeat the analysis of the binary stoichiometries and ask how many stoichiometries\nper element are there for the ternaries.\nThis is shown in Figure~\\ref{fig:art130:tern_stoi_periodic_oxide}.\nHere, also, the similarity of sulfides and selenides is clear.\nIn addition, while there are similarities between the distributions of binary stoichiometries\nper element to the ternary distributions, there are also obvious differences.\nOne might guess that there should be a correlation between the binary and ternary distributions.\nThis is examined in Figure~\\ref{fig:art130:bintern1a}(a).\n\nIt is evident that the correlation between ternary and binary number of stoichiometries is not strong\nbut the minimal number of ternary stoichiometries tends to grow with the number of binary stoichiometries.\nWe check this further in Figure~\\ref{fig:art130:bintern1a}(b), by comparing the number of ternary stoichiometries of\n$A_xB_y$O$_z$ to the product of stoichiometry numbers of $A_x$O$_y$ and $B_x$O$_y$.\nThe general trend obtained is an inverse correlation, \\nobreak\\mbox{\\it i.e.}, as the product of the numbers of binary\nstoichiometries increases, the number of ternaries decreases. This trend can be explained by the following argument: when the two binaries are rich with stable compounds, the ternaries need to compete with more possibilities of\nbinary phases, which makes the formation of a stable ternary more difficult.\nIn Figure~\\ref{fig:art130:bintern1a}(b), this trend is highlighted for vanadium,\nthe element with the most binary stoichiometries, but this pattern repeats itself for most elements.\nWe analyze this behavior for the sulfides and selenides in Section~\\ref{subsec:art130:prev_stoich_supp}, similar trends are found but\nthey are less pronounced due to a smaller number of known compounds.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig015}\n\\mycaption[Analysis of ternary \\nobreak\\mbox{\\it vs.}\\ binary stoichiometry counts for oxides.]\n{(\\textbf{a}) The number of ternary oxide stoichiometries per element as a function\nof the count of its binary stoichiometries.\nThe dashed line marks perfect similarity $(y=x)$, and the dotted line marks the ratio $y=4x$.\n(\\textbf{b}) The number of ternary oxide stoichiometries as a function of the\nproduct of the numbers of the binary stoichiometries of participating elements.\nThe data for vanadium is shown with red crosses, all the rest is shown with blue circles.}\n\\label{fig:art130:bintern1a}\n\\end{figure}\n\n\\boldsection{Composition and Mendeleev maps.}\nThe occurrence of each element in the binary and ternary compound lists has been\ncounted and tabulated.\nThe results are described in Figure~\\ref{fig:art130:mendeleev_distribution_all_in_one}.\nFor the binary oxides a very prominent peak appears at $M=85$, the\nMendeleev number of silicon.\nIt represents the 185 different silicon oxide\nstructures types reported in the {\\small ICSD}\\ database for just a {\\it single} stoichiometry, SiO$_2$.\nSmaller peaks appear for $M=51$ (titanium, 42 structure types, 14 stoichiometries,\nleading stoichiometry is\nTiO$_2$ with 14 structure types), $M=54$ (vanadium, 42 structure types,\n18 stoichiometries, leading stoichiometry is VO$_2$ with 10 structure types),\n$M=56$ (tungsten, 24 structure types, 9 stoichiometries, leading stoichiometry is WO$_3$ with 13 structure types),\nand $M=45$\n(uranium, 22 structure types, 9 stoichiometries, leading stoichiometries are UO$_2$ and U$_3$O$_8$ with 6 structure types each).\nUnlike the silicon peak which is composed of a single stoichiometry,\nthe other leading peaks evidently include multiple stoichiometries, reflecting the different chemistry of those elements.\nThese differences also carry over into the ternary oxide compounds involving those elements.\nFor example, the stoichiometry distribution of silicon ternary oxides is more tilted towards\nthe silicon poor compounds compared to the corresponding distributions of vanadium and titanium ternary oxides,\nas is shown in Figure~\\ref{fig:art130:specific_triangle_stoichiometries}.\n\nThe distribution of the sulfides is generally much lower than\nthat of the oxides, due to the much smaller total number of known binaries, but is also more uniformly structured.\nIt has one major peak\nfor $M=76$ (zinc, 40 structure types, 2 stoichiometries, leading stoichiometry is ZnS with 39 structure types),\nand quite a few smaller ones such as $M=51$ (titanium, 16 structure types, 5 stoichiometries, leading stoichiometry is TiS$_2$ with 9 structure types),\n$M=61$ (iron, 18 structure types, 5 stoichiometries, leading stoichiometry is FeS with 6 structure types),\n$M=67$ (nickel, 16 structure types, 6 stoichiometries, leading stoichiometry is NiS$_2$ with 8 structure types),\n$M=90$ (phosphorus, 13 structure types, 8 stoichiometries, of which\nP$_2$S$_7$, P$_4$S$_9$, P$_4$S$_6$, P$_4$S$_5$ and P$_4$S$_3$ have 2 structure types each).\nThe $M$~=~8--33 region also exhibits a minor concentration of\nparticipating elements.\nThe selenides distribution is yet smaller than that of the sulfides, and\neven more uniform.\nSeveral peaks appear, $M=51$ (titanium, 13 structure types, 9 stoichiometries, leading stoichiometry is\nTiSe with 3 structure types),\n$M=52$ (niobium, 15 structure types, 8 stoichiometries, leading stoichiometry is\nNbSe$_2$ with 8 structure types),\n$M=53$ (tantalum, 15 structure types, 4 stoichiometries, leading stoichiometry is\nTaSe$_2$ with 10 structure types) and\n$M=79$ (indium, 14 structure types, 5 stoichiometries, leading stoichiometry is In$_2$Se$_3$ with 6 structure types).\nAll distributions cover most of the elements except two obvious gaps, one at $M<9$,\nwhich includes the noble gases and the two heaviest alkali metals, cesium and francium, and another\nat $34\\leq M\\leq 42$ which represents the heavy actinides. Another gap appears in the sulfide and selenide distributions at $91\\leq M\\leq 97$,\nwhich reflects the rarity of polonium and astatine compounds and shows that the elements of the 6A column,\nexcept oxygen, do not coexist, in the known compounds, with each other\nor with the heavier halogen iodine.\n\nThe element occurrence distributions for the ternary oxides, sulfides and\nselenides exhibit greater similarity than the\ncorresponding binary distributions. The most apparent difference, however, is the\nmost common component, which is sulfur, $M=90$, in the\noxides, but oxygen itself, $M=101$, in the sulfides and selenides. The\nsulfide and selenide distributions are almost the same, except for\ngenerally lower numbers in the selenides (due to the smaller total\nnumber of compounds) and an apparent lower participation of the\nlanthanides $M$~=~17--35.\n\nMendeleev maps for the ternaries are shown in\nFigures~\\ref{fig:art130:mendeleev_bigger_x_upper_all_in_one}-\\ref{fig:art130:mendeleev_sulfur_selenium_prototypes}.\nFigure~\\ref{fig:art130:mendeleev_bigger_x_upper_all_in_one} shows the cumulated maps for all\nstoichiometries reported for the respective ternary family.\nThey reflect the same major gaps as the binary distributions.\nThe maps show that most of the reported compositions are represented by one or two compounds\nwith just a few hot-spots that include up to 20 compounds in the oxides and\n10 compounds in the sulfides and selenides.\nThe oxides map is obviously denser, reflecting the much richer, currently known, chemistry of the oxides compared\nto the other two elements.\nThe chemistry becomes more constrained as we proceed down the periodic table column from\noxygen to sulfur and then to selenium.\n\nNext, we examine maps of specific stoichiometries.\nMaps of a few notable oxide stoichiometries and their\nleading structure types are shown in Figure~\\ref{fig:art130:mendeleev_oxide_prototypes}.\nThese maps reflect the dominant features of the\nfull ternary oxides map (Figure~\\ref{fig:art130:mendeleev_bigger_x_upper_all_in_one}),\nbut with significant new additional gaps of absent compounds. These gaps are naturally\nwider for less prevalent stoichiometries, \\nobreak\\mbox{\\it i.e.}, the map of the most\nprevalent stoichiometry, $A_1B_1$O$_3$, is denser than the three\nother maps in Figure~\\ref{fig:art130:mendeleev_oxide_prototypes}.\nDifferent structure types in all stoichiometries tend to accumulate at\nwell defined regions of the map. The separation between them\nis not perfect, but is similar to that exhibited by the classical Pettifor maps for\nbinary structure types~\\cite{pettifor:1984,pettifor:1986}.\nA similar picture is obtained for the sulfide and selenide structure types, although more sparse\n(Figure~\\ref{fig:art130:mendeleev_sulfur_selenium_prototypes}).\nIt is interesting to note that the maps of, \\nobreak\\mbox{\\it e.g.},\n$A_1B_2C_4$ ($C=$ O, S, Se), show similar patterns in the map for oxides\n(Figure~\\ref{fig:art130:mendeleev_oxide_prototypes}) and sulfides\/selenides\n(Figure~\\ref{fig:art130:mendeleev_sulfur_selenium_prototypes}) ---\nsuggesting that similar elements tend to form this stoichiometry.\nIn the same manner, the 2:1:1 stoichiometry shows very similar patterns in oxides, sulfides and selenides (see also Figure~\\ref{fig:art130:mend_211_stoichiometries}).\n\n\\boldsection{Symmetries.}\nThe distribution of the compounds and structure types among the 14 Bravais lattices\nis presented in Table~\\ref{table:bravais_lattice_distribution} and\nFigure~\\ref{fig:art130:bravais_combined}.\nIt is interesting to note that in all six cases (binary and ternary oxides, sulfides and selenides)\nthe distribution is double peaked, with the majority of the compounds belonging to the\nmonoclinic and orthorhombic primitive lattices,\nand a smaller local maximum at the hexagonal and tetragonal lattices.\nAll distributions exhibit a local minimum for the orthorhombic face and body centered lattices.\nThe high symmetry cubic lattices are also relatively rare.\nThis reflects the complex spatial arrangement of the compound forming electrons of oxygen, sulfur and selenium,\nwhich does not favor the high symmetry cubic structures or the\ndensely packed face and body centered orthorhombic structures.\n\nFigure~\\ref{fig:art130:symmetry_distribution_of_structures}\nshows a more detailed distribution of the compounds among the different space groups.\nThe binary compounds show a distinct seesaw structure, with a few\nlocal peaks near the highest symmetry groups of each crystal system.\nThe corresponding ternary distributions have three sharp peaks in the triclinic,\nmonoclinic and orthorhombic systems, and much smaller peaks in the hexagonal and cubic groups.\nIt is interesting to note that the three compound families, exhibit distributions of very similar structure.\nThe oxide distributions are the densest, simply due to the existence of more oxide compounds in the database,\nand become sparser in the sulfide and selenide cases.\nThe compounds of all these families are distributed among a rather limited number of space groups,\nwith most space groups represented by just a single compound or not at all.\n\n\\boldsection{Unit cell size.}\nThe distributions of unit cell sizes (\\nobreak\\mbox{\\it i.e.}, the number of atoms per unit cell) for the six compound families we discuss\nare shown in Figure~\\ref{fig:art130:number_of_atoms_distribution}.\nAll of these distributions have strong dense peaks at small cell sizes and decay sharply at sizes above a few tens of atoms.\nHowever, the details of the distributions differ quite significantly from group to group.\nAmong the binaries, the oxides exhibit the highest and widest peak with\nits maximum of 102 binary oxide compounds located at 12 atoms per cell.\n90\\% of the binary oxides have less than 108 atoms in the unit cell and 50\\% of them have less than 24 atoms.\nThe sulfides distribution has a lower and narrower peak of 70 compounds at 8 atoms.\nThe distribution of the selenides has a still lower peak of 60 compounds at 8 atoms.\nThe fact that oxygen has a peak at 12 atoms in the unit cell and not at 8 as the sulfides and selenides,\nis related to the fact that binary oxides prefer the $A$O$_2$ stoichiometry over $A$O,\nwhere as both sulfides and selenides prefer the 1:1 stoichiometry over 1:2.\nThis is probably related to the different chemistry of oxygen \\nobreak\\mbox{\\it vs.}\\ sulfur and selenium.\nAdditional computational analysis would be required to fully understand the effect of the\ndifferent chemistry on the stoichiometry and number of atoms.\nDetailed data for these dense parts of the distributions is tabulated in Tables~\\ref{tab:art130:Number_of_atoms_in_Binaries_unit_cells}-\\ref{tab:art130:Number_of_atoms_in_Binaries_unit_cells_3}.\nThe oxides distribution exhibits the longest tail of the binaries, with the largest\nbinary oxide unit cell including 576 atoms.\nThe largest binary sulfide and selenide unit cells include 376 and 160 atoms, respectively.\n\nThe distributions of the ternary compounds have higher, wider peaks and longer tails than\ntheir binary counterparts.\nThe relative differences between the oxide, sulfide and selenide distributions\nremain similar to the distributions of the binaries.\nThe ternary oxides exhibit a high and wide peak.\nIts maximum of 465 compounds is located at 24 atoms per cell, and\n90\\% of the compounds have less than 92\natoms in the unit cell and 50\\% of the compounds have less than 32 atoms.\nAs in the binary case, the distribution of the ternary sulfides has a lower and narrower peak than the oxides,\nwhere the maximum of 190 compounds at 28 atoms and 90\\% of the compounds have less than 72 atoms in the unit cell.\nThe distribution of the selenides has a still lower and narrower peak, where the corresponding numbers are\n130 compounds at 28 atoms and 90\\% of the compounds having less than 28 atoms in the unit cell.\nDetailed data for these dense parts of the distributions is shown in Tables~\\ref{tab:art130:Number_of_atoms_in_ternary_unit_cells}-\\ref{tab:art130:Number_of_atoms_in_ternary_unit_cells_3}.\nThe ternary oxides distribution exhibits the longest tail of\nthe three types, with the largest ternary oxide unit cell having 1,080 atoms.\nThe largest ternary sulfide and selenide unit cells have 736 and 756\natoms, respectively.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig016}\n\\mycaption[Distributions of (\\textbf{a}) binary and (\\textbf{b}) ternary compounds among the elements.]\n{The binary oxides exhibit a structures distribution with two prominent peaks. The distributions\nof the binary sulfides and selenides are less structured and more similar to each other.\nThe distributions of the ternary compounds have higher, wider peaks than\ntheir binary counterparts. The relative differences between the oxide, sulfide and selenide\ndistributions remain similar to the distributions of the binaries.}\n\\label{fig:art130:mendeleev_distribution_all_in_one}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.375\\linewidth]{fig017}\n\\mycaption[Mendeleev maps of ternary (\\textbf{a}) oxide $A_xB_y$O$_z$, (\\textbf{b}) sulfide $A_xB_y$S$_z$\nand (\\textbf{c}) selenide $A_xB_y$Se$_z$ compounds.]\n{It is assumed that $x\\geq y$ with the $x$-axis indicating $M_A$\nand the $y$-axis $M_B$.\nIf the stoichiometry is such that $x=y$, the compound is counted as $0.5 A_xB_y$O$_z + 0.5 B_xA_y$O$_z$.\nA color scheme is used to represent the compound count for each composition, blue means the minimal number (one)\nand green means the maximal number which is different for each element.}\n\\label{fig:art130:mendeleev_bigger_x_upper_all_in_one}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig018}\n\\mycaption[Mendeleev maps of the three leading structure types in each of the four leading stoichiometries in ternary oxides.]\n{(\\textbf{a}) $A_1B_1$O$_3$,\n(\\textbf{b}) $A_1B_1$O$_4$,\n(\\textbf{c}) $A_1B_2$O$_4$, and\n(\\textbf{d}) $A_2B_2$O$_7$.\nThe legend box appears at a region with no data points.\nThe number in parenthesis is the number of compounds for this structure type, for ``Other'',\nit refers to the total number of compounds with this stoichiometry.}\n\\label{fig:art130:mendeleev_oxide_prototypes}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig019}\n\\mycaption[Mendeleev maps of the three leading structure types in each of the two leading stoichiometries in sulfur and selenium ternaries.]\n{(\\textbf{a}) $A_1B_2$S$_4$,\n(\\textbf{b}) $A_1B_1$S$_2$,\n(\\textbf{c}) $A_1B_2$Se$_4$, and\n(\\textbf{d}) $A_1B_1$Se$_2$.\nThe number in parenthesis is the number of compounds for this structure type, for ``Other'',\nit refers to the total number of compounds with this stoichiometry.}\n\\label{fig:art130:mendeleev_sulfur_selenium_prototypes}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig020}\n\\mycaption[Number of compounds (\\textbf{a} and \\textbf{b}) and\nstructure types (\\textbf{c} and \\textbf{d}) for each Bravais lattice.]\n{Binaries are on the left (\\textbf{a} and \\textbf{c}) and\nternaries on the right (\\textbf{b} and \\textbf{d}).\nOxides are shown in blue, sulfides in light green and selenides in darker green. All six\ndistributions (binary and ternary oxides, sulfides and selenides) are double peaked with a\nlocal minimum for the orthorhombic face and body centered lattices. The high symmetry\ncubic lattices are also relatively rare. This reflects the complex spatial arrangement of\nthe compound forming electrons of the 6A elements, which does not favor the\nhigh symmetry of these\nstructures.}\n\\label{fig:art130:bravais_combined}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig021}\n\\mycaption[Distributions of compounds (\\textbf{a} and \\textbf{b}) and structure types (\\textbf{c} and \\textbf{d}) among the 230 space groups.]\n{Binaries are on the left (\\textbf{a} and \\textbf{c}) and\nternaries on the right (\\textbf{b} and \\textbf{d}).\nCompounds are depicted on the top (\\textbf{a} and \\textbf{b})\nand structure types on the bottom (\\textbf{c} and \\textbf{d}).}\n\\label{fig:art130:symmetry_distribution_of_structures}\n\\end{figure}\n\nIt should be noted that large unit cells, within the tails of all distributions, tend to have very few representatives,\nwith just one compound with a given unit cell size in most cases.\nNotable exceptions are local peaks near $80$ atoms per unit cell in the binary\ndistributions and near 200 atoms per unit cell in the ternary distributions.\nThe oxide distributions exhibit additional peaks, near 300 atoms per unit cell for the\nbinaries and near 600 atoms per unit cell for the ternaries.\nThese minor peaks may indicate preferable arrangements of cluster-based structures.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.9\\linewidth]{fig022}\n\\mycaption[Unit cell size distributions for oxides, sulfides, and selenides.]\n{Binaries are on the left (\\textbf{a}, \\textbf{c} and \\textbf{e}) and\nternaries on the right (\\textbf{b}, \\textbf{d} and \\textbf{f}).\nOxides are at the top (\\textbf{a} and \\textbf{b}),\nsulfides in the middle (\\textbf{c} and \\textbf{d}) and\nselenides at the bottom (\\textbf{e} and \\textbf{f}).\nThe insets show the compounds with up to 50 atoms per unit cell in each case.\nAll distributions exhibit long tails of rare very large unit cells which extend much further in the oxides.\nThe dense cores of the distributions reflects the higher prevalence of oxides and are very similar for the sulfides and selenides.}\n\\label{fig:art130:number_of_atoms_distribution}\n\\end{figure}\n\n\\subsection{Structure sub-types}\nThe definition of structure type by the combination of stoichiometry,\nPearson symbol and symmetry is common in the literature, but it is not\nnecessarily unique. A given structure, according to this definition,\ncan contain few sub-types.\nAs an example, the structure types\n($A_1B_1$O$_3$:oP20:62),\n($A_1B_1$O$_3$:hR10:167) and ($A_1B_1$O$_3$:cP5:221) contain 115, 36, and\n78 unique compounds, respectively, of mostly perovskites. However, the\noP20 also includes the aragonite structure, the MgSeO$_{3}$\nstructure, and others.\nThe hR10 contains also calcite-like\nstructures. The cP5 group, which has a more strict symmetry, contains\nonly perovskites.\nThese three structure types belong to a common parent class, the high symmetry\ncP5, with two different types of symmetry breaking.\nThe different sub-types within each structure type may be discerned by\nexamining relations between structural descriptors, \\nobreak\\mbox{\\it e.g.}, the volume\nas a function of nearest neighbor distance cubed, as shown in Figure~\\ref{fig:art130:structure_types_comparison}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig023}\n\\mycaption{Comparison of different structure types according to volume per atom \\nobreak\\mbox{\\it vs.}\\\nshortest nearest-neighbor distance cubed $\\left(d^{3}_{\\mathrm{n.n.}}\\right)$.}\n\\label{fig:art130:structure_types_comparison}\n\\end{figure}\n\nIt can be easily seen that the ($A_1B_1$O$_3$:cP5:221) group follows a\nperfect linear relation, as is expected from a uniform structure type.\nHowever, both the ($A_1B_1$O$_3$:oP20:62) and the ($A_1B_1$O$_3$:hR10:167) types\ninclude points that are close to the ($A_1B_1$O$_3$:cP5:221) line but also clusters of\npoints that deviate from it.\n\nThose points represent non-perovskite\nstructures, including those that are close to aragonite and calcite.\n\n\\subsection{Ternary stoichiometry triangles}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig024}\n\\mycaption{Comparison of ternary stoichiometries for (\\textbf{a}) oxygen, (\\textbf{b}) sulfur and (\\textbf{c}) selenium compounds.\nAll stoichiometries of $A_xB_yC_z$, $x,y,z \\leq 12$ are shown in (\\textbf{d}).}\n\\label{fig:triangle_stoichiometries}\n\\end{figure}\n\nFigure~\\ref{fig:art130:stoi_periodic_oxide} shows the prevalence of different ternary\nstoichiometries in a triangle shape.\nThe points inside the triangle are defined by\nthe intersection of lines that connect the vertex points with the\ncorresponding binary stoichiometry on the opposing edge.\nFor example, the stoichiometry $A_{u}B_{v}$O$_{w}$ is represented by the\nintersection of three lines, one from the O vertex to the point\n$u\/(u+v)$ on the $AB$ edge,\nanother from the $A$ vertex to the point $v\/(v+w)$ on the $B$O edge, and\nthe third from the $B$ vertex to the point $u\/(u+w)$ on the $A$O edge.\nThe different stoichiometries in Figure~\\ref{fig:art130:stoi_periodic_oxide} are denoted by\ncircles that vary in size according to the number of compounds for\neach stoichiometry.\nFigure~\\ref{fig:triangle_stoichiometries}(a-c) shows the same data but without reference to prevalence,\nshowing just the stoichiometries locations.\nFigure~\\ref{fig:triangle_stoichiometries}(d) shows, for comparison, the locations of all possible stoichiometries up to\n12 atoms per species ($A_xB_yC_z$ where $x,y,z \\leq 12$). The differences in the distributions of the\nreported compositions of the three compound families are clearly apparent.\n\n\\subsection{Prevalence of structure types among the oxide, sulfide and binary and ternary selenide compounds}\n\nNumerical data for the leading 40 structure types of the oxides,\nsulfides and selenides are shown in Tables\n\\ref{tab:art130:oxide_binary_data} through\n\\ref{tab:art130:selenide_binary_data} for the binaries, and in Tables\n\\ref{tab:art130:oxide_ternary_data} through\n\\ref{tab:art130:selenide_ternary_data} for the ternaries.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of the 40 most common structure types among the binary oxide compounds.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{L{2.5cm}|L{2.5cm}|L{2.5cm}|L{5cm}|R{3.25cm}}\nstoichiometry & Pearson & symmetry & name & compounds count \\\\\n\\hline\n1:1& cF8 & 225&NaCl&29\\\\\n3:2& cI80 & 206&Bixbyite-Mn$_{2}$O$_{3}$&24\\\\\n2:1& tP6 & 136&KrF$_{2}$&22\\\\\n3:2& hP5 & 164&La$_{2}$O$_{3}$&19\\\\\n2:1& cF12 & 225&Fluorite-CaF$_{2}$&16\\\\\n3:2& mS30 & 12&Sm$_{2}$O$_{3}$&16\\\\\n3:2& cI80 & 199&Sm$_{2}$O$_{3}$ (c180)&15\\\\\n2:1& cP12 & 205&CO$_{2}$ (cP12)&13\\\\\n2:1& mP12 & 14&Baddeleyite-ZrO$_{2}$ (mP12)&8\\\\\n2:1& oP12 & 60&&8\\\\\n2:1& oP12 & 62&&8\\\\\n2:1& oP6 & 58&&8\\\\\n3:2& hR10 & 167&&8\\\\\n1:1& hP4 & 186&&7\\\\\n2:1& oP24 & 61&&7\\\\\n2:1& tI6 & 139&&7\\\\\n1:1& cF8 & 216&&5\\\\\n1:2& cF12 & 225&&5\\\\\n1:2& cP6 & 224&&5\\\\\n3:2& mP20 & 14&&5\\\\\n3:2& oP20 & 60&&5\\\\\n2:1& aP24 & 1&&4\\\\\n2:1& tP12 & 92&&4\\\\\n3:1& mP16 & 14&&4\\\\\n3:2& mS20 & 12&&4\\\\\n3:2& oP20 & 62&&4\\\\\n5:2& mS28 & 15&&4\\\\\n1:1& tP4 & 129&&3\\\\\n2:1& hP9 & 152&&3\\\\\n2:1& mS6 & 12&&3\\\\\n3:1& cP4 & 221&&3\\\\\n3:2& cF80 & 227&&3\\\\\n3:2& hP5 & 150&&3\\\\\n3:2& oS20 & 63&&3\\\\\n7:4& aP22 & 2&&3\\\\\n12:7& hR19 & 148&&3\\\\\n1:1& cP2 & 221&&2\\\\\n1:1& hP4 & 194&&2\\\\\n1:1& mS4 & 12&&2\\\\\n1:1& mS8 & 15&&2\\\\\n\\end{tabular}}\n\\label{tab:art130:oxide_binary_data}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of the 40 most common structure types among the binary sulfide compounds.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{L{2.5cm}|L{2.5cm}|L{2.5cm}|L{5cm}|R{3.25cm}}\nstoichiometry & Pearson & symmetry & name & compounds count \\\\\n\\hline\n1:1& cF8 & 225&NaCl&32\\\\\n3:2& oP20 & 62&Sb$_{2}$S$_{3}$&20\\\\\n2:1& tP6 & 129&PbClF\/Cu$_{2}$Sb&13\\\\\n2:1& cP12 & 205&Pyrite-Fe$_{2}$S$_{2}$ (cP12)&12\\\\\n2:1& cF24 & 227&Laves(Cub)-Cu$_{2}$Mg&11\\\\\n1:1& hP4 & 194&Nickeline-NiAs&8\\\\\n4:3& cI28 & 220&Th$_{3}$P$_{4}$&8\\\\\n1:1& cF8 & 216&Sphalerite-ZnS (cF8)&7\\\\\n2:1& hP3 & 164&CdI$_{2}$&7\\\\\n2:1& mP12 & 14&CeSe$_{2}$&7\\\\\n1:1& cP2 & 221&&6\\\\\n1:1& hP4 & 186&&6\\\\\n1:2& cF12 & 225&&6\\\\\n2:1& hP6 & 194&&6\\\\\n7:5& mS24 & 12&&6\\\\\n1:1& oP8 & 62&&5\\\\\n2:1& oP6 & 58&&5\\\\\n3:2& hR10 & 167&&5\\\\\n3:2& mP30 & 11&&5\\\\\n1:1& hP2 & 187&&4\\\\\n1:2& hP6 & 194&&4\\\\\n1:2& oP12 & 62&&4\\\\\n2:1& hR3 & 160&&4\\\\\n2:1& oP12 & 62&&4\\\\\n2:1& oP24 & 62&&4\\\\\n3:1& mP8 & 11&&4\\\\\n4:3& cF56 & 227&&4\\\\\n5:2& oP28 & 19&&4\\\\\n1:1& oS8 & 63&&3\\\\\n2:2& hP12 & 189&&3\\\\\n3:2& hP30 & 185&&3\\\\\n3:2& oS20 & 36&&3\\\\\n1:1& hP16 & 186&&2\\\\\n1:1& hP8 & 194&&2\\\\\n1:1& hR2 & 160&&2\\\\\n1:1& hR4 & 166&&2\\\\\n1:1& hR6 & 160&&2\\\\\n1:1& mP8 & 14&&2\\\\\n1:1& mS8 & 5&&2\\\\\n1:1& oS8 & 39&&2\\\\\n\\end{tabular}}\n\\label{tab:art130:sulfide_binary_data}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of the 40 most common structure types among the binary selenide compounds.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{L{2.5cm}|L{2.5cm}|L{2.5cm}|L{5cm}|R{3.25cm}}\nstoichiometry & Pearson & symmetry & name & compounds count \\\\\n\\hline\n1:1& cF8 & 225&NaCl&31\\\\\n2:1& tP6 & 129&PbClF\/Cu$_{2}$Sb&13\\\\\n2:1& cP12 & 205&Pyrite-FeS$_{2}$ (cP12)&11\\\\\n1:1& hP4 & 186&Wurtzite-ZnS(2H)&9\\\\\n1:1& hP4 & 194&Nickeline-NiAs&9\\\\\n2:1& hP3 & 164&CdI$_{2}$&9\\\\\n3:2& oP20 & 62&Sb$_{2}$S$_{3}$&9\\\\\n4:3& cI28 & 220&Th$_{3}$P$_{4}$&9\\\\\n1:1& cF8 & 216&Sphalerite-ZnS (cF8)&8\\\\\n3:2& oF80 & 70&Sc$_{2}$S$_{3}$&8\\\\\n1:1& cP2 & 221&&7\\\\\n1:2& cF12 & 225&&6\\\\\n4:3& mS14 & 12&&6\\\\\n1:1& oP8 & 62&&4\\\\\n2:1& hP6 & 194&&4\\\\\n2:1& mP12 & 14&&4\\\\\n3:1& mP8 & 11&&4\\\\\n2:1& hP12 & 187&&3\\\\\n2:1& hR3 & 160&&3\\\\\n2:1& oP6 & 58&&3\\\\\n3:2& oS20 & 36&&3\\\\\n4:4& mP32 & 14&&3\\\\\n4:5& tI18 & 87&&3\\\\\n5:2& oP28 & 19&&3\\\\\n1:1& hP8 & 187&&2\\\\\n1:1& hP8 & 194&&2\\\\\n1:1& hR4 & 160&&2\\\\\n1:1& mS8 & 12&&2\\\\\n1:2& oP36 & 58&&2\\\\\n2:1& hP12 & 194&&2\\\\\n2:1& oP12 & 62&&2\\\\\n2:1& oP24 & 62&&2\\\\\n2:2& hP12 & 189&&2\\\\\n2:2& mP16 & 14&&2\\\\\n3:1& mP24 & 11&&2\\\\\n3:2& hR5 & 166&&2\\\\\n3:2& mP10 & 11&&2\\\\\n3:2& mS20 & 9&&2\\\\\n4:3& hP14 & 176&&2\\\\\n8:3& hR11 & 148&&2\\\\\n\\end{tabular}}\n\\label{tab:art130:selenide_binary_data}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of the 40 most common structure types among the ternary oxide compounds.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{L{2.5cm}|L{2.5cm}|L{2.5cm}|L{5cm}|R{3.25cm}}\nstoichiometry & Pearson & symmetry & name & compounds count \\\\\n\\hline\n7:2:2& cF88 & 227&Pyrochlore&116\\\\\n3:1:1& oP20 & 62&Perovskite-GdFeO$_{3}$ (mostly)&115\\\\\n3:1:1& cP5 & 221&Perovskite-CaTiO$_{3}$&78\\\\\n2:1:1& hR4 & 166&Delafossite-NaCrS$_{2}$&72\\\\\n4:1:1& tI24 & 141&Zircon-ZrSiO$_{4}$&66\\\\\n4:1:2& cF56 & 227&Spinel-Al$_{2}$MgO$_{4}$&66\\\\\n4:1:1& tI24 & 88&Scheelite-CaWO$_{4}$&47\\\\\n4:1:2& oP28 & 62&CaFe$_{2}$O$_{4}$&44\\\\\n4:1:1& mP24 & 14&AgMnO4&43\\\\\n3:1:1& hR10 & 167&Perovskite-NdAlO$_{3}$&36\\\\\n4:1:1& oP24 & 62&Barite-BaSO$_{4}$&34\\\\\n3:1:1& mP20 & 14&&33\\\\\n7:1:3& oS44 & 63&&33\\\\\n2:1:2& hP5 & 164&&32\\\\\n4:1:2& tI14 & 139&&32\\\\\n2:1:2& tI10 & 139&&31\\\\\n4:1:1& oS24 & 63&&31\\\\\n12:3:5& cI160 & 230&&30\\\\\n3:1:1& hR10 & 148&&28\\\\\n2:1:1& hP8 & 194&&27\\\\\n4:1:1& mP12 & 13&&26\\\\\n1:1:1& tP6 & 129&&25\\\\\n5:1:2& mP32 & 14&&25\\\\\n7:2:2& mS22 & 12&&24\\\\\n6:1:2& tP18 & 136&&22\\\\\n11:2:4& mS68 & 15&&20\\\\\n3:1:1& hR10 & 161&&19\\\\\n4:1:2& mP28 & 14&&19\\\\\n7:2:2& mP44 & 14&&19\\\\\n1:1:3& cP5 & 221&&18\\\\\n3:1:1& hR5 & 160&&18\\\\\n1:2:4& tI14 & 139&&17\\\\\n3:1:1& mP40 & 14&&17\\\\\n6:1:2& hP9 & 162&&16\\\\\n7:2:2& aP22 & 2&&16\\\\\n7:2:2& aP44 & 2&&16\\\\\n9:1:3& mP52 & 14&&16\\\\\n1:4:6& hP22 & 186&&15\\\\\n3:1:1& hP30 & 185&&15\\\\\n5:1:2& oP32 & 55&&15\\\\\n\\end{tabular}}\n\\label{tab:art130:oxide_ternary_data}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of the 40 most common structure types among the ternary sulfide compounds.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{L{2.5cm}|L{2.5cm}|L{2.5cm}|L{5cm}|R{3.25cm}}\nstoichiometry & Pearson & symmetry & name & compounds count \\\\\n\\hline\n2:1:1& hR4 & 166&Delafossite-NaCrS$_{2}$&88\\\\\n4:1:2& oP28 & 62&CaFe$_{2}$O$_{4}$&77\\\\\n4:1:2& cF56 & 227&Spinel-Al$_{2}$MgO$_{4}$&53\\\\\n5:1:2& oP32 & 62&U$_{3}$S$_{5}$&37\\\\\n3:1:1& oP20 & 62&SrZrS$_{3}$&32\\\\\n8:1:6& hR15 & 148&Mo$_{6}$PbS$_{8}$&32\\\\\n1:1:1& mP12 & 14&CeAsS&27\\\\\n2:1:1& hP8 & 194&SnTaS$_{2}$&26\\\\\n6:1:3& mP20 & 11&Tm$_{2}$S$_{3}$&26\\\\\n7:1:4& hP24 & 173&La$_{3}$CuSiS$_{7}$&25\\\\\n1:1:1& tP6 & 129&&23\\\\\n3:1:1& oP20 & 33&&21\\\\\n22:4:11& mS74 & 12&&20\\\\\n1:2:2& hP5 & 164&&19\\\\\n6:1:3& oP40 & 18&&15\\\\\n1:1:1& cP12 & 198&&14\\\\\n2:2:3& hR7 & 166&&14\\\\\n4:1:3& oP32 & 62&&14\\\\\n1:1:1& oP12 & 62&&12\\\\\n4:1:1& tI96 & 142&&12\\\\\n1:1:1& oP24 & 62&&11\\\\\n2:1:1& mP16 & 14&&11\\\\\n2:1:1& oP16 & 62&&11\\\\\n3:1:1& mP40 & 11&&11\\\\\n6:1:3& hP20 & 182&&11\\\\\n12:3:4& hR38 & 161&&11\\\\\n13:4:5& oP44 & 55&&11\\\\\n1:1:4& oP24 & 62&&10\\\\\n2:1:1& hR4 & 160&&10\\\\\n2:1:1& tI16 & 122&&10\\\\\n2:1:2& tI10 & 139&&10\\\\\n3:1:2& oP24 & 62&&10\\\\\n4:1:2& oS28 & 66&&10\\\\\n8:1:5& mS28 & 12&&10\\\\\n3:1:3& oP28 & 62&&9\\\\\n4:1:2& mS14 & 12&&9\\\\\n4:1:2& oF224 & 70&&9\\\\\n4:1:3& cP16 & 223&&9\\\\\n2:1:1& mS64 & 15&&8\\\\\n2:1:2& mP20 & 14&&8\\\\\n\\end{tabular}}\n\\label{tab:art130:sulfide_ternary_data}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of the 40 most structure types among the ternary selenide compounds.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{L{2.5cm}|L{2.5cm}|L{2.5cm}|L{5cm}|R{3.25cm}}\nstoichiometry & Pearson & symmetry & name & compounds count \\\\\n\\hline\n2:1:1& hR4 & 166&Delafossite&51\\\\\n4:1:2& oP28 & 62&CaFe$_{2}$O$_{4}$&48\\\\\n4:1:2& cF56 & 227&Spinel-Al$_{2}$MgO$_{4}$&30\\\\\n8:1:6& hR15 & 148&Mo$_{6}$PbS$_{8}$&25\\\\\n1:1:1& mP12 & 14&CeAsS&22\\\\\n4:1:2& mS14 & 12&CrNb$_{2}$Se$_{4}$-Cr$_{3}$S$_{4}$&18\\\\\n8:1:5& mS28 & 12&Cr$_{5}$CsS$_{8}$&18\\\\\n1:1:1& tP6 & 129&PbClF\/Cu$_{2}$Sb&16\\\\\n1:1:1& cP12 & 198&NiSSb&14\\\\\n3:1:1& oP20 & 62&NH$_{4}$CdCl$_{3}$\/Sn$_{2}$S$_{3}$&14\\\\\n4:1:2& tI14 & 82&&13\\\\\n5:1:2& oP32 & 62&&13\\\\\n4:1:3& oP32 & 62&&12\\\\\n1:2:2& hP5 & 164&&11\\\\\n2:1:1& mP16 & 14&&10\\\\\n2:1:2& tI10 & 139&&10\\\\\n3:1:1& oS20 & 63&&10\\\\\n3:1:3& cP28 & 198&&10\\\\\n6:1:3& hP20 & 182&&10\\\\\n1:1:1& oP12 & 62&&9\\\\\n1:1:3& oP20 & 62&&9\\\\\n2:1:1& tI16 & 122&&9\\\\\n2:1:2& oI20 & 72&&9\\\\\n3:1:3& hP14 & 176&&9\\\\\n4:1:2& oS28 & 66&&9\\\\\n19:2:15& hR72 & 167&&9\\\\\n2:1:1& oP16 & 19&&8\\\\\n6:1:3& oP40 & 58&&8\\\\\n17:1:8& mS52 & 12&&8\\\\\n4:1:6& hP22 & 186&&7\\\\\n6:2:2& mP20 & 14&&7\\\\\n6:2:6& mP28 & 14&&7\\\\\n2:1:1& mS64 & 15&&6\\\\\n2:1:1& tI16 & 140&&6\\\\\n4:1:2& oF224 & 70&&6\\\\\n1:1:4& oS24 & 63&&5\\\\\n2:1:1& hR4 & 160&&5\\\\\n2:1:6& mP18 & 14&&5\\\\\n2:1:12& oF120 & 43&&5\\\\\n2:2:3& hR7 & 166&&5\\\\\n\\end{tabular}}\n\\label{tab:art130:selenide_ternary_data}\n\\end{table}\n\n\\clearpage\n\n\\subsection{Prevalence of stoichiometries}\n\\label{subsec:art130:prev_stoich_supp}\n\nTables~\\ref{tab:art130:Prevalence_of_Binaries_Stoichiometries}-\\ref{tab:art130:Prevalence_of_Binaries_Stoichiometries_3} list all the binary\nstoichiometries among the three compound families examined in this paper.\nAn interesting finding is that the stoichiometry\n$A_1$O$_2$ has 356 unique compounds, a number that is significantly\nlarger than the number of atoms in the periodic table. This is because\na given chemical composition can have many different structure type\nrealizations.\nThe most prominent example is SiO$_{2}$ which has 185 different\nreported structures, representing the majority of the 356 compounds\nand 244 structure types of this stoichiometry.\nIn contrast, SiS$_{2}$ has only two reported structures, and SiSe$_{2}$\nhas only one.\nChecking other atoms from the same column of Si in the periodic\ntable, we find that GeO$_{2}$ has seven structures and CO$_{2}$ has\nnine.\nThe last observation means that, since CO$_{2}$ is\ngaseous in atmospheric conditions, the {\\small ICSD}\\ compounds of\nCO$_{2}$ are not in atmospheric conditions (either temperature or\npressure or both).\nExamining Tables~\\ref{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries}-\\ref{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries_3}\nwe also observe that the $A_xO_y$ set of compounds exhibits\nseveral gaps (missing ratios) along the axis of $y\/(x+y)$. There are\nno reported binary oxides\nfrom 0.51 to (and not including) 0.55, from 0.34 to 0.4,\nfrom 0.26 to 0.3, and from 0.44 to 0.5.\nThose gaps {\\it do not exist} in the sulfides and\nselenides.\nMost of the gaps in the sulfides appear above 0.6, and no\nselenide compounds are reported above 0.65.\nThe maximal ratio for the oxides is 0.84, while the maximal ratio for the sulfides is 0.93.\nTables~\\ref{tab:art130:Elements_stoichiometries}-\\ref{tab:art130:Elements_stoichiometries_3} show the leading stoichiometry for each element\nas well as the number of stoichiometries and unique compounds for this element.\nWhile SiO$_2$ is the {\\it only} stoichiometry of silicon oxide (with 185 structure types),\nvanadium has 18 different stoichiometries and 42 unique compounds,\nVO$_2$ is the stoichiometry with the largest number (10) of structure types.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of binary stoichiometries (1\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:2& 356 & 123 & 79\\\\\n1:1& 99 & 165 & 108\\\\\n2:3& 146 & 58 & 38\\\\\n1:3& 41 & 12 & 10\\\\\n2:1& 27 & 30 & 21\\\\\n2:5& 26 & 8 & 5\\\\\n3:4& 22 & 25 & 21\\\\\n3:1& 9 & 7 & 1\\\\\n6:11& 9 & 0 & 0\\\\\n3:8& 9 & 2 & 2\\\\\n6:1& 4 & 7 & 0\\\\\n3:5& 7 & 2 & 2\\\\\n5:9& 7 & 0 & 0\\\\\n5:7& 0 & 6 & 0\\\\\n4:7& 6 & 1 & 0\\\\\n4:1& 2 & 1 & 5\\\\\n4:3& 2 & 5 & 4\\\\\n6:13& 5 & 0 & 0\\\\\n4:9& 5 & 2 & 1\\\\\n5:4& 0 & 2 & 4\\\\\n12:29& 4 & 0 & 0\\\\\n2:7& 4 & 2 & 0\\\\\n1:4& 4 & 1 & 2\\\\\n8:1& 3 & 1 & 0\\\\\n3:2& 1 & 3 & 2\\\\\n7:8& 0 & 3 & 2\\\\\n4:5& 3 & 3 & 1\\\\\n7:12& 3 & 0 & 0\\\\\n8:15& 3 & 0 & 1\\\\\n4:11& 3 & 0 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Binaries_Stoichiometries}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of binary stoichiometries continued (2\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n21:8& 0 & 2 & 0\\\\\n7:3& 2 & 0 & 0\\\\\n5:3& 0 & 0 & 2\\\\\n9:8& 0 & 2 & 1\\\\\n6:7& 0 & 1 & 2\\\\\n9:11& 0 & 0 & 2\\\\\n11:20& 2 & 0 & 0\\\\\n7:13& 2 & 0 & 0\\\\\n9:17& 2 & 0 & 0\\\\\n8:21& 2 & 0 & 0\\\\\n17:47& 2 & 0 & 0\\\\\n5:14& 2 & 0 & 0\\\\\n8:23& 2 & 0 & 0\\\\\n9:26& 2 & 0 & 0\\\\\n2:9& 1 & 0 & 2\\\\\n61:2& 1 & 0 & 0\\\\\n12:1& 0 & 1 & 0\\\\\n7:1& 1 & 0 & 0\\\\\n16:3& 1 & 0 & 0\\\\\n9:2& 1 & 1 & 1\\\\\n15:4& 0 & 1 & 0\\\\\n11:3& 1 & 0 & 0\\\\\n7:2& 0 & 0 & 1\\\\\n34:11& 0 & 0 & 1\\\\\n45:16& 0 & 0 & 1\\\\\n14:5& 0 & 1 & 0\\\\\n11:4& 0 & 0 & 1\\\\\n8:3& 0 & 1 & 1\\\\\n5:2& 0 & 0 & 1\\\\\n16:7& 0 & 1 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Binaries_Stoichiometries_2}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of binary stoichiometries continued (3\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n31:16& 0 & 1 & 0\\\\\n29:16& 0 & 1 & 0\\\\\n9:5& 0 & 1 & 0\\\\\n7:4& 0 & 1 & 1\\\\\n6:5& 0 & 1 & 1\\\\\n8:7& 0 & 0 & 1\\\\\n17:15& 0 & 1 & 1\\\\\n17:18& 0 & 1 & 0\\\\\n8:9& 0 & 1 & 1\\\\\n5:6& 0 & 1 & 0\\\\\n13:16& 1 & 0 & 0\\\\\n15:19& 0 & 1 & 1\\\\\n8:11& 0 & 1 & 0\\\\\n15:22& 0 & 1 & 0\\\\\n5:8& 1 & 1 & 1\\\\\n9:16& 1 & 0 & 0\\\\\n16:35& 1 & 0 & 0\\\\\n3:7& 1 & 0 & 0\\\\\n5:12& 1 & 0 & 0\\\\\n13:34& 1 & 0 & 0\\\\\n18:49& 1 & 0 & 0\\\\\n25:73& 1 & 0 & 0\\\\\n4:21& 1 & 0 & 0\\\\\n1:8& 0 & 1 & 0\\\\\n1:14& 0 & 1 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Binaries_Stoichiometries_3}\n\\end{table}\n\n\\clearpage\n\nThese differences carry over into the ternary oxide compounds involving those elements,\nwhere the stoichiometry distribution of silicon ternary oxides is much tilted towards the\nsilicon poor compounds than those of vanadium and titanium as shown in Figure~\\ref{fig:art130:specific_triangle_stoichiometries}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig025}\n\\mycaption{Comparison of ternary oxide stoichiometries containing (\\textbf{a}) silicon, (\\textbf{b}) titanium, and (\\textbf{c}) vanadium.}\n\\label{fig:art130:specific_triangle_stoichiometries}\n\\end{figure}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se; top 120) (1\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:1:3& 718 & 147 & 70\\\\\n1:1:4& 428 & 28 & 8\\\\\n1:2:4& 396 & 242 & 158\\\\\n2:2:7& 304 & 2 & 0\\\\\n1:1:2& 269 & 242 & 145\\\\\n1:2:6& 237 & 8 & 8\\\\\n1:2:5& 149 & 57 & 22\\\\\n1:1:1& 113 & 140 & 90\\\\\n1:2:2& 131 & 64 & 38\\\\\n1:2:3& 100 & 55 & 42\\\\\n2:3:8& 87 & 8 & 4\\\\\n1:3:6& 83 & 63 & 22\\\\\n1:4:4& 78 & 17 & 13\\\\\n1:3:9& 78 & 0 & 0\\\\\n1:3:7& 67 & 1 & 0\\\\\n2:2:5& 64 & 32 & 23\\\\\n2:4:9& 62 & 7 & 3\\\\\n1:3:3& 62 & 50 & 39\\\\\n1:3:4& 58 & 54 & 31\\\\\n1:3:8& 55 & 1 & 0\\\\\n1:2:7& 49 & 3 & 2\\\\\n2:4:11& 46 & 8 & 2\\\\\n3:5:12& 46 & 4 & 0\\\\\n2:3:12& 45 & 0 & 0\\\\\n1:4:1& 15 & 44 & 12\\\\\n1:3:1& 43 & 13 & 30\\\\\n1:2:8& 41 & 2 & 3\\\\\n1:6:8& 4 & 39 & 30\\\\\n1:3:5& 37 & 15 & 6\\\\\n1:1:5& 37 & 2 & 2\\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Ternary_Stoichiometries}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se; top 120) continued (2\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n2:3:9& 34 & 2 & 3\\\\\n1:3:2& 33 & 19 & 11\\\\\n2:3:2& 9 & 32 & 11\\\\\n1:4:7& 32 & 29 & 2\\\\\n2:3:6& 31 & 11 & 11\\\\\n2:2:1& 29 & 30 & 17\\\\\n1:5:8& 29 & 14 & 22\\\\\n2:4:7& 28 & 6 & 0\\\\\n1:2:1& 27 & 12 & 5\\\\\n2:3:4& 17 & 27 & 17\\\\\n2:2:9& 26 & 2 & 2\\\\\n1:5:14& 25 & 0 & 0\\\\\n2:2:3& 25 & 8 & 3\\\\\n2:3:7& 24 & 10 & 3\\\\\n1:4:8& 15 & 24 & 11\\\\\n2:4:1& 24 & 15 & 12\\\\\n2:4:13& 22 & 1 & 1\\\\\n4:6:1& 22 & 5 & 3\\\\\n1:5:4& 21 & 6 & 1\\\\\n4:11:22& 0 & 20 & 3\\\\\n2:6:7& 19 & 5 & 4\\\\\n1:4:6& 18 & 4 & 3\\\\\n3:4:10& 17 & 2 & 0\\\\\n1:4:5& 16 & 0 & 0\\\\\n1:5:2& 3 & 1 & 16\\\\\n1:4:11& 15 & 0 & 0\\\\\n2:4:5& 15 & 4 & 4\\\\\n1:4:3& 14 & 8 & 8\\\\\n3:4:12& 11 & 14 & 0\\\\\n2:4:15& 14 & 0 & 0\\\\\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se; top 120) continued (3\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:6:12& 14 & 0 & 0\\\\\n1:5:5& 13 & 3 & 2\\\\\n2:2:11& 13 & 0 & 0\\\\\n4:5:13& 1 & 12 & 0\\\\\n2:12:3& 0 & 12 & 4\\\\\n1:4:9& 12 & 1 & 0\\\\\n1:6:11& 12 & 0 & 0\\\\\n1:4:12& 12 & 1 & 0\\\\\n4:4:11& 11 & 1 & 1\\\\\n3:4:9& 11 & 5 & 3\\\\\n3:5:14& 11 & 0 & 0\\\\\n2:3:10& 11 & 0 & 0\\\\\n1:6:2& 6 & 2 & 11\\\\\n1:6:4& 9 & 10 & 11\\\\\n2:4:3& 5 & 10 & 7\\\\\n1:12:20& 10 & 0 & 0\\\\\n10:14:1& 10 & 1 & 0\\\\\n2:5:13& 10 & 0 & 0\\\\\n2:15:19& 0 & 4 & 10\\\\\n1:8:6& 10 & 6 & 4\\\\\n1:8:14& 10 & 0 & 0\\\\\n1:7:12& 9 & 0 & 0\\\\\n1:5:7& 8 & 1 & 1\\\\\n1:6:6& 8 & 0 & 0\\\\\n2:9:3& 0 & 0 & 8\\\\\n2:6:13& 8 & 1 & 1\\\\\n1:3:12& 8 & 0 & 3\\\\\n1:8:17& 0 & 8 & 8\\\\\n1:12:19& 8 & 0 & 0\\\\\n3:4:8& 8 & 2 & 1\\\\\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se; top 120) continued (4\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:5:6& 7 & 1 & 0\\\\\n2:7:2& 0 & 6 & 7\\\\\n1:5:1& 2 & 7 & 5\\\\\n3:4:4& 7 & 2 & 0\\\\\n2:3:1& 2 & 5 & 7\\\\\n2:5:10& 7 & 0 & 0\\\\\n2:5:12& 7 & 0 & 0\\\\\n2:3:11& 4 & 7 & 2\\\\\n4:6:19& 7 & 0 & 0\\\\\n1:1:6& 7 & 3 & 2\\\\\n4:6:13& 5 & 6 & 3\\\\\n1:7:1& 0 & 5 & 6\\\\\n1:10:14& 0 & 6 & 4\\\\\n4:5:15& 6 & 0 & 0\\\\\n3:3:1& 6 & 0 & 0\\\\\n1:12:2& 0 & 1 & 6\\\\\n1:3:10& 6 & 0 & 0\\\\\n2:6:1& 2 & 6 & 4\\\\\n5:9:5& 6 & 0 & 0\\\\\n2:6:15& 5 & 0 & 0\\\\\n2:9:6& 1 & 5 & 0\\\\\n4:4:3& 1 & 1 & 5\\\\\n4:5:12& 5 & 1 & 0\\\\\n2:8:7& 5 & 0 & 0\\\\\n3:6:1& 0 & 5 & 0\\\\\n1:8:2& 0 & 5 & 1\\\\\n1:7:6& 3 & 5 & 4\\\\\n1:8:8& 1 & 5 & 2\\\\\n2:9:2& 1 & 5 & 1\\\\\n3:5:2& 5 & 3 & 0\\\\\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se, with $M_A>M_B$ when $x\\neq y$; top 120) (1\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:1:3& 718 & 147 & 70\\\\\n1:1:4& 428 & 28 & 8\\\\\n2:2:7& 304 & 2 & 0\\\\\n1:1:2& 269 & 242 & 145\\\\\n2:1:4& 206 & 101 & 70\\\\\n1:2:4& 190 & 141 & 88\\\\\n1:1:1& 113 & 140 & 90\\\\\n2:1:6& 122 & 7 & 3\\\\\n1:2:6& 115 & 1 & 5\\\\\n1:2:5& 90 & 32 & 14\\\\\n2:1:2& 72 & 37 & 24\\\\\n2:2:5& 64 & 32 & 23\\\\\n3:1:9& 62 & 0 & 0\\\\\n2:3:8& 60 & 6 & 2\\\\\n2:1:5& 59 & 25 & 8\\\\\n1:2:2& 59 & 27 & 14\\\\\n2:1:3& 59 & 23 & 22\\\\\n1:3:6& 34 & 54 & 19\\\\\n3:1:6& 49 & 9 & 3\\\\\n4:1:1& 15 & 43 & 12\\\\\n5:3:12& 43 & 2 & 0\\\\\n3:1:3& 42 & 26 & 23\\\\\n1:2:3& 41 & 32 & 20\\\\\n4:1:4& 40 & 8 & 6\\\\\n4:2:9& 40 & 6 & 1\\\\\n1:4:4& 38 & 9 & 7\\\\\n1:1:5& 37 & 2 & 2\\\\\n2:1:7& 36 & 3 & 1\\\\\n3:1:7& 34 & 1 & 0\\\\\n3:1:4& 32 & 33 & 13\\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Ternary_Stoichiometries_2}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se, with $M_A>M_B$ when $x\\neq y$; top 120) continued (2\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:3:7& 33 & 0 & 0\\\\\n4:2:11& 31 & 5 & 0\\\\\n2:2:1& 29 & 30 & 17\\\\\n3:1:1& 30 & 12 & 29\\\\\n3:1:8& 30 & 1 & 0\\\\\n6:1:8& 2 & 29 & 21\\\\\n4:1:7& 28 & 3 & 2\\\\\n3:2:2& 1 & 27 & 7\\\\\n3:2:8& 27 & 2 & 2\\\\\n3:2:9& 27 & 2 & 3\\\\\n2:2:9& 26 & 2 & 2\\\\\n1:3:4& 26 & 21 & 18\\\\\n2:1:8& 26 & 2 & 1\\\\\n1:4:7& 4 & 26 & 0\\\\\n5:1:14& 25 & 0 & 0\\\\\n2:2:3& 25 & 8 & 3\\\\\n1:3:8& 25 & 0 & 0\\\\\n2:3:6& 24 & 8 & 8\\\\\n3:1:2& 24 & 12 & 4\\\\\n1:3:3& 20 & 24 & 16\\\\\n1:3:5& 23 & 9 & 5\\\\\n3:2:12& 23 & 0 & 0\\\\\n2:3:12& 22 & 0 & 0\\\\\n2:4:9& 22 & 1 & 2\\\\\n6:4:1& 18 & 1 & 2\\\\\n5:1:4& 17 & 6 & 1\\\\\n2:3:7& 16 & 10 & 2\\\\\n4:11:22& 0 & 16 & 3\\\\\n4:1:5& 16 & 0 & 0\\\\\n1:3:9& 16 & 0 & 0\\\\\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se, with $M_A>M_B$ when $x\\neq y$; top 120) continued (3\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:2:8& 15 & 0 & 2\\\\\n2:4:11& 15 & 3 & 2\\\\\n1:4:8& 9 & 15 & 9\\\\\n2:4:7& 15 & 4 & 0\\\\\n5:1:8& 15 & 10 & 15\\\\\n3:2:4& 10 & 14 & 9\\\\\n4:2:13& 14 & 1 & 1\\\\\n1:5:8& 14 & 4 & 7\\\\\n3:4:12& 10 & 14 & 0\\\\\n2:1:1& 14 & 1 & 3\\\\\n4:2:1& 14 & 11 & 10\\\\\n3:1:5& 14 & 6 & 1\\\\\n1:2:1& 13 & 11 & 2\\\\\n4:2:7& 13 & 2 & 0\\\\\n4:2:15& 13 & 0 & 0\\\\\n4:1:6& 13 & 0 & 0\\\\\n2:3:4& 7 & 13 & 8\\\\\n1:3:1& 13 & 1 & 1\\\\\n6:1:12& 13 & 0 & 0\\\\\n2:2:11& 13 & 0 & 0\\\\\n1:2:7& 13 & 0 & 1\\\\\n4:1:3& 12 & 5 & 5\\\\\n5:1:5& 12 & 1 & 0\\\\\n5:1:2& 3 & 1 & 12\\\\\n4:1:11& 12 & 0 & 0\\\\\n12:2:3& 0 & 12 & 3\\\\\n2:6:7& 11 & 2 & 0\\\\\n6:1:2& 6 & 2 & 11\\\\\n4:4:11& 11 & 1 & 1\\\\\n5:4:13& 0 & 11 & 0\\\\\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se, with $M_A>M_B$ when $x\\neq y$; top 120) continued (4\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n4:1:12& 11 & 1 & 0\\\\\n6:1:4& 7 & 10 & 11\\\\\n8:1:6& 10 & 2 & 2\\\\\n3:4:10& 10 & 1 & 0\\\\\n1:6:8& 2 & 10 & 9\\\\\n1:12:20& 10 & 0 & 0\\\\\n14:10:1& 10 & 1 & 0\\\\\n2:4:1& 10 & 4 & 2\\\\\n4:1:9& 9 & 1 & 0\\\\\n3:4:9& 9 & 2 & 1\\\\\n1:3:2& 9 & 7 & 7\\\\\n2:4:5& 9 & 2 & 1\\\\\n4:1:8& 6 & 9 & 2\\\\\n3:2:7& 8 & 0 & 1\\\\\n6:2:7& 8 & 3 & 4\\\\\n9:2:3& 0 & 0 & 8\\\\\n2:4:13& 8 & 0 & 0\\\\\n5:2:13& 8 & 0 & 0\\\\\n12:1:19& 8 & 0 & 0\\\\\n2:3:2& 8 & 5 & 4\\\\\n1:8:17& 0 & 8 & 8\\\\\n8:1:14& 8 & 0 & 0\\\\\n3:2:6& 7 & 3 & 3\\\\\n6:1:11& 7 & 0 & 0\\\\\n2:3:9& 7 & 0 & 0\\\\\n3:4:4& 7 & 1 & 0\\\\\n3:2:10& 7 & 0 & 0\\\\\n6:1:6& 7 & 0 & 0\\\\\n5:1:1& 2 & 7 & 4\\\\\n4:3:10& 7 & 1 & 0\\\\\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[Prevalence of binary stoichiometries (1\/3).]\n{The entries for each element column denote the total number of structure types,\ntotal number of unique compounds and then the leading atom with the total\nnumber of structure types of this stoichiometry in which it appears.\nThe second column shows the stoichiometry $(x:y)$ for $A_xZ_y$, $Z=$ O, S, Se, respectively.}\n\\vspace{3mm}\n\\begin{tabular}{l|l|r|r|r}\nratio $y\/(x+y)$ & stoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n0.032 & (2:61) & 1 1 C(1) & & \\\\\n0.077 & (1:12) & & 1 1 B(1) & \\\\\n0.11 & (1:8) & 3 3 V(1) & 1 1 Ag(1) & \\\\\n0.12 & (1:7) & 1 1 Cs(1) & & \\\\\n0.14 & (1:6) & 4 4 Ti(2) & 7 7 F(5) & \\\\\n0.16 & (3:16) & 1 1 V(1) & & \\\\\n0.18 & (2:9) & 1 1 Rb(1) & 1 1 Zr(1) & 1 1 Ti(1)\\\\\n0.2 & (1:4) & 2 2 Ta(1) & 1 1 Pd(1) & 5 5 Cl(2)\\\\\n0.21 & (4:15) & & 1 1 C(1) & \\\\\n0.21 & (3:11) & 1 1 Cs(1) & & \\\\\n0.22 & (2:7) & & & 1 1 Pd(1)\\\\\n0.24 & (11:34) & & & 1 1 Pd(1)\\\\\n0.25 & (1:3) & 8 9 Zr(3) & 7 7 O(3) & 1 1 O(1)\\\\\n0.26 & (16:45) & & & 1 1 Ti(1)\\\\\n0.26 & (5:14) & & 1 1 Nb(1) & \\\\\n0.27 & (4:11) & & & 1 1 Ti(1)\\\\\n0.27 & (3:8) & & 1 1 Ti(1) & 1 1 Ti(1)\\\\\n0.28 & (8:21) & & 1 2 Zr(1) & \\\\\n0.29 & (2:5) & & & 1 1 O(1)\\\\\n0.3 & (3:7) & 2 2 V(2) & & \\\\\n0.3 & (7:16) & & 1 1 Pd(1) & \\\\\n0.33 & (1:2) & 17 27 H(6) & 18 30 Cu(4) & 15 21 O(4)\\\\\n0.34 & (16:31) & & 1 1 Cu(1) & \\\\\n0.36 & (16:29) & & 1 1 Cu(1) & \\\\\n0.36 & (5:9) & & 1 1 Cu(1) & \\\\\n0.36 & (4:7) & & 1 1 Cu(1) & 1 1 Pd(1)\\\\\n0.38 & (3:5) & & & 2 2 Tl(2)\\\\\n0.4 & (2:3) & 1 1 C(1) & 3 3 Ni(2) & 2 2 Ni(1)\\\\\n0.43 & (3:4) & 2 2 Tl(1) & 4 5 P(2) & 4 4 P(1)\\\\\n0.44 & (4:5) & & 2 2 V(1) & 2 4 V(1)\\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[Prevalence of binary stoichiometries continued (2\/3).]\n{The entries for each element column denote the total number of structure types,\ntotal number of unique compounds and then the leading atom with the total\nnumber of structure types of this stoichiometry in which it appears.\nThe second column shows the stoichiometry $(x:y)$ for $A_xZ_y$, $Z=$ O, S, Se, respectively.}\n\\vspace{3mm}\n\\begin{tabular}{l|l|r|r|r}\nratio $y\/(x+y)$ & stoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n0.45 & (5:6) & & 1 1 N(1) & 1 1 Ni(1)\\\\\n0.47 & (7:8) & & & 1 1 Bi(1)\\\\\n0.47 & (15:17) & & 1 1 Rh(1) & 1 1 Pd(1)\\\\\n0.47 & (8:9) & & 2 2 Ni(1) & 1 1 Co(1)\\\\\n0.5 & (1:1) & 51 99 Mg(12) & 88 165 Zn(39) & 38 108 Ga(5)\\\\\n0.51 & (18:17) & & 1 1 Ni(1) & \\\\\n0.53 & (9:8) & & 1 1 As(1) & 1 1 Bi(1)\\\\\n0.53 & (8:7) & & 3 3 Fe(3) & 2 2 Fe(2)\\\\\n0.54 & (7:6) & & 1 1 In(1) & 2 2 In(2)\\\\\n0.55 & (6:5) & & 1 1 Cr(1) & \\\\\n0.55 & (11:9) & & & 2 2 Mo(2)\\\\\n0.55 & (16:13) & 1 1 V(1) & & \\\\\n0.56 & (5:4) & 3 3 Ti(1) & 2 3 P(2) & 1 1 P(1)\\\\\n0.56 & (19:15) & & 1 1 Mo(1) & 1 1 Mo(1)\\\\\n0.57 & (4:3) & 18 22 Fe(8) & 13 25 Fe(3) & 7 21 Ti(2)\\\\\n0.58 & (11:8) & & 1 1 Tm(1) & \\\\\n0.58 & (7:5) & & 1 6 Y(1) & \\\\\n0.59 & (22:15) & & 1 1 Tm(1) & \\\\\n0.6 & (3:2) & 43 146 Bi(16) & 23 58 Yb(6) & 18 38 In(6)\\\\\n0.62 & (8:5) & 1 1 Mn(1) & 1 1 Cr(1) & 1 1 Cr(1)\\\\\n0.62 & (5:3) & 6 7 V(4) & 2 2 U(2) & 2 2 U(2)\\\\\n0.63 & (12:7) & 1 3 Tb(1) & & \\\\\n0.64 & (7:4) & 4 6 Ti(3) & 1 1 P(1) & \\\\\n0.64 & (16:9) & 1 1 Pr(1) & & \\\\\n0.64 & (9:5) & 6 7 Ti(3) & & \\\\\n0.65 & (20:11) & 1 2 Tb(1) & & \\\\\n0.65 & (11:6) & 7 9 Ti(4) & & \\\\\n0.65 & (13:7) & 1 2 V(1) & & \\\\\n0.65 & (15:8) & 2 3 Ti(2) & & 1 1 Gd(1)\\\\\n0.65 & (17:9) & 1 2 V(1) & & \\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries_2}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[Prevalence of binary stoichiometries continued (3\/3).]\n{The entries for each element column denote the total number of structure types,\ntotal number of unique compounds and then the leading atom with the total\nnumber of structure types of this stoichiometry in which it appears.\nThe second column shows the stoichiometry $(x:y)$ for $A_xZ_y$, $Z=$ O, S, Se, respectively.}\n\\vspace{3mm}\n\\begin{tabular}{l|l|r|r|r}\nratio $y\/(x+y)$ & stoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n0.67 & (2:1) & 244 356 Si(185) & 50 123 Ti(9) & 34 79 Ta(10)\\\\\n0.68 & (13:6) & 5 5 V(5) & & \\\\\n0.69 & (35:16) & 1 1 U(1) & & \\\\\n0.69 & (9:4) & 5 5 V(2) & 2 2 P(2) & 1 1 Ge(1)\\\\\n0.7 & (7:3) & 1 1 V(1) & & \\\\\n0.71 & (12:5) & 1 1 Cr(1) & & \\\\\n0.71 & (29:12) & 4 4 Nb(4) & & \\\\\n0.71 & (5:2) & 20 26 Nb(6) & 4 8 U(1) & 3 5 Th(1)\\\\\n0.72 & (34:13) & 1 1 U(1) & & \\\\\n0.72 & (21:8) & 2 2 W(1) & & \\\\\n0.73 & (8:3) & 8 9 U(6) & 2 2 Ir(1) & 1 2 Rh(1)\\\\\n0.73 & (49:18) & 1 1 W(1) & & \\\\\n0.73 & (11:4) & 3 3 Mo(3) & & \\\\\n0.73 & (47:17) & 2 2 W(1) & & \\\\\n0.74 & (14:5) & 2 2 W(1) & & \\\\\n0.74 & (23:8) & 2 2 Mo(2) & & \\\\\n0.74 & (26:9) & 2 2 Mo(2) & & \\\\\n0.74 & (73:25) & 1 1 W(1) & & \\\\\n0.75 & (3:1) & 33 41 W(13) & 8 12 Ti(2) & 6 10 Ta(3)\\\\\n0.78 & (7:2) & 4 4 Tc(1) & 2 2 P(2) & \\\\\n0.8 & (4:1) & 3 4 Ru(2) & 1 1 V(1) & 2 2 Nb(1)\\\\\n0.82 & (9:2) & 1 1 P(1) & & 2 2 V(1)\\\\\n0.84 & (21:4) & 1 1 U(1) & & \\\\\n0.89 & (8:1) & & 1 1 O(1) & \\\\\n0.93 & (14:1) & & 1 1 C(1) & \\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries_3}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[Prevalence of stoichiometries for the elements in the binary compounds, $A_xB_y,~B=$ O, S, Se (1\/3).]\n{The data presented is:\n$y:x(n_1)$, $n_2$, $n_3$, where $y:x$ is the leading stoichiometry, $n_1$ is number of compounds\nfor this stoichiometry, $n_2$ is number of stoichiometries and\n$n_3$ is number of unique compounds for this element.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\natom & oxides & sulfides & selenides \\\\\n\\hline\nAc & 3:2(1),1,1&&\\\\\nAg & 1:1(5),5,11&1:2(3),2,4&1:2(2),2,3\\\\\nAl & 3:2(7),3,9&3:2(3),1,3&3:2(1),1,1\\\\\nAs & 3:2(5),3,8&1:1(4),6,10&1:1(2),3,5\\\\\nAu & 3:2(1),1,1&1:2(1),1,1&1:1(3),1,3\\\\\nB & 3:2(2),3,4&1:12(1),3,3&\\\\\nBa & 1:1(3),2,5&3:1(2),4,5&1:1(2),3,4\\\\\nBe & 1:1(4),1,4&1:1(1),1,1&1:1(2),1,2\\\\\nBi & 3:2(16),4,20&3:2(1),1,1&1:1(4),6,11\\\\\nBr & 1:2(1),2,2&1:1(1),1,1&1:1(2),2,3\\\\\nC & 2:1(9),4,13&4:15(1),5,5&2:1(1),1,1\\\\\nCa & 1:1(2),2,3&1:1(1),1,1&1:1(2),1,2\\\\\nCd & 1:1(2),2,3&1:1(4),2,5&1:1(3),2,4\\\\\nCe & 2:1(5),5,12&2:1(4),4,8&2:1(3),3,6\\\\\nCl & 1:2(1),4,4&1:2(1),2,2&1:4(2),2,3\\\\\nCo & 1:1(5),3,10&8:9(1),4,4&4:3(2),4,6\\\\\nCr & 2:1(5),8,12&1:1(2),5,7&1:1(2),5,7\\\\\nCs & 1:7(1),8,8&1:2(1),5,5&1:2(2),4,5\\\\\nCu & 1:1(3),5,9&1:2(4),7,14&1:2(2),4,7\\\\\nDy & 3:2(4),1,4&2:1(2),5,6&1:1(2),4,5\\\\\nEr & 3:2(4),1,4&3:2(3),4,8&2:1(2),3,4\\\\\nEu & 3:2(4),4,7&1:1(3),3,6&1:1(2),2,3\\\\\nF & &1:6(5),1,5&1:4(1),1,1\\\\\nFe & 4:3(8),4,19&1:1(6),5,18&1:1(4),4,9\\\\\nGa & 3:2(3),1,3&1:1(2),2,3&1:1(5),3,7\\\\\nGd & 3:2(4),3,6&2:1(3),4,6&1:1(1),5,5\\\\\nGe & 2:1(7),1,7&2:1(5),2,7&2:1(5),3,8\\\\\nH & 1:2(6),2,7&1:2(3),1,3&1:2(1),1,1\\\\\nHf & 2:1(5),1,5&1:2(1),3,3&2:1(1),2,2\\\\\nHg & 1:1(5),2,7&1:1(4),1,4&1:1(3),1,3\\\\\n\\end{tabular}\n\\label{tab:art130:Elements_stoichiometries}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[Prevalence of stoichiometries for the elements in the binary compounds, $A_xB_y,~B=$ O, S, Se continued (2\/3).]\n{The data presented is:\n$y:x(n_1)$, $n_2$, $n_3$, where $y:x$ is the leading stoichiometry, $n_1$ is number of compounds\nfor this stoichiometry, $n_2$ is number of stoichiometries and\n$n_3$ is number of unique compounds for this element.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\natom & oxides & sulfides & selenides \\\\\n\\hline\nHo & 3:2(4),1,4&1:1(2),4,7&1:1(1),3,3\\\\\nI & 3:1(2),3,4&&\\\\\nIn & 3:2(5),1,5&1:1(2),5,7&3:2(6),5,14\\\\\nIr & 2:1(1),1,1&2:1(2),3,4&2:1(1),2,2\\\\\nK & 2:1(2),4,5&1:2(2),4,6&1:2(1),4,4\\\\\nLa & 3:2(4),1,4&1:1(5),4,12&1:1(4),3,7\\\\\nLi & 1:2(2),4,6&1:2(3),2,4&1:2(1),1,1\\\\\nLu & 3:2(4),1,4&3:2(3),3,6&1:1(1),3,3\\\\\nMg & 1:1(12),2,13&1:1(2),1,2&1:1(3),2,4\\\\\nMn & 3:2(4),6,14&1:1(4),2,6&1:1(4),2,5\\\\\nMo & 2:1(3),7,15&4:3(2),4,6&11:9(2),4,6\\\\\nN & 2:1(4),5,8&1:1(4),3,6&1:1(2),1,2\\\\\nNa & 2:1(2),4,5&1:1(4),4,9&1:2(1),3,3\\\\\nNb & 5:2(6),6,18&2:1(7),6,14&2:1(7),8,15\\\\\nNd & 3:2(4),2,5&2:1(3),4,6&2:1(2),3,4\\\\\nNi & 1:1(4),2,6&2:1(8),6,16&1:1(2),5,6\\\\\nO & &1:3(3),3,5&1:2(4),3,6\\\\\nOs & 4:1(2),2,3&2:1(1),1,1&2:1(1),1,1\\\\\nP & 5:2(3),5,7&3:4(2),8,13&3:4(1),4,4\\\\\nPa & 2:1(2),2,3&&\\\\\nPb & 1:1(4),5,14&1:1(13),1,13&1:1(3),2,4\\\\\nPd & 1:1(4),3,6&1:4(1),5,5&1:1(2),7,8\\\\\nPm & 3:2(3),1,3&&\\\\\nPr & 3:2(3),7,12&2:1(3),4,6&2:1(2),3,4\\\\\nPt & 2:1(5),3,9&1:1(2),2,3&4:5(1),2,2\\\\\nPu & 3:2(2),3,4&2:1(2),2,3&1:1(1),3,3\\\\\nRb & 3:2(2),7,8&1:2(3),4,7&1:2(1),4,4\\\\\nRe & 3:1(4),3,8&2:1(2),1,2&2:1(2),1,2\\\\\nRh & 3:2(4),2,5&15:17(1),4,4&2:1(2),4,5\\\\\nRu & 2:1(4),2,6&2:1(1),1,1&2:1(1),1,1\\\\\n\\end{tabular}\n\\label{tab:art130:Elements_stoichiometries_2}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[Prevalence of stoichiometries for the elements in the binary compounds, $A_xB_y,~B=$ O, S, Se continued (3\/3).]\n{The data presented is:\n$y:x(n_1)$, $n_2$, $n_3$, where $y:x$ is the leading stoichiometry, $n_1$ is number of compounds\nfor this stoichiometry, $n_2$ is number of stoichiometries and\n$n_3$ is number of unique compounds for this element.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\natom & oxides & sulfides & selenides \\\\\n\\hline\nS & 3:1(3),3,5&&\\\\\nSb & 3:2(5),3,11&3:2(1),1,1&3:2(1),1,1\\\\\nSc & 3:2(4),1,4&3:2(2),2,3&1:1(1),2,2\\\\\nSe & 2:1(4),3,6&&\\\\\nSi & 2:1(185),1,185&2:1(2),1,2&2:1(1),1,1\\\\\nSm & 3:2(4),2,5&1:1(2),3,4&1:1(2),4,5\\\\\nSn & 2:1(8),3,12&1:1(5),3,8&1:1(3),2,4\\\\\nSr & 1:1(2),2,3&1:1(2),3,4&1:1(1),1,1\\\\\nTa & 5:2(3),6,9&2:1(7),6,13&2:1(10),4,15\\\\\nTb & 3:2(4),5,9&2:1(4),4,7&1:1(2),3,4\\\\\nTc & 2:1(1),2,2&2:1(1),1,1&\\\\\nTe & 2:1(9),4,12&&\\\\\nTh & 2:1(2),1,2&1:1(1),4,4&1:1(2),5,6\\\\\nTi & 2:1(14),14,42&2:1(9),5,16&1:1(3),9,13\\\\\nTl & 1:2(2),3,5&1:1(6),4,9&1:1(3),3,6\\\\\nTm & 3:2(4),1,4&3:2(6),6,12&1:1(1),3,3\\\\\nU & 8:3(6),9,22&5:3(2),7,9&5:3(2),6,8\\\\\nV & 2:1(10),18,42&2:1(3),6,11&4:5(1),5,5\\\\\nW & 3:1(13),9,24&2:1(2),1,2&2:1(1),1,1\\\\\nXe & 3:1(1),1,1&&\\\\\nY & 3:2(5),1,5&3:2(2),4,6&1:1(1),2,2\\\\\nYb & 3:2(2),3,4&3:2(6),4,12&1:1(2),4,5\\\\\nZn & 1:1(4),2,5&1:1(39),2,40&1:1(2),2,3\\\\\nZr & 2:1(7),4,12&1:1(2),7,8&1:2(1),3,3\\\\\n\\end{tabular}\n\\label{tab:art130:Elements_stoichiometries_3}\n\\end{table}\n\n\\clearpage\n\n\\subsection{Correlation between ternary and binary stoichiometries for sulfides and selenides}\nIn this section we analyze the correlation between ternary and binary stoichiometries for sulfides and selenides.\nFigure~\\ref{fig:art130:tern_bin_stoichiometries} shows that, like in the oxides, in both the sulfides and selenides\nwe see a quite scattered pattern. However unlike in the oxides many atoms show points below the line $y=4x$.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig026}\n\\mycaption[The number of ternary (\\textbf{a}) sulfide and (\\textbf{b}) selenide stoichiometries\nper element as a function of the count of its respective binary stoichiometries.]\n{The dashed line marks perfect similarity $y=x$, and the dotted line marks the ratio $y=4x$.}\n\\label{fig:art130:tern_bin_stoichiometries}\n\\end{figure}\n\nWe next analyze in Figure~\\ref{fig:art130:tern_binproduct_stoichiometries} the number of ternary stoichiometries\nas a function of the product of the numbers of the binary stoichiometries of participating atoms.\nAs for the oxides (Figure~\\ref{fig:art130:mendeleev_distribution_all_in_one})\nwe see a trend of inverse correlation, \\nobreak\\mbox{\\it i.e.}, as the product of the numbers of binary\nstoichiometries increases, the number of ternaries decreases.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig027}\n\\mycaption[The number of ternary (\\textbf{a}) sulfide and (\\textbf{b}) selenide stoichiometries\nas a function of the product of the number of the binary stoichiometries of participating elements.]\n{The element with the most binary sulfide\/selenide stoichiometries (P\/Ti) is shown with red ``x'' symbols.\nAll other compounds are shown with blue circles.}\n\\label{fig:art130:tern_binproduct_stoichiometries}\n\\end{figure}\n\n\\subsection{Prevalence of unit cell sizes}\nIn Tables~\\ref{tab:art130:Number_of_atoms_in_Binaries_unit_cells} and\n\\ref{tab:art130:Number_of_atoms_in_ternary_unit_cells}, the number of atoms per unit cell in\nbinary and ternary compounds is shown for systems of up to 100 atoms in the unit cell.\nIn the binary oxides, there is higher prevalence for numbers that are multiples of\n4, 6, and also 12 ---\nfor example --- 12(102), 24(58), 80(47) and 72(20). In addition, 5(24) and a few of its multiples are also common.\nPrime numbers of atoms per unit cell above 10 are very rare ---\n11(2), 19(3), 29(1), 31(2), 67(1) and all the rest do not appear at all.\nIn the ternary oxides, we see a similar behavior: a high prevalence for numbers that are multiple of\n4, 6 and 12 --- for example --- 12 (119), 18 (140), 24 (465), 30(106), 72(102), 80(83), 88(178), 96(51).\nPrime numbers, between 10 to 20 do appear --- 11(15), 13(30), 17(6), 19(15), but those\nabove 20 are very rare.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of unit cell sizes among the binary compounds (1\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nnumber of atoms & oxides & sulfides & selenides \\\\\n\\hline\n1 & 0 & 0 & 0\\\\\n2 & 7 & 12 & 8\\\\\n3 & 7 & 15 & 13\\\\\n4 & 24 & 20 & 22\\\\\n5 & 24 & 1 & 5\\\\\n6 & 60 & 48 & 24\\\\\n7 & 0 & 3 & 2\\\\\n8 & 63 & 70 & 60\\\\\n9 & 10 & 1 & 1\\\\\n10 & 15 & 9 & 6\\\\\n11 & 2 & 1 & 2\\\\\n12 & 102 & 63 & 50\\\\\n13 & 0 & 1 & 0\\\\\n14 & 19 & 11 & 9\\\\\n15 & 2 & 1 & 0\\\\\n16 & 23 & 22 & 11\\\\\n17 & 0 & 0 & 1\\\\\n18 & 12 & 6 & 5\\\\\n19 & 3 & 0 & 0\\\\\n20 & 37 & 38 & 20\\\\\n21 & 0 & 2 & 0\\\\\n22 & 8 & 2 & 3\\\\\n23 & 0 & 0 & 0\\\\\n24 & 58 & 39 & 14\\\\\n25 & 1 & 0 & 0\\\\\n26 & 2 & 1 & 2\\\\\n27 & 0 & 1 & 0\\\\\n28 & 27 & 24 & 18\\\\\n29 & 1 & 0 & 0\\\\\n30 & 17 & 10 & 2\\\\\n31 & 2 & 0 & 0\\\\\n32 & 18 & 14 & 10\\\\\n33 & 0 & 0 & 0\\\\\n34 & 2 & 1 & 0\\\\\n35 & 0 & 0 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Number_of_atoms_in_Binaries_unit_cells}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of unit cell sizes among the binary compounds continued (2\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nnumber of atoms & oxides & sulfides & selenides \\\\\n\\hline\n36 & 25 & 8 & 4\\\\\n37 & 0 & 0 & 0\\\\\n38 & 4 & 0 & 0\\\\\n39 & 0 & 0 & 1\\\\\n40 & 15 & 6 & 2\\\\\n41 & 0 & 0 & 0\\\\\n42 & 0 & 2 & 0\\\\\n43 & 0 & 0 & 0\\\\\n44 & 5 & 5 & 1\\\\\n45 & 0 & 2 & 3\\\\\n46 & 2 & 1 & 0\\\\\n47 & 0 & 0 & 0\\\\\n48 & 27 & 10 & 3\\\\\n49 & 0 & 0 & 0\\\\\n50 & 0 & 0 & 0\\\\\n51 & 0 & 0 & 0\\\\\n52 & 3 & 3 & 2\\\\\n53 & 0 & 0 & 0\\\\\n54 & 2 & 0 & 0\\\\\n55 & 0 & 0 & 0\\\\\n56 & 9 & 9 & 2\\\\\n57 & 0 & 0 & 0\\\\\n58 & 0 & 2 & 0\\\\\n59 & 0 & 0 & 0\\\\\n60 & 5 & 1 & 0\\\\\n61 & 0 & 0 & 0\\\\\n62 & 1 & 0 & 0\\\\\n63 & 0 & 0 & 0\\\\\n64 & 2 & 4 & 1\\\\\n65 & 0 & 0 & 0\\\\\n66 & 0 & 0 & 0\\\\\n67 & 1 & 0 & 0\\\\\n68 & 7 & 3 & 2\\\\\n69 & 0 & 0 & 0\\\\\n70 & 0 & 0 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Number_of_atoms_in_Binaries_unit_cells_2}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of unit cell sizes among the binary compounds continued (3\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nnumber of atoms & oxides & sulfides & selenides \\\\\n\\hline\n71 & 0 & 0 & 0\\\\\n72 & 20 & 2 & 1\\\\\n73 & 0 & 0 & 0\\\\\n74 & 0 & 1 & 0\\\\\n75 & 0 & 0 & 0\\\\\n76 & 3 & 2 & 0\\\\\n77 & 0 & 0 & 0\\\\\n78 & 0 & 0 & 0\\\\\n79 & 0 & 0 & 0\\\\\n80 & 47 & 6 & 12\\\\\n81 & 0 & 0 & 0\\\\\n82 & 2 & 0 & 0\\\\\n83 & 0 & 0 & 0\\\\\n84 & 2 & 0 & 0\\\\\n85 & 0 & 0 & 0\\\\\n86 & 0 & 0 & 0\\\\\n87 & 0 & 0 & 0\\\\\n88 & 1 & 3 & 2\\\\\n89 & 0 & 0 & 0\\\\\n90 & 0 & 0 & 2\\\\\n91 & 0 & 0 & 0\\\\\n92 & 1 & 0 & 0\\\\\n93 & 0 & 0 & 0\\\\\n94 & 1 & 0 & 0\\\\\n95 & 0 & 0 & 0\\\\\n96 & 22 & 1 & 0\\\\\n97 & 0 & 0 & 0\\\\\n98 & 1 & 0 & 0\\\\\n99 & 0 & 0 & 0\\\\\n100 & 0 & 0 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Number_of_atoms_in_Binaries_unit_cells_3}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of unit cell sizes among the ternary compounds (1\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nnumber of atoms & oxides & sulfides & selenides \\\\\n\\hline\n1 & 0 & 0 & 0\\\\\n2 & 0 & 0 & 0\\\\\n3 & 1 & 1 & 2\\\\\n4 & 81 & 112 & 64\\\\\n5 & 173 & 36 & 16\\\\\n6 & 62 & 35 & 23\\\\\n7 & 10 & 29 & 16\\\\\n8 & 64 & 48 & 18\\\\\n9 & 38 & 8 & 4\\\\\n10 & 186 & 33 & 35\\\\\n11 & 15 & 2 & 0\\\\\n12 & 119 & 104 & 76\\\\\n13 & 30 & 5 & 1\\\\\n14 & 116 & 44 & 60\\\\\n15 & 12 & 40 & 30\\\\\n16 & 143 & 106 & 69\\\\\n17 & 6 & 3 & 0\\\\\n18 & 140 & 31 & 26\\\\\n19 & 15 & 0 & 0\\\\\n20 & 363 & 179 & 88\\\\\n21 & 10 & 1 & 0\\\\\n22 & 142 & 25 & 20\\\\\n23 & 1 & 0 & 1\\\\\n24 & 465 & 146 & 57\\\\\n25 & 7 & 0 & 0\\\\\n26 & 65 & 26 & 14\\\\\n27 & 16 & 1 & 0\\\\\n28 & 287 & 190 & 130\\\\\n29 & 1 & 0 & 0\\\\\n30 & 106 & 22 & 12\\\\\n31 & 0 & 0 & 0\\\\\n32 & 181 & 96 & 67\\\\\n33 & 4 & 0 & 0\\\\\n34 & 38 & 16 & 8\\\\\n35 & 0 & 1 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Number_of_atoms_in_ternary_unit_cells}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of unit cell sizes among the ternary compounds continued (2\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nnumber of atoms & oxides & sulfides & selenides \\\\\n\\hline\n36 & 211 & 67 & 46\\\\\n37 & 5 & 0 & 0\\\\\n38 & 26 & 19 & 3\\\\\n39 & 1 & 2 & 0\\\\\n40 & 216 & 65 & 30\\\\\n41 & 2 & 0 & 0\\\\\n42 & 40 & 4 & 5\\\\\n43 & 3 & 0 & 0\\\\\n44 & 193 & 40 & 22\\\\\n45 & 6 & 2 & 2\\\\\n46 & 24 & 2 & 5\\\\\n47 & 1 & 0 & 0\\\\\n48 & 118 & 28 & 17\\\\\n49 & 6 & 0 & 0\\\\\n50 & 12 & 0 & 0\\\\\n51 & 0 & 0 & 0\\\\\n52 & 114 & 27 & 16\\\\\n53 & 0 & 0 & 0\\\\\n54 & 17 & 8 & 2\\\\\n55 & 1 & 0 & 0\\\\\n56 & 171 & 109 & 56\\\\\n57 & 6 & 0 & 0\\\\\n58 & 14 & 8 & 3\\\\\n59 & 1 & 0 & 0\\\\\n60 & 104 & 31 & 10\\\\\n61 & 1 & 0 & 0\\\\\n62 & 7 & 0 & 3\\\\\n63 & 5 & 0 & 0\\\\\n64 & 86 & 31 & 31\\\\\n65 & 0 & 0 & 0\\\\\n66 & 17 & 0 & 1\\\\\n67 & 0 & 0 & 0\\\\\n68 & 99 & 28 & 14\\\\\n69 & 0 & 0 & 0\\\\\n70 & 6 & 0 & 2\\\\\n\\end{tabular}\n\\label{tab:art130:Number_of_atoms_in_ternary_unit_cells_2}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of unit cell sizes among the ternary compounds continued (3\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nnumber of atoms & oxides & sulfides & selenides \\\\\n\\hline\n71 & 0 & 0 & 0\\\\\n72 & 102 & 48 & 39\\\\\n73 & 0 & 0 & 0\\\\\n74 & 2 & 21 & 3\\\\\n75 & 0 & 0 & 0\\\\\n76 & 48 & 6 & 5\\\\\n77 & 0 & 0 & 0\\\\\n78 & 8 & 1 & 0\\\\\n79 & 0 & 0 & 0\\\\\n80 & 83 & 8 & 8\\\\\n81 & 0 & 0 & 0\\\\\n82 & 3 & 1 & 0\\\\\n83 & 0 & 0 & 0\\\\\n84 & 30 & 17 & 7\\\\\n85 & 0 & 0 & 0\\\\\n86 & 7 & 0 & 0\\\\\n87 & 1 & 0 & 0\\\\\n88 & 178 & 12 & 20\\\\\n89 & 0 & 0 & 0\\\\\n90 & 9 & 2 & 1\\\\\n91 & 0 & 0 & 0\\\\\n92 & 20 & 8 & 6\\\\\n93 & 0 & 0 & 0\\\\\n94 & 3 & 0 & 0\\\\\n95 & 0 & 0 & 0\\\\\n96 & 51 & 23 & 6\\\\\n97 & 0 & 0 & 0\\\\\n98 & 1 & 1 & 2\\\\\n99 & 3 & 0 & 0\\\\\n100 & 14 & 3 & 2\\\\\n\\end{tabular}\n\\label{tab:art130:Number_of_atoms_in_ternary_unit_cells_3}\n\\end{table}\n\n\\clearpage\n\n\\subsection{Additional Mendeleev plots}\nThe Mendeleev map for the 1:1:2 stoichiometry are shown in~\\ref{fig:art130:mend_211_stoichiometries}.\nThe maps of the sulfides and selenides cover nearly identical regions, while that of the oxides\nincludes an additional row for hydrogen (Mendeleev number 103).\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig028}\n\\mycaption[Comparison of Mendeleev maps for the 211 (\\textbf{a}) oxide, (\\textbf{b}) sulfide and (\\textbf{c}) selenide stoichiometries.]\n{The number in parenthesis is the number of compounds for this structure type, for ``Other'',\nit refers to the total number of compounds with this stoichiometry.}\n\\label{fig:art130:mend_211_stoichiometries}\n\\end{figure}\n\n\\subsection{Summary}\nWe present a comprehensive analysis of the statistics of the binary\nand ternary compounds of oxygen, sulfur and selenium. This analysis and the visualization tools presented here are\nvaluable to finding trends as well as exceptions and peculiar phenomena.\n\nOxygen has a higher electronegativity (3.44) than sulfur\n(2.58) and selenium (2.55), which are similar to each other.\nTherefore, one can expect that oxygen will form compounds with a stronger ionic character.\nOxygen is 1000 times more abundant than sulfur, and more than $10^6$ times than selenium~\\cite{wedepohl1995composition},\nhowever, it has less than two times the number of binary compounds compared to sulfur and $2.5$ that of selenium.\nHence, the abundance of those elements plays a little role in the relative numbers of their known compounds.\nThese important differences are reflected in our analysis by\nthe significantly larger fraction of oxygen rich compounds compared to\nthose that are sulfur or selenium rich.\nStructure type classification also shows that there is little overlap between the oxygen\nstructure types to sulfur or selenium structure types, while\nsulfur and selenium present a much higher overlap. The gaps in these overlaps, especially between\nthe sulfides and selenides, indicate that favorable candidates for new compounds\nmay be obtained by simple element substitution in the corresponding structures.\nIn particular, structures that are significantly more common in one family,\nsuch as KrF$_{2}$ in the oxides, may be good candidates for new compounds in another.\nComparison of these three 6A elements binary and ternary\ncompounds shows significant differences but also some similarities in the symmetry\ndistributions among the various Bravais lattices and their\ncorresponding space groups. In particular, the majority of structure types in all three families have a few or single\ncompound realizations. This prevalence of unique structure types suggests a ripe field for identification of currently unknown compounds,\nby substitution of elements of similar chemical characteristics.\nIn addition, the analysis of the distribution of known compounds among symmetry space groups and,\nin particular, their apparent concentration in specific hot spots of this\nsymmetry space may be serve as a useful insight for searches of potential new compounds.\n\nAn important observation is the existence of different gaps (missing stoichiometries) in the stoichiometry distribution of the oxide\nbinary compounds compared to the sulfides and selenides (Figures~\\ref{fig:triangle_stoichiometries} and \\ref{fig:art130:specific_triangle_stoichiometries},\nand Tables~\\ref{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries}-\\ref{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries_3}).\nStoichiometries such as 5:7 appear in the oxides but are missing in the sulfides and selenides.\nMore rare are non-overlapping gaps between the selenides and sulfides, \\nobreak\\mbox{\\it e.g.}, 6:1 and 5:7.\nThese should be prime candidates for new compounds by element substitution between the two families.\nFuture work would be directed at exploiting these discrepancies to search for new compounds within different subsets of those compound families.\n\nSpecific elements tend to present very different stoichiometry distributions, for example, silicon forms only\none oxide stoichiometry (SiO$_2$) while transition metals such as titanium and vanadium present\n14 and 18 different stoichiometries respectively.\nThese differences clearly reflect the different chemistry of those elements, while the large number of reported\nSiO$_2$ structures might reflect research bias into silicon compounds.\n\nAnother important finding is that there is an inverse correlation between the number of ternary stoichiometries\nto the product of binary stoichiometries of participating elements.\nThis can be caused by the fact that there are too many binary phases and hence it becomes\ndifficult to create a stable ternary that competes with all of them.\n\nA Mendeleev analysis of the common structure types of these\nfamilies shows accumulation of different structures at\nwell defined regions of their respective maps, similar to the well-known Pettifor maps of binary structure types.\nFurthermore, at least for some of the stoichiometries, similarity of the maps for a\ngiven stoichiometry is demonstrated across all three elements.\nThese maps should therefore prove useful for predictive purposes regarding the existence\nof yet unknown compounds of the corresponding structure types.\nFuture work will be directed at exploiting identified non-overlapping gaps in the\nMendeleev maps for a directed search of new compounds in these families.\nComplementary properties (\\nobreak\\mbox{\\it e.g.}, partial charges, bond analysis, electronic properties)\nshould be incorporated in the analysis to reveal additional insights of the aforementioned trends among the three elements.\n\\clearpage\n\\section{AFLOW Standard for High-Throughput Materials Science Calculations}\n\\label{sec:art104}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art104},\nwhich was awarded with Comput. Mater. Sci. Editor's Choice.\n\n\\subsection{Introduction} \\label{subsec:art104:intro}\n\nThe emergence of computational materials science over the last two decades has been inextricably linked to the\ndevelopment of complex quantum-mechanical codes that enable accurate evaluation of the electronic and\nthermodynamic properties of a wide range of materials. The continued advancement of this field entails the\nconstruction of large open databases of materials properties that can be easily reproduced and extended.\nOne obstacle to the reproducibility of the data is the unavoidable complexity of the codes used to obtain\nit. Published data usually includes basic information about the underlying calculations that allows rough\nreproduction. However, exact duplication depends on many details, that are seldom reported, and is therefore\ndifficult to achieve.\n\nThese difficulties might limit the utility of the databases currently being created by high-throughput frameworks,\nsuch as {\\small AFLOW}~\\cite{aflowPAPER, aflowlibPAPER, aflowAPI} and the Materials Project~\\cite{APL_Mater_Jain2013,CMS_Ong2012b}.\nFor maximal impact, the data stored in these repositories must be generated and represented in a consistent and robust manner,\nand shared through standardized calculation and communication protocols. Following these guidelines would promote\noptimal use of the results generated by the entire community.\n\nThe {\\small AFLOW}\\ (Automatic FLOW) code is a framework for high-throughput computational materials\ndiscovery~\\cite{aflowPAPER, aflowlibPAPER, aflowAPI, aflowlib.org},\nusing separate {\\small DFT}\\ packages to calculate electronic\nstructure and optimize the atomic geometry. The {\\small AFLOW}\\ framework works with the\n{\\small VASP}~\\cite{vasp_prb1996}\\ {\\small DFT}\\ package,\nand integration with the \\textsc{Quantum {\\small ESPRESSO}}\\ software~\\cite{quantum_espresso_2009}\nis currently in progress.\nThe {\\small AFLOW}\\ framework includes\npreprocessing functions for generating input files for the {\\small DFT}\\ package; obtaining the initial geometric structures\nby extracting the relevant data from crystallographic information files or by generating them using inbuilt prototype\ndatabases, and then transforming them into standard forms which are easiest to calculate. It then runs and monitors\nthe {\\small DFT}\\ calculations automatically, detecting and responding to calculation failures, whether they are due to insufficient\nhardware resources or to runtime errors of the {\\small DFT}\\ calculation itself. Finally, {\\small AFLOW}\\ contains postprocessing\nroutines to extract specific properties from the results of one or more of the {\\small DFT}\\ calculations, such as the band\nstructure or thermal properties~\\cite{curtarolo:art96}.\n\nThe {\\sf \\AFLOW.org}\\ repository~\\cite{aflowlibPAPER, aflowAPI, aflowlib.org} was built according to these principles of consistency and reproducibility,\nand the data it contains can be easily accessed through a representational state transfer application programming\ninterface ({\\small REST-API})~\\cite{aflowAPI}. In this study we present a detailed description of the {\\small AFLOW}\\ standard\nfor high-throughput (HT) materials science calculations by which the data in this repository was created.\n\n\\subsection{AFLOW calculation types} \\label{subsec:art104:AFLOWtypes}\nThe {\\sf \\AFLOW.org}\\ repository~\\cite{aflowlibPAPER} is divided into databases containing calculated\nproperties of over 625,000 materials:\nthe Binary Alloy Project, the Electronic Structure database, the Heusler database, and the Elements database.\nThese are freely accessible online via the {\\sf \\AFLOW.org}~\\cite{aflowlib.org}, as well as through the\n{\\small API}~\\cite{aflowAPI}. The Electronic Structure database consists of entries found in the Inorganic Crystal\nStructures Database, {\\small ICSD}~\\cite{ICSD, ICSD3}, and will thus be referred to as ``{\\small ICSD}'' throughout this publication.\nThe Heusler database consists of ternary compounds, primarily based on the Heusler structure but with other\nstructure types now being added.\n\nThe high-throughput construction of these materials databases relies on a pre-defined set of standard {\\textit{calculation\ntypes}}. These are designed to accommodate the interest in various properties of a given material (\\nobreak\\mbox{\\it e.g.}, the ground\nstate ionic configuration, thermodynamic quantities, electronic and\nmagnetic properties), the program flow of the HT framework that\nenvelopes the {\\small DFT}\\ portions of the calculations, as well as the practical\nneed for computational robustness. The {\\small AFLOW}\\ standard thus deals with the parameters involved in the following\ncalculation types:\n\n\\begin{enumerate}\n \\item {{\\verb!RELAX!}.} Geometry optimizations using algorithms implemented within the {\\small DFT}\\ package. This calculation\n type is concerned with obtaining the ionic configuration and cell\n shape and volume that correspond to a minimum in the\n total energy. It consists of two sequential relaxation steps. The starting point for the first step, {\\verb!RELAX1!},\n can be an entry taken from an external source, such as a library of alloy\n prototypes~\\cite{Massalski, curtarolo:calphad_2005_monster}, the {\\small ICSD}\\ database, or the Pauling\n File~\\cite{PaulingFile}. These initial entries are preprocessed by\n {\\small AFLOW}, and cast into a unit cell that is most convenient\n for calculation, usually the standard primitive cell, in the format appropriate for the {\\small DFT}\\ package in use. The second step, {\\verb!RELAX2!},\n uses the final ionic positions from the first step as its starting point, and serves as a type of annealing step.\n This is used for jumping out of possible local minima resulting from wavefunction artifacts.\n \\item {{\\verb!STATIC!}.} A single-point energy calculation. The starting point is the set of final ionic positions,\n as produced by the {\\verb!RELAX2!} step. The outcome of this calculation is used in the determination of most\n of the thermodynamic and electronic properties included in the various {\\sf \\AFLOW.org}\\ database.\n It therefore applies a more demanding set of parameters than those used on the {\\verb!RELAX!}\n set of runs.\n \\item {{\\verb!BANDS!}.} Electronic band structure generation. The converged {\\verb!STATIC!} charge\n density and ionic positions are used as the starting points, and the wavefunctions are reoptimized along standardized\n high symmetry lines connecting special {\\bf k}-points in the irreducible Brillouin zone (IBZ)~\\cite{aflowBZ}.\n\\end{enumerate}\n\nThese calculation types are performed in the order shown above (\\nobreak\\mbox{\\it i.e.}, {\\verb!RELAX1!} $\\rightarrow$ {\\verb!RELAX2!}\n$\\rightarrow$ {\\verb!STATIC!} $\\rightarrow$ {\\verb!BANDS!}) on all materials found in the Elements,\n{\\small ICSD}, and Heusler databases. Those found in the Binary Alloy database contain data produced only by the two\n{\\verb!RELAX!} calculations.\nSets of these calculation types can be combined to describe more complex\nphenomena than can be obtained from a single calculation. For\nexample, sets of {\\verb!RELAX!} and {\\verb!STATIC!} calculations for different cell\nvolumes and\/or atomic configurations are used to calculate\nthermal and mechanical properties by the {Automatic Gibbs Library}, {\\small AGL}~\\cite{curtarolo:art96},\nand {Automatic Phonon Library}, {\\small APL}~\\cite{aflowPAPER}, methods\nimplemented within the {\\small AFLOW}\\ framework.\nIn the following, we describe the parameter sets used to address the\nparticular challenges of the calculations included in each {\\sf \\AFLOW.org}\\ repository.\n\n\\subsection{The AFLOW Standard parameter set} \\label{subsec:art104:AFLOWstandard}\nThe standard parameters described in this work are classified according to the wide variety of tasks that a typical solid\nstate {\\small DFT}\\ calculation involves: Brillouin zone sampling, Fourier transform meshes, basis sets, potentials,\nself-interaction error (SIE) corrections, electron spin, algorithms guiding SCF convergence and ionic relaxation, and\noutput options.\n\nDue to the intrinsic complexity of the {\\small DFT}\\ codes it is impractical to\nspecify the full set of {\\small DFT}\\ calculation parameters within an HT framework. Therefore, the {\\small AFLOW}\\ standard\nadopts many, but not all, of the internal defaults set by the {\\small DFT}\\ software package. This is most notable in the description of the\nFourier transform meshes, which rely on a discretization scheme that depends on the applied basis and crystal\ngeometry for its specification. Those internal default settings are cast aside when\nerror corrections of failed {\\small DFT}\\ runs, an integral part of {\\small AFLOW}{}'s functionality, take place. The settings\ndescribed in this work are nevertheless prescribed as fully as is practicable, in the interest of providing as\nmuch information as possible to anyone interested in reproducing or building on our results.\n\n\\subsubsection{{\\bf k}-point sampling} \\label{subsubsec:art104:kpointgrid}\nTwo approaches are used when sampling the IBZ: the first consists of uniformly distributing a large number\nof {\\bf k}-points in the IBZ, while the second relies on the construction of paths connecting high symmetry (special)\n{\\bf k}-points in the IBZ. Within {\\small AFLOW}, the second sampling method corresponds to the {\\verb!BANDS!}\ncalculation type, whereas the other calculation types (non-{\\verb!BANDS!}) are performed using the first sampling\nmethod.\n\nSampling in non-{\\verb!BANDS!} calculations is obtained by defining and setting $N_{\\mathrm{KPPRA}}$, the number of\n{\\bf k}-points per atom. This quantity determines the total number of {\\bf k}-points in the IBZ,\ntaking into account the {\\bf k}-points density along each reciprocal lattice vector as well as the number of atoms\nin the simulation cell, via the relation:\n\\begin{equation} \\label{eq:art104:kppra}\n { N_{\\mathrm{KPPRA}} \\leq \\min \\left[ \\prod\\limits_{i=1}^3 N_i \\right] \\times N_{\\mathrm{a} } }\n\\end{equation}\n$N_{\\mathrm{a}}$ is the number of atoms in the cell, and the $N_{i}$ factors correspond to the number\nof sampling points along each reciprocal lattice vector,\n$\\vec{b_{i}}$, respectively. These factors define the grid resolution,\n${\\it \\delta} k{_i} {\\| \\vec{b{_i}} \\|}\/{N{_i} }$, which is made as uniform as possible\nunder the constraint of Equation~\\ref{eq:art104:kppra}. The {\\bf k}-point meshes are then\ngenerated within the Monkhorst-Pack scheme~\\cite{Monkhorst1976}, unless the material belongs to the\n{\\textit{hP}}, or {\\textit{hR}} Bravais lattices, in which case the hexagonal symmetry is preserved by centering the mesh\nat the $\\Gamma$-point.\n\nDefault $N_{\\mathrm{KPPRA}}$ values depend on the calculation type and the\ndatabase. The $N_{\\mathrm{KPPRA}}$ values used for the entries in the Elements\ndatabase are material specific and set manually due to convergence of\nthe total energy calculation. The defaults applied to the\n{\\verb!RELAX!} and {\\verb!STATIC!} calculations are summarized in\nTable~\\ref{tab:art104:kgridnonbands}.\nThese defaults ensure proper convergence of the calculations. They\nmay be too stringent for some cases but enable reliable\napplication within the HT framework, thus presenting a practicable\nbalance between accuracy and calculation cost.\n\n\\begin{table}[tp]\\centering\n\\cprotect\\mycaption{Default $N_{\\mathrm{KPPRA}}$ values used in non-{\\verb!BANDS!} calculations.}\n\\vspace{3mm}\n\\begin{tabular}{l | r r}\n database & {\\verb!STATIC!} & {\\verb!RELAX!} \\\\\n \\hline\n binary alloy & N.A. & 6000 \\\\\n Heusler & 10000 & 6000 \\\\\n {\\small ICSD}\\ & 10000 & 8000 \\\\\n\\end{tabular}\n\\label{tab:art104:kgridnonbands}\n\\end{table}\n\nFor {\\verb!BANDS!} calculations {\\small AFLOW}\\ generates Brillouin zone integration\npaths in the manner described in a previous\npublication~\\cite{aflowBZ}.\nThe {\\bf k}-point sampling density is the {\\textit{line\ndensity}} of {\\bf k}-points along each of the straight-line\nsegments of the path in the IBZ. The default setting\nof {\\small AFLOW}\\ is 128 {\\bf k}-points along each segment connecting high-symmetry {\\bf k}-points in\nthe IBZ for single element structures, and 20 {\\textit\n k}-points for compounds.\n\nThe occupancies at the Fermi edge in all non-{\\verb!RELAX!} type runs are handled via the tetrahedron method with\nBl{\\\"o}chl corrections~\\cite{Bloechl1994a}. This involves the $N_{\\mathrm{KPPRA}}$ parameter, as described above. In\n{\\verb!RELAX!} type calculations, where the determination of accurate forces is important, some type of\nsmearing must be performed. In cases where the material is assumed to be a metal, the\nMethfessel-Paxton approach~\\cite{Methfessel_prb_1989} is adopted, with a smearing width of 0.10~eV.\nGaussian smearing is used in all other types of materials, with a smearing width of 0.05~eV.\n\n\\subsubsection{Potentials and basis set} \\label{subsubsec:art104:pseudopot}\n\nThe interactions involving the valence electron shells are handled with the potentials provided with the {\\small DFT}\\ software\npackage. In {\\small VASP}, these include ultra-soft pseudopotentials (USPP)~\\cite{Vanderbilt, vasp_JPCM_1994} and\nprojector-augmented wavefunction ({\\small PAW}) potentials~\\cite{PAW,kresse_vasp_paw}, which are constructed according to the Local\nDensity Approximation ({\\small LDA})~\\cite{Ceperley_prl_1980, Perdew_prb_1981}, and the Generalized Gradient Approximation\n({\\small GGA}) PW91~\\cite{VASP_PW91_1,VASP_PW91_2} and {\\small PBE}~\\cite{PBE, PBE2} exchange-correlation (XC) functionals.\nThe {\\small ICSD}, Binary Alloy and Heusler databases built according to the {\\small AFLOW}\\ standard use the {\\small PBE}\\ functional combined with\nthe {\\small PAW}\\ potential as the default. The {\\small PBE}\\ functional is among the best studied {\\small GGA}\\ functionals used in crystalline systems, while the {\\small PAW}\\ potentials\nare preferred due to their advantages over the USPP methodology. Nevertheless, defaults have been defined for a number of potential \/ XC functional\ncombinations, and in the case of the Elements database, results are available for {\\small LDA}, {\\small GGA}-PW91 and {\\small GGA}-{\\small PBE}\\ functionals with both USPP and {\\small PAW}\\ potentials.\nAdditionally, there are a small number of entries in the {\\small ICSD}\\ and Binary Alloy databases (less than 1\\% of the total) which have been calculated with the {\\small GGA}-PW91\nfunctional using either the USPP or {\\small PAW}\\ potential. The exact combination of exchange-correlation functional and potential used for a specific entry\nin the {\\sf \\AFLOW.org}\\ database can always be determined by querying the keyword \\verb|dft_type| using the {\\small AFLOW}\\\n{\\small REST-API}~\\cite{aflowAPI}.\n\n{\\small DFT}\\ packages often provide more than one potential of each type per element. The {\\small AFLOW}\\\nstandardized lists of {\\small PAW}\\ and USPP potentials are presented in\nTables~\\ref{tab:art104:tab:pot_paw} and \\ref{tab:art104:pot_uspp}, respectively.\nThe ``Label'' column in these tables corresponds to the naming convention adopted\nby {\\small VASP}. The checksum of each file listed in the tables is included in the accompanying supplement\nfor verification purposes.\n\nEach potential provided with the {\\small VASP}\\ package has two recommended plane-wave kinetic energy cut-off ($E_{\\mathrm{cut}}$)\nvalues, the smaller of which ensures the reliability of a calculation to within a well-defined error. Additionally,\nmaterials with more than one element type will have two or more sets of recommended $E_{\\mathrm{cut}}$ values.\nIn the {\\small AFLOW}\\ standard, the applied $E_{\\mathrm{cut}}$ value is the largest found among the recommendations for all\nspecies involved in the calculation, increased by a factor of 1.4.\n\nIt is possible to evaluate the the non-local parts of the potentials in real space, rather than in the more computationally\nintensive reciprocal space. This approach is prone to aliasing errors, and requires the optimization of real-space\nprojectors if these are to be avoided. The real-space projection scheme is most appropriate for larger systems, \\nobreak\\mbox{\\it e.g.}, surfaces,\nand is therefore not used in the construction of the databases found in the {\\sf \\AFLOW.org}\\ repository.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Projector-Augmented Wavefunction ({\\small PAW}) potentials, parameterized for the {\\small LDA}, PW91, and {\\small PBE}\\\nfunctionals, included in the {\\small AFLOW}\\ standard.]\n{The {\\small PAW}-{\\small PBE}\\ combination is used as the default for {\\small ICSD}\\, Binary Alloy and Heusler databases.\n$\\dagger$: {\\small PBE}\\ potentials only.\n$\\ddagger$: {\\small LDA}\\ and PW91 potentials only.}\n\\vspace{3mm}\n{\\small\n\\begin{tabular}{l r | l r | l r}\n element & label & element & label & element & label \\\\\n \\hline\n H & H & Se & Se & Gd $\\ddagger$ & Gd\\_3 \\\\\n He & He & Br & Br & Tb & Tb\\_3 \\\\\n Li & Li\\_sv & Kr & Kr & Dy & Dy\\_3 \\\\\n Be & Be\\_sv & Rb & Rb\\_sv & Ho & Ho\\_3 \\\\\n B & B\\_h & Sr & Sr\\_sv & Er & Er\\_3 \\\\\n C & C & Y & Y\\_sv & Tm & Tm \\\\\n N & N & Zr & Zr\\_sv & Yb & Yb \\\\\n O & O & Nb & Nb\\_sv & Lu & Lu \\\\\n F & F & Mo & Mo\\_pv & Hf & Hf \\\\\n Ne & Ne & Tc & Tc\\_pv & Ta & Ta\\_pv \\\\\n Na & Na\\_pv & Ru & Ru\\_pv & W & W\\_pv \\\\\n Mg & Mg\\_pv & Rh & Rh\\_pv & Re & Re\\_pv \\\\\n Al & Al & Pd & Pd\\_pv & Os & Os\\_pv \\\\\n Si & Si & Ag & Ag & Ir & Ir \\\\\n P & P & Cd & Cd & Pt & Pt \\\\\n S & S & In & In\\_d & Au & Au \\\\\n Cl & Cl & Sn & Sn & Hg & Hg \\\\\n Ar & Ar & Sb & Sb & Tl & Tl\\_d \\\\\n K & K\\_sv & Te & Te & Pb & Pb\\_d \\\\\n Ca & Ca\\_sv & I & I & Bi & Bi\\_d \\\\\n Sc & Sc\\_sv & Xe & Xe & Po & Po \\\\\n Ti & Ti\\_sv & Cs & Cs\\_sv & At & At \\\\\n V & V\\_sv & Ba & Ba\\_sv & Rn & Rn \\\\\n Cr & Cr\\_pv & La & La & Fr & Fr \\\\\n Mn & Mn\\_pv & Ce & Ce & Ra & Ra \\\\\n Fe & Fe\\_pv & Pr & Pr & Ac & Ac \\\\\n Co & Co & Nd & Nd & Th & Th\\_s \\\\\n Ni & Ni\\_pv & Pm & Pm & Pa & Pa \\\\\n Cu & Cu\\_pv & Sm $\\dagger$ & Sm & U & U \\\\\n Zn & Zn & Sm $\\ddagger$ & Sm\\_3 & Np & Np\\_s \\\\\n Ga & Ga\\_h & Eu & Eu & Pu & Pu\\_s \\\\\n As & As & Gd $\\dagger$ & Gd & & \\\\\n\\end{tabular}}\n\\label{tab:art104:tab:pot_paw}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption{Ultra-Soft Pseudopotentials (USPP), parameterized for\nthe {\\small LDA}\\ and PW91 functionals, included in the {\\small AFLOW}\\ standard.}\n\\vspace{3mm}\n{\\small\n\\begin{tabular}{l r | l r | l r}\n element & label & element & label & element & label \\\\\n \\hline\n H & H\\_soft & As & As & Tb & Tb\\_3 \\\\\n He & He & Se & Se & Dy & Dy\\_3 \\\\\n Li & Li\\_pv & Br & Br & Ho & Ho\\_3 \\\\\n Be & Be & Kr & Kr & Er & Er\\_3 \\\\\n B & B & Rb & Rb\\_pv & Tm & Tm \\\\\n C & C & Sr & Sr\\_pv & Yb & Yb \\\\\n N & N & Y & Y\\_pv & Lu & Lu \\\\\n O & O & Zr & Zr\\_pv & Hf & Hf \\\\\n F & F & Nb & Nb\\_pv & Ta & Ta \\\\\n Ne & Ne & Mo & Mo\\_pv & W & W \\\\\n Na & Na\\_pv & Tc & Tc & Re & Re \\\\\n Mg & Mg\\_pv & Ru & Ru & Os & Os \\\\\n Al & Al & Rh & Rh & Ir & Ir \\\\\n Si & Si & Pd & Pd & Pt & Pt \\\\\n P & P & Ag & Ag & Au & Au \\\\\n S & S & Cd & Cd & Hg & Hg \\\\\n Cl & Cl & In & In\\_d & Tl & Tl\\_d \\\\\n Ar & Ar & Sn & Sn & Pb & Pb \\\\\n K & K\\_pv & Sb & Sb & Bi & Bi \\\\\n Ca & Ca\\_pv & Te & Te & Po & Po \\\\\n Sc & Sc\\_pv & I & I & At & At \\\\\n Ti & Ti\\_pv & Xe & Xe & Rn & Rn \\\\\n V & V\\_pv & Cs & Cs\\_pv & Fr & Fr \\\\\n Cr & Cr & Ba & Ba\\_pv & Ra & Ra \\\\\n Mn & Mn & La & La & Ac & Ac \\\\\n Fe & Fe & Ce & Ce & Th & Th\\_s \\\\\n Co & Co & Pr & Pr & Pa & Pa \\\\\n Ni & Ni & Nd & Nd & U & U \\\\\n Cu & Cu & Pm & Pm & Np & Np\\_s \\\\\n Zn & Zn & Sm & Sm\\_3 & Pu & Pu\\_s \\\\\n Ga & Ga\\_d & Eu & Eu & & \\\\\n Ge & Ge & Gd & Gd & & \\\\\n\\end{tabular}}\n\\label{tab:art104:pot_uspp}\n\\end{table}\n\n\\subsubsection{Fourier transform meshes} \\label{subsubsec:art104:fftmesh}\n\nAs mentioned previously, it is not practical to describe the precise default settings that are applied by the {\\small AFLOW}\\\nstandard in the specification of the Fourier transform meshes. We\nshall just note that they are defined in terms of the grid\nspacing along each of the reciprocal lattice vectors, $\\vec{b}_i$. These are obtained from the set of real space lattice\nvectors, $\\vec{a}_i$, via $ [\\vec{b}_1 \\vec{b}_2 \\vec{b}_3]^T = 2 \\pi [\\vec{a}_1 \\vec{a}_2 \\vec{a}_3]^{-1} $. A distance\nin reciprocal space is then defined by $d_i={\\|\\vec{b{_i}}\\|} \/n_i$, where the set of $n_i$ are the number\nof grid points along each reciprocal lattice vector, and where the total number of points in the simulation is\n$n_1 \\times n_2 \\times n_3$.\n\nThe {\\small VASP}\\ package relies primarily on the so-called {\\textit{dual grid technique}}, which consists of two overlapping\nmeshes with different coarseness. The least dense of the two is directly dependent on the applied plane-wave basis, $E_{\\mathrm{cut}}$,\nwhile the second is a finer mesh onto which the charge density is mapped. The {\\small AFLOW}\\ standard relies on placing\nsufficient points in the finer mesh such that wrap-around (``aliasing'') errors are avoided. In terms of the quantity $d_i$,\ndefined above, the finer grid is characterized by $d_i \\approx 0.10${\\textit{ \\r{A}}$^{-1}$}, while the coarse grid results\nin $d_i \\approx 0.15${\\textit{ \\r{A}}$^{-1}$}. These two values are approximate, as there is significant dispersion in\nthese quantities across the various databases.\n\n\\subsubsection{DFT$+U$ corrections} \\label{subsubsec:art104:Hubbard}\n\nExtended systems containing {\\textit d} and {\\textit f} block elements are often poorly represented within {\\small DFT}\\ due to\nthe well known self interaction error (SIE)~\\cite{Perdew_prb_1981}. The influence that the SIE has on the energy gap of\ninsulators has long been recognized, and several methods that account for it are available. These include the\n{\\textit{GW}} approximation~\\cite{Hedin_GW_1965}, the rotationally invariant approach introduced by\nDudarev~\\cite{Dudarev_dftu} and Liechtenstein~\\cite{Liechtenstein1995} (denoted here as {\\small DFT}$+U$), as well as the recently\ndeveloped ACBN0 pseudo-hybrid density functional~\\cite{curtarolo:art93}.\n\nThe {\\small DFT}$+U$ approach is currently the best suited for high-throughput investigations, and is therefore included in\nthe {\\small AFLOW}\\ standard for the entire {\\small ICSD}\\ database, and is also used for certain entries in the Heusler\ndatabase containing the elements O, S, Se, and F. It is not used for the Binary Alloy database.\nThis method has a significant dependence on parameters, as each atom is associated with\ntwo numbers, the screened Coulomb parameter, $U$, and the Stoner exchange parameter, $J$. These are usually reported\nas a single factor, combined via $U_{\\mathrm{eff}}=U-J$. The set of $U_{\\mathrm{eff}}$ values associated with the\n{\\textit d} block elements~\\cite{aflowBZ,curtarolo:art68} are presented in Table~\\ref{tab:art104:Ud}, to which the\nelements In and Sn have been added.\n\nA subset of the {\\textit f}-block elements can be found among the systems included in\nthe {\\sf \\AFLOW.org}\\ databases. We are not aware of the existence of a systematic search for the best set\nof $U$ and $J$ parameters for this region of the periodic table, so we have relied on an in-house\nparameterization~\\cite{aflowBZ} in the construction of the databases. The values used are reproduced\nin Table~\\ref{tab:art104:Uf}. Note that by construction the SIE correction must be applied to a pre-selected value of the\n$\\ell$-quantum number, and all elements listed in Table~\\ref{tab:art104:Ud} correspond to $\\ell=2$, while those\nfound in Table~\\ref{tab:art104:Uf} correspond to $\\ell=3$.\n\n\\begin{table}[tp]\\centering\n\\mycaption{$U_{\\mathrm{eff}}$ parameters applied to {\\textit d} orbitals.}\n\\vspace{3mm}\n \\begin{tabular}{l r | l r}\n element & $U_{\\mathrm{eff}}$ & element & $U_{\\mathrm{eff}}$ \\\\\n \\hline\n Sc~\\cite{ScUJ} & 2.9 & W~\\cite{NbUJ} & 2.2 \\\\\n Ti~\\cite{TiUJ} & 4.4 & Tc~\\cite{NbUJ} & 2.7 \\\\\n V~\\cite{VUJ} & 2.7 & Ru~\\cite{NbUJ} & 3.0 \\\\\n Cr~\\cite{CrUJ} & 3.5 & Rh~\\cite{NbUJ} & 3.3 \\\\\n Mn~\\cite{CrUJ} & 4.0 & Pd~\\cite{NbUJ} & 3.6 \\\\\n Fe~\\cite{FeUJ} & 4.6 & Ag~\\cite{AgUJ} & 5.8 \\\\\n Co~\\cite{VUJ} & 5.0 & Cd~\\cite{ZnUJ} & 2.1 \\\\\n Ni~\\cite{VUJ} & 5.1 & In~\\cite{ZnUJ} & 1.9 \\\\\n Cu~\\cite{CrUJ} & 4.0 & Sn~\\cite{SnUJ} & 3.5 \\\\\n Zn~\\cite{ZnUJ} & 7.5 & Ta~\\cite{NbUJ} & 2.0 \\\\\n Ga~\\cite{GaUJ} & 3.9 & Re~\\cite{NbUJ} & 2.4 \\\\\n Sn~\\cite{SnUJ} & 3.5 & Os~\\cite{NbUJ} & 2.6 \\\\\n Nb~\\cite{NbUJ} & 2.1 & Ir~\\cite{NbUJ} & 2.8 \\\\\n Mo~\\cite{NbUJ} & 2.4 & Pt~\\cite{NbUJ} & 3.0 \\\\\n Ta~\\cite{SnUJ} & 2.0 & Au & 4.0 \\\\\n\\end{tabular}\n\\label{tab:art104:Ud}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption{$U$ and $J$ parameters applied to selected {\\textit f}-block elements.}\n\\vspace{3mm}\n \\begin{tabular}{l r r | l r r}\n element & {\\textit U} & {\\textit J} & element & {\\textit U} & {\\textit J} \\\\\n \\hline\n La~\\cite{LaUJ} & 8.1 & 0.6 & Dy~\\cite{DyUJ} & 5.6 & 0.0 \\\\\n Ce~\\cite{CeUJ} & 7.0 & 0.7 & Tm~\\cite{TmUJ} & 7.0 & 1.0 \\\\\n Pr~\\cite{PrUJ} & 6.5 & 1.0 & Yb~\\cite{YbUJ} & 7.0 & 0.67 \\\\\n Nd~\\cite{aflowSCINT}& 7.2 & 1.0 & Lu~\\cite{LaUJ} & 4.8 & 0.95 \\\\\n Sm~\\cite{aflowSCINT}& 7.4 & 1.0 & Th~\\cite{ThUJ} & 5.0 & 0.0 \\\\\n Eu~\\cite{aflowSCINT}& 6.4 & 1.0 & U~\\cite{UUJ} & 4.0 & 0.0 \\\\\n Gd~\\cite{GdUJ} & 6.7 & 0.1 & & & \\\\\n\\end{tabular}\n\\label{tab:art104:Uf}\n\\end{table}\n\n\\subsubsection{Spin polarization} \\label{subsubsec:art104:SpinPol}\n\nThe first of the two {\\verb!RELAX!} calculations is always performed in a collinear spin-polarized fashion.\nThe initial magnetic moments in this step are set to the number of atoms in the system, \\nobreak\\mbox{\\it e.g.}, 1.0 $\\mu B\/$atom. If\nthe magnetization resulting from the {\\verb!RELAX1!} step is found to be below 0.025 $\\mu B\/$atom, {\\small AFLOW}\\\neconomizes computational resources by turning spin polarization off in all ensuing calculations. Spin-orbit coupling\nis not used in the current {\\small AFLOW}\\ standard, since it is still\ntoo expensive to include in a HT framework.\n\n\\subsubsection{Calculation methods and convergence criteria} \\label{subsubsec:art104:convergence}\n\nTwo nested loops are involved in the {\\small DFT}\\ calculations used by {\\small AFLOW}\\ in the construction of the databases.\nThe inner loop contains routines that iteratively optimize the electronic degrees of freedom (EDOF), and features\na number of algorithms that are concerned with diagonalizing the Kohn-Sham (KS) Hamiltonian at each iteration.\nThe outer loop performs adjustments to the system geometry (ionic degrees of freedom, IDOF) until the forces acting\non the system are minimized.\n\nThe convergence condition for each loop has been defined in terms of an energy difference, $\\delta E$. If successive\nenergies resulting from the completion of a loop are denoted as $E_{i-1}$ and $E_i$, then\nconvergence is met when the condition $\\delta E \\geqslant E_i - E_{i-1}$ is fulfilled. Note that $E_i$ can either be\nthe electronic energy resulting from the inner loop, or the configurational energy resulting from the outer loop.\nThe electronic convergence criteria will be denoted as $\\delta E_{\\mathrm{elec}}$, and the ionic criteria as $\\delta E_{\\mathrm{ion}}$.\nThe {\\small AFLOW}\\ standard relies on $\\delta E_{\\mathrm{elec}} = 10^{-5}$~eV and $\\delta E_{\\mathrm{ion}} = 10^{-4}$~eV for entries in the\nElements database. All other databases include calculations performed with $\\delta E_{\\mathrm{elec}} = 10^{-3}$~eV and $\\delta E_{\\mathrm{ion}} = 10^{-2}$~eV.\n\nOptimizations of the EDOF depend on sets of parameters that fall under three general themes: initial guesses, diagonalization\nmethods, and charge mixing. The outer loop (optimizations of the IDOF) is concerned with the lattice vectors and the ionic\npositions, and is not as dependent on user input as the inner\nloops. These are described in the following paragraphs.\n\n\\boldsection{Electronic degrees of freedom.}\nThe first step in the process of optimizing the EDOF consists of choosing a trial charge density and a trial\nwavefunction. In the case of the non-{\\verb!BANDS!}-type calculations, the trial wavefunctions are initialized\nusing random numbers, while the trial charge density is obtained from the superposition\nof atomic charge densities. The {\\verb!BANDS!} calculations are not self-consistent, and thus do not feature\na charge density optimization. In these cases the charge density obtained from the previously performed {\\verb!STATIC!}\ncalculation is used in the generation of the starting wavefunctions.\n\nTwo iterative methods are used for diagonalizing the KS Hamiltonian: the Davidson blocked scheme\n(DBS)~\\cite{Liu_rep_1978,Davidson_1983}, and the preconditioned residual minimization method -- direct inversion in\nthe iterative subspace (RMM--DIIS)~\\cite{vasp_prb1996}. Of the two, DBS is known to be the slower and more stable option.\nAdditionally, the subspace rotation matrix is always optimized. These methods are applied in a manner that is dependent\non the calculation type:\n\n\\begin{enumerate}\n \\item {\\verb!RELAX!} calculations. Geometry optimizations contain at least one determination of the system\n forces. The initial determination consists of 5 initial DBS steps,\n followed by as many RMM-DIIS steps as needed to\n fulfill the $\\delta E_{\\mathrm{elec}}$ condition. Later determinations of\n system forces are performed by a similar\n sequence, but only a single DBS step is applied at the outset of the process. Across all\n databases the minimum of number of electronic iterations for {\\verb!RELAX!} calculations is 2. The maximum number is set\n to 120 for entries in the {\\small ICSD}, and 60 for all others.\n \\item non-{\\verb!RELAX!} calculations. In {\\verb!STATIC!} or\n {\\verb!BANDS!} calculations, the diagonalizations are always performed using RMM--DIIS. The minimum number of electronic\n iterations performed during non-{\\verb!RELAX!} calculations is 2, and the maximum is 120.\n\\end{enumerate}\n\nIf the number of iterations in the inner loop somehow exceed the limits listed above, the calculation breaks\nout of this loop, and the system forces and energy are determined. If the $\\delta E_{\\mathrm{ion}}$ convergence condition is\nnot met the calculation re-enters the inner loop, and proceeds normally.\n\nCharge mixing is performed via Pulay's method~\\cite{Pulay_cpl_1980}. The implementation of this charge mixing\napproach in the {\\small VASP}\\ package depends on a series of parameters, of which all but the maximum $\\ell$-quantum number\nhandled by the mixer have been left in their default state. This parameter is modified\nonly in systems included in the {\\small ICSD}\\ database which contain the elements\nlisted in Tables~\\ref{tab:art104:Ud} and \\ref{tab:art104:Uf}. In practical terms, the value applied in these cases is the maximum\n$\\ell$-quantum number found in the {\\small PAW}\\ potential, multiplied by 2.\n\n\\boldsection{Ionic degrees of freedom and lattice vectors.}\nThe {\\verb!RELAX!} calculation type contains determinations of the forces acting on the ions, as well as the full system\nstress tensor. The applied algorithm is the conjugate gradients (CG) approach~\\cite{press1992numerical}, which depends on\nthese quantities for the full optimization of the system geometry, \\nobreak\\mbox{\\it i.e.}, the ionic positions, the lattice vectors, as well\nas modifications of the cell volume. The implementation of CG in {\\small VASP}\\ requires minimal\nuser input, where the only independent parameter is the initial scaling factor which is always left at its\ndefault value. Convergence of the IDOF, as stated above, depends on the value for the $\\delta E_{\\mathrm{ion}}$ parameter,\nas applied across the various databases. The adopted $E_{\\mathrm{cut}}$ (see discussion on ``Potentials and basis set'',\nsection~\\ref{subsubsec:art104:pseudopot}) makes corrections for Pulay stresses unnecessary.\n\nForces acting on the ions and stress tensor are subjected to Harris-Foulkes~\\cite{Harris_prb_1985} corrections.\nMolecular dynamics based relaxations are not performed in the construction of the databases found in the\n{\\sf \\AFLOW.org}\\ repository, so any related settings are not applicable to this work.\n\n\\subsubsection{Output options} \\label{subsubsec:art104:output}\n\nThe reproduction of the results presented on {\\sf \\AFLOW.org}\\ also depends on a select few parameters that\ngovern the output of the {\\small DFT}\\ package. The density of states plots are generated from the {\\verb!STATIC!}\ncalculation. States are plotted with a range of -30~eV to 45~eV, and with a resolution of 5000 points. The band\nstructures are plotted according to the paths of {\\bf k}-points generated for a {\\verb!BANDS!}\ncalculation~\\cite{aflowBZ}. All bands found between -10~eV and 10~eV are included in the plots.\n\n\\subsection{Conclusion} \\label{subsec:art104:conclusion}\n\nThe {\\small AFLOW}\\ standard described here has been applied in the automated creation of the {\\sf \\AFLOW.org}\\ database of\nmaterial properties in a consistent and reproducible manner. The use of standardized parameter sets facilitates\nthe direct comparison of properties between different materials, so that specific trends can be identified to assist\nin the formulation of design rules for accelerated materials development. Following this {\\small AFLOW}\\ standard should\nallow materials science researchers to reproduce the results reported by the {\\small AFLOW}\\ consortium, as well as to\nextend on the database and make meaningful comparisons with their own results.\n\\clearpage\n\\section{Combining the AFLOW GIBBS and Elastic Libraries for Efficiently and Robustly Screening Thermomechanical Properties of Solids}\n\\label{sec:art115}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art115}.\n\n\\subsection{Introduction}\n\nCalculating the thermal and elastic properties of materials is\nimportant for predicting the thermodynamic and mechanical stability of structural\nphases~\\cite{Greaves_Poisson_NMat_2011, Poirier_Earth_Interior_2000,Mouhat_Elastic_PRB_2014, curtarolo:art106}\nand assessing their importance for a variety of applications.\nElastic and mechanical properties such as the shear and bulk moduli are important for predicting the\nhardness of materials~\\cite{Chen_hardness_Intermetallics_2011}, and thus their resistance to\nwear and distortion.\nThermal properties, such as specific heat capacity and lattice thermal conductivity, are important for applications including thermal barrier coatings,\nthermoelectrics~\\cite{zebarjadi_perspectives_2012, aflowKAPPA, Garrity_thermoelectrics_PRB_2016}, and heat sinks~\\cite{Watari_MRS_2001, Yeh_2002}.\n\n\\boldsection{Elasticity.} There are two main methods for calculating the elastic constants,\nbased on the response of either the stress tensor or the total energy to a set of\napplied strains~\\cite{Mehl_TB_Elastic_1996, Mehl_Elastic_1995, Golesorkhtabar_ElaStic_CPC_2013, curtarolo:art100, Silveira_Elastic_CPC_2008, Silveira_Elastic_CPC_2008, Silva_Elastic_PEPI_2007}.\nIn this study, we obtain the elastic constants from the calculated stress tensors for a set of independent deformations of the crystal lattice.\nThis method is implemented within the {\\small AFLOW}\\ framework for\ncomputational materials design\n\\cite{aflowPAPER,curtarolo:art49,monsterPGM}, where it is referred to as the\n\\underline{A}utomatic \\underline{E}lasticity \\underline{L}ibrary ({\\small AEL}).\n{A similar} implementation within the Materials\nProject~\\cite{curtarolo:art100} {allows} extensive\nscreening studies by combining data from these two large\nrepositories of computational materials data.\n\n\\boldsection{Thermal properties.} The determination of the thermal conductivity of materials from first principles requires either calculation of anharmonic\n\\underline{i}nteratomic \\underline{f}orce \\underline{c}onstants (IFCs) for use in the\n\\underline{B}oltzmann \\underline{T}ransport \\underline{E}quation (BTE)~\\cite{Broido2007, Wu_PRB_2012, ward_ab_2009, ward_intrinsic_2010,\nZhang_JACS_2012, Li_PRB_2012, Lindsay_PRL_2013, Lindsay_PRB_2013}, {or molecular dynamics} simulations in combination with\nthe Green-Kubo formula~\\cite{Green_JCP_1954,Kubo_JPSJ_1957}, both of\nwhich are highly demanding computationally even within multiscale approaches~\\cite{curtarolo:art12}.\nThese methods are unsuitable for rapid generation and screening of large databases of materials properties in order to identify trends\nand simple descriptors~\\cite{nmatHT}.\nPreviously, we have implemented the ``{\\small GIBBS}'' quasi-harmonic Debye model\n\\cite{Blanco_CPC_GIBBS_2004, Blanco_jmolstrthch_1996} within both the\n\\underline{A}utomatic \\underline{{\\small G}}{\\small IBBS} \\underline{L}ibrary ({\\small AGL})~\\cite{curtarolo:art96} of the\n{\\small AFLOW}~\\cite{aflowPAPER, aflowlibPAPER, aflowAPI, curtarolo:art104,curtarolo:art110} and\nMaterials Project~\\cite{materialsproject.org,APL_Mater_Jain2013,CMS_Ong2012b} frameworks.\nThis approach does not require large supercell calculations since it\nrelies merely on first-principles calculations of the energy as a function of unit cell volume. It is thus\nmuch more tractable computationally and eminently suited to investigating the thermal properties of\nentire classes of materials in a highly-automated {fashion\nto identify} promising candidates for more in-depth experimental and computational analysis.\n\nThe data set of computed thermal and elastic properties\nproduced for this study is available in the {\\small AFLOW}\\\n\\cite{aflowlibPAPER} online data repository, either using the {\\small AFLOW}\\\n\\underline{RE}presentational \\underline{S}tate \\underline{T}ransfer \\underline{A}pplication \\underline{P}rogramming \\underline{I}nterface\n({\\small REST-API})~\\cite{aflowAPI} or via the {\\sf \\AFLOW.org}\\ web portal~\\cite{aflowlibPAPER,aflowBZ}.\n\n\\subsection{The AEL-AGL methodology}\n\nThe {\\small AEL}-{\\small AGL}\\ methodology combines elastic constants calculations, in\nthe Automatic Elasticity Library ({\\small AEL}), with the calculation of\nthermal properties within the Automatic {\\small GIBBS}\\ Library ({\\small AGL}\\\n\\cite{curtarolo:art96}) - ``{\\small GIBBS}''~\\cite{Blanco_CPC_GIBBS_2004} implementation of the Debye model.\nThis integrated software library includes automatic {error correction} to facilitate high-throughput\ncomputation of thermal and elastic materials properties within the\n{\\small AFLOW}\\ framework~\\cite{aflowPAPER, aflowlibPAPER, aflowAPI, curtarolo:art104,curtarolo:art53,curtarolo:art57,curtarolo:art63,curtarolo:art67,curtarolo:art54}.\nThe principal ingredients of the calculation are described in the following Sections.\n\n\\subsubsection{Elastic properties}\n\\label{subsubsec:art115:aelmethod}\n\nThe elastic constants are evaluated from the stress-strain relations\n\\begin{equation}\n\\left( \\begin{array}{l} s_{11} \\\\ s_{22} \\\\ s_{33} \\\\ s_{23} \\\\ s_{13} \\\\ s_{12} \\end{array} \\right) =\n\\left( \\begin{array}{l l l l l l} c_{11}\\ c_{12}\\ c_{13}\\ c_{14}\\ c_{15}\\ c_{16} \\\\\nc_{12}\\ c_{22}\\ c_{23}\\ c_{24}\\ c_{25}\\ c_{26} \\\\\nc_{13}\\ c_{23}\\ c_{33}\\ c_{34}\\ c_{35}\\ c_{36} \\\\\nc_{14}\\ c_{24}\\ c_{34}\\ c_{44}\\ c_{45}\\ c_{46} \\\\\nc_{15}\\ c_{25}\\ c_{35}\\ c_{45}\\ c_{55}\\ c_{56} \\\\\nc_{16}\\ c_{26}\\ c_{36}\\ c_{46}\\ c_{56}\\ c_{66} \\end{array} \\right)\n\\left( \\begin{array}{c} \\epsilon_{11} \\\\ \\epsilon_{22} \\\\ \\epsilon_{33} \\\\ 2\\epsilon_{23} \\\\ 2\\epsilon_{13} \\\\ 2\\epsilon_{12} \\end{array} \\right)\n\\end{equation}\nwith stress tensor elements $s_{ij}$ calculated\nfor a set of independent normal and shear strains $\\epsilon_{ij}$. The elements of the\nelastic stiffness tensor $c_{ij}$, written in the 6x6 Voigt notation using the mapping~\\cite{Poirier_Earth_Interior_2000}:\n$11 \\mapsto 1$, $22 \\mapsto 2$, $33 \\mapsto 3$, $23 \\mapsto 4$, $13 \\mapsto 5$, $12 \\mapsto 6$;\nare derived from polynomial fits for each independent strain, where the polynomial degree\nis automatically set to be less than the number of strains applied in each independent {direction to} avoid overfitting.\nThe elastic constants are then used to compute the bulk and shear\nmoduli, using either the Voigt approximation\n\\begin{equation}\n\\label{eq:art115:bulkmodvoigt}\nB_{{\\substack{\\scalebox{0.6}{Voigt}}}} = \\frac{1}{9} \\left[ (c_{11} + c_{22} + c_{33}) + 2 (c_{12} + c_{23} + c_{13}) \\right]\n\\end{equation}\nfor the bulk modulus, and\n\\begin{multline}\n\\label{eq:art115:shearmodvoigt}\nG_{{\\substack{\\scalebox{0.6}{Voigt}}}} = \\frac{1}{15} \\left[ (c_{11} + c_{22} + c_{33}) - (c_{12} + c_{23} + c_{13}) \\right]\n+ \\frac{1}{5} (c_{44} + c_{55} + c_{66})\n\\end{multline}\nfor the shear modulus; or the Reuss approximation, which uses the elements of the compliance tensor $s_{ij}$ (the inverse of the stiffness tensor),\nwhere the bulk modulus is given by\n\\begin{equation}\n\\label{eq:art115:bulkmodreuss}\n\\frac{1}{B_{{\\substack{\\scalebox{0.6}{Reuss}}}}} = (s_{11} + s_{22} + s_{33}) + 2 (s_{12} + s_{23} + s_{13})\n\\end{equation}\nand the shear modulus is\n\\begin{multline}\n\\label{eq:art115:shearmodreuss}\n\\frac{15}{G_{{\\substack{\\scalebox{0.6}{Reuss}}}}} = 4(s_{11} + s_{22} + s_{33}) - 4 (s_{12} + s_{23} + s_{13})\n+ 3 (s_{44} + s_{55} + s_{66}).\n\\end{multline}\nFor polycrystalline materials, the Voigt approximation {corresponds to assuming that the strain is uniform and that the stress is supported by the individual grains in parallel, giving} the upper bound on the elastic moduli{;} while the Reuss approximation {assumes that the stress is uniform and that the strain is the sum of the strains of the individual grains in series, giving} the lower bound {on the elastic moduli~\\cite{Poirier_Earth_Interior_2000}}.\nThe two approximations can be combined in the \\underline{V}oigt-\\underline{R}euss-\\underline{H}ill ({\\small VRH})~\\cite{Hill_elastic_average_1952} averages for the bulk modulus\n\\begin{equation}\n\\label{eq:art115:bulkmodvrh}\nB_{{\\substack{\\scalebox{0.6}{VRH}}}} = \\frac{B_{{\\substack{\\scalebox{0.6}{Voigt}}}} + B_{{\\substack{\\scalebox{0.6}{Reuss}}}}}{2};\n\\end{equation}\nand the shear modulus\n\\begin{equation}\n\\label{eq:art115:shearmodvrh}\nG_{{\\substack{\\scalebox{0.6}{VRH}}}} = \\frac{G_{{\\substack{\\scalebox{0.6}{Voigt}}}} + G_{{\\substack{\\scalebox{0.6}{Reuss}}}}}{2}.\n\\end{equation}\nThe Poisson ratio $\\sigma$ is then obtained by:\n\\begin{equation}\n\\label{eq:art115:Poissonratio}\n\\sigma = \\frac{3 B_{{\\substack{\\scalebox{0.6}{VRH}}}} - 2 G_{{\\substack{\\scalebox{0.6}{VRH}}}}}{6 B_{{\\substack{\\scalebox{0.6}{VRH}}}} + 2 G_{{\\substack{\\scalebox{0.6}{VRH}}}}}\n\\end{equation}\n\nThese elastic moduli can also be used to compute the speed of sound for the transverse and longitudinal waves, as well as the\naverage speed of sound in the material~\\cite{Poirier_Earth_Interior_2000}.\nThe speed of sound for the longitudinal waves is\n\\begin{equation}\n\\label{eq:art115:longitudinalsoundspeed}\nv_{\\substack{\\scalebox{0.6}{L}}} = \\left(\\frac{B + \\frac{4}{3}G}{\\rho}\\right)^{\\frac{1}{2}}\\!\\!\\!,\n\\end{equation}\nand for the transverse waves\n\\begin{equation}\n\\label{eq:art115:transversesoundspeed}\nv_{\\substack{\\scalebox{0.6}{T}}} = \\left(\\frac{G}{\\rho}\\right)^{\\frac{1}{2}}\\!\\!\\!,\n\\end{equation}\nwhere $\\rho$ is the mass density of the material. The average speed of\nsound is then evaluated by\n\\begin{equation}\n\\label{eq:art115:speedsound}\n{\\overline v} = \\left[\\frac{1}{3} \\left( \\frac{2}{v_{\\substack{\\scalebox{0.6}{T}}}^3} + \\frac{1}{v_{\\substack{\\scalebox{0.6}{L}}}^3} \\right) \\right]^{-\\frac{1}{3}}\\!\\!\\!.\n\\end{equation}\n\n\\subsubsection{The {\\small AGL}\\ quasi-harmonic Debye-Gr{\\\"u}neisen model}\n\nThe Debye temperature of a solid can be written as~\\cite{Poirier_Earth_Interior_2000}\n\\begin{equation}\n\\label{eq:art115:debyetempv}\n\\theta_{\\substack{\\scalebox{0.6}{D}}} = \\frac{\\hbar}{k_{\\substack{\\scalebox{0.6}{B}}}}\\left[\\frac{6 \\pi^2 n}{V}\\right]^{1\/3} \\!\\! {\\overline v},\n\\end{equation}\nwhere $n$ is the number of atoms in the cell, $V$ is its volume, and\n${\\overline v}$ is the average speed of sound of Equation~\\ref{eq:art115:speedsound}.\nIt can be shown by combining Equations~\\ref{eq:art115:Poissonratio}, \\ref{eq:art115:longitudinalsoundspeed}, \\ref{eq:art115:transversesoundspeed} and \\ref{eq:art115:speedsound}\nthat ${\\overline v}$ is equivalent to~\\cite{Poirier_Earth_Interior_2000}\n\\begin{equation}\n\\label{eq:art115:speedsoundB}\n{\\overline v} = \\sqrt{\\frac{B_{\\substack{\\scalebox{0.6}{S}}}}{\\rho}} f(\\sigma).\n\\end{equation}\nwhere $B_{\\substack{\\scalebox{0.6}{S}}}$ is the adiabatic bulk modulus, $\\rho$ is the density, and $f(\\sigma)$ is a function of the Poisson ratio $\\sigma$:\n\\begin{equation}\n\\label{eq:art115:fpoisson}\nf(\\sigma) = \\left\\{ 3 \\left[ 2 \\left( \\frac{2}{3} \\!\\cdot\\! \\frac{1 + \\sigma}{1 - 2 \\sigma} \\right)^{3\/2} \\!\\!\\!\\!\\!\\!\\!+ \\left( \\frac{1}{3} \\!\\cdot\\! \\frac{1 + \\sigma}{1 - \\sigma} \\right)^{3\/2} \\right]^{-1} \\right\\}^{\\frac{1}{3}}\\!\\!\\!\\!,\n\\end{equation}\nIn an earlier version of {\\small AGL}~\\cite{curtarolo:art96}, the Poisson ratio in Equation~\\ref{eq:art115:fpoisson} was assumed to have the {constant\nvalue $\\sigma = 0.25$ which} is the ratio for a Cauchy solid. This was found to be a reasonable approximation, producing\ngood correlations with experiment.\nThe {\\small AEL}\\ approach, Equation~\\ref{eq:art115:Poissonratio}, directly evaluates $\\sigma$ assuming only that it is independent of temperature and pressure.\nSubstituting Equation~\\ref{eq:art115:speedsoundB} into Equation~\\ref{eq:art115:debyetempv}, the\nDebye temperature is obtained as\n\\begin{equation}\n\\label{eq:art115:debyetemp}\n\\theta_{\\substack{\\scalebox{0.6}{D}}} = \\frac{\\hbar}{k_{\\substack{\\scalebox{0.6}{B}}}}[6 \\pi^2 V^{1\/2} n]^{1\/3} f(\\sigma) \\sqrt{\\frac{B_{\\substack{\\scalebox{0.6}{S}}}}{M}},\n\\end{equation}\nwhere $M$ is the mass of the unit cell.\nThe bulk modulus $B_{\\substack{\\scalebox{0.6}{S}}}$ is obtained from a set of DFT calculations for different volume cells, either by fitting the resulting $E_{\\substack{\\scalebox{0.6}{DFT}}}(V)$\ndata to a phenomenological equation of state or by taking the numerical second derivative of\na polynomial fit\n\\begin{eqnarray}\n\\label{eq:art115:bulkmod}\nB_{\\substack{\\scalebox{0.6}{S}}} (V) &\\approx& B_{\\mathrm{static}} (\\vec{x}) \\approx B_{\\mathrm{static}}(\\vec{x}_{\\substack{\\scalebox{0.6}{opt}}}(V)) \\\\ \\nonumber\n &=&V \\left( \\frac{\\partial^2 E(\\vec{x}_{\\substack{\\scalebox{0.6}{opt}}} (V))}{\\partial V^2} \\right) = V \\left( \\frac{\\partial^2 E(V)}{\\partial V^2} \\right).\n\\end{eqnarray}\nInserting Equation~\\ref{eq:art115:bulkmod} into Equation~\\ref{eq:art115:debyetemp} gives the Debye temperature as a function of volume $\\theta_{\\substack{\\scalebox{0.6}{D}}}(V)$, for each value of\npressure, $p$, and temperature, $T$.\n\nThe equilibrium volume at any particular $(p, T)$ point is obtained by minimizing the Gibbs free energy with\nrespect to volume. First, the vibrational Helmholtz free energy, $F_{\\substack{\\scalebox{0.6}{vib}}}(\\vec{x}; T)$, is calculated in the quasi-harmonic approximation\n\\begin{equation}\nF_{\\substack{\\scalebox{0.6}{vib}}}(\\vec{x}; T) \\!=\\!\\! \\int_0^{\\infty} \\!\\!\\left[\\frac{\\hbar \\omega}{2} \\!+\\! k_{\\substack{\\scalebox{0.6}{B}}} T\\ \\mathrm{log}\\!\\left(1\\!-\\!{\\mathrm e}^{- \\hbar \\omega \/ k_{\\substack{\\scalebox{0.6}{B}}} T}\\right)\\!\\right]\\!g(\\vec{x}; \\omega) d\\omega,\n\\end{equation}\nwhere $g(\\vec{x}; \\omega)$ is the phonon density of states and $\\vec{x}$ describes the geometrical configuration of the system. In the Debye-Gr{\\\"u}neisen model, $F_{\\substack{\\scalebox{0.6}{vib}}}$ can be expressed\nin terms of the Debye temperature $\\theta_{\\substack{\\scalebox{0.6}{D}}}$\n\\begin{equation}\n\\label{eq:art115:helmholtzdebye}\nF_{\\substack{\\scalebox{0.6}{vib}}}(\\theta_{\\substack{\\scalebox{0.6}{D}}}; T) \\!=\\! n k_{\\substack{\\scalebox{0.6}{B}}} T \\!\\left[ \\frac{9}{8} \\frac{\\theta_{\\substack{\\scalebox{0.6}{D}}}}{T} \\!+\\! 3\\ \\mathrm{log}\\!\\left(1 \\!-\\! {\\mathrm e}^{- \\theta_{\\substack{\\scalebox{0.6}{D}}} \/ T}\\!\\right) \\!\\!-\\!\\! D\\left(\\frac{\\theta_{\\substack{\\scalebox{0.6}{D}}}}{T}\\right)\\!\\!\\right],\n\\end{equation}\nwhere $D(\\theta_{\\substack{\\scalebox{0.6}{D}}} \/ T)$ is the Debye integral\n\\begin{equation}\nD \\left(\\theta_{\\substack{\\scalebox{0.6}{D}}}\/T \\right) = 3 \\left( \\frac{T}{\\theta_{\\substack{\\scalebox{0.6}{D}}}} \\right)^3 \\int_0^{\\theta_{\\substack{\\scalebox{0.6}{D}}}\/T} \\frac{x^3}{e^x - 1} dx.\n\\end{equation}\nThe Gibbs free energy is calculated as\n\\begin{equation}\n\\label{eq:art115:gibbsdebye}\n{\\sf G}(V; p, T) = E_{\\substack{\\scalebox{0.6}{DFT}}}(V) + F_{\\substack{\\scalebox{0.6}{vib}}} (\\theta_{\\substack{\\scalebox{0.6}{D}}}(V); T) + pV,\n\\end{equation}\nand fitted by a polynomial of $V$. The equilibrium volume, $V_{\\mathrm{eq}}$, is that which minimizes ${\\sf G}(V; p, T)$.\n\nOnce $V_{\\mathrm{eq}}$ has been determined, $\\theta_{\\substack{\\scalebox{0.6}{D}}}$ can be determined, and then other thermal properties including the Gr{\\\"u}neisen parameter and thermal\nconductivity can be calculated as described in the following Sections.\n\n\\subsubsection{Equations of state}\n\\label{subsubsec:art115:eqnsofstate}\n\nWithin {\\small AGL}\\, the bulk modulus can be determined either numerically from the second derivative of the polynomial fit of $E_{\\substack{\\scalebox{0.6}{DFT}}}(V)$,\nEquation~\\ref{eq:art115:bulkmod}, or by fitting the $(p,V)$ data to a\nphenomenological equation of state ({\\small EOS}). Three different analytic {\\small EOS}\\ have been implemented within\n{\\small AGL}: the Birch-Murnaghan {\\small EOS}~\\cite{Birch_Elastic_JAP_1938, Poirier_Earth_Interior_2000, Blanco_CPC_GIBBS_2004}; the Vinet {\\small EOS}~\\cite{Vinet_EoS_JPCM_1989, Blanco_CPC_GIBBS_2004};\nand the Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez spinodal {\\small EOS}~\\cite{Baonza_EoS_PRB_1995, Blanco_CPC_GIBBS_2004}.\n\nThe Birch-Murnaghan {\\small EOS}\\ is\n\\begin{equation}\n\\label{eq:art115:birch}\n\\frac{p}{3 f (1 + 2 f)^\\frac{5}{2}} = \\sum_{i=0}^2 a_i f^i ,\n\\end{equation}\nwhere $p$ is the pressure, $a_i$ are polynomial coefficients, and $f$ is the ``compression'' given by\n\\begin{equation}\n\\label{eq:art115:birchf}\nf = \\frac{1}{2} \\left[\\left(\\frac{V}{V_0} \\right)^{-\\frac{2}{3}}- 1 \\right].\n\\end{equation}\nThe zero pressure bulk modulus is equal to the coefficient $a_0$.\n\nThe Vinet {\\small EOS}\\ is~\\cite{Vinet_EoS_JPCM_1989, Blanco_CPC_GIBBS_2004}\n\\begin{equation}\n\\label{eq:art115:vinet}\n\\log \\left[ \\frac{p x^2}{3 (1 - x)} \\right] = \\log B_0 + a (1 - x),\n\\end{equation}\nwhere $a$ and $\\log B_0$ are fitting parameters and\n\\begin{equation}\n\\label{eq:art115:vinetx}\nx = \\left(\\frac{V}{V_0} \\right)^{\\frac{1}{3}}\\!\\!\\!, \\\na = 3 (B_0' - 1) \/ 2.\n\\end{equation}\nThe isothermal bulk modulus $B_{\\substack{\\scalebox{0.6}{T}}}$ is given by~\\cite{Vinet_EoS_JPCM_1989, Blanco_CPC_GIBBS_2004}\n\\begin{equation}\n\\label{eq:art115:vinetBT}\nB_{\\substack{\\scalebox{0.6}{T}}} = - x^{-2} B_0 e^{a(1-x)} f(x),\n\\end{equation}\nwhere\n\\begin{equation*}\n\\label{eq:art115:vinetfx}\nf(x) = x - 2 - ax (1 - x).\n\\end{equation*}\n\nThe Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez spinodal equation of state has the form~\\cite{Baonza_EoS_PRB_1995, Blanco_CPC_GIBBS_2004}\n\\begin{equation}\n\\label{eq:art115:bcn}\nV = V_{\\mathrm{sp}} \\exp \\left[ - \\left(\\frac{K^*}{1 - \\beta} \\right) (p - p_{\\mathrm{sp}})^{1 - \\beta} \\right],\n\\end{equation}\nwhere $K^*$, $p_{\\mathrm{sp}}$ and $\\beta$ are the fitting parameters, and $V_{\\mathrm{sp}} $ is given by\n\\begin{equation*}\nV_{\\mathrm{sp}} = V_0 \\exp \\left[ \\frac{\\beta}{\\left(1 - \\beta \\right) B_0'} \\right],\n\\end{equation*}\nwhere $B_0 = [K^*]^{-1} (-p_{\\mathrm{sp}})^{\\beta}$ and $B_0' = (-p_{\\mathrm{sp}})^{-1}\\beta B_0$.\nThe isothermal bulk modulus $B_{\\substack{\\scalebox{0.6}{T}}}$ is then given by~\\cite{Baonza_EoS_PRB_1995, Blanco_CPC_GIBBS_2004}\n\\begin{equation}\n\\label{eq:art115:bcnBT}\nB_{\\substack{\\scalebox{0.6}{T}}} = \\frac{(p - p_{\\mathrm{sp}})^{\\beta}}{K^*}.\n\\end{equation}\n\n{Note that {\\small AGL}\\ uses $B_{\\substack{\\scalebox{0.6}{T}}}$ instead of $B_{\\substack{\\scalebox{0.6}{S}}}$ in Equation~\\ref{eq:art115:debyetemp} when one of these phenomenological {\\small EOS}\\\nis selected. $B_{\\substack{\\scalebox{0.6}{S}}}$ can then be calculated as\n\\begin{equation}\n\\label{eq:art115:BsBT}\nB_{\\substack{\\scalebox{0.6}{S}}} = B_{\\substack{\\scalebox{0.6}{T}}}(1 + \\alpha \\gamma T),\n\\end{equation}\nwhere $\\gamma$ is the Gr{\\\"u}neisen parameter (described in Section~\\ref{subsubsec:gruneisen} below), and $\\alpha$ is the thermal expansion\n\\begin{equation}\n\\label{eq:art115:thermal_expansion}\n\\alpha = \\frac{\\gamma C_{\\substack{\\scalebox{0.6}{V}}}}{B_{\\substack{\\scalebox{0.6}{T}}} V},\n\\end{equation}\nwhere $C_{\\substack{\\scalebox{0.6}{V}}}$ is the heat capacity at constant volume, given by\n\\begin{equation}\n \\label{eq:art115:heat_capacity}\nC_{\\substack{\\scalebox{0.6}{V}}} = 3 n k_{\\substack{\\scalebox{0.6}{B}}} \\left[4 D\\left(\\frac{\\theta_{\\substack{\\scalebox{0.6}{D}}}}{T}\\right) - \\frac{3 \\theta_{\\substack{\\scalebox{0.6}{D}}} \/ T}{\\exp(\\theta_{\\substack{\\scalebox{0.6}{D}}} \/ T) - 1} \\right].\n\\end{equation}\n}\n\n\\subsubsection{The Gr{\\\"u}neisen parameter}\n\\label{subsubsec:gruneisen}\n\nThe Gr{\\\"u}neisen parameter describes the variation of the thermal properties of a material with the unit cell size, and contains\ninformation about higher order phonon scattering which is important\nfor calculating the lattice thermal conductivity\n\\cite{Leibfried_formula_1954, slack, Morelli_Slack_2006, Madsen_PRB_2014, curtarolo:art96},\nand thermal expansion~\\cite{Poirier_Earth_Interior_2000, Blanco_CPC_GIBBS_2004, curtarolo:art114}.\nIt is defined as the phonon frequencies dependence on the unit cell volume\n\\begin{equation}\n\\label{eq:art115:gamma_micro}\n\\gamma_i = - \\frac{V}{\\omega_i} \\frac{\\partial \\omega_i}{\\partial V}.\n\\end{equation}\nDebye's theory assumes that the volume dependence of all mode\nfrequencies is the same as that of the cut-off Debye frequency, so the Gr{\\\"u}neisen parameter can be expressed in terms of $\\theta_{\\substack{\\scalebox{0.6}{D}}}$\n\\begin{equation}\n\\label{eq:art115:gruneisen_theta}\n\\gamma = - \\frac{\\partial \\ \\mathrm{log} (\\theta_{\\substack{\\scalebox{0.6}{D}}}(V))}{\\partial \\ \\mathrm{log} V}.\n\\end{equation}\n\nThis macroscopic definition of the Debye temperature is a weighted\naverage of Equation~\\ref{eq:art115:gamma_micro} with the heat capacities for each branch of the phonon spectrum\n\\begin{equation}\n\\gamma = \\frac{\\sum_i \\gamma_i C_{V, i}} {\\sum_i C_{V,i}}.\n\\end{equation}\n\n{\nWithin {\\small AGL}~\\cite{curtarolo:art96}, the Gr{\\\"u}neisen parameter can\nbe calculated in several different ways, including direct evaluation of Equation~\\ref{eq:art115:gruneisen_theta},\nby using the more stable Mie-Gr{\\\"u}neisen equation~\\cite{Poirier_Earth_Interior_2000},\n\\begin{equation}\n\\label{eq:art115:miegruneisen}\np - p_{T=0} = \\gamma \\frac{U_{\\substack{\\scalebox{0.6}{vib}}}}{V},\n\\end{equation}\nwhere $U_{\\substack{\\scalebox{0.6}{vib}}}$ is the vibrational internal energy~\\cite{Blanco_CPC_GIBBS_2004}\n\\begin{equation}\n\\label{eq:art115:Uvib}\nU_{\\substack{\\scalebox{0.6}{vib}}} = n k_{\\substack{\\scalebox{0.6}{B}}} T\\left[ \\frac{9}{8} \\frac{\\theta_{\\substack{\\scalebox{0.6}{D}}}}{T} + 3D \\left( \\frac{\\theta_{\\substack{\\scalebox{0.6}{D}}}}{T} \\right)\\right].\n\\end{equation}\nThe ``Slater gamma'' expression~\\cite{Poirier_Earth_Interior_2000}\n\\begin{equation}\n\\label{eq:art115:slatergamma}\n\\gamma = - \\frac{1}{6} + \\frac{1}{2} \\frac{\\partial B_{\\substack{\\scalebox{0.6}{S}}}}{\\partial p}\n\\end{equation}\nis the default method in the automated workflow used\nfor the {\\small AFLOW}\\ database.\n}\n\n\\subsubsection{Thermal conductivity}\n\nIn the {\\small AGL}\\ framework, the thermal conductivity is calculated using the\nLeibfried-Schl{\\\"o}mann equation~\\cite{Leibfried_formula_1954, slack, Morelli_Slack_2006}\n\\begin{eqnarray}\n\\label{eq:art115:thermal_conductivity}\n\\kappa_{\\mathrm l} (\\theta_{\\mathrm{a}}) &=& \\frac{0.849 \\times 3 \\sqrt[3]{4}}{20 \\pi^3(1 - 0.514\\gamma_{\\mathrm{a}}^{-1} + 0.228\\gamma_{\\mathrm{a}}^{-2})}\n \\left( \\frac{k_{\\substack{\\scalebox{0.6}{B}}} \\theta_{\\mathrm{a}}}{\\hbar} \\right)^2 \\frac{k_{\\substack{\\scalebox{0.6}{B}}} m V^{\\frac{1}{3}}}{\\hbar \\gamma_{\\mathrm{a}}^2}.\n\\end{eqnarray}\nwhere $V$ is the volume of the unit cell and $m$ is the average atomic mass.\nIt should be noted that the Debye temperature and Gr{\\\"u}neisen parameter in this formula, $\\theta_{\\mathrm{a}}$ and $\\gamma_{\\mathrm{a}}$, are slightly\ndifferent {from} the traditional Debye temperature, $\\theta_{\\substack{\\scalebox{0.6}{D}}}$, calculated in Equation~\\ref{eq:art115:debyetemp} and Gr{\\\"u}neisen parameter, $\\gamma$, obtained from\nEquation~\\ref{eq:art115:slatergamma}. Instead, $\\theta_{\\mathrm{a}}$ and $\\gamma_{\\mathrm{a}}$ are obtained by only considering the acoustic modes, based on the assumption that the optical\nphonon modes in crystals do not contribute to heat transport~\\cite{slack}. This $\\theta_{\\mathrm{a}}$ is referred to as the ``acoustic'' Debye temperature\n\\cite{slack, Morelli_Slack_2006}. It can be derived directly from the phonon DOS by integrating only over the acoustic modes~\\cite{slack,\nWee_Fornari_TiNiSn_JEM_2012}. Alternatively, it can be calculated from the traditional Debye temperature $\\theta_{\\substack{\\scalebox{0.6}{D}}}$~\\cite{slack, Morelli_Slack_2006}\n\\begin{equation}\n\\label{eq:art115:acousticdebyetemp}\n\\theta_{\\mathrm{a}} = \\theta_{\\substack{\\scalebox{0.6}{D}}} n^{-\\frac{1}{3}}.\n\\end{equation}\n\n{There is no simple way to extract the ``acoustic'' Gr{\\\"u}neisen parameter from the traditional Gr{\\\"u}neisen parameter.}\nInstead, it must be calculated from Equation~\\ref{eq:art115:gamma_micro} for each phonon branch separately and summed over the acoustic branches~\\cite{curtarolo:art114, curtarolo:art119}.\nThis requires using the quasi-harmonic phonon approximation which involves calculating the full phonon spectrum for different\nvolumes~\\cite{Wee_Fornari_TiNiSn_JEM_2012, curtarolo:art114, curtarolo:art119}, and is therefore too computationally demanding to be used for\nhigh-throughput screening, particularly for large, low symmetry systems. Therefore, we use the approximation\n$\\gamma_{\\mathrm{a}} = \\gamma$ in the {\\small AEL}-{\\small AGL}\\ approach to {calculate} the thermal conductivity. The dependence of the expression in\nEquation~\\ref{eq:art115:thermal_conductivity} on $\\gamma$ is weak~\\cite{curtarolo:art96, Morelli_Slack_2006}, thus\nthe evaluation of $\\kappa_l$ using the traditional Gr{\\\"u}neisen parameter introduces just a small systematic error which is insignificant for\nscreening purposes~\\cite{curtarolo:art119}.\n\nThe thermal conductivity at temperatures other than $\\theta_{\\mathrm{a}}$ is estimated by~\\cite{slack, Morelli_Slack_2006, Madsen_PRB_2014}:\n\\begin{equation}\n\\label{eq:art115:kappa_temperature}\n\\kappa_{\\mathrm l} (T) = \\kappa_{\\mathrm l}(\\theta_{\\mathrm{a}}) \\frac{\\theta_{\\mathrm{a}}}{T}.\n\\end{equation}\n\n\\subsubsection{{\\small DFT}\\ calculations and workflow details}\n\nThe {\\small DFT}\\ calculations to obtain $E(V)$ and the strain tensors were performed using\nthe {\\small VASP}\\ software~\\cite{kresse_vasp} with projector-augmented-wave\npseudopotentials~\\cite{PAW} and the {\\small PBE}\\ parameterization of the\ngeneralized gradient approximation to the exchange-correlation\nfunctional~\\cite{PBE}, using the {parameters described} in the {\\small AFLOW}\\\nStandard~\\cite{curtarolo:art104}. The energies were calculated at zero\ntemperature and pressure, with spin polarization and without zero-point motion or lattice\nvibrations. The initial crystal structures were fully relaxed (cell\nvolume and shape and the basis atom coordinates inside the cell).\n\nFor the {\\small AEL}\\ calculations, 4 strains were applied in each independent lattice direction\n(two compressive and two expansive) with a maximum strain of 1\\% in each direction,\nfor a total of 24 configurations~\\cite{curtarolo:art100}. For cubic systems,\nthe crystal symmetry was used to reduce the number of required strain configurations\nto 8. For each configuration, two ionic positions {\\small AFLOW}\\ Standard {\\verb!RELAX!}~\\cite{curtarolo:art104}\ncalculations at fixed cell volume and shape were followed by a single {\\small AFLOW}\\ Standard {\\verb!STATIC!}~\\cite{curtarolo:art104}\ncalculation.\nThe elastic constants are then calculated by fitting the elements of stress tensor obtained for each independent strain.\nThe stress tensor from the zero-strain configuration\n(\\nobreak\\mbox{\\it i.e.}, the initial unstrained relaxed structure) can also be {included in the set of fitted strains}, although this was found to have negligible effect on the results.\nOnce these calculations are complete, it is verified that the eigenvalues of the stiffness tensor are all positive,\nthat the stiffness tensor obeys the appropriate symmetry rules for the lattice type~\\cite{Mouhat_Elastic_PRB_2014}, and\nthat the applied strain is still within the linear regime, using the method described by de Jong~\\nobreak\\mbox{\\it et al.}~\\cite{curtarolo:art100}.\nIf any of these conditions fail, the calculation is repeated with\nadjusted applied strain.\n\nThe {\\small AGL}\\ calculation of $E(V)$ is fitted to the energy at 28 different\nvolumes of the unit cell obtained by increasing or decreasing the relaxed lattice parameters in fractional\nincrements of 0.01, with a single {\\small AFLOW}\\ Standard\n{\\verb!STATIC!}~\\cite{curtarolo:art104} calculation at each volume.\nThe resulting $E(V)$ data is checked for convexity and to verify that the minimum energy is at the\ninitial volume (\\nobreak\\mbox{\\it i.e.}, at the properly relaxed cell size). If any of these\nconditions fail, the calculation is repeated with adjusted parameters,\n\\nobreak\\mbox{\\it e.g.}, increased k-point grid density.\n\n\\subsubsection{Correlation analysis}\n\nPearson and Spearman correlations {are used to}\nanalyze the results for entire sets of materials. The {Pearson coefficient} $r$ is a measure of the linear\ncorrelation between two variables, $X$ and $Y$. It is calculated by\n\\begin{equation}\n\\label{eq:art115:Pearson}\nr = \\frac{\\sum_{i=1}^{n} \\left(X_i - \\overline{X} \\right) \\left(Y_i - \\overline{Y} \\right) }{ \\sqrt{\\sum_{i=1}^{n} \\left(X_i - \\overline{X} \\right)^2} \\sqrt{\\sum_{i=1}^{n} \\left(Y_i - \\overline{Y} \\right)^2}},\n\\end{equation}\nwhere $\\overline{X}$ and $\\overline{Y}$ are the mean values of $X$ and $Y$.\n\nThe {Spearman coefficient} $\\rho$ is a measure of the monotonicity of the relation between two variables.\nThe raw values of the two variables $X_i$ and $Y_i$ are sorted in ascending order, and are assigned rank values $x_i$ and $y_i$ which\nare equal to their position in the sorted list. If there is more than one variable with the same value, the average of the position values\nare assigned to {all duplicate entries}. The correlation coefficient is then given by\n\\begin{equation}\n\\label{eq:art115:Spearman}\n\\rho = \\frac{\\sum_{i=1}^{n} \\left(x_i - \\overline{x} \\right) \\left(y_i - \\overline{y} \\right) }{ \\sqrt{\\sum_{i=1}^{n} \\left(x_i - \\overline{x} \\right)^2} \\sqrt{\\sum_{i=1}^{n} \\left(y_i - \\overline{y} \\right)^2}}.\n\\end{equation}\nIt is useful for determining how well the ranking order of the values of one variable predict the ranking order of the values of the other variable.\n\nThe discrepancy between the {\\small AEL}-{\\small AGL}\\ predictions and experiment is\nevaluated in terms normalized root-mean-square relative deviation\n\\begin{equation}\n\\label{eq:art115:RMSD}\n{\\mathrm{RMSrD}} = \\sqrt{\\frac{ \\sum_{i=1}^{n} \\left( \\frac{X_i - Y_i}{X_i} \\right)^2 }{N - 1}} ,\n\\end{equation}\n{In contrast} to the correlations described above, lower values of the {\\small RMSrD}\\ indicate better agreement with experiment. This measure is particularly useful for\ncomparing predictions of the same property using different\nmethodologies that may have very similar correlations with, but different\ndeviations from, the experimental results.\n\n\\subsection{Results}\n\nWe used the {\\small AEL}-{\\small AGL}\\ methodology to calculate the mechanical and thermal properties, including the bulk modulus,\nshear modulus, Poisson ratio, Debye temperature, Gr{\\\"u}neisen parameter and thermal conductivity for a set of 74 materials\nwith structures including diamond, zincblende, rocksalt, wurtzite, rhombohedral and body-centered tetragonal.\nThe results have been compared to experimental values (where available), and the correlations between the calculated and\nexperimental values were deduced.\nIn cases where multiple experimental values are present in the literature, we used the most recently reported\nvalue, unless otherwise specified.\n\nIn Section~\\ref{subsubsec:art115:aelmethod}, three different approximations for the bulk and shear moduli are described: Voigt (Equations~\\ref{eq:art115:bulkmodvoigt}, \\ref{eq:art115:shearmodvoigt}),\nReuss (Equations~\\ref{eq:art115:bulkmodreuss}, \\ref{eq:art115:shearmodreuss}), and the Voigt-Reuss-Hill ({\\small VRH}) average (Equations~\\ref{eq:art115:bulkmodvrh}, \\ref{eq:art115:shearmodvrh}).\nThese approximation{s give very similar values for the\nbulk modulus} for the set of materials included in this work, particularly those with cubic symmetry.\nTherefore only {$B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$}\nis explicitly cited in the following listed results\n(the values obtained for all three approximations are available in the {\\small AFLOW}\\ database entries for\nthese materials). The values for the shear modulus in these three\napproximations exhibit larger variations, and are therefore all listed and compared to experiment.\nIn several cases, the experimental values of the bulk and shear moduli have been calculated\nfrom the measured elastic constants using Equations~\\ref{eq:art115:bulkmodvoigt} through \\ref{eq:art115:shearmodvrh}, and an experimental Poisson ratio $\\sigma^{\\mathrm{exp}}$\nwas calculated from these values using Equation~\\ref{eq:art115:Poissonratio}.\n\nAs described in Section~\\ref{subsubsec:art115:eqnsofstate}, the bulk modulus in {\\small AGL}\\ can be calculated from a polynomial fit of the $E(V)$ data as shown in Equation~\\ref{eq:art115:bulkmod},\nor by fitting the $E(V)$ data to one of three empirical equations\nof state: Birch-Murnaghan (Equation~\\ref{eq:art115:birch}), Vinet (Equation~\\ref{eq:art115:vinet}), and the Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez\n(Equation~\\ref{eq:art115:bcn}). We compare the results of these four methods, labeled $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$, $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$, $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$, and\n$B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$, respectively, with the experimental values $B^{\\mathrm{exp}}$ and those obtained from the\nelastic calculations $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$.\nThe Debye temperatures, Gr{\\\"u}neisen parameters and thermal conductivities depend on the calculated bulk modulus and are\ntherefore also cited below for each of the equations of state.\nAlso included are the Debye temperatures derived from the calculated\nelastic constants and speed of sound as given by Equation~\\ref{eq:art115:speedsound}.\nThe Debye temperatures, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$\n(Equation~\\ref{eq:art115:birch}), $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ (Equation~\\ref{eq:art115:vinet}),\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ (Equation~\\ref{eq:art115:bcn}), calculated using the Poisson ratio $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ obtained from\nEquation~\\ref{eq:art115:Poissonratio}, are compared to $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$, obtained from the numerical fit\nof $E(V)$ (Equation~\\ref{eq:art115:bulkmod}) using both $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ and the approximation $\\sigma =\n0.25$ used in Reference~\\onlinecite{curtarolo:art96}, to\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, calculated with the speed of sound obtained\nusing Equation~\\ref{eq:art115:speedsound},\nand to the experimental values $\\theta^{\\mathrm{exp}}$.\nThe values of the acoustic Debye temperature ($\\theta_{\\mathrm{a}}$, Equation~\\ref{eq:art115:acousticdebyetemp})\nare shown, where available, in parentheses below the traditional Debye temperature value.\n\nThe experimental Gr{\\\"u}neisen parameter, $\\gamma^{\\mathrm{exp}}$, is compared to $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ (Equation~\\ref{eq:art115:bulkmod}), obtained using the numerical\npolynomial fit of $E(V)$ and both values of the Poisson ratio\n($\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ and the approximation $\\sigma = 0.25$ from\nReference~\\onlinecite{curtarolo:art96}), and to $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$\n(Equation~\\ref{eq:art115:birch}), $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ (Equation~\\ref{eq:art115:vinet}), and $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ (Equation~\\ref{eq:art115:bcn}), calculated\nusing $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ only. Similarly, the experimental lattice thermal\nconductivity $\\kappa^{\\mathrm{exp}}$ is compared to $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ (Equation~\\ref{eq:art115:bulkmod}),\nobtained using the numerical polynomial fit and both the calculated\nand approximated values of $\\sigma$, and to $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$\n(Equation~\\ref{eq:art115:birch}), $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ (Equation~\\ref{eq:art115:vinet}), and $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$\n(Equation~\\ref{eq:art115:bcn}), calculated using only $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$.\n\nThe {\\small AEL}\\ method has been been previously implemented in the Materials Project framework for calculating\nelastic constants~\\cite{curtarolo:art100}. {Data from} the Materials Project database are included\nin the tables below for comparison {for the bulk modulus $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, shear modulus $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, and Poisson ratio $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$.}\n\n\\subsubsection{Zincblende and diamond structure materials}\n\nThe mechanical and thermal properties were calculated for a set of materials with the\nzincblende(spacegroup: $F\\overline{4}3m$,\\ $\\#$216; Pearson symbol: cF8; {\\small AFLOW}\\ prototype: {\\sf AB\\_cF8\\_216\\_c\\_a}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/AB_cF8_216_c_a.html}})\nand diamond ($Fd\\overline{3}m$,\\ $\\#$227; cF8; {\\sf A\\_cF8\\_227\\_a}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/A_cF8_227_a.html}}) structures.\nThis {is the same set of materials\nas} in Table I of Reference~\\onlinecite{curtarolo:art96}, which in {turn are from} Table II of\nReference~\\onlinecite{slack} and Table 2.2 of Reference~\\onlinecite{Morelli_Slack_2006}.\n\nThe elastic {properties bulk modulus}, shear modulus and Poisson {ratio calculated} using {\\small AEL}\\ and {\\small AGL}\\ are shown\nin Table~\\ref{tab:art115:zincblende_elastic} and Figure~\\ref{fig:art115:zincblende_thermal_elastic}, together\nwith experimental values from the literature where available. As can be seen\nfrom the results in Table~\\ref{tab:art115:zincblende_elastic} and Figure~\\ref{fig:art115:zincblende_thermal_elastic}(a), the $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ values are\ngenerally closest to experiment as shown by the {\\small RMSrD}\\ value of $0.13$, producing an underestimate of the order of 10\\%. The {\\small AGL}\\ values from both the numerical\nfit and the empirical equations of state are generally very similar to each other, while being slightly less than the $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$\nvalues.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Bulk modulus, shear modulus and Poisson ratio of\nzincblende and diamond structure semiconductors.]\n{The zincblende structure is designated {\\small AFLOW}\\ prototype {\\sf AB\\_cF8\\_216\\_c\\_a}~\\cite{aflowANRL}\nand the diamond structure {\\sf A\\_cF8\\_227\\_a}~\\cite{aflowANRL}.\n``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ in {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ \\\\\n\\hline\nC & 442~\\cite{Semiconductors_BasicData_Springer,\n Lam_BulkMod_PRB_1987, Grimsditch_ElasticDiamond_PRB_1975} & 434 & N\/A & 408 & 409 & 403 & 417 & 534~\\cite{Semiconductors_BasicData_Springer, Grimsditch_ElasticDiamond_PRB_1975} & 520 & 516 & 518 & N\/A & 0.069~\\cite{Semiconductors_BasicData_Springer, Grimsditch_ElasticDiamond_PRB_1975} & 0.073 & N\/A \\\\\nSiC & 248~\\cite{Strossner_ElasticSiC_SSC_1987} & 212 & 211 & 203 & 207 & 206 & 206 & 196~\\cite{Fate_ShearSiC_JACeramS_1974} & 195 & 178 & 187 & 187 & 0.145~\\cite{Lam_BulkMod_PRB_1987, Fate_ShearSiC_JACeramS_1974} & 0.160 & 0.16 \\\\\n & 211~\\cite{Semiconductors_BasicData_Springer, Lam_BulkMod_PRB_1987} & & & & & & & 170~\\cite{Semiconductors_BasicData_Springer} & & & & & 0.183~\\cite{Semiconductors_BasicData_Springer} & & \\\\\nSi & 97.8~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 89.1 & 83.0 & 84.2 & 85.9 & 85.0 & 86.1 & 66.5~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 64 & 61 & 62.5 & 61.2 & 0.223~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 0.216 & 0.2 \\\\\n & 98~\\cite{Lam_BulkMod_PRB_1987} & & & & & & & & & & & & & \\\\\nGe & 75.8~\\cite{Semiconductors_BasicData_Springer, Bruner_ElasticGe_PRL_1961} & 61.5 & 59.0 & 54.9 & 55.7 & 54.5 & 56.1 & 55.3~\\cite{Semiconductors_BasicData_Springer, Bruner_ElasticGe_PRL_1961} & 47.7 & 44.8 & 46.2 & 45.4 & 0.207~\\cite{Semiconductors_BasicData_Springer, Bruner_ElasticGe_PRL_1961} & 0.199 & 0.19 \\\\\n & 77.2~\\cite{Lam_BulkMod_PRB_1987} & & & & & & & & & & & & & \\\\\nBN & 367.0~\\cite{Lam_BulkMod_PRB_1987} & 372 & N\/A & 353 & 356 & 348 & 359 & N\/A & 387 & 374 & 380 & N\/A & N\/A & 0.119 & N\/A \\\\\nBP & 165.0~\\cite{Semiconductors_BasicData_Springer, Lam_BulkMod_PRB_1987} & 162 & 161 & 155 & 157 & 156 & 157 & 136~\\cite{Semiconductors_BasicData_Springer, Wettling_ElasticBP_SSC_1984} & 164 & 160 & 162 & 162 & 0.186~\\cite{Semiconductors_BasicData_Springer, Wettling_ElasticBP_SSC_1984} & 0.125 & 0.12 \\\\\n & 267~\\cite{Semiconductors_BasicData_Springer, Suzuki_ElasticBP_JAP_1983} & & & & & & & & & & & & & \\\\\n & 172~\\cite{Semiconductors_BasicData_Springer, Wettling_ElasticBP_SSC_1984} & & & & & & & & & & & & & \\\\\nAlP & 86.0~\\cite{Lam_BulkMod_PRB_1987} & 82.9 & 85.2 & 78.9 & 80.4 & 79.5 & 80.4 & N\/A & 48.6 & 44.2 & 46.4 & 47.2 & N\/A & 0.264 & 0.27 \\\\\nAlAs & 77.0~\\cite{Lam_BulkMod_PRB_1987} & 67.4 & 69.8 & 63.8 & 65.1 & 64.0 & 65.3 & N\/A & 41.1 & 37.5 & 39.3 & 39.1 & N\/A & 0.256 & 0.26 \\\\\n & 74~\\cite{Greene_ElasticAlAs_PRL_1994} & & & & & & & & & & & & & \\\\\nAlSb & 58.2~\\cite{Lam_BulkMod_PRB_1987, Semiconductors_BasicData_Springer, Bolef_ElasticAlSb_JAP_1960, Weil_ElasticAlSb_JAP_1972} & 49.4 & 49.2 & 46.5 & 47.8 & 46.9 & 47.8 & 31.9~\\cite{Semiconductors_BasicData_Springer, Bolef_ElasticAlSb_JAP_1960, Weil_ElasticAlSb_JAP_1972} & 29.7 & 27.4 & 28.5 & 29.6 & 0.268~\\cite{Semiconductors_BasicData_Springer, Bolef_ElasticAlSb_JAP_1960, Weil_ElasticAlSb_JAP_1972} & 0.258 & 0.25 \\\\\nGaP & 88.7~\\cite{Lam_BulkMod_PRB_1987} & 78.8 & 76.2 & 71.9 & 73.4 & 72.2 & 73.8 & 55.3~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 53.5 & 49.1 & 51.3 & 51.8 & 0.244~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 0.232 & 0.22 \\\\\n & 89.8~\\cite{Boyle_ElasticGaPSb_PRB_1975} & & & & & & & & & & & & & \\\\\nGaAs & 74.8~\\cite{Lam_BulkMod_PRB_1987} & 62.7 & 60.7 & 56.8 & 57.7 & 56.6 & 58.1 & 46.6~\\cite{Bateman_ElasticGaAs_JAP_1975} & 42.6 & 39.1 & 40.8 & 40.9 & 0.244~\\cite{Bateman_ElasticGaAs_JAP_1975} & 0.233 & 0.23 \\\\\n & 75.5~\\cite{Bateman_ElasticGaAs_JAP_1975} & & & & & & & & & & & & & \\\\\nGaSb & 57.0~\\cite{Lam_BulkMod_PRB_1987} & 47.0 & 44.7 & 41.6 & 42.3 & 41.2 & 42.6 & 34.2~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 30.8 & 28.3 & 29.6 & 30.0 & 0.248~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 0.240 & 0.23 \\\\\n & 56.3~\\cite{Boyle_ElasticGaPSb_PRB_1975} & & & & & & & & & & & & & \\\\\nInP & 71.1~\\cite{Lam_BulkMod_PRB_1987, Nichols_ElasticInP_SSC_1980} & 60.4 & N\/A & 56.4 & 57.6 & 56.3 & 57.8 & 34.3~\\cite{Nichols_ElasticInP_SSC_1980} & 33.6 & 29.7 & 31.6 & N\/A & 0.292~\\cite{Nichols_ElasticInP_SSC_1980} & 0.277 & N\/A \\\\\nInAs & 60.0~\\cite{Lam_BulkMod_PRB_1987} & 50.1 & 49.2 & 45.7 & 46.6 & 45.4 & 46.9 & 29.5~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 27.3 & 24.2 & 25.7 & 25.1 & 0.282~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 0.281 & 0.28 \\\\\n & 57.9~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & & & & & & & & & & & & & \\\\\nInSb & 47.3~\\cite{Lam_BulkMod_PRB_1987, DeVaux_ElasticInSb_PR_1956} & 38.1 & N\/A & 34.3 & 35.0 & 34.1 & 35.2 & 22.1~\\cite{DeVaux_ElasticInSb_PR_1956} & 21.3 & 19.0 & 20.1 & N\/A & 0.298~\\cite{DeVaux_ElasticInSb_PR_1956} & 0.275 & N\/A \\\\\n & 48.3~\\cite{Semiconductors_BasicData_Springer, Slutsky_ElasticInSb_PR_1959} & & & & & & & 23.7~\\cite{Semiconductors_BasicData_Springer, Slutsky_ElasticInSb_PR_1959} & & & & & 0.289~\\cite{Semiconductors_BasicData_Springer, Slutsky_ElasticInSb_PR_1959} & \\\\\n & 46.5~\\cite{Vanderborgh_ElasticInSb_PRB_1990} & & & & & & & & & & & & & \\\\\nZnS & 77.1~\\cite{Lam_BulkMod_PRB_1987} & 71.2 & 68.3 & 65.8 & 66.1 & 65.2 & 66.6 & 30.9~\\cite{Semiconductors_BasicData_Springer} & 36.5 & 31.4 & 33.9 & 33.2 & 0.318~\\cite{Semiconductors_BasicData_Springer} & 0.294 & 0.29 \\\\\n & 74.5~\\cite{Semiconductors_BasicData_Springer} & & & & & & & & & & & & & \\\\\nZnSe & 62.4~\\cite{Lam_BulkMod_PRB_1987, Lee_ElasticZnSeTe_JAP_1970} & 58.2 & 58.3 & 53.3 & 53.8 & 52.8 & 54.1 & 29.1~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 29.5 & 25.6 & 27.5 & 27.5 & 0.298~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 0.296 & 0.3\\\\\nZnTe & 51.0~\\cite{Lam_BulkMod_PRB_1987, Lee_ElasticZnSeTe_JAP_1970} & 43.8 & 46.0 & 39.9 & 40.5 & 39.4 & 40.7 & 23.4~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 23.3 & 20.8 & 22.1 & 22.4 & 0.30~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 0.284 & 0.29 \\\\\nCdSe & 53.0~\\cite{Lam_BulkMod_PRB_1987} & 46.7 & 44.8 & 41.5 & 42.1 & 41.1 & 42.3 & N\/A & 16.2 & 13.1 & 14.7 & 15.3 & N\/A & 0.358 & 0.35 \\\\\nCdTe & 42.4~\\cite{Lam_BulkMod_PRB_1987} & 36.4 & 35.3 & 32.2 & 32.7 & 31.9 & 32.8 & N\/A & 14.2 & 11.9 & 13.0 & 13.6 & N\/A & 0.340 & 0.33 \\\\\nHgSe & 50.0~\\cite{Lam_BulkMod_PRB_1987} & 43.8 & 41.2 & 39.0 & 39.7 & 38.5 & 39.9 & 14.8~\\cite{Lehoczky_ElasticHgSe_PR_1969} & 15.6 & 11.9 & 13.7 & 13.3 & 0.361~\\cite{Lehoczky_ElasticHgSe_PR_1969} & 0.358 & 0.35 \\\\\n & 48.5~\\cite{Lehoczky_ElasticHgSe_PR_1969} & & & & & & & & & & & & & \\\\\nHgTe & 42.3~\\cite{Lam_BulkMod_PRB_1987, Semiconductors_BasicData_Springer, Cottam_ElasticHgTe_JPCS_1975} & 35.3 & N\/A & 31.0 & 31.6 & 30.8 & 31.9 & 14.7~\\cite{Semiconductors_BasicData_Springer, Cottam_ElasticHgTe_JPCS_1975} & 14.4 & 11.6 & 13.0 & N\/A & 0.344~\\cite{Semiconductors_BasicData_Springer, Cottam_ElasticHgTe_JPCS_1975} & 0.335 & N\/A \\\\\n\\end{tabular}}\n\\label{tab:art115:zincblende_elastic}\n\\end{table}\n\nFor the shear modulus, the experimental values $G^{\\mathrm{exp}}$ are compared to the {\\small AEL}\\ values $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$,\n$G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$. As can be seen from the values in\nTable~\\ref{tab:art115:zincblende_elastic} and Figure~\\ref{fig:art115:zincblende_thermal_elastic}(b), the agreement with the experimental values is generally\ngood with a very low {\\small RMSrD}\\ of 0.111 for $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, with the Voigt approximation tending to overestimate and the Reuss approximation tending to underestimate, as would be\nexpected. The experimental values of the Poisson ratio $\\sigma^{\\mathrm{exp}}$ and the {\\small AEL}\\ values $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ (Equation~\\ref{eq:art115:Poissonratio}) are\nalso shown in Table~\\ref{tab:art115:zincblende_elastic} and Figure~\\ref{fig:art115:zincblende_thermal_elastic}(c), and the values are generally in good\nagreement. The Pearson (\\nobreak\\mbox{\\it i.e.}, linear, Equation~\\ref{eq:art115:Pearson}) and Spearman (\\nobreak\\mbox{\\it i.e.}, rank order, Equation~\\ref{eq:art115:Spearman}) correlations between all of\nthe {\\small AEL}-{\\small AGL}\\ elastic property values and experiment are shown in Table~\\ref{tab:art115:zincblende_correlation}, and are generally\nvery high for all of these properties, ranging from 0.977 and 0.982 respectively for $\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$, up to 0.999\nand 0.992 for $B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$. These very high correlation values demonstrate the validity of using the {\\small AEL}-{\\small AGL}\\\nmethodology to predict the elastic and mechanical properties of\nmaterials.\n\nThe Materials Project values of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ for diamond and zincblende structure materials are also shown in\nTable~\\ref{tab:art115:zincblende_elastic}, where available. The Pearson correlations values for the experimental results with the available values of\n$B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ were calculated to be 0.995, 0.987 and 0.952, respectively, while the respective Spearman correlations\nwere 0.963, 0.977 and 0.977, and the {\\small RMSrD}\\ values were 0.149, 0.116 and 0.126. For comparison, the corresponding Pearson correlations for the same\nsubset of materials for $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ are 0.997, 0.987, and 0.957 respectively, while the respective Spearman correlations\nwere 0.982, 0.977 and 0.977, and the {\\small RMSrD}\\ values were 0.129, 0.114 and 0.108. These correlation values are very similar, and the general close agreement\n{for $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ with $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$}\ndemonstrate that the small differences in the parameters used for the {\\small DFT}\\ calculations make little difference to the results,\nindicating that the parameter set used here is robust for high-throughput calculations.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.98\\linewidth]{fig029}\n\\mycaption[\n({\\bf a}) Bulk modulus,\n({\\bf b}) shear modulus,\n({\\bf c}) Poisson ratio,\n({\\bf d}) lattice thermal conductivity at 300~K,\n({\\bf e}) acoustic Debye temperature and\n({\\bf f}) Gr{\\\"u}neisen parameter of zincblende and\ndiamond structure semiconductors.]\n{The zincblende structure is designated {\\small AFLOW}\\ prototype {\\sf AB\\_cF8\\_216\\_c\\_a}~\\cite{aflowANRL}\nand the diamond structure {\\sf A\\_cF8\\_227\\_a}~\\cite{aflowANRL}.}\n\\label{fig:art115:zincblende_thermal_elastic}\n\\end{figure}\n\nThe thermal {properties Debye} temperature, Gr{\\\"u}neisen parameter and thermal conductivity calculated using {\\small AGL}\\ for this set of materials are\ncompared to the experimental values taken from the literature in Table~\\ref{tab:art115:zincblende_thermal} and are also plotted in Figure~\\ref{fig:art115:zincblende_thermal_elastic}.\nFor the Debye temperature, the experimental values $\\theta^{\\mathrm{exp}}$ are compared {to\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ and $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$} in Figure~\\ref{fig:art115:zincblende_thermal_elastic}(e), while {the values} for\nthe empirical equations of state are provided in Table~\\ref{tab:art115:zincblende_thermal_eos}.\nNote that the $\\theta^{\\mathrm{exp}}$ values taken from Reference~\\onlinecite{slack} and\nReference~\\onlinecite{Morelli_Slack_2006} are for $\\theta_{\\mathrm{a}}$, and generally are in good agreement with the $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ values. The\nvalues obtained using the numerical $E(V)$ fit and the three different equations of state are also in good agreement with each other, whereas\nthe values of $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ calculated using different $\\sigma$ values differ significantly, indicating that for this property the value\nof $\\sigma$ used is far more important than the equation of state used. The correlation\nbetween $\\theta^{\\mathrm{exp}}$ and the various {\\small AGL}\\ values is also very high,\nof the order of 0.999, and the {\\small RMSrD}\\ is low, of the order of 0.13.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Thermal properties lattice thermal conductivity at\n300~K, Debye temperature and Gr{\\\"u}neisen parameter of\nzincblende and diamond structure semiconductors, comparing the effect of using the\ncalculated value of the Poisson ratio to the previous approximation of $\\sigma = 0.25$.]\n{The zincblende structure is designated {\\small AFLOW}\\ prototype {\\sf AB\\_cF8\\_216\\_c\\_a}~\\cite{aflowANRL}\nand the diamond structure {\\sf A\\_cF8\\_227\\_a}~\\cite{aflowANRL}.\nThe values listed for $\\theta^{\\mathrm{exp}}$ are\n$\\theta_{\\mathrm{a}}$, except 141K for HgTe which is $\\theta_{\\mathrm D}$~\\cite{Snyder_jmatchem_2011}.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ \\\\\n & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & & & \\\\\n & & ($\\sigma = 0.25$)\\cite{curtarolo:art96} & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & & ($\\sigma = 0.25$)\\cite{curtarolo:art96} & \\\\\n\\hline\nC & 3000~\\cite{Morelli_Slack_2006} & 169.1 & 419.9 & 1450~\\cite{slack, Morelli_Slack_2006} & 1536 & 2094 & 2222 & 0.75~\\cite{Morelli_Slack_2006} & 1.74 & 1.77 \\\\\n & & & & & (1219) & (1662) & & 0.9~\\cite{slack} & & \\\\\nSiC & 360~\\cite{Ioffe_Inst_DB} & 67.19 & 113.0 & 740~\\cite{slack} & 928 & 1106 & 1143 & 0.76~\\cite{slack} & 1.84 & 1.85\t\\\\\n & & & & & (737) & (878) & & & & \\\\\nSi & 166~\\cite{Morelli_Slack_2006} & 20.58 & 26.19 & 395~\\cite{slack, Morelli_Slack_2006} & 568 & 610 & 624 & 1.06~\\cite{Morelli_Slack_2006} & 2.09 & 2.06\t \\\\\n & & & & & (451) & (484) & & 0.56~\\cite{slack} & & \\\\\nGe & 65~\\cite{Morelli_Slack_2006} & 6.44 & 8.74 & 235~\\cite{slack, Morelli_Slack_2006} & 296 & 329 & 342 & 1.06~\\cite{Morelli_Slack_2006} & 2.3 & 2.31 \t \\\\\n & & & & & (235) & (261) & & 0.76~\\cite{slack} & & \\\\\nBN & 760~\\cite{Morelli_Slack_2006} & 138.4 & 281.6 & 1200~\\cite{Morelli_Slack_2006} & 1409 & 1793 & 1887 & 0.7~\\cite{Morelli_Slack_2006} & 1.73 & 1.75\t\\\\\n & & & & & (1118) & (1423) & & & & \\\\\nBP & 350~\\cite{Morelli_Slack_2006} & 52.56 & 105.0 & 670~\\cite{slack, Morelli_Slack_2006} & 811 & 1025 & 1062 & 0.75~\\cite{Morelli_Slack_2006} & 1.78 & 1.79\t\\\\\n & & & & & (644) & (814) & & & & \\\\\nAlP & 90~\\cite{Landolt-Bornstein, Spitzer_JPCS_1970} & 21.16 & 19.34 & 381~\\cite{Morelli_Slack_2006} & 542 & 525 & 531 & 0.75~\\cite{Morelli_Slack_2006} & 1.96 & 1.96\t \\\\\n & & & & & (430) & (417) & & & & \\\\\nAlAs & 98~\\cite{Morelli_Slack_2006} & 12.03 & 11.64 & 270~\\cite{slack, Morelli_Slack_2006} & 378 & 373 & 377 & 0.66~\\cite{slack, Morelli_Slack_2006} & 2.04 & 2.04\t \\\\\n & & & & & (300) & (296) & & & & \\\\\nAlSb & 56~\\cite{Morelli_Slack_2006} & 7.22 & 6.83 & 210~\\cite{slack, Morelli_Slack_2006} & 281 & 276 & 277 & 0.6~\\cite{slack, Morelli_Slack_2006} & 2.12 & 2.13 \t \\\\\n & & & & & (223) & (219) & & & & \\\\\nGaP & 100~\\cite{Morelli_Slack_2006} & 11.76 & 13.34 & 275~\\cite{slack, Morelli_Slack_2006} & 396 & 412 & 423 & 0.75~\\cite{Morelli_Slack_2006} & 2.15 & 2.15 \t\\\\\n & & & & & (314) & (327) & & 0.76~\\cite{slack} & & \\\\\nGaAs & 45~\\cite{Morelli_Slack_2006} & 7.2 & 8.0 & 220~\\cite{slack, Morelli_Slack_2006} & 302 & 313\t& 322 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.24 \\\\\n & & & & & (240) & (248) & & & & \\\\\nGaSb & 40~\\cite{Morelli_Slack_2006} & 4.62 & 4.96 & 165~\\cite{slack, Morelli_Slack_2006} & 234 & 240 & 248 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.27 & 2.28 \t \\\\\n & & & & & (186) & (190) & & & & \\\\\nInP & 93~\\cite{Morelli_Slack_2006} & 7.78 & 6.53 & 220~\\cite{slack, Morelli_Slack_2006} & 304 & 286 & 287 & 0.6~\\cite{slack, Morelli_Slack_2006} & 2.22 & 2.21 \t \\\\\n & & & & & (241) & (227) & & & & \\\\\nInAs & 30~\\cite{Morelli_Slack_2006} & 5.36 & 4.33 & 165~\\cite{slack, Morelli_Slack_2006} & 246 & 229 & 231 & 0.57~\\cite{slack, Morelli_Slack_2006} & 2.26 & 2.26\t \\\\\n & & & & & (195) & (182) & & & & \\\\\nInSb & 20~\\cite{Morelli_Slack_2006} & 3.64 & 3.02 & 135~\\cite{slack, Morelli_Slack_2006} & 199 & 187 & 190 & 0.56~\\cite{slack, Morelli_Slack_2006} & 2.3 & 2.3 \t \\\\\n & 16.5~\\cite{Snyder_jmatchem_2011} & & & & (158) & (148) & & & & \\\\\nZnS & 27~\\cite{Morelli_Slack_2006} & 11.33 & 8.38 & 230~\\cite{slack, Morelli_Slack_2006} & 379 & 341 & 346 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.01 & 2.00 \t \\\\\n & & & & & (301) & (271) & & & & \\\\\nZnSe & 19~\\cite{Morelli_Slack_2006} & 7.46 & 5.44 & 190~\\cite{slack, Morelli_Slack_2006} & 290 & 260\t& 263 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.07 & 2.06 \t\\\\\n & 33~\\cite{Snyder_jmatchem_2011} & & & & (230) & (206) & & & & \\\\\nZnTe & 18~\\cite{Morelli_Slack_2006} & 4.87 & 3.83 & 155~\\cite{slack, Morelli_Slack_2006} & 228 & 210 & 212 & 0.97~\\cite{slack, Morelli_Slack_2006} & 2.14 & 2.13 \\\\\n & & & & & (181) & (167) & & & & \\\\\nCdSe & 4.4~\\cite{Snyder_jmatchem_2011} & 4.99 & 2.04 & 130~\\cite{Morelli_Slack_2006} & 234 & 173 & 174 & 0.6~\\cite{Morelli_Slack_2006} & 2.19 & 2.18 \\\\\n & & & & & (186) & (137) & & & & \\\\\nCdTe & 7.5~\\cite{Morelli_Slack_2006} & 3.49 & 1.71 & 120~\\cite{slack, Morelli_Slack_2006} & 191 & 150 & 152 & 0.52~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.22\t \\\\\n & & & & & (152) & (119) & & & & \\\\\nHgSe & 3~\\cite{Whitsett_PRB_1973} & 3.22 & 1.32 & 110~\\cite{slack} & 190 & 140\t& 140 & 0.17~\\cite{slack} & 2.4 & 2.38\t \\\\\n & & & & & (151) & (111) & & & & \\\\\nHgTe & 2.5~\\cite{Snyder_jmatchem_2011} & 2.36 & 1.21 & 141~\\cite{Snyder_jmatchem_2011} & 162 & 129 & 130 & 1.9~\\cite{Snyder_jmatchem_2011} & 2.46 & 2.45 \\\\\n & & & & (100)~\\cite{slack} & (129) & (102) & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:zincblende_thermal}\n\\end{table}\n\nThe experimental values $\\gamma^{\\mathrm{exp}}$ of the Gr{\\\"u}neisen parameter are plotted {against\n$\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$, $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$, $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ and $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$} in Figure~\\ref{fig:art115:zincblende_thermal_elastic}(f), and the values\nare listed in Table~\\ref{tab:art115:zincblende_thermal} and in Table~\\ref{tab:art115:zincblende_thermal_eos}.\nThe very high {\\small RMSrD}\\ values (see Table~\\ref{tab:art115:zincblende_correlation}) show that {\\small AGL}\\ has problems accurately predicting\nthe Gr{\\\"u}neisen parameter for this set of materials, as the calculated value is often 2 to 3 times larger than the experimental one.\nNote also that there are quite large differences between the values obtained for different equations of state, with $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ generally\nhaving the lowest values while $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ has the highest values.\nOn the other hand, in contrast to the case of $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$, the value of $\\sigma$ used makes little difference to the value\nof $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$. The {correlations} between $\\gamma^{\\mathrm{exp}}$ and the {\\small AGL}\\ values, as shown in Table~\\ref{tab:art115:zincblende_correlation},\nare also quite poor, with no value higher than 0.2 for the Pearson correlations, and {negative Spearman} correlations.\n\nThe experimental thermal conductivity $\\kappa^{\\mathrm{exp}}$ is compared in Figure~\\ref{fig:art115:zincblende_thermal_elastic}(d) to the\nthermal conductivities calculated with {\\small AGL}\\ using the Leibfried-Schl{\\\"o}mann equation (Equation~\\ref{eq:art115:thermal_conductivity}): $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$, $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$, $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ and $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$,\nwhile the values are listed in Table~\\ref{tab:art115:zincblende_thermal} and in Table~\\ref{tab:art115:zincblende_thermal_eos}.\nThe absolute agreement between the {\\small AGL}\\ values and $\\kappa^{\\mathrm{exp}}$ is quite poor, with {\\small RMSrD}\\ values of the order of 0.8 and discrepancies of tens, or even hundreds, of percent\nquite common. Considerable disagreements also exist between different experimental reports of these properties, in\nalmost all cases where they exist. Unfortunately, the scarcity of experimental data from different sources on the thermal properties of these materials\nprevents reaching definite conclusions regarding the true values of these properties. The available data can thus only be considered as a rough indication\nof their order of magnitude.\n\n{The Pearson} correlations between the {\\small AGL}\\ calculated thermal conductivity values and the experimental\nvalues are high, ranging from $0.871$ to $0.932$, while the Spearman correlations are even higher, ranging from $0.905$\nto $0.954$, as shown in Table~\\ref{tab:art115:zincblende_correlation}. In particular, note that using the $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ in the {\\small AGL}\\ calculations\nimproves the correlations by about 5\\%, from $0.878$ to $0.927$ and from $0.905$ to $0.954$. For the different equations of state,\n$\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ and $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ appear to correlate better with $\\kappa^{\\mathrm{exp}}$ than $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ and $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ for this set of\nmaterials.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations and deviations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for zincblende and diamond structure semiconductors.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.878 & 0.905 & 0.776 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.927 & 0.95 & 0.796 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.871 & 0.954 & 0.787 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.908 & 0.954 & 0.815 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.932 & 0.954 & 0.771 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.995 & 0.984 & 0.200 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.999 & 0.998 & 0.132 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.999 & 0.998 & 0.132 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.999 & 0.998 & 0.127 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.999 & 0.998 & 0.136 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.137 & -0.187 & 3.51 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.145 & -0.165 & 3.49 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.169 & -0.178 & 3.41 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.171 & -0.234 & 3.63 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.144 & -0.207 & 3.32 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.999 & 0.992 & 0.130 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.999 & 0.986 & 0.201 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.999 & 0.986 & 0.189 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.999 & 0.986 & 0.205 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.999 & 0.986 & 0.185 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 0.980 & 0.111 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 0.980 & 0.093 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 0.980 & 0.152 \\\\\n$\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.977 & 0.982 & 0.095 \\\\\n\\end{tabular}\n\\label{tab:art115:zincblende_correlation}\n\\end{table}\n\nAs we noted in our previous work on {\\small AGL}~\\cite{curtarolo:art96}, some of the inaccuracy in the thermal conductivity results may be due to the inability of the Leibfried-Schl{\\\"o}mann equation to fully\ndescribe effects such as the suppression of phonon-phonon scattering due to large gaps between the branches of\nthe phonon dispersion~\\cite{Lindsay_PRL_2013}. This can be seen from the thermal conductivity values shown in Table 2.2 of Reference~\\onlinecite{Morelli_Slack_2006}\ncalculated using the experimental values of $\\theta_{\\mathrm{a}}$ and $\\gamma$ in the Leibfried-Schl{\\\"o}mann equation. There are large discrepancies in certain cases such as diamond,\nwhile the Pearson and Spearman correlations of $0.932$ and $0.941$ respectively are very similar to the correlations we calculated using the {\\small AGL}\\ evaluations of\n$\\theta_{\\mathrm{a}}$ and $\\gamma$.\n\nThus, the unsatisfactory quantitative reproduction of these quantities by the Debye quasi-harmonic model\nhas little impact on its effectiveness as a screening tool for identifying high or\nlow thermal conductivity materials. The model can be used when these\nexperimental values are unavailable to help determine the relative values of these quantities and for\nranking {materials conductivity}.\n\n\\subsubsection{Rocksalt structure materials}\n\nThe mechanical and thermal properties were calculated for a set of materials with the rocksalt structure\n(spacegroup: $Fm\\overline{3}m$,\\ $\\#$225; Pearson symbol: cF8;\n{\\small AFLOW}\\ prototype: {\\sf AB\\_cF8\\_225\\_a\\_b}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/AB_cF8_225_a_b.html}}).\n This {is the same set of materials\nas} in Table II of Reference~\\onlinecite{curtarolo:art96}, which in turn {are from} the\nsets in Table III of Reference~\\onlinecite{slack} and Table 2.1 of Reference~\\onlinecite{Morelli_Slack_2006}.\n\nThe elastic properties of bulk modulus, shear modulus and Poisson ratio, as calculated using {\\small AEL}\\ and {\\small AGL}\\ are shown\nin Table~\\ref{tab:art115:rocksalt_elastic} and Figure~\\ref{fig:art115:rocksalt_thermal_elastic}, together\nwith experimental values from the literature where available. As can be seen\nfrom the results in Table~\\ref{tab:art115:rocksalt_elastic} and Figure~\\ref{fig:art115:rocksalt_thermal_elastic}(a), for this set of materials the\n$B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ values are closest to experiment, with an {\\small RMSrD}\\ of 0.078. The {\\small AGL}\\ values from both the numerical\nfit and the empirical equations of state are generally very similar to each other, while being slightly less than the $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$\nvalues.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Mechanical properties bulk modulus, shear modulus\nand Poisson ratio of rocksalt structure semiconductors.]\n{The rocksalt structure is designated {\\small AFLOW}\\ Prototype {\\sf AB\\_cF8\\_225\\_a\\_b}~\\cite{aflowANRL}.\n``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ in {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ \\\\\n\\hline\nLiH & 33.7~\\cite{Laplaze_ElasticLiH_SSC_1976} & 37.7 & 36.1 & 29.5 & 29.0 & 27.7 & 31.4 & 36.0~\\cite{Laplaze_ElasticLiH_SSC_1976} & 43.4 & 42.3 & 42.8 & 42.9 & 0.106~\\cite{Laplaze_ElasticLiH_SSC_1976} & 0.088 & 0.07 \\\\\nLiF & 69.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 70.4 & 69.9 & 58.6 & 59.9 & 57.5 & 61.2 & 48.8~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 46.4 & 45.8 & 46.1 & 50.9 & 0.216~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.231 & 0.21 \\\\\nNaF & 48.5~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 46.9 & 47.6 & 38.7 & 38.6 & 36.8 & 39.3 & 31.2~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 29.5 & 28.4 & 28.9 & 30.0 & 0.236~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.244 & 0.24 \\\\\nNaCl & 25.1~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 24.9 & 22.6 & 20.0 & 20.5 & 19.2 & 20.7 & 14.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 14.0 & 12.9 & 13.5 & 14.3 & 0.255~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.271 & 0.24 \\\\\nNaBr & 20.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 20.5 & 27.1 & 16.3 & 16.9 & 15.7 & 16.9 & 11.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 11.0 & 9.9 & 10.4 & 11.6 & 0.264~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.283 & 0.31 \\\\\nNaI & 15.95~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 16.4 & 15.8 & 12.6 & 13.2 & 12.2 & 13.1 & 8.59~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 8.35 & 7.31 & 7.83 & 8.47 & 0.272~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.295 & 0.27 \\\\\nKF & 31.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 29.9 & 28.9 & 25.1 & 24.2 & 22.9 & 24.7 & 16.7~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 16.5 & 15.4 & 15.9 & 16.5 & 0.275~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.274 & 0.26 \\\\\nKCl & 18.2~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 16.7 & 15.8 & 13.8 & 13.7 & 12.7 & 13.6 & 9.51~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 10.1 & 8.51 & 9.30 & 9.24 & 0.277~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.265 & 0.26 \\\\\nKBr & 15.4~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 13.8 & 21.6 & 11.1 & 11.4 & 10.5 & 11.2 & 7.85~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 8.14 & 6.46 & 7.30 & 7.33 & 0.282~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.276 & 0.35 \\\\\nKI & 12.2~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 10.9 & 9.52 & 8.54 & 9.03 & 8.28 & 8.84 & 5.96~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 6.05 & 4.39 & 5.22 & 5.55 & 0.290~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.294 & 0.26 \\\\\nRbCl & 16.2~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 14.3 & 14.6 & 12.1 & 11.8 & 11.0 & 11.8 & 7.63~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 8.06 & 6.41 & 7.24 & 7.67 & 0.297~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.284 & 0.28 \\\\\nRbBr & 13.8~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 12.6 & 13.8 & 10.3 & 9.72 & 9.06 & 9.67 & 6.46~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 7.12 & 5.24 & 6.18 & 6.46 & 0.298~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.289 & 0.3 \\\\\nRbI & 11.1~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 9.90 & 9.66 & 8.01 & 7.74 & 7.12 & 7.54 & 5.03~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 5.50 & 3.65 & 4.57 & 4.63 & 0.303~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.300 & 0.29 \\\\\nAgCl & 44.0~\\cite{Hughes_ElasticAgCl_PRB_1996} & 40.6 & N\/A & 33.7 & 34.1 & 33.0 & 34.7 & 8.03~\\cite{Hughes_ElasticAgCl_PRB_1996} & 8.68 & 8.66 & 8.67 & N\/A & 0.414~\\cite{Hughes_ElasticAgCl_PRB_1996} & 0.400 & N\/A \\\\\nMgO & 164~\\cite{Sumino_ElasticMgO_JPE_1976} & 152 & 152 & 142 & 142 & 140 & 144 & 131~\\cite{Sumino_ElasticMgO_JPE_1976} & 119 & 115 & 117 & 119 & 0.185~\\cite{Sumino_ElasticMgO_JPE_1976} & 0.194 & 0.19 \\\\\nCaO & 113~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 105 & 105 & 99.6 & 100 & 98.7 & 101 & 81.0~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 73.7 & 73.7 & 73.7 & 74.2 & 0.210~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 0.216 & 0.21 \\\\\nSrO & 91.2~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 84.7 & 87.4 &80.0 & 80.2 & 79.1 & 80.8 & 58.7~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 55.1 & 55.0 & 55.1 & 56.0 & 0.235~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 0.233 & 0.24 \\\\\nBaO & 75.4~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 69.1 & 68.4 & 64.6 & 64.3 & 63.0 & 64.6 & 35.4~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 36.4 & 36.4 & 36.4 & 37.8 & 0.297~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 0.276 & 0.27 \\\\\nPbS & 52.9~\\cite{Semiconductors_BasicData_Springer, Peresada_ElasticPbS_PSSa_1976} & 53.5 & N\/A & 49.9 & 50.8 & 50.0 & 51.0 & 27.9~\\cite{Semiconductors_BasicData_Springer, Peresada_ElasticPbS_PSSa_1976} & 34.0 & 26.8 & 30.4 & N\/A & 0.276~\\cite{Semiconductors_BasicData_Springer, Peresada_ElasticPbS_PSSa_1976} & 0.261 & N\/A \\\\\nPbSe & 54.1~\\cite{Semiconductors_BasicData_Springer, Lippmann_ElasticPbSe_PSSa_1971} & 47.7 & N\/A & 43.9 & 44.8 & 43.9 & 44.9 & 26.2~\\cite{Semiconductors_BasicData_Springer, Lippmann_ElasticPbSe_PSSa_1971} & 31.7 & 23.6 & 27.6 & N\/A & 0.291~\\cite{Semiconductors_BasicData_Springer, Lippmann_ElasticPbSe_PSSa_1971} & 0.257 & N\/A \\\\\nPbTe & 39.8~\\cite{Semiconductors_BasicData_Springer, Miller_ElasticPbTe_JPCSS_1981} & 39.5 & N\/A & 36.4 & 36.6 & 35.8 & 36.8 & 23.1~\\cite{Semiconductors_BasicData_Springer, Miller_ElasticPbTe_JPCSS_1981} & 28.7 & 19.8 & 24.3 & N\/A & 0.256~\\cite{Semiconductors_BasicData_Springer, Miller_ElasticPbTe_JPCSS_1981} & 0.245 & N\/A \\\\\nSnTe & 37.8~\\cite{Semiconductors_BasicData_Springer, Seddon_ElasticSnTe_SSC_1976} & 40.4 & 39.6 & 38.1 & 38.4 & 37.6 & 38.6 & 20.8~\\cite{Semiconductors_BasicData_Springer, Seddon_ElasticSnTe_SSC_1976} & 31.4 & 22.0 & 26.7 & 27.6 & 0.267~\\cite{Semiconductors_BasicData_Springer, Seddon_ElasticSnTe_SSC_1976} & 0.229 & 0.22 \\\\\n\\end{tabular}}\n\\label{tab:art115:rocksalt_elastic}\n\\end{table}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.98\\linewidth]{fig030}\n\\mycaption[\n({\\bf a}) Bulk modulus,\n({\\bf b}) shear modulus,\n({\\bf c}) Poisson ratio,\n({\\bf d}) lattice thermal conductivity at 300~K,\n({\\bf e}) Debye temperature and\n({\\bf f}) Gr{\\\"u}neisen parameter of rocksalt structure\nsemiconductors.]\n{The rocksalt structure is designated {\\small AFLOW}\\ Prototype {\\sf AB\\_cF8\\_225\\_a\\_b}~\\cite{aflowANRL}.\nThe Debye temperatures plotted in ({\\bf b}) are\n$\\theta_{\\mathrm{a}}$, except for SnTe where $\\theta_{\\mathrm D}$ is\nquoted in Reference~\\onlinecite{Snyder_jmatchem_2011}.}\n\\label{fig:art115:rocksalt_thermal_elastic}\n\\end{figure}\n\nFor the shear modulus, the experimental values $G^{\\mathrm{exp}}$ are compared to the {\\small AEL}\\ values $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$,\n$G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$. As can be seen from the values in\nTable~\\ref{tab:art115:rocksalt_elastic} and Figure~\\ref{fig:art115:rocksalt_thermal_elastic}(b), the agreement with the experimental values is generally\ngood with an {\\small RMSrD}\\ of 0.105 for $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, with the Voigt approximation tending to overestimate and the Reuss approximation tending to underestimate, as would be\nexpected. The experimental values of the Poisson ratio $\\sigma^{\\mathrm{exp}}$ and the {\\small AEL}\\ values $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ (Equation~\\ref{eq:art115:Poissonratio}) are\nalso shown in Table~\\ref{tab:art115:rocksalt_elastic} and Figure~\\ref{fig:art115:rocksalt_thermal_elastic}(c), and the values are generally in good\nagreement. The Pearson (\\nobreak\\mbox{\\it i.e.}, linear, Equation~\\ref{eq:art115:Pearson}) and Spearman (\\nobreak\\mbox{\\it i.e.}, rank order, Equation~\\ref{eq:art115:Spearman}) correlations between all of\nthe the {\\small AEL}-{\\small AGL}\\ elastic property values and experiment are shown in Table~\\ref{tab:art115:rocksalt_correlation}, and are generally\nvery high for all of these properties, ranging from 0.959 and 0.827 respectively for $\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$, up to 0.998\nand 0.995 for $B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$. These very high correlation values demonstrate the validity of using the {\\small AEL}-{\\small AGL}\\\nmethodology to predict the elastic and mechanical properties of materials.\n\n{The values} of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ for rocksalt structure materials are also shown in\nTable~\\ref{tab:art115:rocksalt_elastic}, where available. The Pearson {correlations for} the experimental results with the available values of\n$B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ were calculated to be 0.997, 0.994 and 0.890, respectively, while the respective Spearman correlations\nwere 0.979, 0.998 and 0.817, and the {\\small RMSrD}\\ values were 0.153, 0.105 and 0.126. For comparison, the corresponding Pearson correlations for the same\nsubset of materials for $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ are 0.998, 0.995, and 0.951 respectively, while the respective Spearman correlations\nwere 0.996, 1.0 and 0.843, and the {\\small RMSrD}\\ values were 0.079, 0.111 and 0.071. These correlation values are very similar, and the general close agreement\nfor the results for the values of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ with those of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$\ndemonstrate that the small differences in the parameters used for the {\\small DFT}\\ calculations make little difference to the results,\nindicating that the parameter set used here is robust for high-throughput calculations.\n\nThe thermal properties of Debye temperature, Gr{\\\"u}neisen parameter and thermal conductivity calculated using {\\small AGL}\\ are\ncompared to the experimental values taken from the literature in Table~\\ref{tab:art115:rocksalt_thermal} and are also plotted in Figure~\\ref{fig:art115:rocksalt_thermal_elastic}.\nFor the Debye temperature, the experimental values $\\theta^{\\mathrm{exp}}$ are compared {to\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$}, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ and $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ in Figure~\\ref{fig:art115:rocksalt_thermal_elastic}(e), while the actual values for\nthe empirical equations of state are provided in Table~\\ref{tab:art115:rocksalt_thermal_eos}.\nNote that the $\\theta^{\\mathrm{exp}}$ values taken from Reference~\\onlinecite{slack} and\nReference~\\onlinecite{Morelli_Slack_2006} are for $\\theta_{\\mathrm{a}}$, and generally are in good agreement with the $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ values. The\nvalues obtained using the numerical $E(V)$ fit and the three different equations of state are also in good agreement with each other, whereas\nthe values of $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ calculated using different $\\sigma$ values differ significantly, indicating that, as in the case of the zincblende\nand diamond structures, the value of $\\sigma$ used is far more important for this property than the equation of state used. The correlation\nbetween $\\theta^{\\mathrm{exp}}$ and the various {\\small AGL}\\ values is also quite high, of the order of 0.98 for the Pearson correlation and 0.92 for the Spearman\ncorrelation.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Thermal properties lattice thermal conductivity at 300~K, Debye temperature and Gr{\\\"u}neisen parameter of rocksalt\nstructure semiconductors, comparing the effect of using the\ncalculated value of the Poisson ratio to the previous approximation of $\\sigma = 0.25$.]\n{The rocksalt structure is designated {\\small AFLOW}\\ Prototype {\\sf AB\\_cF8\\_225\\_a\\_b}~\\cite{aflowANRL}.\nThe values listed for $\\theta^{\\mathrm{exp}}$ are\n$\\theta_{\\mathrm{a}}$, except 155K for SnTe which is $\\theta_{\\mathrm D}$~\\cite{Snyder_jmatchem_2011}.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ \\\\\n & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & & & & \\\\\n & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & \\\\\n\\hline\nLiH & 15~\\cite{Morelli_Slack_2006} & 8.58 & 18.6 & 615~\\cite{slack, Morelli_Slack_2006} & 743 & 962 & 1175 & 1.28~\\cite{slack, Morelli_Slack_2006} & 1.62 & 1.66 \\\\\n & & & & & (590) & (764) & & & & \\\\\nLiF & 17.6~\\cite{Morelli_Slack_2006} & 8.71 & 9.96 & 500~\\cite{slack, Morelli_Slack_2006} & 591 & 617 & 681 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.02 & 2.03 \t \\\\\n & & & & & (469) & (490) & & & & \\\\\nNaF & 18.4~\\cite{Morelli_Slack_2006} & 4.52 & 4.67 & 395~\\cite{slack, Morelli_Slack_2006} & 411 & 416 & 455 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.2 & 2.21 \t \\\\\n & & & & & (326) & (330) & & & & \\\\\nNaCl & 7.1~\\cite{Morelli_Slack_2006} & 2.43 & 2.12 & 220~\\cite{slack, Morelli_Slack_2006} & 284 & 271 & 289 & 1.56~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.23 \t \\\\\n & & & & & (225) & (215) & & & & \\\\\nNaBr & 2.8~\\cite{Morelli_Slack_2006} & 1.66 & 1.33 & 150~\\cite{slack, Morelli_Slack_2006} & 203 & 188 & 198 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.22 & 2.22 \t \\\\\n & & & & & (161) & (149) & & & &\\\\\nNaI & 1.8~\\cite{Morelli_Slack_2006} & 1.17 & 0.851 & 100~\\cite{slack, Morelli_Slack_2006} & 156 & 140 & 147 & 1.56~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.23 \t \\\\\n & & & & & (124) & (111) & & & &\\\\\nKF & N\/A & 2.68 & 2.21 & 235~\\cite{slack, Morelli_Slack_2006} & 305 & 288 & 309\t& 1.52~\\cite{slack, Morelli_Slack_2006} & 2.29 & 2.32 \t\\\\\n & & & & & (242) & (229) & & & &\\\\\nKCl & 7.1~\\cite{Morelli_Slack_2006} & 1.4 & 1.25 & 172~\\cite{slack, Morelli_Slack_2006} & 220 & 213 & 226 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.38 & 2.40 \t \\\\\n & & & & & (175) & (169) & & & &\\\\\nKBr & 3.4~\\cite{Morelli_Slack_2006} & 1.0 & 0.842 & 117~\\cite{slack, Morelli_Slack_2006} & 165 & 156 & 162 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.37 & 2.37 \\\\\n & & & & & (131) & (124) & & & &\\\\\nKI & 2.6~\\cite{Morelli_Slack_2006} & 0.72 & 0.525 & 87~\\cite{slack, Morelli_Slack_2006} & 129 & 116 & 120 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.35 & 2.35 \t \\\\\n & & & & & (102) & (92) & & & &\\\\\nRbCl & 2.8~\\cite{Morelli_Slack_2006} & 1.09 & 0.837 & 124~\\cite{slack, Morelli_Slack_2006} & 168 & 155 & 160 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.34 & 2.37 \t \\\\\n & & & & & (133) & (123) & & & &\\\\\nRbBr & 3.8~\\cite{Morelli_Slack_2006} & 0.76 & 0.558 & 105~\\cite{slack, Morelli_Slack_2006} & 134 & 122 & 129 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.40 & 2.43 \\\\\n & & & & & (106) & (97) & & & &\\\\\nRbI & 2.3~\\cite{Morelli_Slack_2006} & 0.52 & 0.368 & 84~\\cite{slack, Morelli_Slack_2006} & 109 & 97 & 102 & 1.41~\\cite{slack, Morelli_Slack_2006} & 2.47 & 2.47 \t \\\\\n & & & & & (87) & (77) & & & &\\\\\nAgCl & 1.0~\\cite{Landolt-Bornstein, Maqsood_IJT_2003} & 2.58 & 0.613 & 124~\\cite{slack} & 235 & 145 & 148 & 1.9~\\cite{slack} & 2.5 & 2.49 \t \\\\\n & & & & & (187) & (115) & & & &\\\\\nMgO & 60~\\cite{Morelli_Slack_2006} & 31.9 & 44.5 & 600~\\cite{slack, Morelli_Slack_2006} & 758 & 849 & 890\t& 1.44~\\cite{slack, Morelli_Slack_2006} & 1.95 & 1.96 \\\\\n & & & & & (602) & (674) & & & &\\\\\nCaO & 27~\\cite{Morelli_Slack_2006} & 19.5 & 24.3 & 450~\\cite{slack, Morelli_Slack_2006} & 578 & 620 & 638 & 1.57~\\cite{slack, Morelli_Slack_2006} & 2.07 & 2.06 \t \\\\\n & & & & & (459) & (492) & & & &\\\\\nSrO & 12~\\cite{Morelli_Slack_2006} & 12.5 & 13.4 & 270~\\cite{slack, Morelli_Slack_2006} & 399 & 413 & 421 & 1.52~\\cite{slack, Morelli_Slack_2006} & 2.09 & 2.13 \t \\\\\n & & & & & (317) & (328) & & & &\\\\\nBaO & 2.3~\\cite{Morelli_Slack_2006} & 8.88 & 7.10 & 183~\\cite{slack, Morelli_Slack_2006} & 305 & 288 & 292 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.09 & 2.14 \\\\\n & & & & & (242) & (229) & & & &\\\\\nPbS & 2.9~\\cite{Morelli_Slack_2006} & 6.48 & 6.11 & 115~\\cite{slack, Morelli_Slack_2006} & 226 & 220 & 221 & 2.0~\\cite{slack, Morelli_Slack_2006} & 2.02 & 2.00 \t\\\\\n & & & & & (179) & (175) & & & &\\\\\nPbSe & 2.0~\\cite{Morelli_Slack_2006} & 4.88 & 4.81 & 100~\\cite{Morelli_Slack_2006} & 197 & 194 & 196 & 1.5~\\cite{Morelli_Slack_2006} & 2.1 & 2.07 \t \\\\\n & & & & & (156) & (154) & & & &\\\\\nPbTe & 2.5~\\cite{Morelli_Slack_2006} & 4.15 & 4.07 & 105~\\cite{slack, Morelli_Slack_2006} & 170 & 172 & 175 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.04 & 2.09 \t \\\\\n & & & & & (135) & (137) & & & &\\\\\nSnTe & 1.5~\\cite{Snyder_jmatchem_2011} & 4.46 & 5.24 & 155~\\cite{Snyder_jmatchem_2011} & 202 & 210 & 212 & 2.1~\\cite{Snyder_jmatchem_2011} & 2.15 & 2.11 \t \\\\\n & & & & & (160) & (167) & & & &\\\\\n\\end{tabular}}\n\\label{tab:art115:rocksalt_thermal}\n\\end{table}\n\nThe experimental values $\\gamma^{\\mathrm{exp}}$ of the Gr{\\\"u}neisen parameter are plotted {against\n$\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$}, $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$, $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ and $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ in Figure~\\ref{fig:art115:rocksalt_thermal_elastic}(f), and the values\nare listed in Table~\\ref{tab:art115:rocksalt_thermal} and in Table~\\ref{tab:art115:rocksalt_thermal_eos}.\nThese results show that {\\small AGL}\\ has problems accurately predicting the Gr{\\\"u}neisen parameter for this set of materials as well, as the calculated values\nare often 30\\% to 50\\% larger than the experimental ones and the {\\small RMSrD}\\ values are of the order of 0.5. Note also that there are quite large differences between the values\nobtained for different equations of state, with $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ generally having the lowest values while\n$\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ has the highest values, a similar pattern to that seen above for the zincblende and diamond structure materials. On the other hand, in contrast to the case of $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$,\nthe value of $\\sigma$ used makes little difference to the value of $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$. The correlation values between $\\gamma^{\\mathrm{exp}}$\nand the {\\small AGL}\\ values, as shown in Table~\\ref{tab:art115:rocksalt_correlation}, are also quite poor, with values ranging from -0.098 to\n0.118 for the Pearson correlations, and negative values for the Spearman correlations.\n\nThe experimental thermal conductivity $\\kappa^{\\mathrm{exp}}$ is compared in Figure~\\ref{fig:art115:rocksalt_thermal_elastic}(d) to the\nthermal conductivities calculated with {\\small AGL}\\ using the Leibfried-Schl{\\\"o}mann equation (Equation~\\ref{eq:art115:thermal_conductivity}): $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$, $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$, $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ and $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$,\nwhile the values are listed in Table~\\ref{tab:art115:rocksalt_thermal} and in Table~\\ref{tab:art115:rocksalt_thermal_eos}.\nThe linear correlation between the {\\small AGL}\\ values and $\\kappa^{\\mathrm{exp}}$ is somewhat better than for the zincblende materials set, with a Pearson\ncorrelation as high as $0.94$, although the Spearman correlations are somewhat lower, ranging from $0.445$\nto $0.556$. In particular, note that using the $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ in the {\\small AGL}\\ calculations improves the correlations by about\n2\\% to 8\\%, from $0.910$ to $0.932$ and from $0.445$ to $0.528$. For the different equations of state, the results for $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$\nappear to correlate best with $\\kappa^{\\mathrm{exp}}$ for this set of materials.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for rocksalt structure semiconductors.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.910 & 0.445 & 1.093 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.932 & 0.528 & 1.002 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.940 & 0.556 & 1.038 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.933 & 0.540 & 0.920 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.930 & 0.554 & 1.082 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.985 & 0.948 & 0.253 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.978 & 0.928 & 0.222 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.980 & 0.926 & 0.222 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.979 & 0.925 & 0.218 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.978 & 0.929 & 0.225 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.118 & -0.064 & 0.477 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.036 & -0.110 & 0.486 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & -0.019 & -0.088 & 0.462 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & -0.098 & -0.086 & 0.591 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.023 & -0.110 & 0.443 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 0.995 & 0.078 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.998 & 0.993 & 0.201 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.997 & 0.993 & 0.199 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.997 & 0.990 & 0.239 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.998 & 0.993 & 0.197 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.994 & 0.997 & 0.105 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.991 & 0.990 & 0.157 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.995 & 0.995 & 0.142 \\\\\n$\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.959 & 0.827 & 0.070 \\\\\n\\end{tabular}\n\\label{tab:art115:rocksalt_correlation}\n\\end{table}\n\nAs in the case of the diamond and zincblende structure materials discussed in the previous Section,\nReference~\\onlinecite{Morelli_Slack_2006} includes values of the thermal conductivity at 300~K for rocksalt structure materials,\ncalculated using the experimental values of $\\theta_{\\mathrm{a}}$ and $\\gamma$ in the Leibfried-Schl{\\\"o}mann equation, in Table 2.1.\nThe correlation values of $0.986$ and $0.761$ with experiment are\nbetter than those obtained for the {\\small AGL}\\ results by a larger margin than for the zincblende materials.\nNevertheless, the Pearson correlation between the calculated and\nexperimental conductivities is high in both calculations, indicating that the {\\small AGL}\\\napproach may be used as a screening tool for high or low conductivity\ncompounds in cases where gaps exist in the experimental data for these\nmaterials.\n\n\\subsubsection{Hexagonal structure materials}\n\nThe experimental data for this set of materials appears in Table III of Reference~\\onlinecite{curtarolo:art96}, taken from Table 2.3 of\nReference~\\onlinecite{Morelli_Slack_2006}. Most of these materials have the wurtzite structure ($P6_3mc$,\\ $\\#$186;\nPearson symbol: hP4; {\\small AFLOW}\\ prototype: {\\sf AB\\_hP4\\_186\\_b\\_b}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/AB_hP4_186_b_b.html}}) except InSe which is $P6_3mmc$,\\ $\\#$194,\nPearson symbol: hP8.\n\nThe calculated elastic properties are shown in Table~\\ref{tab:art115:wurzite_elastic} and Figure~\\ref{fig:art115:wurzite_thermal_elastic}. The bulk moduli\nvalues obtained from a direct calculation of the elastic tensor, $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, are usually slightly higher than those obtained from the\n$E(V)$ curve and are also closer to experiment (Table~\\ref{tab:art115:wurzite_elastic} and Figure~\\ref{fig:art115:wurzite_thermal_elastic}(a)), with the exception of\nInSe where it is noticeably lower.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Bulk modulus, shear modulus and Poisson ratio of hexagonal structure semiconductors.]\n{``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ in {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ \\\\\n\\hline\nSiC & 219~\\cite{Arlt_ELasticSiC_JAAcS_1965} & 213 & 213 & 204 & 208 & 207 & 207 & 198~\\cite{Arlt_ELasticSiC_JAAcS_1965} & 188 & 182 & 185 & 187 & 0.153~\\cite{Arlt_ELasticSiC_JAAcS_1965} & 0.163 & 0.16 \\\\\nAlN & 211~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993} & 195 & 194 & 187 & 190 & 189 & 189 & 135~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993} & 123 & 122 & 122 & 122 & 0.237~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993} & 0.241 & 0.24 \\\\\n & 200~\\cite{Dodd_BulkmodAlN_JMS_2001} & & & & & & & 130~\\cite{Dodd_BulkmodAlN_JMS_2001} & & & & & 0.234~\\cite{Dodd_BulkmodAlN_JMS_2001} &\\\\\nGaN & 195~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978} & 175 & 172 & 166 & 167 & 166 & 168 & 51.6~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978} & 107 & 105 & 106 & 105 & 0.378~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978} & 0.248 & 0.25 \\\\\n & 210~\\cite{Polian_ElasticGaN_JAP_1996} & & & & & & & 123~\\cite{Polian_ElasticGaN_JAP_1996} & & & & & 0.255~\\cite{Polian_ElasticGaN_JAP_1996} & \\\\\nZnO & 143~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 137 & 130 & 128 & 129 & 127 & 129 & 49.4~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 51.7 & 51.0 & 51.4 & 41.2 & 0.345~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 0.334 & 0.36 \\\\\nBeO & 224.4~\\cite{Cline_JAP_1967} & 206 & 208 & 195 & 195 & 192 & 198 & 168~\\cite{Cline_JAP_1967} & 157 & 154 & 156 & 156 & 0.201~\\cite{Cline_JAP_1967} & 0.198 & 0.2 \\\\\nCdS & 60.7~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 55.4 & 53.3 & 49.7 & 50.3 & 49.4 & 50.6 & 18.2~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 17.6 & 17.0 & 17.3 & 17.6 & 0.364~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 0.358 & 0.35 \\\\\nInSe & 37.1~\\cite{Gatulle_ElasticInSe_PSSb_1983} & 19.2 & N\/A & 39.8 & 40.8 & 39.7 & 41.0 & 14.8~\\cite{Gatulle_ElasticInSe_PSSb_1983} & 14.9 & 12.3 & 13.6 & N\/A & 0.324~\\cite{Gatulle_ElasticInSe_PSSb_1983} & 0.214 & N\/A \\\\\nInN & 126~\\cite{Ueno_BulkmodInN_PRB_1994} & 124 & N\/A & 118 & 120 & 119 & 119 & N\/A & 55.4 & 54.4 & 54.9 & N\/A & N\/A & 0.308 & N\/A \\\\\n\\end{tabular}}\n\\label{tab:art115:wurzite_elastic}\n\\end{table}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.98\\linewidth]{fig031}\n\\mycaption[\n({\\bf a}) Bulk modulus,\n({\\bf b}) shear modulus,\n({\\bf c}) Poisson ratio,\n({\\bf d}) lattice thermal conductivity,\n({\\bf e}) Debye temperature and\n({\\bf f}) Gr{\\\"u}neisen parameter of hexagonal structure\nsemiconductors.]\n{The Debye temperatures plotted in ({\\bf e}) are\n$\\theta_{\\mathrm{a}}$, except for InSe and InN where $\\theta_{\\mathrm D}$\nvalues are quoted in References~\\onlinecite{Snyder_jmatchem_2011, Ioffe_Inst_DB, Krukowski_jphyschemsolids_1998}.}\n\\label{fig:art115:wurzite_thermal_elastic}\n\\end{figure}\n\nFor the shear modulus, the experimental values $G^{\\mathrm{exp}}$ are compared to the {\\small AEL}\\ values $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$,\n$G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$. As can be seen in\nTable~\\ref{tab:art115:wurzite_elastic} and Figure~\\ref{fig:art115:wurzite_thermal_elastic}(b), the agreement with the experimental values is very\ngood. Similarly good agreement is obtained for the Poisson ratio of most materials (Table~\\ref{tab:art115:wurzite_elastic}\nand Figure~\\ref{fig:art115:wurzite_thermal_elastic}(c)), with\na single exception for InSe where the calculation deviates significantly from the experiment.\nThe Pearson (\\nobreak\\mbox{\\it i.e.}, linear, Equation~\\ref{eq:art115:Pearson}) and Spearman (\\nobreak\\mbox{\\it i.e.}, rank order, Equation~\\ref{eq:art115:Spearman}) correlations between the calculated\nelastic properties and their experimental values are generally\nquite high (Table~\\ref{tab:art115:wurzite_correlation}), ranging from 0.851 and 0.893 respectively for $\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$, up to 0.998\nand 1.0 for $G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$.\n\nThe Materials Project values of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ for hexagonal structure materials are also shown in\nTable~\\ref{tab:art115:wurzite_elastic}, where available. The Pearson correlations values for the experimental results with the available values of\n$B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ were calculated to be 0.984, 0.998 and 0.993, respectively, while the respective Spearman correlations\nwere 0.943, 1.0 and 0.943, and the {\\small RMSrD}\\ values were 0.117, 0.116 and 0.034. For comparison, the corresponding Pearson correlations for the same\nsubset of materials for $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ are 0.986, 0.998, and 0.998 respectively, while the respective Spearman correlations\nwere 0.943, 1.0 and 1.0, and the {\\small RMSrD}\\ values were 0.100, 0.091 and 0.036. These correlation values are very similar, and the general close agreement\nfor the results for the values of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ with those of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$\ndemonstrate that the small differences in the parameters used for the {\\small DFT}\\ calculations make little difference to the results,\nindicating that the parameter set used here is robust for high-throughput calculations.\n\nThe thermal properties calculated using {\\small AGL}\\ are\ncompared to the experimental values in Table~\\ref{tab:art115:wurzite_thermal} and are also plotted in Figure~\\ref{fig:art115:wurzite_thermal_elastic}.\nFor the Debye temperature, the $\\theta^{\\mathrm{exp}}$ values taken from Reference~\\onlinecite{Morelli_Slack_2006} are for $\\theta_{\\mathrm{a}}$,\nand are mostly in good agreement with the calculated $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ values. As in the case of the other materials sets,\nthe values obtained using the numerical $E(V)$ fit and the three different\nequations of state are very similar to each other, whereas $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ calculated using $\\sigma=0.25$ differs significantly.\nIn fact, the values of $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ calculated with $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ have a lower the correlation with $\\theta^{\\mathrm{exp}}$ than the values calculated with\n$\\sigma = 0.25$ do, although the {\\small RMSrD}\\ values are lower when $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ is used. However, most of this discrepancy appears to be due to the clear\noutlier value for the material InN. When the values for this material are removed from the data set, the Pearson correlation values become very similar\nwhen both the $\\sigma = 0.25$ and $\\sigma = \\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ values are used, increasing to 0.995 and 0.994 respectively.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity, Debye temperature and Gr{\\\"u}neisen parameter\nof hexagonal structure semiconductors, comparing the effect of using the\ncalculated value of the Poisson ratio to the previous approximation of $\\sigma = 0.25$.]\n{The values listed for $\\theta^{\\mathrm{exp}}$ are $\\theta_{\\mathrm{a}}$,\nexcept 190K for InSe~\\cite{Snyder_jmatchem_2011} and 660K for InN~\\cite{Ioffe_Inst_DB,Krukowski_jphyschemsolids_1998}\nwhich are $\\theta_{\\mathrm D}$.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ \\\\\n & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & & & & \\\\\n & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & \\\\\n\\hline\nSiC & 490~\\cite{Morelli_Slack_2006} & 42.49 & 70.36 & 740~\\cite{Morelli_Slack_2006} & 930 & 1103 & 1138 & 0.75~\\cite{Morelli_Slack_2006} & 1.86 & 1.86 \\\\\n & & & & & (586) & (695) & & & & \\\\\nAlN & 350~\\cite{Morelli_Slack_2006} & 36.73 & 39.0 & 620~\\cite{Morelli_Slack_2006} & 880 & 898 & 917 & 0.7~\\cite{Morelli_Slack_2006} & 1.85 & 1.85 \t \\\\\n & & & & & (554) & (566) & & & & \\\\\nGaN & 210~\\cite{Morelli_Slack_2006} & 18.17 & 18.54 & 390~\\cite{Morelli_Slack_2006} & 592 & 595 & 606 & 0.7~\\cite{Morelli_Slack_2006} & 2.07 & 2.08 \t \\\\\n & & & & & (373) & (375) & & & & \\\\\nZnO & 60~\\cite{Morelli_Slack_2006} & 14.10 & 7.39 & 303~\\cite{Morelli_Slack_2006} & 525 & 422 & 427 & 0.75~\\cite{Morelli_Slack_2006} & 1.97 & 1.94 \t \\\\\n & & & & & (331) & (266) & & & & \\\\\nBeO & 370~\\cite{Morelli_Slack_2006} & 39.26 & 53.36 & 809~\\cite{Morelli_Slack_2006} & 1065 & 1181 & 1235 & 1.38~\\cite{Slack_JAP_1975, Cline_JAP_1967, Morelli_Slack_2006} & 1.76 & 1.76 \t \\\\\n & & & & & (671) & (744) & & & & \\\\\nCdS & 16~\\cite{Morelli_Slack_2006} & 4.40 & 1.76 & 135~\\cite{Morelli_Slack_2006} & 287 & 211 & 213 & 0.75~\\cite{Morelli_Slack_2006} & 2.14 & 2.14 \t \\\\\n & & & & & (181) & (133) & & & & \\\\\nInSe & 6.9~\\cite{Snyder_jmatchem_2011} & 1.84 & 2.34 & 190~\\cite{Snyder_jmatchem_2011} & 230 & 249 & 168 & 1.2~\\cite{Snyder_jmatchem_2011} & 2.24 & 2.24 \t \\\\\n & & & & & (115) & (125) & & & & \\\\\nInN & 45~\\cite{Ioffe_Inst_DB, Krukowski_jphyschemsolids_1998} & 10.44 & 6.82 & 660~\\cite{Ioffe_Inst_DB, Krukowski_jphyschemsolids_1998} & 426 & 369 & 370 & 0.97~\\cite{Krukowski_jphyschemsolids_1998} & 2.17 & 2.18 \t \\\\\n & & & & & (268) & (232) & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:wurzite_thermal}\n\\end{table}\n\nThe experimental and calculated values of the Gr{\\\"u}neisen parameter are listed in Table~\\ref{tab:art115:wurzite_thermal}\nand in Table~\\ref{tab:art115:wurzite_thermal_eos}, and are plotted in Figure~\\ref{fig:art115:wurzite_thermal_elastic}(f).\nAgain, the Debye model does not reproduce the experimental data, as the calculated values\nare often 2 to 3 times too large and the {\\small RMSrD}\\ is larger than 1.5.\nThe corresponding correlation, shown in Table~\\ref{tab:art115:wurzite_correlation}, are also quite poor, with no value higher than 0.160 for\nthe Spearman correlations, and negative values for the Pearson correlations.\n\nThe comparison between the experimental thermal conductivity $\\kappa^{\\mathrm{exp}}$ and the calculated values is also quite poor\n(Figure~\\ref{fig:art115:wurzite_thermal_elastic}(d) and Table~\\ref{tab:art115:wurzite_thermal}), with {\\small RMSrD}\\ values of the order of 0.9.\nConsiderable disagreements also exist between different experimental reports for most materials.\nNevertheless, the Pearson correlations between the {\\small AGL}\\ calculated thermal conductivity values and the experimental\nvalues are high, ranging from $0.974$ to $0.980$, while the Spearman correlations are even higher, ranging from $0.976$\nto $1.0$.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for hexagonal structure semiconductors.}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.977 & 1.0 & 0.887 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.980 & 0.976 & 0.911 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.974 & 0.976 & 0.904 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.980 & 0.976 & 0.926 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.980 & 0.976 & 0.895 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.960 & 0.976 & 0.233 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.921 & 0.929 & 0.216 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.921 & 0.929 & 0.217 \\\\\n$\\theta_{\\mathrm{a}}{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.920 & 0.929 & 0.218 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.921 & 0.929 & 0.216 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & -0.039 & 0.160 & 1.566 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & -0.029 & 0.160 & 1.563 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & -0.124 & -0.233 & 1.547 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & -0.043 & 0.012 & 1.677 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ & -0.054 & 0.098 & 1.467 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.990 & 0.976 & 0.201 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.990 & 0.976 & 0.138 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.988 & 0.976 & 0.133 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.988 & 0.976 & 0.139 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.990 & 0.976 & 0.130 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 1.0 & 0.090 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 1.0 & 0.076 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 1.0 & 0.115 \\\\\n$\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.851 & 0.893 & 0.143 \\\\\n\\end{tabular}\n\\label{tab:art115:wurzite_correlation}\n\\end{table}\n\nAs for the rocksalt and zincblende material sets, Reference~\\onlinecite{Morelli_Slack_2006} (Table 2.3) includes\nvalues of the thermal conductivity at 300~K for wurtzite structure materials, calculated using the\nexperimental values of the Debye temperature and Gr{\\\"u}neisen parameter in the Leibfried-Schl{\\\"o}mann equation.\nThe Pearson and Spearman correlations are $0.996$ and $1.0$ respectively, which are slightly higher than the correlations obtained using\nthe {\\small AGL}\\ calculated quantities. The difference is insignificant since all of these\ncorrelations are very high and\ncould reliably serve as a screening tool of the thermal conductivity.\nHowever, as we noted in our previous work on {\\small AGL}~\\cite{curtarolo:art96}, the high correlations calculated with the\nexperimental $\\theta_{\\mathrm{a}}$ and $\\gamma$ were obtained using\n$\\gamma=0.75$ for BeO. Table 2.3 of\nReference~\\onlinecite{Morelli_Slack_2006} also cites an alternative value\nof $\\gamma=1.38$ for BeO (Table~\\ref{tab:art115:wurzite_thermal}). Using this outlier\nvalue would severely degrade the results down to $0.7$, for the\nPearson correlation, and $0.829$, for the Spearman correlation.\nThese values are too low for a reliable screening tool. This\ndemonstrates the ability of the\n{\\small AEL}-{\\small AGL}\\ calculations to compensate for anomalies in the\nexperimental data when\nthey exist and still provide a reliable screening method for the\nthermal conductivity.\n\n\\subsubsection{Rhombohedral materials}\n\nThe elastic properties of a few materials with rhombohedral structures\n(spacegroups: $R\\overline{3}mR$,\\ $\\#$166, $R\\overline{3}mH$,\\ $\\#$166; Pearson symbol: hR5; {\\small AFLOW}\\ prototype: {\\sf A2B3\\_hR5\\_166\\_c\\_ac}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/A2B3_hR5_166_c_ac.html}};\nand spacegroup: $R\\overline{3}cH$,\\ $\\#$167; Pearson symbol: hR10; {\\small AFLOW}\\ prototype: {\\sf A2B3\\_hR10\\_167\\_c\\_e}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/A2B3_hR10_167_c_e.html}})\nare shown in Table~\\ref{tab:art115:rhombo_elastic} (we have left out the material Fe$_2$O$_3$ which was included in\nthe data set in Table IV of Reference~\\onlinecite{curtarolo:art96}, due to convergence issues with some of the\nstrained structures required for the calculation of the elastic tensor).\nThe comparison between experiment and calculation is qualitatively reasonable, but the scarcity of experimental results\ndoes not allow for a proper correlation analysis.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Bulk modulus, shear modulus and Poisson ratio of rhombohedral semiconductors.]\n{``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ in {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ \\\\\n\\hline\nBi$_2$Te$_3$ & 37.0~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972} & 28.8 & 15.0 & 43.7 & 44.4 & 43.3 & 44.5 & 22.4~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972} & 23.5 & 16.3 & 19.9 & 10.9 & 0.248~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972} & 0.219 & 0.21 \\\\\nSb$_2$Te$_3$ & N\/A & 22.9 & N\/A & 45.3 & 46.0 & 45.2 & 46.0 & N\/A & 20.6 & 14.5 & 17.6 & N\/A & N\/A & 0.195 & N\/A \\\\\nAl$_2$O$_3$ & 254~\\cite{Goto_ElasticAl2O3_JGPR_1989} & 231 & 232 & 222 & 225 & 224 & 224 & 163.1~\\cite{Goto_ElasticAl2O3_JGPR_1989} & 149 & 144 & 147 & 147 & 0.235~\\cite{Goto_ElasticAl2O3_JGPR_1989} & 0.238 & 0.24 \\\\\nCr$_2$O$_3$ & 234~\\cite{Alberts_ElasticCr2O3_JMMM_1976} & 203 & 203 & 198 & 202 & 201 & 201 & 129~\\cite{Alberts_ElasticCr2O3_JMMM_1976} & 115 & 112 & 113 & 113 & 0.266~\\cite{Alberts_ElasticCr2O3_JMMM_1976} & 0.265 & 0.27 \\\\\nBi$_2$Se$_3$ & N\/A & 93.9 & N\/A & 57.0 & 57.5 & 56.4 & 57.9 & N\/A & 53.7 & 28.0 & 40.9 & N\/A & N\/A & 0.310 & N\/A \\\\\n\\end{tabular}}\n\\label{tab:art115:rhombo_elastic}\n\\end{table}\n\nThe thermal properties calculated using {\\small AGL}\\ are\ncompared to the experimental values in Table~\\ref{tab:art115:rhombo_thermal} and the thermal conductivity is also plotted in\nFigure~\\ref{fig:art115:mixed_thermal}(a).\nThe experimental Debye temperatures are $\\theta_{\\substack{\\scalebox{0.6}{D}}}$ for Bi$_2$Te$_3$ and Sb$_2$Te$_3$, and\n$\\theta_{\\mathrm{a}}$ for Al$_2$O$_3$. The values obtained using the numerical $E(V)$ fit and the three different equations of state\n(see Table~\\ref{tab:art115:rhombo_thermal_eos})\nare very similar, but just roughly reproduce the experiments.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity, Debye temperatures and Gr{\\\"u}neisen parameter of rhombohedral\nsemiconductors, comparing the effect of using the\ncalculated value of the Poisson ratio to the previous approximation of $\\sigma = 0.25$.]\n{The experimental Debye temperatures are $\\theta_{\\mathrm D}$ for\nBi$_2$Te$_3$ and Sb$_2$Te$_3$, and $\\theta_{\\mathrm{a}}$ for Al$_2$O$_3$.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ \\\\\n & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & & & & \\\\\n & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & \\\\\n\\hline\nBi$_2$Te$_3$ & 1.6~\\cite{Snyder_jmatchem_2011} & 2.79 & 3.35 & 155~\\cite{Snyder_jmatchem_2011} & 191 & 204 & 161 & 1.49~\\cite{Snyder_jmatchem_2011} & 2.13 & 2.14 \t \\\\\n & & & & & (112) & (119) & & & & \\\\\nSb$_2$Te$_3$ & 2.4~\\cite{Snyder_jmatchem_2011} & 2.90 & 4.46 & 160~\\cite{Snyder_jmatchem_2011} & 217 & 243 & 170 & 1.49~\\cite{Snyder_jmatchem_2011} & 2.2 & 2.11 \t\\\\\n & & & & & (127) & (142) & & & & \\\\\nAl$_2$O$_3$ & 30~\\cite{Slack_PR_1962} & 20.21 & 21.92 & 390~\\cite{slack} & 927 & 952 & 975 & 1.32~\\cite{slack} & 1.91 & 1.91 \t \\\\\n & & & & & (430) & (442) & & & & \\\\\nCr$_2$O$_3$ & 16~\\cite{Landolt-Bornstein, Bruce_PRB_1977} & 10.87 & 12.03 & N\/A & 733 & 717 & 720 & N\/A & 2.26 & 2.10 \t\\\\\n & & & & & (340) & (333) & & & & \\\\\nBi$_2$Se$_3$ & 1.34~\\cite{Landolt-Bornstein} & 3.60 & 2.41 & N\/A & 223 & 199 & 241 & N\/A & 2.08 & 2.12 \t\\\\\n & & & & & (130) & (116) & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:rhombo_thermal}\n\\end{table}\n\nThe calculated Gr{\\\"u}neisen parameters are about 50\\% larger than the experimental ones, and\nthe value of $\\sigma$ used makes a little difference in the calculation.\nThe absolute agreement between the {\\small AGL}\\ values and $\\kappa^{\\mathrm{exp}}$ is also quite poor (Figure~\\ref{fig:art115:mixed_thermal}(a)).\nHowever, despite all these discrepancies,\nthe Pearson correlations between the calculated thermal conductivities and the experimental\nvalues are all high, of the order of $0.998$, while the Spearman correlations range from $0.7$ to $1.0$,\nwith all of the different equations of state having very similar correlations with experiment.\nUsing the calculated $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$, \\nobreak\\mbox{\\it vs.}\\ the rough Cauchy approximation, improves the Spearman correlation from $0.7$ to $1.0$.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig032}\n\\mycaption{\n({\\bf a}) Lattice thermal conductivity of rhombohedral semiconductors at 300~K.\n({\\bf b}) Lattice thermal conductivity of body-centered tetragonal semiconductors at 300~K.\n}\n\\label{fig:art115:mixed_thermal}\n\\end{figure}\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for rhombohedral structure semiconductors.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.997 & 0.7 & 0.955 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.998 & 1.0 & 0.821 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.997 & 1.0 & 0.931 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.998 & 1.0 & 0.741 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.997 & 1.0 & 1.002 \\\\\n\\end{tabular}\n\\label{tab:art115:rhombo_correlation}\n\\end{table}\n\n\\subsubsection{Body-centered tetragonal materials}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Bulk modulus, shear modulus and Poisson ratio of body-centered tetragonal semiconductors.]\n{Note that there appears to be an error in Table 1 of Reference~\\onlinecite{Fernandez_ElasticCuInTe_PSSa_1990}\nwhere the bulk modulus values are stated to be in units of $10^{12}$ Pa.\nThis seems unlikely, as that would give a bulk modulus for CuInTe$_2$ an order of magnitude larger than\nthat for diamond.\nAlso, units of $10^{12}$ Pa would be inconsistent with the experimental results listed in\nReference~\\onlinecite{Neumann_ElasticCuInTe_PSSa_1986},\nso therefore it seems that these values are in units of\n$10^{10}$ Pa, which are the values shown here.\n``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ {in} {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ \\\\\n\\hline\nCuGaTe$_2$ & N\/A & 47.0 & N\/A & 42.5 & 43.2 & 42.0 & 43.5 & N\/A & 25.1 & 22.1 & 23.6 & N\/A & N\/A & 0.285 & N\/A \\\\\nZnGeP$_2$ & N\/A & 73.1 & 74.9 & 70.1 & 71.1 & 70.0 & 71.4 & N\/A & 50.5 & 46.2 & 48.4 & 48.9 & N\/A & 0.229 & 0.23 \\\\\nZnSiAs$_2$ & N\/A & 67.4 & 65.9 & 63.4 & 64.3 & 63.1 & 64.6 & N\/A & 44.4 & 40.4 & 42.4 & 42.2 & N\/A & 0.240 & 0.24 \\\\\nCuInTe$_2$ & 36.0~\\cite{Neumann_ElasticCuInTe_PSSa_1986} & 53.9 & N\/A & 38.6 & 39.2 & 38.2 & 39.4 & N\/A & 20.4 & 17.2 & 18.8 & N\/A & 0.313~\\cite{Fernandez_ElasticCuInTe_PSSa_1990} & 0.344 & N\/A \\\\\n & 45.4~\\cite{Fernandez_ElasticCuInTe_PSSa_1990} & & & & & & & & & & & & & \\\\\nAgGaS$_2$ & 67.0~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975} & 70.3 & N\/A & 56.2 & 57.1 & 56.0 & 57.4 & 20.8~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975} & 20.7 & 17.4 & 19.1 & N\/A & 0.359~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975} & 0.375 & N\/A \\\\\nCdGeP$_2$ & N\/A & 65.3 & 65.2 & 60.7 & 61.6 & 60.4 & 61.9 & N\/A & 37.7 & 33.3 & 35.5 & 35.0 & N\/A & 0.270 & 0.27 \\\\\nCdGeAs$_2$ & 69.9~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982} & 52.6 & N\/A & 49.2 & 49.6 & 48.3 & 49.9 & 29.5~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982} & 30.9 & 26.2 & 28.6 & N\/A & 0.315~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982} & 0.270 & N\/A \\\\\nCuGaS$_2$ & 94.0~\\cite{Bettini_ElasticCuGaS_SSC_1975} & 73.3 & N\/A & 69.0 & 69.9 & 68.7 & 70.6 & N\/A & 37.8 & 32.4 & 35.1 & N\/A & N\/A & 0.293 & N\/A \\\\\nCuGaSe$_2$ & N\/A & 69.9 & N\/A & 54.9 & 55.6 & 54.4 & 56.0 & N\/A & 30.3 & 26.0 & 28.1 & N\/A & N\/A & 0.322 & N\/A \\\\\nZnGeAs$_2$ & N\/A & 59.0 & N\/A & 56.2 & 56.7 & 55.5 & 57.1 & N\/A & 39.0 & 35.6 & 37.3 & N\/A & N\/A & 0.239 & N\/A \\\\\n\\end{tabular}}\n\\label{tab:art115:bct_elastic}\n\\end{table}\n\nThe mechanical properties of the body-centered tetragonal materials (spacegroup:\n$I\\overline{4}2d$,\\ $\\#$122; Pearson symbol: tI16; {\\small AFLOW}\\ prototype: {\\sf ABC2\\_tI16\\_122\\_a\\_b\\_d}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/ABC2_tI16_122_a_b_d.html}})\nof Table V of Reference~\\onlinecite{curtarolo:art96} are reported in Table~\\ref{tab:art115:bct_elastic}.\nThe calculated bulk moduli miss considerably the few available experimental results, while the shear moduli\nare well reproduced. Reasonable estimates are also obtained for the Poisson ratio.\n\nThe thermal properties are reported in Table~\\ref{tab:art115:bct_thermal} and Figure~\\ref{fig:art115:mixed_thermal}(b).\nThe $\\theta^{\\mathrm{exp}}$ values are all for $\\theta_{\\substack{\\scalebox{0.6}{D}}}$, and in most cases are in good agreement with the values obtained\nwith the {\\small AEL}\\ calculated $\\sigma$. The\nvalues from the numerical $E(V)$ fit and the three different equations of state are again very similar, but differ significantly\nfrom {$\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$} calculated with $\\sigma=0.25$.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity at 300~K, Debye temperatures and Gr{\\\"u}neisen parameter of body-centered tetragonal\nsemiconductors, comparing the effect of using the\ncalculated value of the Poisson ratio to the previous approximation of $\\sigma = 0.25$.]\n{``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ \\\\\n & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & & & & \\\\\n & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & \\\\\n\\hline\nCuGaTe$_2$ & 2.2~\\cite{Snyder_jmatchem_2011} & 1.77 & 1.36 & 226~\\cite{Snyder_jmatchem_2011} & 234 & 215 & 218 & 1.46~\\cite{Snyder_jmatchem_2011} & 2.32 & 2.32 \t \\\\\n & & & & & (117) & (108) & & & & \\\\\nZnGeP$_2$ & 35~\\cite{Landolt-Bornstein, Beasley_AO_1994} & 4.45 & 5.07 & 500~\\cite{Landolt-Bornstein} & 390 & 408 & 411 & N\/A & 2.13 & 2.14 \t \\\\\n & 36~\\cite{Landolt-Bornstein, Beasley_AO_1994} & & & 428~\\cite{Abrahams_JCP_1975} & (195) & (204) & & & & \\\\\n & 18~\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & & & & & & & & & \\\\\nZnSiAs$_2$ & 14\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & 3.70 & 3.96 & 347~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1981} & 342 & 350 & 354 & N\/A & 2.15 & 2.15 \t \\\\\n & & & & & (171) & (175) & & & & \\\\\nCuInTe$_2$ & 10\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 1.55 & 0.722 & 185~\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 215 & 166 & 185 & 0.93~\\cite{Rincon_PSSa_1995} & 2.33 & 2.32 \t \\\\\n & & & & 195~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1982} & (108) & (83) & & & &\\\\\nAgGaS$_2$ & 1.4\\cite{Landolt-Bornstein, Beasley_AO_1994} & 2.97 & 0.993 & 255~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 324 & 224 & 237 & N\/A & 2.20 & 2.20 \t\\\\\n & & & & & (162) & (112) & & & & \\\\\nCdGeP$_2$ & 11~\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & 3.40 & 2.96 & 340~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 335 & 320 & 324 & N\/A & 2.20 & 2.21 \\\\\n & & & & & (168) & (160) & & & & \\\\\nCdGeAs$_2$ & 42~\\cite{Landolt-Bornstein, Shay_1975} & 2.44 & 2.11 & 241~\\cite{Bohnhammel_PSSa_1981} & 266 & 254 & 255 & N\/A & 2.20 & 2.20 \t\\\\\n & & & & & (133) & (127) & & & &\\\\\nCuGaS$_2$ & 5.09~\\cite{Landolt-Bornstein} & 3.78 & 2.79 & 356~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 387 & 349 & 349 & N\/A & 2.24 & 2.24 \t \\\\\n & & & & & (194) & (175) & & & &\\\\\nCuGaSe$_2$ & 12.9~\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 2.54 & 1.46 & 262~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1982} & 294 & 244 & 265 & N\/A & 2.27 & 2.26 \t \\\\\n & & & & & (147) & (122) & & & &\\\\\nZnGeAs$_2$ & 11\\cite{Landolt-Bornstein, Shay_1975} & 2.95 & 3.18 & N\/A & 299 & 307 & 308 & N\/A & 2.16 & 2.17 \t \\\\\n & & & & & (150) & (154) & & & &\\\\\n\\end{tabular}}\n\\label{tab:art115:bct_thermal}\n\\end{table}\n\nThe comparison of the experimental thermal conductivity $\\kappa^{\\mathrm{exp}}$ to the calculated values, in Figure~\\ref{fig:art115:mixed_thermal}(b),\nshows poor reproducibility. The available data can thus only be considered a rough indication of their order of magnitude.\nThe Pearson and Spearman correlations are also quite low for all types of calculation,\nbut somewhat better when the calculated $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ is used instead of the Cauchy approximation.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for body-centered tetragonal structure semiconductors.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.265 & 0.201 & 0.812 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.472 & 0.608 & 0.766 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.467 & 0.608 & 0.750 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.464 & 0.608 & 0.778 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.460 & 0.608 & 0.741 \\\\\n\\end{tabular}\n\\label{tab:art115:bct_correlation}\n\\end{table}\n\n\\subsubsection{Miscellaneous materials}\n\nIn this Section we consider materials with various other structures, as in Table VI of Reference~\\onlinecite{curtarolo:art96}:\nCoSb$_3$ and IrSb$_3$\n(spacegroup: $Im\\overline{3}$,\\ $\\#$204; Pearson symbol: cI32; {\\small AFLOW}\\ prototype: {\\sf A3B\\_cI32\\_204\\_g\\_c}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/A3B_cI32_204_g_c.html}}),\nZnSb ($Pbca$,\\ $\\#$61; oP16; {\\small AFLOW}\\ prototype: {\\sf AB\\_oP16\\_61\\_c\\_c}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/AB_oP16_61_c_c.html}}),\nSb$_2$O$_3$ ($Pccn$,\\ $\\#$56; oP20), InTe ($Pm\\overline{3}m$,\\ $\\#$221; cP2; {\\small AFLOW}\\ prototype: {\\sf AB\\_cP2\\_221\\_b\\_a}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/AB_cP2_221_b_a.html}},\nand $I4\/mcm$,\\ $\\#$140; tI16), Bi$_2$O$_3$ ($P121\/c1,\\ \\#14$; mP20); and SnO$_2$ ($P42\/mnm,\\ \\#136$; tP6; {\\sf A2B\\_tP6\\_136\\_f\\_a}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/A2B_tP6_136_f_a.html}}).\nTwo different structures are listed for InTe. In Reference~\\onlinecite{curtarolo:art96}, we\nconsidered its simple cubic structure, but this is a high-pressure phase~\\cite{Chattopadhyay_BulkModInTe_JPCS_1985}, while the ambient\npressure phase is body-centered tetragonal. It appears that the thermal conductivity results should be for the body-centered tetragonal\nphase~\\cite{Spitzer_JPCS_1970}, therefore both sets of results are reported here. The correlation values shown in the tables below\nwere calculated for the body-centered tetragonal structure.\n\nThe elastic properties are shown\nin Table~\\ref{tab:art115:misc_elastic}. Large discrepancies appear between the results of all calculations\nand the few available experimental results.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Bulk modulus, shear modulus and Poisson ratio of materials with various\nstructures.]\n{``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ {in} {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & Pearson & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ \\\\\n\\hline\nCoSb$_3$ & $cI32$ & N\/A & 78.6 & 82.9 & 75.6 & 76.1 & 75.1 & 76.3 & N\/A & 57.2 & 55.1 & 56.2 & 57.0 & N\/A & 0.211 & 0.22 \\\\\nIrSb$_3$ & $cI32$ & N\/A & 97.5 & 98.7 & 94.3 & 94.8 & 93.8 & 95.5 & N\/A & 60.9 & 59.4 & 60.1 & 59.7& N\/A & 0.244 & 0.25 \\\\\nZnSb & $oP16$ & N\/A & 47.7 & 47.8 & 46.7 & 47.0 & 46.0 & 47.7 & N\/A & 29.2 & 27.0 & 28.1 & 28.2 & N\/A & 0.253 & 0.25 \\\\\nSb$_2$O$_3$ & $oP20$ & N\/A & 16.5 & 19.1 & 97.8 & 98.7 & 97.8 & 98.7 & N\/A & 22.8 & 16.4 & 19.6 & 20.4 & N\/A & 0.0749 & 0.11 \\\\\nInTe & $cP2$ & 90.2~\\cite{Chattopadhyay_BulkModInTe_JPCS_1985} & 41.7 & N\/A & 34.9 & 34.4 & 33.6 & 34.7 & N\/A & 8.41 & 8.31 & 8.36 & N\/A& N\/A & 0.406 & N\/A \\\\\nInTe & $tI16$ & 46.5~\\cite{Chattopadhyay_BulkModInTe_JPCS_1985} & 20.9 & N\/A & 32.3 & 33.1 & 32.2 & 33.2 & N\/A & 13.4 & 13.0 & 13.2 & N\/A & N\/A & 0.239 & N\/A \\\\\nBi$_2$O$_3$ & $mP20$ & N\/A & 48.0 & 54.5 & 108 & 110 & 109 & 109 & N\/A & 30.3 & 25.9 & 28.1 & 29.9 & N\/A & 0.255 & 0.27 \\\\\nSnO$_2$ & $tP6$ & 212~\\cite{Chang_ElasticSnO2_JGPR_1975} & 159 & N\/A & 158 & 162 & 161 & 161 & 106~\\cite{Chang_ElasticSnO2_JGPR_1975} & 86.7 & 65.7 & 76.2 & N\/A & 0.285~\\cite{Chang_ElasticSnO2_JGPR_1975} & 0.293 & N\/A \\\\\n\\end{tabular}}\n\\label{tab:art115:misc_elastic}\n\\end{table}\n\nThe thermal properties are\ncompared to the experimental values in Table~\\ref{tab:art115:misc_thermal}.\nThe experimental Debye temperatures are for $\\theta_{\\substack{\\scalebox{0.6}{D}}}$, except ZnSb for which it is $\\theta_{\\mathrm{a}}$. Good agreement\nis found between calculation and the few available experimental values. Again, the numerical $E(V)$ fit and the three different\nequations of state give similar results.\nFor the Gr{\\\"u}neisen parameter, experiment and calculations again differ considerably, while the changes due to the different\nvalues of $\\sigma$ used in the\ncalculations are negligible.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity at 300~K, Debye temperatures\nand Gr{\\\"u}neisen parameter of materials with various structures, comparing the effect of using the\ncalculated value of the Poisson ratio to the previous approximation of $\\sigma = 0.25$.]\n{The experimental Debye temperatures are $\\theta_{\\mathrm D}$,\nexcept ZnSb for which it is $\\theta_{\\mathrm{a}}$.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & Pearson & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ \\\\\n & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & & & & \\\\\n & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & \\\\\n\\hline\nCoSb$_3$ & $cI32$ & 10~\\cite{Snyder_jmatchem_2011} & 1.60 & 2.60 & 307~\\cite{Snyder_jmatchem_2011} & 284 & 310 & 312 & 0.95~\\cite{Snyder_jmatchem_2011} & 2.63 & 2.33 \\\\\n & & & & & & (113) & (123) & & & & \\\\\nIrSb$_3$ & $cI32$ & 16~\\cite{Snyder_jmatchem_2011} & 2.64 & 2.73 & 308~\\cite{Snyder_jmatchem_2011} & 283 & 286 & 286 & 1.42~\\cite{Snyder_jmatchem_2011} & 2.34 & 2.34 \\\\\n & & & & & & (112) & (113) & & & & \\\\\nZnSb & $oP16$ & 3.5~\\cite{Madsen_PRB_2014, Bottger_JEM_2010} & 1.24 & 1.23 & 92~\\cite{Madsen_PRB_2014} & 244 & 242 & 237 & 0.76~\\cite{Madsen_PRB_2014, Bottger_JEM_2010} & 2.24 & 2.23 \t \\\\\n & & & & & & (97) & (96) & & & & \\\\\nSb$_2$O$_3$ & $oP20$ & 0.4~\\cite{Landolt-Bornstein} & 3.45 & 8.74 & N\/A & 418 & 572 & 238 & N\/A & 2.13 & 2.12 \t\\\\\n & & & & & & (154) & (211) & & & & \\\\\nInTe & $cP2$ & N\/A & 3.12 & 0.709 & N\/A & 191 & 113 & 116 & N\/A & 2.28 & 2.19 \t\\\\\n & & & & & & (152) & (90) & & & & \\\\\nInTe & $tP16$ & 1.7~\\cite{Snyder_jmatchem_2011, Spitzer_JPCS_1970} & 1.32 & 1.40 & 186~\\cite{Snyder_jmatchem_2011} & 189 & 193 & 150 & 1.0~\\cite{Snyder_jmatchem_2011} & 2.23 & 2.24 \t\\\\\n & & & & & & (95) & (97) & & & & \\\\\nBi$_2$O$_3$ & $mP20$ & 0.8~\\cite{Landolt-Bornstein} & 3.04 & 2.98 & N\/A & 345 & 342 & 223 & N\/A & 2.10 & 2.10 \t \\\\\n & & & & & & (127) & (126) & & & & \\\\\nSnO$_2$ & $tP6$ & 98~\\cite{Turkes_jpcss_1980} & 9.56 & 6.98 & N\/A & 541 & 487 & 480 & N\/A & 2.48 & 2.42 \t \\\\\n & & 55~\\cite{Turkes_jpcss_1980} & & & & (298) & (268) & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:misc_thermal}\n\\end{table}\n\nThe experimental thermal conductivity $\\kappa^{\\mathrm{exp}}$ is compared in Table~\\ref{tab:art115:misc_thermal} to the thermal conductivity\ncalculated with {\\small AGL}\\ using the\nLeibfried-Schl{\\\"o}mann equation (Equation~\\ref{eq:art115:thermal_conductivity}) for $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$, while the values obtained for $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$, $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$\nand $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ are listed in Table~\\ref{tab:art115:misc_thermal_eos}.\nThe absolute agreement between the {\\small AGL}\\ values and $\\kappa^{\\mathrm{exp}}$ is quite poor.\nThe scarcity of\nexperimental data from different sources\non the thermal properties of these materials prevents reaching definite conclusions regarding the true values of these\nproperties. The available data can thus\nonly be considered as a rough indication of their order of magnitude.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for materials with miscellaneous structures.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.937 & 0.071 & 3.38 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.438 & -0.143 & 8.61 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.498 & -0.143 & 8.81 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.445 & 0.0 & 8.01 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.525 & -0.143 & 9.08 \\\\\n\\end{tabular}\n\\label{tab:art115:misc_correlation}\n\\end{table}\n\nFor these materials, the Pearson correlation between the calculated\nand experimental values of the thermal conductivity ranges from $0.438$ to $0.937$, while the corresponding\nSpearman correlations range from $-0.143$ to $0.071$. In this case, using $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ in the {\\small AGL}\\\ncalculations does not improve the correlations, instead actually lowering the values somewhat.\nHowever, it should be noted that the Pearson correlation is heavily influenced by the values for SnO$_2$.\nWhen this entry is removed from the list, the Pearson correlation values fall to $-0.471$ and $-0.466$\nwhen the $\\sigma = 0.25$ and $\\sigma = \\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ values are used, respectively.\nThe low correlation values, particularly for the Spearman correlation, for this set of materials demonstrates the\nimportance of the information about the material structure when interpreting results obtained using the {\\small AGL}\\ method\nin order to identify candidate materials for specific thermal applications. This is partly due to the fact that the Gr{\\\"u}neisen\nparameter values tend to be similar for materials with the same\nstructure. Therefore, the effect of the Gr{\\\"u}neisen parameter on the ordinal ranking of\nthe lattice thermal conductivity of materials with the same structure\nis small.\n\n\\subsubsection{Thermomechanical properties from LDA}\n\n{\nThe thermomechanical properties of a randomly-selected subset of the materials investigated in this work were calculated using {\\small LDA}\\\nin order to check the impact of the choice of exchange-correlation functional on the results. For the {\\small LDA}\\ calculations, all structures were\nfirst re-relaxed using the {\\small LDA}\\ exchange-correlation functional with {\\small VASP}\\ using the appropriate parameters and potentials as\ndescribed in the {\\small AFLOW}\\ standard~\\cite{curtarolo:art104}, and then the appropriate strained structures were calculated using {\\small LDA}.\nThese calculations were restricted to a subset of materials to limit the total number of additional first-principles calculations required, and the materials were\nselected randomly from each of the sets in the previous sections so as to cover as wide a range of different structure types as possible, given the available experimental data.\nResults for elastic properties obtained using {\\small LDA}, {\\small GGA}\\ and experimental measurements are shown in Table~\\ref{tab:art115:LDA_elastic}, while the thermal properties are shown in\nTable~\\ref{tab:art115:LDA_thermal}. All thermal properties listed in Table~\\ref{tab:art115:LDA_thermal} were calculated using $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ in the expression\nfor the Debye temperature.}\n\n{\nIn general, the {\\small LDA}\\ values for elastic and thermal properties are slightly higher than the {\\small GGA}\\ values, as would be generally expected\ndue to their relative tendencies to overbind and underbind, respectively~\\cite{He_GGA_LDA_PRB_2014, Saadaoui_GGA_LDA_EPJB_2015}.\nThe correlations and {\\small RMSrD}\\ of both the {\\small LDA}\\ and {\\small GGA}\\ results with experiment for this set of materials are listed in Table~\\ref{tab:art115:LDA_correlation}.\nThe Pearson and Spearman correlation values for {\\small LDA}\\ and {\\small GGA}\\ are very close to each other for most of the listed properties. The {\\small RMSrD}\\ values show\ngreater differences, although it isn't clear that one of the exchange-correlation functionals consistently gives better predictions than the other.\nTherefore, the choice of exchange-correlation functional will make little difference to the predictive capability of the workflow, so we choose to\nuse {\\small GGA}-{\\small PBE}\\ as it is the functional used for performing the structural relaxation for the entries in the {\\small AFLOW}\\ data repository.\n}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Bulk modulus, shear modulus and Poisson ratio of a subset of the materials investigated in this work,\ncomparing the effect of using different exchange-correlation functionals.]\n{``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ in {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{GGA}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{LDA}}}$ \\\\\n\\hline\nSi & 97.8~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 89.1& 96.9 & 84.2 & 92.1 & 66.5~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 64 & 65 & 61 & 61.9 & 62.5 & 63.4 & 0.223~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 0.216 & 0.231 \\\\\nBN & 367.0~\\cite{Lam_BulkMod_PRB_1987} & 372 & 402 & 353 & 382 & N\/A & 387 & 411 & 374 & 395 & 380 & 403 & N\/A & 0.119 & 0.124 \\\\\nGaSb & 57.0~\\cite{Lam_BulkMod_PRB_1987} & 47.0 & 58.3 & 41.6 & 52.3 & 34.2~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 30.8 & 35.3 & 28.3 & 32.2 & 29.6 & 33.7 & 0.248~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 0.240 & 0.258 \\\\\nInAs & 60.0~\\cite{Lam_BulkMod_PRB_1987} & 50.1 & 62.3 & 45.7 & 57.4 & 29.5~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 27.3 & 30.1 & 24.2 & 26.4 & 25.7 & 28.2 & 0.282~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 0.281 & 0.303 \\\\\nZnS & 77.1~\\cite{Lam_BulkMod_PRB_1987} & 71.2 & 88.4 & 65.8 & 83.3 & 30.9~\\cite{Semiconductors_BasicData_Springer} & 36.5 & 42.1 & 31.4 & 35.7 & 33.9 & 38.9 & 0.318~\\cite{Semiconductors_BasicData_Springer} & 0.294 & 0.308 \\\\\nNaCl & 25.1~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 24.9 & 33.3 & 20.0 & 27.6 & 14.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 14.0 & 19.8 & 12.9 & 16.6 & 13.5 & 18.2 & 0.255~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.271 & 0.269 \\\\\nKI & 12.2~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 10.9 & 16.3 & 8.54 & 13.3 & 5.96~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 6.05 & 9.39 & 4.39 & 5.3 & 5.22 & 7.35 & 0.290~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.294 & 0.305 \\\\\nRbI & 11.1~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 9.90 & 14.8 & 8.01 & 12.1 & 5.03~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 5.50 & 8.54 & 3.65 & 3.94 & 4.57 & 6.24 & 0.303~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.300 & 0.315 \\\\\nMgO & 164~\\cite{Sumino_ElasticMgO_JPE_1976} & 152 & 164 & 142 & 163 & 131~\\cite{Sumino_ElasticMgO_JPE_1976} & 119 & 138 & 115 & 136 & 117 & 137 & 0.185~\\cite{Sumino_ElasticMgO_JPE_1976} & 0.194 & 0.173 \\\\\nCaO & 113~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 105 & 129 & 99.6 & 122 & 81.0~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 73.7 & 87.4 & 73.7 & 86.3 & 73.7 & 86.9 & 0.210~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 0.216 & 0.225 \\\\\nGaN & 195~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978} & 175 & 202 & 166 & 196 & 51.6~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978} & 107 & 116 & 105 & 113 & 106 & 114 & 0.378~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978} & 0.248 & 0.262 \\\\\n & 210~\\cite{Polian_ElasticGaN_JAP_1996} & & & & & 123~\\cite{Polian_ElasticGaN_JAP_1996} & & & & & & & 0.255~\\cite{Polian_ElasticGaN_JAP_1996} & & \\\\\nCdS & 60.7~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 55.4 & 68.2 & 49.7 & 64.1 & 18.2~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 17.6 & 18.4 & 17.0 & 17.8 & 17.3 & 18.1 & 0.364~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 0.358 & 0.378 \\\\\nAl$_2$O$_3$ & 254~\\cite{Goto_ElasticAl2O3_JGPR_1989} & 231 & 259 & 222 & 250 & 163.1~\\cite{Goto_ElasticAl2O3_JGPR_1989} & 149 & 166 & 144 & 163 & 147 & 165 & 0.235~\\cite{Goto_ElasticAl2O3_JGPR_1989} & 0.238 & 0.238 \\\\\nCdGeP$_2$ & N\/A & 65.3 & 78.4 & 60.7 & 74.5 & N\/A & 37.7 & 42.1 & 33.3 & 36.8 & 35.5 & 39.4 & N\/A & 0.270 & 0.285 \\\\\nCuGaSe$_2$ & N\/A & 69.9 & 76.4 & 54.9 & 72.1 & N\/A & 30.3 & 34.7 & 26.0 & 30.0 & 28.1 & 32.3 & N\/A & 0.322 & 0.315 \\\\\nCoSb$_3$ & N\/A & 78.6 & 99.6 & 75.6 & 96.1 & N\/A & 57.2 & 67.1 & 55.1 & 64.2 & 56.2 & 65.7 & N\/A & 0.211 & 0.23 \\\\\n\\end{tabular}}\n\\label{tab:art115:LDA_elastic}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Thermal properties lattice thermal conductivity at\n300~K, Debye temperature and Gr{\\\"u}neisen parameter of\na subset of materials, comparing the effect of using different exchange-correlation functionals.]\n{The values listed for $\\theta^{\\mathrm{exp}}$ are $\\theta_{\\mathrm{a}}$, except 340~K for CdGeP$_2$~\\cite{Landolt-Bornstein, Abrahams_JCP_1975}, 262K for CuGaSe$_2$\n\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1982} and 307~K for CoSb$_3$~\\cite{Snyder_jmatchem_2011} which are $\\theta_{\\mathrm D}$.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{GGA}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{LDA}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{GGA}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{LDA}}}$ \\\\\n & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{GGA}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{LDA}}}$) & & \\\\\n\\hline\nSi & 166~\\cite{Morelli_Slack_2006} & 26.19 & 27.23 & 395~\\cite{slack, Morelli_Slack_2006} & 610 & 614 & 1.06~\\cite{Morelli_Slack_2006} & 2.06 & 2.03\t \\\\\n & & & & & (484) & (487) & 0.56~\\cite{slack} & \\\\\nBN & 760~\\cite{Morelli_Slack_2006} & 281.6 & 312.9 & 1200~\\cite{Morelli_Slack_2006} & 1793 & 1840 & 0.7~\\cite{Morelli_Slack_2006} & 1.75 & 1.72\t\\\\\n & & & & & (1423) & (1460) & & & \\\\\nGaSb & 40~\\cite{Morelli_Slack_2006} & 4.96 & 5.89 & 165~\\cite{slack, Morelli_Slack_2006} & 240 & 254 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.28 & 2.25 \t \\\\\n & & & & & (190) & (202) & & & \\\\\nInAs & 30~\\cite{Morelli_Slack_2006} & 4.33 & 4.92 & 165~\\cite{slack, Morelli_Slack_2006} & 229 & 238 & 0.57~\\cite{slack, Morelli_Slack_2006} & 2.26 & 2.22\t \\\\\n & & & & & (182) & (189) & & & \\\\\nZnS & 27~\\cite{Morelli_Slack_2006} & 8.38 & 9.58 & 230~\\cite{slack, Morelli_Slack_2006} & 341 & 363 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.00 & 2.02 \t \\\\\n & & & & & (271) & (288) & & & \\\\\nNaCl & 7.1~\\cite{Morelli_Slack_2006} & 2.12 & 2.92 & 220~\\cite{slack, Morelli_Slack_2006} & 271 & 312 & 1.56~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.29 \t \\\\\n & & & & & (215) & (248) & & & \\\\\nKI & 2.6~\\cite{Morelli_Slack_2006} & 0.525 & 0.811 & 87~\\cite{slack, Morelli_Slack_2006} & 116 & 137 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.35 & 2.37 \t \\\\\n & & & & & (92) & (109) & & & \\\\\nRbI & 2.3~\\cite{Morelli_Slack_2006} & 0.368 & 0.593 & 84~\\cite{slack, Morelli_Slack_2006} & 97 & 115 & 1.41~\\cite{slack, Morelli_Slack_2006} & 2.47 & 2.45 \t \\\\\n & & & & & (77) & (91) & & & \\\\\nMgO & 60~\\cite{Morelli_Slack_2006} & 44.5 & 58.4 & 600~\\cite{slack, Morelli_Slack_2006} & 849 & 935 & 1.44~\\cite{slack, Morelli_Slack_2006} & 1.96 & 1.95 \\\\\n & & & & & (674) & (742) & & & \\\\\nCaO & 27~\\cite{Morelli_Slack_2006} & 24.3 & 28.5 & 450~\\cite{slack, Morelli_Slack_2006} & 620 & 665 & 1.57~\\cite{slack, Morelli_Slack_2006} & 2.06 & 2.09 \t \\\\\n & & & & & (492) & (528) & & & \\\\\nGaN & 210~\\cite{Morelli_Slack_2006} & 18.54 & 21.34 & 390~\\cite{Morelli_Slack_2006} & 595 & 619 & 0.7~\\cite{Morelli_Slack_2006} & 2.08 & 2.04 \t \\\\\n & & & & & (375) & (390) & & & \\\\\nCdS & 16~\\cite{Morelli_Slack_2006} & 1.76 & 1.84 & 135~\\cite{Morelli_Slack_2006} & 211 & 217 & 0.75~\\cite{Morelli_Slack_2006} & 2.14 & 2.14 \t \\\\\n & & & & & (133) & (137) & & & \\\\\nAl$_2$O$_3$ & 30~\\cite{Slack_PR_1962} & 21.92 & 25.36 & 390~\\cite{slack} & 952 & 1002 & 1.32~\\cite{slack} & 1.91 & 1.91 \t \\\\\n & & & & & (442) & (465) & & & \\\\\nCdGeP$_2$ & 11~\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & 2.96 & 3.47 & 340~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 320 & 337 & N\/A & 2.21 & 2.18 \\\\\n & & & & & (160) & (169) & & & \\\\\nCuGaSe$_2$ & 12.9~\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 1.46 & 2.23 & 262~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1982} & 244 & 281 & N\/A & 2.26 & 2.23 \t \\\\\n & & & & & (122) & (141) & & & \\\\\nCoSb$_3$ & 10~\\cite{Snyder_jmatchem_2011} & 2.60 & 3.25 & 307~\\cite{Snyder_jmatchem_2011} & 310 & 332 & 0.95~\\cite{Snyder_jmatchem_2011} & 2.33 & 2.28 \\\\\n & & & & & (123) & (132) & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:LDA_thermal}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties comparing the {\\small LDA}\\ and {\\small GGA}\\ exchange-correlation functionals\nfor this subset of materials.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.963 & 0.867 & 0.755 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.959 & 0.848 & 0.706 \\\\\n$\\theta^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.996 & 0.996 & 0.119 \\\\\n$\\theta^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.996 & 0.996 & 0.174 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.172 & 0.130 & 1.514 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.265 & 0.296 & 1.490 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.995 & 1.0 & 0.111 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.996 & 1.0 & 0.185 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.996 & 1.0 & 0.205 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.998 & 1.0 & 0.072 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.999 & 0.993 & 0.108 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.997 & 0.986 & 0.153 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.998 & 0.993 & 0.096 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.996 & 0.986 & 0.315 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.999 & 0.993 & 0.163 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.997 & 0.993 & 0.111 \\\\\n$\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.982 & 0.986 & 0.037 \\\\\n$\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.983 & 0.993 & 0.052 \\\\\n\\end{tabular}\n\\label{tab:art115:LDA_correlation}\n\\end{table}\n\n\\subsubsection{AGL predictions for thermal conductivity}\n\nThe {\\small AEL}-{\\small AGL}\\ methodology has been applied for\nhigh-throughput screening of the elastic and thermal properties of\nover 3000 materials included in the {\\small AFLOW}\\ database~\\cite{aflowAPI}.\nTables~\\ref{tab:art115:highkappa} and \\ref{tab:art115:lowkappa} {list those} found\nto have the highest and lowest thermal conductivities, respectively.\nThe high conductivity list is unsurprisingly dominated by various phases of elemental\ncarbon{, boron nitride, boron carbide and boron carbon nitride,} while {all other}\nhigh-conductivity materials also contain at least one of the elements C, B or N.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Materials from {\\small AFLOW}\\ database with highest thermal conductivities as predicted using\nthe {\\small AEL}-{\\small AGL}\\ methodology.]\n{The {\\small AFLOW}\\ \\underline{u}nique \\underline{id}entifier ({\\small AUID}) is a permanent, server-independent identifier for each entry in the {\\small AFLOW}\\ database~\\cite{aflowAPI}.\nThis identifier allows any of these entries to be retrieved from the repository, and ensures the retrievability and reproducibility of the data\nirrespective of changes in the underlying database structure or hosting location.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r|r}\ncomp. & Pearson & space group \\# & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & {\\small AUID} \\\\\n\\hline\nC & cF8 & 227 & 420 & 3ab7e139e1c29c9f \\\\\nBN & cF8 & 216 & 282 & fd5539a4f79db51c \\\\\nC & hP4 & 194 & 272 & 440c4eee274b61b6 \\\\\nC & tI8 & 139 & 206 & b2688e84030188b8 \\\\\nBC$_2$N & oP4 & 25 & 188 & c0e7523ff8d34297 \\\\\nBN & hP4 & 186 & 178 & 56d00a95d21b5c3a \\\\\nC & hP8 & 194 & 167 & c42dc8ec018245e5 \\\\\nC & cI16 & 206 & 162 & c969067f8a3bbde9 \\\\\nC & oS16 & 65 & 147 & bdc82cca41c811c6 \\\\\nC & mS16 & 12 & 145 & a59baaad49eb5ab9 \\\\\nBC$_7$ & tP8 & 115 & 145 & 0401731cb29df494 \\\\\nBC$_5$ & oI12 & 44 & 137 & f759c5600121a9e9 \\\\\nBe$_2$C & cF12 & 225 & 129 & 378e092c24555651 \\\\\nCN$_2$ & tI6 & 119 & 127 & 6852d98ddee59417 \\\\\nC & hP12 & 194 & 127 &\tbd79f9fa8154aa95 \\\\\nBC$_7$ & oP8 & 25 & 125 & 4d13f06b9fe563ef \\\\\nB$_2$C$_4$N$_2$ & oP8 & 17 & 120 & 9e325d34d65bd890\\\\\nMnB$_2$ & hP3 & 191 & 117 & 0e5997687be5d3dc \\\\\nC & hP4 & 194 & 117 & 2be120d88682ee01 \\\\\nSiC & cF8 & 216 & 113 & 2cab0c35952c733f \\\\\nTiB$_2$ & hP3 & 191 & 110 & 32d72b1701a0a640 \\\\\nAlN & cF8 & 225 & 107 & 06c4f5b0f1a49096 \\\\\nBP & cF8 & 216 & 105 & 598a7a7328a47d85 \\\\\nC & hP16 & 194 & 105 &\tc9d6a8b917d502f0 \\\\\nVN & hP2 & 187 & 101 &\taa89372868af03a8 \\\\\n\\end{tabular}\n\\label{tab:art115:highkappa}\n\\end{table}\n\nThe low thermal conductivity list tends to contain materials\nwith large unit cells and heavier elements such as Hg, Tl, Pb and Au.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Materials from {\\small AFLOW}\\ database with lowest thermal conductivities as predicted using\nthe {\\small AEL}-{\\small AGL}\\ methodology.]\n{The {\\small AFLOW}\\ \\underline{u}nique \\underline{id}entifier ({\\small AUID}) is a permanent, server-independent identifier for each entry in the {\\small AFLOW}\\ database~\\cite{aflowAPI}.\nThis identifier allows any of these entries to be retrieved from the repository, and ensures the retrievability and reproducibility of the data\nirrespective of changes in the underlying database structure or hosting location.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r|r}\ncomp. & Pearson & space group \\# & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & {\\small AUID} \\\\\n\\hline\nHg$_{33}$Rb$_3$ & cP36 & 221 & 0.0113 & 3a84e674e05ac4e6 \\\\\nHg$_{33}$K$_3$ & cP36 & 221 & 0.0116 & ac7610d35123f5c5 \\\\\nCs$_6$Hg$_{40}$ & cP46 & 223 & 0.0136 & 978182b72d30a019 \\\\\nCa$_{16}$Hg$_{36}$ & cP52 & 215 & 0.0751 & fe8eeb1e2af8df90 \\\\\nCrTe & cF8 & 216 & 0.081 & 53c8683bd5648144 \\\\\nHg$_4$K$_2$ & oI12 & 74 & 0.086 & 50b2883feb14cd6e \\\\\nSb$_6$Tl$_{21}$ & cI54 & 229 & 0.089 & f7933008a130dc06 \\\\\nSe & cF24 & 227 & 0.093 & 7d6a2e6c742211e5 \\\\\nCs$_8$I$_{24}$Sn$_4$ & cF36 & 225 & 0.104 & 460691dc51cf5b5d \\\\\nAg$_2$Cr$_4$Te$_8$ & cF56 & 227 & 0.107 & a30bbe2831fa8a18 \\\\\nAsCdLi & cF12 & 216 &\t0.116 & f818510c8952b114 \\\\\nAu$_{36}$In$_{16}$ & cP52 & 215 & 0.117 & bda82cdcf87fa384 \\\\\nCd$_3$In & cP4 & 221 & 0.128 & 3bc3fc68c58fdd1f \\\\\nAuLiSb & cF12 & 216 & 0.130 & bdab7ec2c162ee22 \\\\\nK$_5$Pb$_{24}$ & cI58 & 217 & 0.135 & 58f4471901eff079 \\\\\nK$_8$Sn$_{46}$ & cP54 & 223 & 0.142 & 6b4795df74caacfc \\\\\nAu$_7$Cd$_{16}$Na$_6$ & cF116 & 225 & 0.145 & ec21f32abca24cbd \\\\\nCs & cI2 & 229 & 0.148 & 5acbf212d1783298 \\\\\nCs$_8$Pb$_4$Cl$_{24}$ & cF36 & 225 & 0.157 & 84738cad161f83b3 \\\\\nAu$_{4}$In$_8$Na$_{12}$ & cF96 & 227 & 0.158 & 0393c62d375f5ec6\\\\\nSeTl & cP2 & 221 & 0.164 & 5ebc0f014499d22b \\\\\nCd$_{33}$Na$_6$ & cP39 & 200 & 0.166 & 0e4a5c866567f309 \\\\\nAu$_{18}$In$_{15}$Na$_6$ & cP39 & 200 & 0.168 & f7355e2e7474fb1c \\\\\nCd$_{26}$Cs$_2$ & cF112 & 226 & 0.173 & cfe1448550ccd1d1 \\\\\nAg$_2$I$_2$ & hP4 & 186 & 0.192 & d611e813a85efcb0 \\\\\n\\end{tabular}\n\\label{tab:art115:lowkappa}\n\\end{table}\n\nBy combining the {\\small AFLOW}\\ search for thermal conductivity values with other properties such as chemical, electronic or structural factors,\ncandidate materials for specific engineering applications can be rapidly identified for further in-depth analysis using more accurate\ncomputational methods and for experimental examination. {The full set of thermomechanical properties calculated using\n{\\small AEL}-{\\small AGL}\\ for over 3500 entries can be accessed online at {\\sf \\AFLOW.org}~\\cite{aflowlib.org}, which incorporates search and sort functionality to\ngenerate customized lists of materials.}\n\n\\subsubsection{Results for different equations of state}\n\nThis section includes the results for the thermal conductivity, Debye temperature and the Gr{\\\"u}neisen parameter for the set of 74 materials listed in this work as\ncalculated using the Birch-Murnaghan~\\cite{Birch_Elastic_JAP_1938, Poirier_Earth_Interior_2000, Blanco_CPC_GIBBS_2004}, Vinet~\\cite{Vinet_EoS_JPCM_1989, Blanco_CPC_GIBBS_2004},\nand Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez~\\cite{Baonza_EoS_PRB_1995, Blanco_CPC_GIBBS_2004} equations of state. The experimental values for the lattice thermal conductivity\n$\\kappa^{\\mathrm{exp}}$ are compared to the {\\small AGL}\\ values obtained using the numerical polynomial fit $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$, and the three empirical equations of state:\nBirch-Murnaghan, $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$; Vinet, $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$; and Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez, $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$. The experimental values for the Debye temperature\n$\\theta^{\\mathrm{exp}}$ are compared to the {\\small AGL}\\ values obtained using the numerical polynomial fit $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$, and the three empirical equations of state:\nBirch-Murnaghan, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$; Vinet, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$; and Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$. The {\\small AGL}\\ values listed are for\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}$, while the values for $\\theta_{\\mathrm{a}}$ are listed underneath in parentheses. The experimental values for the Gr{\\\"u}neisen parameter\n$\\gamma^{\\mathrm{exp}}$ are compared to the {\\small AGL}\\ values obtained using the numerical polynomial fit $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$, and the three empirical equations of state:\nBirch-Murnaghan, $\\gamma_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$; Vinet, $\\gamma_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$; and Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez, $\\gamma_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$. The results for the\ndiamond and zincblende structure set of materials are listed in Table~\\ref{tab:art115:zincblende_thermal_eos}, the results for the rocksalt structure set of materials\nare listed in Table~\\ref{tab:art115:rocksalt_thermal_eos}, the results for the hexagonal structure set of materials are listed in Table~\\ref{tab:art115:wurzite_thermal_eos}, the results\nfor the rhombohedral structure set of materials are listed in Table~\\ref{tab:art115:rhombo_thermal_eos}, the results for the body-centered tetragonal structure set of materials\nare listed in Table~\\ref{tab:art115:bct_thermal_eos}, and the results for the miscellaneous structure materials are listed in Table~\\ref{tab:art115:misc_thermal_eos}.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Thermal conductivities, Debye temperatures and Gr{\\\"u}neisen parameters of\nzincblende and diamond structure semiconductors, calculated using the different equations of state.]\n{The values listed for $\\theta^{\\mathrm{exp}}$ are\n$\\theta_{\\mathrm{a}}$, except 141K for HgTe which is $\\theta_{\\mathrm D}$~\\cite{Snyder_jmatchem_2011}.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ \\\\\n& & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$) & & & & & \\\\\n\\hline\nC & 3000~\\cite{Morelli_Slack_2006} & 419.9 & 298.5 & 307.0 & 466.8 & 1450~\\cite{slack, Morelli_Slack_2006} & 2094 & 2051 & 2056 & 2103 & 0.75~\\cite{Morelli_Slack_2006} & 1.77 & 2.01 &\t1.99 & 1.69 \\\\\n & & & & & & & (1662) & (1628) & (1632) & (1669) & 0.9~\\cite{slack} & & & & \\\\\nSiC & 360~\\cite{Ioffe_Inst_DB} & 113.0 & 120.7 & 101.3 & 125.4 & 740~\\cite{slack} & 1106 & 1108 & 1100 & 1110 & 0.76~\\cite{slack} & 1.85 & 1.80 & 1.93 & 1.78 \t\\\\\n& & & & & & & (878) & (879) & (873) & (881) & & & & & \\\\\nSi & 166~\\cite{Morelli_Slack_2006} & 26.19 & 28.61 & 26.23 & 31.39 & 395~\\cite{slack, Morelli_Slack_2006} & 610 & 611 & 609 & 614 & 1.06~\\cite{Morelli_Slack_2006} & 2.06 & 1.99 & 2.06 & 1.92\t \\\\\n & & & & & & & (484) & (485) & (483) & (487) & 0.56~\\cite{slack} & & & & \\\\\nGe & 65~\\cite{Morelli_Slack_2006} & 8.74 & 9.61 & 8.54 & 10.12 & 235~\\cite{slack, Morelli_Slack_2006} & 329 & 330 & 329 & 331 & 1.06~\\cite{Morelli_Slack_2006} & 2.31 & 2.22 & 2.34 & 2.18\t \\\\\n& & & & & & & (261) & (262) & (261) & (263) & 0.76~\\cite{slack} & & & & \\\\\nBN & 760~\\cite{Morelli_Slack_2006} & 281.6 & 243.1 & 220.5 & 303.4 & 1200~\\cite{Morelli_Slack_2006} & 1793 & 1777 & 1769 & 1798 & 0.7~\\cite{Morelli_Slack_2006} & 1.75 & 1.85 & 1.92 & 1.70\t\\\\\n& & & & & & & (1423) & (1410) & (1404) & (1427) & & & & & \\\\\nBP & 350~\\cite{Morelli_Slack_2006} & 105.0 & 108.8 & 89.95 & 117.8 & 670~\\cite{slack, Morelli_Slack_2006} & 1025 & 1025 & 1016 & 1029 & 0.75~\\cite{Morelli_Slack_2006} & 1.79 & 1.76 & 1.90 & 1.71 \t\\\\\n& & & & & & & (814) & (814) & (806) & (817) & & & & & \\\\\nAlP & 90~\\cite{Landolt-Bornstein, Spitzer_JPCS_1970} & 19.34 & 20.48 & 18.79 & 22.49 & 381~\\cite{Morelli_Slack_2006} & 525 & 526 & 524 & 528 & 0.75~\\cite{Morelli_Slack_2006} & 1.96 & 1.92 & 1.98 & 1.84 \t \\\\\n& & & & & & & (417) & (417) & (416) & (419) & & & & & \\\\\nAlAs & 98~\\cite{Morelli_Slack_2006} & 11.64 & 12.84 & 11.59 & 13.64 & 270~\\cite{slack, Morelli_Slack_2006} & 373 & 374 & 373 & 375 & 0.66~\\cite{slack, Morelli_Slack_2006} & 2.04 & 1.96 & 2.04 & 1.91 \t \\\\\n& & & & & & & (296) & (297) & (296) & (298) & & & & & \\\\\nAlSb & 56~\\cite{Morelli_Slack_2006} & 6.83 & 7.84 & 6.85 & 8.34 & 210~\\cite{slack, Morelli_Slack_2006} & 276 & 277 & 276 & 278 & 0.6~\\cite{slack, Morelli_Slack_2006} & 2.13 & 2.01 & 2.12 & 1.96\t \\\\\n& & & & & & & (219) & (220) & (219) & (221) & & & & & \\\\\nGaP & 100~\\cite{Morelli_Slack_2006} & 13.34 & 15.09 & 13.49 & 15.74 & 275~\\cite{slack, Morelli_Slack_2006} & 412 & 414 & 412 & 414 & 0.75~\\cite{Morelli_Slack_2006} & 2.15 & 2.04 & 2.14 & 2.0\t\\\\\n & & & & & & & (327) & (329) & (327) & (329) & 0.76~\\cite{slack} & & & & \\\\\nGaAs & 45~\\cite{Morelli_Slack_2006} & 8.0 & 8.95 & 7.85 & 9.30 & 220~\\cite{slack, Morelli_Slack_2006} & 313 & 315 & 313 & 315 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.24 & 2.15 & 2.26 & 2.11\t \\\\\n& & & & & & & (248) & (250) & (248) & (250) & & & & & \\\\\nGaSb & 40~\\cite{Morelli_Slack_2006} & 4.96 & 5.49 & 4.68 & 5.69 & 165~\\cite{slack, Morelli_Slack_2006} & 240 & 241 & 239 & 241 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.28 & 2.19 & 2.33 & 2.15 \\\\\n& & & & & & & (190) & (191) & (190) & (191) & & & & & \\\\\nInP & 93~\\cite{Morelli_Slack_2006} & 6.53 & 7.40 & 6.57 & 7.71 & 220~\\cite{slack, Morelli_Slack_2006} & 286 & 287 & 286 & 287 & 0.6~\\cite{slack, Morelli_Slack_2006} & 2.21 & 2.1 & 2.2 & 2.06 \t \\\\\n& & & & & & & (227) & (228) & (227) & (228) & & & & & \\\\\nInAs & 30~\\cite{Morelli_Slack_2006} & 4.33 & 4.80 & 4.20 & 4.93 & 165~\\cite{slack, Morelli_Slack_2006} & 229 & 230 & 229 & 230 & 0.57~\\cite{slack, Morelli_Slack_2006} & 2.26 & 2.17 & 2.29 & 2.14\t \\\\\n& & & & & & & (182) & (183) & (182) & (183) & & & & & \\\\\nInSb & 20~\\cite{Morelli_Slack_2006} & 3.02 & 3.33 & 2.76 & 3.44 & 135~\\cite{slack, Morelli_Slack_2006} & 187 & 188 & 186 & 188 & 0.56~\\cite{slack, Morelli_Slack_2006} & 2.3 & 2.22 & 2.38 & 2.18\t \\\\\n & & & & & & & (148) & (149) & (148) & (149) & 16.5~\\cite{Snyder_jmatchem_2011} & & & & \\\\\nZnS & 27~\\cite{Morelli_Slack_2006} & 8.38 & 8.40 & 7.67 & 8.96 & 230~\\cite{slack, Morelli_Slack_2006} & 341 & 341 & 340 & 342 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.0 & 1.99 & 2.07 & 1.94\t \\\\\n& & & & & & & (271) & (271) & (270) & (271) & & & & & \\\\\nZnSe & 19~\\cite{Morelli_Slack_2006} & 5.44 & 5.55 & 4.93 & 5.80 & 190~\\cite{slack, Morelli_Slack_2006} & 260\t& 260 & 259 & 261 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.06 & 2.04 & 2.14 & 2.01\t\\\\\n & 33~\\cite{Snyder_jmatchem_2011} & & & & & & (206) & (206) & (206) & (207) & & & & & \\\\\nZnTe & 18~\\cite{Morelli_Slack_2006} & 3.83 & 3.95 & 3.44 & 4.10 & 155~\\cite{slack, Morelli_Slack_2006} & 210 & 210 & 209 & 210 & 0.97~\\cite{slack, Morelli_Slack_2006} & 2.13 & 2.1 & 2.23 & 2.07 \\\\\n& & & & & & & (167) & (167) & (166) & (167) & & & & & \\\\\nCdSe & 4.4~\\cite{Snyder_jmatchem_2011} & 2.04 & 2.11 & 1.84 & 2.16 & 130~\\cite{Morelli_Slack_2006} & 173 & 173 & 172 & 173 & 0.6~\\cite{Morelli_Slack_2006} & 2.18 & 2.15 & 2.27 & 2.12 \\\\\n& & & & & & & (137) & (137) & (137) & (137) & & & & & \\\\\nCdTe & 7.5~\\cite{Morelli_Slack_2006} & 1.71 & 1.77 & 1.50 & 1.81 & 120~\\cite{slack, Morelli_Slack_2006} & 150 & 150 & 149 & 150 & 0.52~\\cite{slack, Morelli_Slack_2006} & 2.22 & 2.19 & 2.34 & 2.16\t \\\\\n& & & & & & & (119) & (119) & (118) & (119) & & & & & \\\\\nHgSe & 3~\\cite{Whitsett_PRB_1973} & 1.32 & 1.36 & 1.22 & 1.41 & 110~\\cite{slack} & 140\t& 140 & 140 & 140\t& 0.17~\\cite{slack} & 2.38 & 2.35 & 2.47 & 2.31 \\\\\n& & & & & & & (111) & (111) & (111) & (111) & & & & & \\\\\nHgTe & 2.5~\\cite{Snyder_jmatchem_2011} & 1.21 & 1.30 & 1.10 & 1.34 & 141~\\cite{Snyder_jmatchem_2011} & 129\t& 130 & 129 & 130\t& 1.9~\\cite{Snyder_jmatchem_2011} & 2.45 & 2.40 & 2.56 & 2.36 \\\\\n & & & & & & (100)~\\cite{slack} & (102) & (103) & (102) & (103) & 0.46\\cite{slack} & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:zincblende_thermal_eos}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Thermal properties lattice thermal conductivity at 300~K, Debye temperature and Gr{\\\"u}neisen parameter of rocksalt\nstructure semiconductors, calculated using the different equations of state.]\n{The values listed for $\\theta^{\\mathrm{exp}}$ are\n$\\theta_{\\mathrm{a}}$, except 155K for SnTe which is $\\theta_{\\mathrm D}$~\\cite{Snyder_jmatchem_2011}.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ \\\\\n& & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$) & & & & & \\\\\n\\hline\nLiH & 15~\\cite{Morelli_Slack_2006} & 18.6 & 13.54 & 12.37 & 20.98 & 615~\\cite{slack, Morelli_Slack_2006} & 962 & 931 & 927 & 968 & 1.28~\\cite{slack, Morelli_Slack_2006} & 1.66 & 1.84 & 1.90 & 1.58 \\\\\n& & & & & & & (764) & (739) & (734) & (768) & & & & & \\\\\nLiF & 17.6~\\cite{Morelli_Slack_2006} & 9.96 & 10.19 & 8.68 & 11.45 & 500~\\cite{slack, Morelli_Slack_2006} & 617 &\t617 & 610 & 623 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.03 & 2.00 & 2.13 & 1.92\t \\\\\n& & & & & & & (490) & (490) & (485) & (494) & & & & & \\\\\nNaF & 18.4~\\cite{Morelli_Slack_2006} & 4.67 & 4.65 & 3.82 & 4.91 & 395~\\cite{slack, Morelli_Slack_2006} & 416 & 416 & 411 & 417 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.21 & 2.21 & 2.39 & 2.16\t \\\\\n& & & & & & & (330) & (330) & (326) & (331) & & & & & \\\\\nNaCl & 7.1~\\cite{Morelli_Slack_2006} & 2.12 & 2.27 & 1.74 & 2.28 & 220~\\cite{slack, Morelli_Slack_2006} & 271 & 273 & 268 & 272 & 1.56~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.18 & 2.40 & 2.16\t \\\\\n& & & & & & & (215) & (217) & (213) & (216) & & & & & \\\\\nNaBr & 2.8~\\cite{Morelli_Slack_2006} & 1.33 & 1.42 & 1.08 & 1.40 & 150~\\cite{slack, Morelli_Slack_2006} & 188 & 189 & 186 & 188 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.22 & 2.17 & 2.40 & 2.16 \t \\\\\n& & & & & & & (149) & (150) & (148) & (149) & & & & & \\\\\nNaI & 1.8~\\cite{Morelli_Slack_2006} & 0.851 & 0.922 & 0.679 & 0.892 & 100~\\cite{slack, Morelli_Slack_2006} & 140 & 141\t& 138 & 140 & 1.56~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.17 & 2.43 & 2.18\t \\\\\n& & & & & & & (111) & (112) & (110) & (111) & & & & & \\\\\nKF & N\/A & 2.21 & 2.07 & 1.62 & 2.22 & 235~\\cite{slack, Morelli_Slack_2006} & 288 &\t287\t& 281 & 288 & 1.52~\\cite{slack, Morelli_Slack_2006} & 2.32 & 2.38 & 2.60 & 2.32 \t\\\\\n& & & & & & & (229) & (228) & (224) & (229) & & & & & \\\\\nKCl & 7.1~\\cite{Morelli_Slack_2006} & 1.25 & 1.42 & 1.04 & 1.40 & 172~\\cite{slack, Morelli_Slack_2006} & 213 & 215 & 210 & 214 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.40 & 2.29 & 2.57 & 2.29 \t \\\\\n& & & & & & & (169) & (171) & (167) & (170) & & & & & \\\\\nKBr & 3.4~\\cite{Morelli_Slack_2006} & 0.842 & 0.949 & 0.682 & 0.928 & 117~\\cite{slack, Morelli_Slack_2006} & 156 & 157 & 153 & 156 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.37 & 2.26 & 2.55 & 2.27 \\\\\n& & & & & & & (124) & (125) & (121) & (124) & & & & & \\\\\nKI & 2.6~\\cite{Morelli_Slack_2006} & 0.525 & 0.624 & 0.451 & 0.603 & 87~\\cite{slack, Morelli_Slack_2006} & 116 & 118 & 115 & 117 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.35 & 2.23 & 2.50 & 2.23\t \\\\\n& & & & & & & (92) & (94) & (91) & (93) & & & & & \\\\\nRbCl & 2.8~\\cite{Morelli_Slack_2006} & 0.837 & 0.886 & 0.638 & 0.878 & 124~\\cite{slack, Morelli_Slack_2006} & 155 & 156 & 152 & 155 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.37 & 2.33 & 2.62 & 2.32\t \\\\\n& & & & & & & (123) & (124) & (121) & (123) & & & & & \\\\\nRbBr & 3.8~\\cite{Morelli_Slack_2006} & 0.558 & 0.606 & 0.459 & 0.606 & 105~\\cite{slack, Morelli_Slack_2006} & 122 & 123 & 121 & 123 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.43 & 2.36 & 2.62 & 2.36 \\\\\n& & & & & & & (97) & (98) & (96) & (98) & & & & & \\\\\nRbI & 2.3~\\cite{Morelli_Slack_2006} & 0.368 & 0.434 & 0.320 & 0.415 & 84~\\cite{slack, Morelli_Slack_2006} & 97 & 98 & 96 & 97 & 1.41~\\cite{slack, Morelli_Slack_2006} & 2.47 & 2.32 & 2.60 & 2.34 \t \\\\\n& & & & & & & (77) & (78) & (76) & (77) & & & & & \\\\\nAgCl & 1.0~\\cite{Landolt-Bornstein, Maqsood_IJT_2003} & 0.613 & 0.612 & 0.535 & 0.663 & 124~\\cite{slack} & 145 & 145 & 144 & 146 & 1.9~\\cite{slack} & 2.49 & 2.49 & 2.63 & 2.43 \t \\\\\n& & & & & & & (115) & (115) & (114) & (116) & & & & & \\\\\nMgO & 60~\\cite{Morelli_Slack_2006} & 44.5 & 44.7 & 38.5 & 47.1 & 600~\\cite{slack, Morelli_Slack_2006} & 849 & 848 & 842 & 851\t& 1.44~\\cite{slack, Morelli_Slack_2006} & 1.96 & 1.95 & 2.07 & 1.91 \\\\\n& & & & & & & (674) & (673) & (668) & (675) & & & & & \\\\\nCaO & 27~\\cite{Morelli_Slack_2006} & 24.3 & 24.7 & 22.5 & 25.7 & 450~\\cite{slack, Morelli_Slack_2006} & 620 & 620 & 618 & 621 & 1.57~\\cite{slack, Morelli_Slack_2006} & 2.06 & 2.05 & 2.13 & 2.02\t \\\\\n& & & & & & & (492) & (492) & (491) & (493) & & & & & \\\\\nSrO & 12~\\cite{Morelli_Slack_2006} & 13.4 & 13.3 & 12.2 & 14.0 & 270~\\cite{slack, Morelli_Slack_2006} & 413 & 413 & 412 & 414 & 1.52~\\cite{slack, Morelli_Slack_2006} & 2.13 & 2.13 & 2.21\t& 2.09 \t \\\\\n& & & & & & & (328) & (328) & (327) & (329) & & & & & \\\\\nBaO & 2.3~\\cite{Morelli_Slack_2006} & 7.10 & 6.73 & 6.10 & 6.98 & 183~\\cite{slack, Morelli_Slack_2006} & 288 & 288 & 287 & 288 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.14 & 2.20 & 2.29 & 2.16 \\\\\n& & & & & & & (229) & (229) & (228) & (229) & & & & & \\\\\nPbS & 2.9~\\cite{Morelli_Slack_2006} & 6.11 & 6.77 & 5.99 & 7.02 & 115~\\cite{slack, Morelli_Slack_2006} & 220 & 221 & 220 & 221 & 2.0~\\cite{slack, Morelli_Slack_2006} & 2.00 & 1.92 & 2.02 & 1.89\t\\\\\n& & & & & & & (175) & (175) & (175) & (175) & & & & & \\\\\nPbSe & 2.0~\\cite{Morelli_Slack_2006} & 4.81 & 5.29 & 4.63 & 5.44 & 100~\\cite{Morelli_Slack_2006} & 194 & 195 & 194 & 195 & 1.5~\\cite{Morelli_Slack_2006} & 2.07 & 2.00 & 2.11 & 1.97\t \\\\\n& & & & & & & (154) & (155) & (154) & (155) & & & & & \\\\\nPbTe & 2.5~\\cite{Morelli_Slack_2006} & 4.07 & 4.11 & 3.50 & 4.32 & 105~\\cite{slack, Morelli_Slack_2006} & 172 & 172 & 171 & 173 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.09 & 2.08 & 2.22 & 2.05 \t \\\\\n& & & & & & & (137) & (137) & (136) & (137) & & & & & \\\\\nSnTe & 1.5~\\cite{Snyder_jmatchem_2011} & 5.24 & 5.59 & 4.64 & 5.78 & 155~\\cite{Snyder_jmatchem_2011} & 210 & 211 & 209 & 211 & 2.1~\\cite{Snyder_jmatchem_2011} & 2.11 & 2.06 & 2.22 & 2.03 \t \\\\\n& & & & & & & (167) & (167) & (166) & (167) & & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:rocksalt_thermal_eos}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity, Debye temperature and Gr{\\\"u}neisen parameter of hexagonal\nstructure semiconductors, calculated using the different equations of state.]\n{The values listed for $\\theta^{\\mathrm{exp}}$ are\n$\\theta_{\\mathrm{a}}$, except 190K for InSe~\\cite{Snyder_jmatchem_2011} and 660K for InN~\\cite{Ioffe_Inst_DB, Krukowski_jphyschemsolids_1998}\nwhich are $\\theta_{\\mathrm D}$.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ \\\\\n& & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$) & & & & & \\\\\n\\hline\nSiC & 490~\\cite{Morelli_Slack_2006} & 70.36 & 75.17 & 62.86 & 77.82 & 740~\\cite{Morelli_Slack_2006} & 1103 & 1105 & 1096 & 1106 & 0.75~\\cite{Morelli_Slack_2006} & 1.86 & 1.80 & 1.94 & 1.78 \\\\\n& & & & & & & (695) & (696) & (690) & (697) & & & & & \\\\\nAlN & 350~\\cite{Morelli_Slack_2006} & 39.0 & 40.53 & 34.49 & 42.3 & 620~\\cite{Morelli_Slack_2006} & 898 & 899 & 893 & 900 & 0.7~\\cite{Morelli_Slack_2006} & 1.85 & 1.82 & 1.95 & 1.79 \t \\\\\n& & & & & & & (566) & (566) & (563) & (567) & & & & & \\\\\nGaN & 210~\\cite{Morelli_Slack_2006} & 18.53 & 16.33 & 16.21 & 20.15 & 390~\\cite{Morelli_Slack_2006} & 595 & 590 & 591 & 596 & 0.7~\\cite{Morelli_Slack_2006} & 2.08 & 2.18 & 2.19 & 2.01\t \\\\\n& & & & & & & (375) & (372) & (372) & (375) & & & & & \\\\\nZnO & 60~\\cite{Morelli_Slack_2006} & 7.39 & 7.72 & 6.80 & 8.06 & 303~\\cite{Morelli_Slack_2006} & 422 & 422 & 420 & 423 & 0.75~\\cite{Morelli_Slack_2006} & 1.94 & 1.91 & 2.01 & 1.87 \t \\\\\n& & & & & & & (266) & (266) & (265) & (266) & & & & & \\\\\nBeO & 370~\\cite{Morelli_Slack_2006} & 53.36 & 54.41 & 46.97 & 56.95 & 809~\\cite{Morelli_Slack_2006} & 1181 & 1182 & 1173 & 1184 & 1.38~\\cite{Slack_JAP_1975, Cline_JAP_1967, Morelli_Slack_2006} & 1.76 & 1.74 & 1.85 & 1.71\t \\\\\n& & & & & & & (744) & (745) & (739) & (746) & & & & & \\\\\nCdS & 16~\\cite{Morelli_Slack_2006} & 1.76 & 1.89 & 1.66 & 1.93 & 135~\\cite{Morelli_Slack_2006} & 211 & 212 & 211 & 212 & 0.75~\\cite{Morelli_Slack_2006} & 2.14 & 2.08 & 2.19 & 2.06\t \\\\\n& & & & & & & (133) & (134) & (133) & (134) & & & & & \\\\\nInSe & 6.9~\\cite{Snyder_jmatchem_2011} & 2.34 & 2.61 & 2.23 & 2.69 & 190~\\cite{Snyder_jmatchem_2011} & 249 & 250 & 248\t& 250 & 1.2~\\cite{Snyder_jmatchem_2011} & 2.24 & 2.14 & 2.28 & 2.11 \t \\\\\n& & & & & & & (125) & (125) & (124) & (125) & & & & & \\\\\nInN & 45~\\cite{Ioffe_Inst_DB, Krukowski_jphyschemsolids_1998} & 6.82 & 6.97 & 5.59 & 7.49 & 660~\\cite{Ioffe_Inst_DB, Krukowski_jphyschemsolids_1998} & 369 & 369 & 365 & 370 & 0.97~\\cite{Krukowski_jphyschemsolids_1998} & 2.18 & 2.15 & 2.35 & 2.09 \t \\\\\n& & & & & & & (232) & (232) & (230) & (233) & & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:wurzite_thermal_eos}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity, Debye temperatures and Gr{\\\"u}neisen parameter of rhombohedral\nsemiconductors, calculated using the different equations of state.]\n{The experimental Debye temperatures are $\\theta_{\\mathrm D}$ for\nBi$_2$Te$_3$ and Sb$_2$Te$_3$, and $\\theta_{\\mathrm{a}}$ for Al$_2$O$_3$.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ \\\\\n& & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$) & & & & & \\\\\n\\hline\nBi$_2$Te$_3$ & 1.6~\\cite{Snyder_jmatchem_2011} & 3.35 & 3.63 & 3.17 & 3.73 & 155~\\cite{Snyder_jmatchem_2011} & 204 & 205 & 204 & 205 & 1.49~\\cite{Snyder_jmatchem_2011} & 2.14 & 2.08 & 2.20 & 2.05\t \\\\\n& & & & & & & (119) & (120) & (119) & (120) & & & & & \\\\\nSb$_2$Te$_3$ & 2.4~\\cite{Snyder_jmatchem_2011} & 4.46 & 4.76 & 4.07 & 4.99 & 160~\\cite{Snyder_jmatchem_2011} & 243 & 244 & 242 & 244 & 1.49~\\cite{Snyder_jmatchem_2011} & 2.11 & 2.06 & 2.19 & 2.02\t\\\\\n& & & & & & & (142) & (143) & (142) & (143) & & & & & \\\\\nAl$_2$O$_3$ & 30~\\cite{Slack_PR_1962} & 21.92 & 23.36 & 19.51 & 23.19 & 390~\\cite{slack} & 952 & 954 & 947 & 954 & 1.32~\\cite{slack} & 1.91 & 1.86 & 2.00 & 1.87\t \\\\\n& & & & & & & (442) & (443) & (440) & (443) & & & & & \\\\\nCr$_2$O$_3$ & 16~\\cite{Landolt-Bornstein, Bruce_PRB_1977} & 12.03 & 12.61 & 10.78 & 12.92 & N\/A & 718 & 717 & 713 & 718 & N\/A & 2.10 & 2.05 & 2.19 & 2.04 \t\\\\\n& & & & & & & (333) & (333) & (331) & (333) & & & & & \\\\\nBi$_2$Se$_3$ & 1.34~\\cite{Landolt-Bornstein} & 2.41 & 2.54 & 2.31 & 2.68 & N\/A & 199 & 199 & 199 & 200 & N\/A & 2.12 & 2.07 & 2.16 & 2.03 \t\\\\\n& & & & & & & (116) & (116) & (116) & (117) & & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:rhombo_thermal_eos}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity at 300~K, Debye temperatures and Gr{\\\"u}neisen parameter of body-centered tetragonal\nsemiconductors, calculated using the different equations of state.]\n{``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ \\\\\n& & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$) & & & & & \\\\\n\\hline\nCuGaTe$_2$ & 2.2~\\cite{Snyder_jmatchem_2011} & 1.36 & 1.49 & 1.30 & 1.53 & 226~\\cite{Snyder_jmatchem_2011} & 215 & 216 & 215 & 216 & 1.46~\\cite{Snyder_jmatchem_2011} & 2.32 & 2.23 & 2.36 & 2.21 \t \\\\\n& & & & & & & (108) & (108) & (108) & (108) & & & & &\\\\\nZnGeP$_2$ & 35~\\cite{Landolt-Bornstein, Beasley_AO_1994} & 5.07 & 5.54 & 4.95 & 5.73 & 500~\\cite{Landolt-Bornstein} & 408 & 410 & 408 & 410 & N\/A & 2.14 & 2.07 & 2.17 & 2.04 \t \\\\\n& 36~\\cite{Landolt-Bornstein, Beasley_AO_1994} & & & & & & (204) & (205) & (204) & (205) & & & & & \\\\\n& 18~\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & & & & & & & & & & & & & & \\\\\nZnSiAs$_2$ & 14\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & 3.96 & 4.19 & 3.76 & 4.43 & 347~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1981} & 350 & 350 & 349 & 351 & N\/A & 2.15 & 2.10 & 2.20 & 2.05\t \\\\\n& & & & & & & (175) & (175) & (175) & (176) & & & & &\\\\\nCuInTe$_2$ & 10\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 0.722 & 0.797 & 0.693 & 0.812 & 185~\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 166 & 167\t& 166 & 167 & 0.93~\\cite{Rincon_PSSa_1995} & 2.32 & 2.23 & 2.36 & 2.21 \t \\\\\n& & & & & & 195~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1982} & (83) & (84) & (83) & (84) & & & & &\\\\\nAgGaS$_2$ & 1.4\\cite{Landolt-Bornstein, Beasley_AO_1994} & 0.993 & 1.04 & 0.92 & 1.08 & 255~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 224 & 224 & 223 & 224 & N\/A & 2.20 & 2.14 & 2.26 & 2.11\t\\\\\n& & & & & & & (112) & (112) & (112) & (112) & & & & &\\\\\nCdGeP$_2$ & 11~\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & 2.96 & 3.18 & 2.85 & 3.31 & 340~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 320 & 321 & 320 & 321 & N\/A & 2.21 & 2.14 & 2.25 & 2.10 \t \\\\\n& & & & & & & (160) & (161) & (160) & (161) & & & & &\\\\\nCdGeAs$_2$ & 42~\\cite{Landolt-Bornstein, Shay_1975} & 2.11 & 2.17 & 1.92 & 2.24 & N\/A & 254 & 254 & 253 & 254 & N\/A & 2.20 & 2.17 & 2.29 & 2.14 \t\\\\\n& & & & & & & (127) & (127) & (127) & (127) & & & & &\\\\\nCuGaS$_2$ & 5.09~\\cite{Landolt-Bornstein} & 2.79 & 2.99 & 2.67 & 3.11 & 356~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 349 & 350 & 348 & 350 & N\/A & 2.24 & 2.18 & 2.28 & 2.14 \t \\\\\n& & & & & & & (175) & (175) & (174) & (175) & & & & &\\\\\nCuGaSe$_2$ & 12.9~\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 1.46 & 1.53 & 1.37 & 1.61 & 262~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1982} & 244 & 244 & 243 & 245 & N\/A & 2.26 & 2.21 & 2.32 & 2.17\t \\\\\n& & & & & & & (122) & (122) & (122) & (123) & & & & &\\\\\nZnGeAs$_2$ & 11\\cite{Landolt-Bornstein, Shay_1975} & 3.18 & 3.29 & 2.93 & 3.45 & N\/A & 307 & 307 & 306 & 308 & N\/A & 2.17 & 2.13 & 2.24 & 2.10\t \\\\\n& & & & & & & (154) & (154) & (153) & (154) & & & & &\\\\\n\\end{tabular}}\n\\label{tab:art115:bct_thermal_eos}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity at 300~K, Debye temperatures and Gr{\\\"u}neisen parameter of materials with various\nstructures, calculated using the different equations of state.]\n{The experimental Debye temperatures are $\\theta_{\\mathrm D}$,\nexcept ZnSb for which it is $\\theta_{\\mathrm{a}}$.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & Pearson & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ \\\\\n& & & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$) & & & & & \\\\\n\\hline\nCoSb$_3$ & cI32 & 10~\\cite{Snyder_jmatchem_2011} & 2.60 & 2.58 & 2.38 & 2.78 & 307~\\cite{Snyder_jmatchem_2011} & 310 & 310 & 309 & 311 & 0.95~\\cite{Snyder_jmatchem_2011} & 2.33 & 2.33 & 2.42 & 2.27 \\\\\n& & & & & & & & (123) & (123) & (123) & (123) & & & & \\\\\nIrSb$_3$ & cI32 & 16~\\cite{Snyder_jmatchem_2011} & 2.73 & 2.89 & 2.67 & 3.01 & 308~\\cite{Snyder_jmatchem_2011} & 286 & 287 & 286 & 287 & 1.42~\\cite{Snyder_jmatchem_2011} & 2.34 & 2.29 & 2.37 & 2.25 \\\\\n& & & & & & & & (113) & (114) & (113) & (114) & & & & \\\\\nZnSb & oP16 & 3.5~\\cite{Madsen_PRB_2014, Bottger_JEM_2010} & 1.23 & 1.29 & 1.13 & 1.36 & 92~\\cite{Madsen_PRB_2014} & 242 & 242 & 241 & 243 & 0.76~\\cite{Madsen_PRB_2014, Bottger_JEM_2010} & 2.23 & 2.18 & 2.30 & 2.14 \t \\\\\n& & & & & & & & (96) & (96) & (96) & (96) & & & & \\\\\nSb$_2$O$_3$ & oP20 & 0.4~\\cite{Landolt-Bornstein} & 8.74 & 8.93 & 8.18 & 9.20 & N\/A & 572 & 573 & 571 & 573 & N\/A & 2.12 & 2.10 & 2.18 & 2.07\t\\\\\n& & & & & & & & (211) & (211) & (210) & (211) & & & & \\\\\nInTe & cP2 & N\/A & 0.709 & 0.602 & 0.524 & 0.626 & N\/A & 113 & 112 & 111 & 112 & N\/A & 2.19 & 2.33 & 2.45 & 2.29 \\\\\n& & & & & & & & (90) & (89) & (88) & (89) & & & & \\\\\nInTe & tP16 & 1.7~\\cite{Snyder_jmatchem_2011} & 1.40 & 1.53 & 1.27 & 1.55 & 186~\\cite{Snyder_jmatchem_2011} & 193 & 194 & 192 & 194 & 1.0~\\cite{Snyder_jmatchem_2011} & 2.24 & 2.16 & 2.32 & 2.14\t\\\\\n& & & & & & & & (97) & (97) & (96) & (97) & & & & \\\\\nBi$_2$O$_3$ & mP20 & 0.8~\\cite{Landolt-Bornstein} & 2.98 & 3.05 & 2.49 & 3.14 & N\/A & 342 & 342 & 339 & 342 & N\/A & 2.10 & 2.08 & 2.26 & 2.05\t \\\\\n& & & & & & & & (126) & (126) & (125) & (126) & & & & \\\\\nSnO$_2$ & tP6 & 98\\cite{Turkes_jpcss_1980} & 6.98 & 7.76 & 6.52 & 8.31 & N\/A & 487 & 489 & 485 & 490 & N\/A & 2.42 & 2.32 & 2.48 & 2.25\t \\\\\n& & 55~\\cite{Turkes_jpcss_1980} & & & & & & (268) & (269) & (267) & (270) & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:misc_thermal_eos}\n\\end{table}\n\n\\subsubsection{Elastic constant values}\n\nThe elastic constant values in the 6x6 Voigt notation are shown for zincblende and diamond structure materials in Table~\\ref{tab:art115:zincblende_elastic_supp}, for rocksalt structure materials in Table~\\ref{tab:art115:rocksalt_elastic_supp}, for hexagonal structure materials in Table~\\ref{tab:art115:wurzite_elastic_supp}, for rhombohedral structure materials in Table~\\ref{tab:art115:rhombo_elastic_supp}, for body-centered tetragonal\nternary materials in Table~\\ref{tab:art115:bct_elastic_supp}, for body-centered cubic and simple cubic materials in Table~\\ref{tab:art115:bcc_elastic}, for orthorhombic structures in Table~\\ref{tab:art115:orc_elastic}, and for tetragonal\nstructure materials in Table~\\ref{tab:art115:tet_elastic}.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$ and $c_{44}$ of\nzincblende and diamond structure semiconductors.]\n{The zincblende structure is designated {\\small AFLOW}\\ prototype {\\sf AB\\_cF8\\_216\\_c\\_a}~\\cite{aflowANRL}\nand the diamond structure {\\sf A\\_cF8\\_227\\_a}~\\cite{aflowANRL}.\n``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r}\ncomp. & $c_{11}^{\\mathrm{exp}}$ & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\mathrm{exp}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\mathrm{exp}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n\\hline\nC & 1076.4~\\cite{Semiconductors_BasicData_Springer} & 1048 & 125.2~\\cite{Semiconductors_BasicData_Springer} & 127 & 577.4~\\cite{Semiconductors_BasicData_Springer} & 560 \\\\\nSiC & 352.3~\\cite{Semiconductors_BasicData_Springer} & 384 & 140.4~\\cite{Semiconductors_BasicData_Springer} & 127 & 232.9~\\cite{Semiconductors_BasicData_Springer} & 240 \\\\\nSi & 165.64~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 153 & 63.94~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 57.1 & 79.51~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 74.6 \\\\\nGe & 129.9~\\cite{Semiconductors_BasicData_Springer, Bruner_ElasticGe_PRL_1961} & 107 & 48.73~\\cite{Semiconductors_BasicData_Springer, Bruner_ElasticGe_PRL_1961} & 38.8 & 68.0~\\cite{Semiconductors_BasicData_Springer, Bruner_ElasticGe_PRL_1961} & 56.7 \\\\\nBN & N\/A & 777 & N\/A & 170 & N\/A & 442 \\\\\nBP & 315.0~\\cite{Semiconductors_BasicData_Springer, Wettling_ElasticBP_SSC_1984} & 339 & 100~\\cite{Semiconductors_BasicData_Springer, Wettling_ElasticBP_SSC_1984} & 73.3 & 160~\\cite{Semiconductors_BasicData_Springer, Wettling_ElasticBP_SSC_1984} & 185 \\\\\nAlP & N\/A & 125 & N\/A & 61.6 & N\/A & 59.7 \\\\\nAlAs & N\/A & 104 & N\/A & 49.3 & N\/A & 50.4 \\\\\nAlSb & 87.69~\\cite{Semiconductors_BasicData_Springer, Bolef_ElasticAlSb_JAP_1960, Weil_ElasticAlSb_JAP_1972} & 76.3 & 43.41~\\cite{Semiconductors_BasicData_Springer, Bolef_ElasticAlSb_JAP_1960, Weil_ElasticAlSb_JAP_1972} & 36.0 & 40.76~\\cite{Semiconductors_BasicData_Springer, Bolef_ElasticAlSb_JAP_1960, Weil_ElasticAlSb_JAP_1972} & 36.0 \\\\\nGaP & 141.4~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 127 & 63.98~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 54.9 & 70.28~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 65.2 \\\\\nGaAs & 188.8~\\cite{Bateman_ElasticGaAs_JAP_1975} & 101 & 53.8~\\cite{Bateman_ElasticGaAs_JAP_1975} & 43.7 & 59.4~\\cite{Bateman_ElasticGaAs_JAP_1975} & 51.9 \\\\\nGaSb & 88.34~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 74.6 & 40.23~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 33.2 & 43.22~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 37.6 \\\\\nInP & 101.1~\\cite{Nichols_ElasticInP_SSC_1980} & 87.7 & 56.1~\\cite{Nichols_ElasticInP_SSC_1980} & 46.7 & 45.6~\\cite{Nichols_ElasticInP_SSC_1980} & 42.3 \\\\\nInAs & 83.29~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 72.4 & 45.26~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 38.9 & 39.59~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 34.3 \\\\\nInSb & 66.0~\\cite{DeVaux_ElasticInSb_PR_1956} & 55.8 & 38.0~\\cite{DeVaux_ElasticInSb_PR_1956} & 29.3 & 30.0~\\cite{DeVaux_ElasticInSb_PR_1956} & 26.7 \\\\\nZnS & 98.1~\\cite{Semiconductors_BasicData_Springer} & 99.2 & 62.7~\\cite{Semiconductors_BasicData_Springer} & 57.2 & 44.83~\\cite{Semiconductors_BasicData_Springer} & 46.9 \\\\\nZnSe & 85.9~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 81.4 & 50.6~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 46.6 & 40.6~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 37.5 \\\\\nZnTe & 71.1~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 63.2 & 40.7~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 34.1 & 31.3~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 29.2 \\\\\nCdSe & N\/A & 57.7 & N\/A & 41.1 & N\/A & 21.5 \\\\\nCdTe & N\/A & 46.7 & N\/A & 31.2 & N\/A & 18.5 \\\\\nHgSe & 59.5~\\cite{Lehoczky_ElasticHgSe_PR_1969} & 53.3 & 43.07~\\cite{Lehoczky_ElasticHgSe_PR_1969} & 39.0 & 22.015~\\cite{Lehoczky_ElasticHgSe_PR_1969} & 21.2 \\\\\nHgTe & 53.61~\\cite{Semiconductors_BasicData_Springer, Cottam_ElasticHgTe_JPCS_1975} & 45.0 & 36.6~\\cite{Semiconductors_BasicData_Springer, Cottam_ElasticHgTe_JPCS_1975} & 30.4 & 21.23~\\cite{Semiconductors_BasicData_Springer, Cottam_ElasticHgTe_JPCS_1975} & 19.2 \\\\\n\\end{tabular}}\n\\label{tab:art115:zincblende_elastic_supp}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$ and $c_{44}$ of\nrocksalt structure semiconductors.]\n{The rocksalt structure is designated {\\small AFLOW}\\ Prototype {\\sf AB\\_cF8\\_225\\_a\\_b}~\\cite{aflowANRL}.\n``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r|r|r|r}\ncomp. & $c_{11}^{\\mathrm{exp}}$ & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\mathrm{exp}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\mathrm{exp}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n\\hline\nLiH & 67.1~\\cite{Laplaze_ElasticLiH_SSC_1976} & 84.8 & 17.0~\\cite{Laplaze_ElasticLiH_SSC_1976} & 14.2 & 46.0~\\cite{Laplaze_ElasticLiH_SSC_1976} & 48.8 \\\\\nLiF & 113.55~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 124 & 47.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 43.7 & 63.5~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 50.6 \\\\\nNaF & 97.0~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 96.1 & 24.3~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 22.3 & 28.1~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 24.6 \\\\\nNaCl & 49.36~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 50.5 & 12.9~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 12.1 & 12.65~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 10.6 \\\\\nNaBr & 40.12~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 41.2 & 10.9~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 10.2 & 9.9~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 7.97 \\\\\nNaI & 30.25~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 32.7 & 8.8~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 8.3 & 7.4~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 5.77 \\\\\nKF & 65.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 59.3 & 14.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 15.3 & 12.5~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 12.8 \\\\\nKCl & 40.78~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 37.2 & 6.9~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 6.39 & 6.33~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 6.55 \\\\\nKBr & 34.76~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 31.3 & 5.7~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 5.1 & 5.07~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 4.83 \\\\\nKI & 27.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 24.8 & 4.5~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 4.02 & 3.7~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 3.17 \\\\\nRbCl & 36.34~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 31.6 & 6.15~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 5.68 & 4.65~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 4.8 \\\\\nRbBr & 31.57~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 28.7 & 4.95~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 4.5 & 3.8~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 3.8 \\\\\nRbI & 25.83~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 23.1 & 3.7~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 3.3 & 2.78~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 2.57 \\\\\nAgCl & 59.6~\\cite{Hughes_ElasticAgCl_PRB_1996} & 52.7 & 36.2~\\cite{Hughes_ElasticAgCl_PRB_1996} & 34.6 & 6.21~\\cite{Hughes_ElasticAgCl_PRB_1996} & 8.4 \\\\\nMgO & 297.8~\\cite{Sumino_ElasticMgO_JPE_1976} & 276 & 97.0~\\cite{Sumino_ElasticMgO_JPE_1976} & 90.7 & 156.3~\\cite{Sumino_ElasticMgO_JPE_1976} & 137 \\\\\nCaO & 221.89~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 202 & 57.81~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 57.0 & 80.32~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 74.6 \\\\\nSrO & 175.47~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 161 & 49.08~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 46.7 & 55.87~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 53.8 \\\\\nBaO & 126.14~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 118 & 50.03~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 44.8 & 33.68~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 36.4 \\\\\nPbS & 126.15~\\cite{Semiconductors_BasicData_Springer, Peresada_ElasticPbS_PSSa_1976} & 127 & 16.24~\\cite{Semiconductors_BasicData_Springer, Peresada_ElasticPbS_PSSa_1976} & 16.9 & 17.09~\\cite{Semiconductors_BasicData_Springer, Peresada_ElasticPbS_PSSa_1976} & 20.0 \\\\\nPbSe & 123.7~\\cite{Semiconductors_BasicData_Springer, Lippmann_ElasticPbSe_PSSa_1971} & 119 & 19.3~\\cite{Semiconductors_BasicData_Springer, Lippmann_ElasticPbSe_PSSa_1971} & 12.2 & 15.91~\\cite{Semiconductors_BasicData_Springer, Lippmann_ElasticPbSe_PSSa_1971} & 17.2 \\\\\nPbTe & 105.3~\\cite{Semiconductors_BasicData_Springer, Miller_ElasticPbTe_JPCSS_1981} & 107 & 7.0~\\cite{Semiconductors_BasicData_Springer, Miller_ElasticPbTe_JPCSS_1981} & 5.63 & 13.22~\\cite{Semiconductors_BasicData_Springer, Miller_ElasticPbTe_JPCSS_1981} & 14.1 \\\\\nSnTe & 109.3~\\cite{Semiconductors_BasicData_Springer, Seddon_ElasticSnTe_SSC_1976} & 114 & 2.1~\\cite{Semiconductors_BasicData_Springer, Seddon_ElasticSnTe_SSC_1976} & 3.72 & 9.69~\\cite{Semiconductors_BasicData_Springer, Seddon_ElasticSnTe_SSC_1976} & 15.7 \\\\\n\\end{tabular}\n\\label{tab:art115:rocksalt_elastic_supp}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$, $c_{13}$, $c_{33}$, $c_{44}$ and $c_{66}$ of hexagonal structure semiconductors.]\n{Experimental values, where available, are shown in parentheses underneath the calculated values.\n``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ in {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r}\ncomp. & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{13}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{33}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{66}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n& ($c_{11}^{\\mathrm{exp}}$) & ($c_{12}^{\\mathrm{exp}}$) & ($c_{13}^{\\mathrm{exp}}$) & ($c_{33}^{\\mathrm{exp}}$) & ($c_{44}^{\\mathrm{exp}}$) & ($c_{66}^{\\mathrm{exp}}$) \\\\\n\\hline\nSiC & 494 & 102 & 48.7 & 534 & 151 & 196 \\\\\n& (500~\\cite{Arlt_ELasticSiC_JAAcS_1965}) & (92~\\cite{Arlt_ELasticSiC_JAAcS_1965}) & (55.8~\\cite{Arlt_ELasticSiC_JAAcS_1965}) & (564~\\cite{Arlt_ELasticSiC_JAAcS_1965}) & (168~\\cite{Arlt_ELasticSiC_JAAcS_1965}) & (204~\\cite{Arlt_ELasticSiC_JAAcS_1965}) \\\\\nAlN & 377 & 123 & 97.7 & 356 & 113 & 124 \\\\\n& (410.5~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993}) & (148.5~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993}) & (98.9~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993}) & (388.5~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993}) & (124.6~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993}) & (131.0~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993}) \\\\\nGaN & 329 & 115 & 80.5 & 362 & 90.3 & 109 \\\\\n& (296~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978}) & (130.0~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978}) & (158.0~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978}) & (267~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978}) & (24.0~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978}) & (83.0~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978}) \\\\\n& (390~\\cite{Polian_ElasticGaN_JAP_1996}) & (145.0~\\cite{Polian_ElasticGaN_JAP_1996}) & (106.0~\\cite{Polian_ElasticGaN_JAP_1996}) & (398~\\cite{Polian_ElasticGaN_JAP_1996}) & (105.0~\\cite{Polian_ElasticGaN_JAP_1996}) & (123.0~\\cite{Polian_ElasticGaN_JAP_1996}) \\\\\nZnO & 210 & 109 & 93.2 & 220 & 46.4 & 51.4 \\\\\n & (207~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (117.7~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (106.1~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (209.5~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (44.8~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (44.6~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) \\\\\nBeO & 427 & 110 & 79.4 & 464 & 138 & 158 \\\\\n & (460.6~\\cite{Cline_JAP_1967}) & (126.5~\\cite{Cline_JAP_1967}) & (88.48~\\cite{Cline_JAP_1967}) & (491.6~\\cite{Cline_JAP_1967}) & (147.7~\\cite{Cline_JAP_1967}) & (167.0~\\cite{Cline_JAP_1967}) \\\\\nCdS & 80.9 & 47.2 & 39.4 & 87.2 & 14.6 & 17.6 \\\\\n & (83.1~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (50.4~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (46.2~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (94.8~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (15.33~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (16.3~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) \\\\\nInSe & 58.95 & 18.0 & 7.5 & 19.6 & 9.95 & 20.5 \\\\\n & (73.0~\\cite{Gatulle_ElasticInSe_PSSb_1983}) & (27.0~\\cite{Gatulle_ElasticInSe_PSSb_1983}) & (30.0~\\cite{Gatulle_ElasticInSe_PSSb_1983}) & (36.0~\\cite{Gatulle_ElasticInSe_PSSb_1983}) & (11.7~\\cite{Gatulle_ElasticInSe_PSSb_1983}) & (23.0~\\cite{Gatulle_ElasticInSe_PSSb_1983}) \\\\\nInN & 205 & 94.7 & 77.2 & 213 & 48.1 & 55.4 \\\\\n& N\/A & N\/A & N\/A & N\/A & N\/A & N\/A \\\\\n\\end{tabular}}\n\\label{tab:art115:wurzite_elastic_supp}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$, $c_{13}$, $c_{14}$, $c_{33}$, $c_{44}$ and $c_{66}$ of rhombohedral\nsemiconductors.]\n{Experimental values, where available, are shown in parentheses underneath the calculated values.\n``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r}\ncomp. & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{13}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{14}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{33}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{66}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n& ($c_{11}^{\\mathrm{exp}}$) & ($c_{12}^{\\mathrm{exp}}$) & ($c_{13}^{\\mathrm{exp}}$) & ($c_{14}^{\\mathrm{exp}}$) & ($c_{33}^{\\mathrm{exp}}$) & ($c_{44}^{\\mathrm{exp}}$) & ($c_{66}^{\\mathrm{exp}}$) \\\\\n\\hline\nBi$_2$Te$_3$ & 67.6 & 16.6 & 22.05 & 13.9 & 32.7 & 29.25 & 24.6 \\\\\n& (68.47~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) & (21.77~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) & (27.04~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) & (13.25~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) & (47.68~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) & (27.38~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) & (23.35~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) \\\\\nSb$_2$Te$_3$ & 67.8 & 11.2 & 19.1 & 9.92 & 23.2 & 21.35 & 28.8 \\\\\n & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nAl$_2$O$_3$ & 458 & 133 & 123 & -22.2 & 437 & 138 & 145 \\\\\n& (197.3~\\cite{Goto_ElasticAl2O3_JGPR_1989}) & (162.8~\\cite{Goto_ElasticAl2O3_JGPR_1989}) & (116.0~\\cite{Goto_ElasticAl2O3_JGPR_1989}) & (-21.9~\\cite{Goto_ElasticAl2O3_JGPR_1989}) & (500.9~\\cite{Goto_ElasticAl2O3_JGPR_1989}) & (146.8~\\cite{Goto_ElasticAl2O3_JGPR_1989}) & (17.25~\\cite{Goto_ElasticAl2O3_JGPR_1989}) \\\\\nCr$_2$O$_3$ & 350 & 145 & 131 & 17.1 & 325 & 128 & 111.5 \\\\\n& (374~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) & (148~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) & (175~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) & (-19~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) & (362~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) & (159~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) & (113~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) \\\\\nBi$_2$Se$_3$ & 135 & 85.2 & 69.4 & 43.7 & 145 & 64.7 & 82.9 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\n\\end{tabular}}\n\\label{tab:art115:rhombo_elastic_supp}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$, $c_{13}$, $c_{33}$, $c_{44}$ and $c_{66}$ of body-centered tetragonal semiconductors.]\n{Experimental values, where available, are shown in parentheses underneath the calculated values.\n``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r}\ncomp. & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{13}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{33}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{66}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n& ($c_{11}^{\\mathrm{exp}}$) & ($c_{12}^{\\mathrm{exp}}$) & ($c_{13}^{\\mathrm{exp}}$) & ($c_{33}^{\\mathrm{exp}}$) & ($c_{44}^{\\mathrm{exp}}$) & ($c_{66}^{\\mathrm{exp}}$) \\\\\n\\hline\nCuGaTe$_2$ & 67.6 & 36.3 & 37.0 & 66.8 & 32.1 & 31.1 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nZnGeP$_2$ & 118 & 48.95 & 51.7 & 117 & 62.1 & 61.05 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nZnSiAs$_2$ & 108 & 44.7 & 49.3 & 103 & 55.3 & 53.0 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nCuInTe$_2$ & 71.4 & 42.1 & 49.55 & 64.5 & 26.6 & 26.7 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nAgGaS$_2$ & 93.95 & 61.9 & 62.8 & 75.7 & 25.1 & 28.0 \\\\\n& (87.9~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975}) & (58.4~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975}) & (59.2~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975}) & (75.8~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975}) & (24.1~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975}) & (30.8~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975}) \\\\\nCdGeP$_2$ & 102 & 46.25 & 50.6 & 88.2 & 48.0 & 44.9 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nCdGeAs$_2$ & 80.15 & 38.7 & 41.6 & 69.8 & 36.1 & 46.4 \\\\\n& (94.5~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982}) & (59.6~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982}) & (59.7~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982}) & (83.4~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982}) & (42.1~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982}) & (40.8~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982}) \\\\\nCuGaS$_2$ & 102 & 57.0 & 60.5 & 104 & 48.9 & 47.9 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nCuGaSe$_2$ & 93.65 & 57.5 & 58.8 & 92.75 & 39.3 & 37.95 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nZnGeAs$_2$ & 93.7 & 40.65 & 42.6 & 92.6 & 48.2 & 47.1 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\n\\end{tabular}}\n\\label{tab:art115:bct_elastic_supp}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$ and $c_{44}$ of materials with BCC and simple\ncubic structures.]\n{``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r|r|r|r|r}\ncomp. & Pearson & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{11}^{\\mathrm{exp}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\mathrm{exp}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\mathrm{exp}}$ \\\\\n\\hline\nCoSb$_3$ & cI32 & 173 & N\/A & 31.2 & N\/A & 48.0 & N\/A \\\\\nIrSb$_3$ & cI32 & 195 & N\/A & 48.9 & N\/A & 52.85 & N\/A \\\\\nInTe & cP2 & 54.4 & N\/A & 35.3 & N\/A & 7.65 & N\/A \\\\\n\\end{tabular}\n\\label{tab:art115:bcc_elastic}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$, $c_{13}$, $c_{23}$, $c_{33}$, $c_{44}$, $c_{55}$ and $c_{66}$ of materials with orthorhombic structures.]\n{Experimental values, where available, are shown in parentheses underneath the calculated values.\n``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & Pearson & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{13}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{22}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{23}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{33}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{55}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{66}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n& & ($c_{11}^{\\mathrm{exp}}$) & ($c_{12}^{\\mathrm{exp}}$) & ($c_{13}^{\\mathrm{exp}}$) & ($c_{22}^{\\mathrm{exp}}$) & ($c_{23}^{\\mathrm{exp}}$) & ($c_{33}^{\\mathrm{exp}}$) & ($c_{44}^{\\mathrm{exp}}$) & ($c_{55}^{\\mathrm{exp}}$) & ($c_{66}^{\\mathrm{exp}}$) \\\\\n\\hline\nZnSb & oP16 & 84.1 & 30.5 & 28.4 & 93.1 & 25.3 & 83.2 & 16.9 & 39.3 & 31.4 \\\\\n& & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nSb$_2$O$_3$ & oP20 & 17.4 & 7.17 & 0.0 & 82.7 & -7.08 & 79.35 & 24.9 & 18.4 & 11.1 \\\\\n& & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\n\\end{tabular}}\n\\label{tab:art115:orc_elastic}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$, $c_{13}$, $c_{33}$, $c_{44}$ and $c_{66}$ of materials with tetragonal structures.]\n{Experimental values, where available, are shown in parentheses underneath the calculated values.\n``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & Pearson & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{13}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{33}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{66}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n& & ($c_{11}^{\\mathrm{exp}}$) & ($c_{12}^{\\mathrm{exp}}$) & ($c_{13}^{\\mathrm{exp}}$) & ($c_{33}^{\\mathrm{exp}}$) & ($c_{44}^{\\mathrm{exp}}$) & ($c_{66}^{\\mathrm{exp}}$) \\\\\n\\hline\nInTe & tI16 & 32.4 & 11.55 & 13.4 & 52.8 & 13.4 & 13.45 \\\\\n& & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nSnO$_2$ & tP6 & 191 & 128 & 123 & 346 & 73.8 & 168 \\\\\n& & (261.7~\\cite{Chang_ElasticSnO2_JGPR_1975}) & (177.2~\\cite{Chang_ElasticSnO2_JGPR_1975}) & (155.5~\\cite{Chang_ElasticSnO2_JGPR_1975}) & (449.6~\\cite{Chang_ElasticSnO2_JGPR_1975}) & (103.07~\\cite{Chang_ElasticSnO2_JGPR_1975}) & (207.4~\\cite{Chang_ElasticSnO2_JGPR_1975}) \\\\\n\\end{tabular}}\n\\label{tab:art115:tet_elastic}\n\\end{table}\n\n\\subsection{Conclusions}\n\nWe have implemented the ``Automatic Elasticity Library'' framework for \\nobreak\\mbox{\\it ab-initio}\\\nelastic constant calculations, and integrated it with the ``Automatic {\\small GIBBS}\\ Library'' implementation of the {\\small GIBBS}\\ quasi-harmonic Debye model within\nthe {\\small AFLOW}\\ and Materials Project ecosystems.\nWe used it\nto automatically calculate the bulk modulus, shear modulus, Poisson ratio, thermal conductivity, Debye temperature and Gr{\\\"u}neisen parameter of materials with\nvarious structures and compared them with available experimental results.\n\nA major aim of high-throughput calculations is to identify useful\nproperty descriptors for screening large datasets of structures~\\cite{nmatHT}.\nHere, we have examined whether the {\\it inexpensive} Debye model, despite its well known deficiencies, can be usefully leveraged for estimating thermal properties of materials by analyzing\ncorrelations between calculated and corresponding experimental quantities.\n\nIt is found that the {\\small AEL}\\ calculation of the elastic moduli\nreproduces the experimental results quite well, within 5\\% to 20\\%,\nparticularly for materials with cubic and\nhexagonal structures. The {\\small AGL}\\ method, using an isotropic approximation\nfor the bulk modulus, tends to provide a slightly worse quantitative\nagreement but still reproduces trends equally well.\nThe correlations are very high, often above $~0.99$.\nUsing different values of the Poisson ratio mainly affects Debye temperatures,\nwhile having very little effect on Gr{\\\"u}neisen parameters.\nSeveral different numerical and empirical equations of state have also been investigated. The differences\nbetween the results obtained from them are\nsmall, but in some cases they are found to introduce an additional\nsource of error compared to a direct evaluation of the bulk modulus\nfrom the elastic tensor or from the $E(V)$ curve.\nUsing the different equations of state has very little effect on Debye temperatures,\nbut has more of an effect on Gr{\\\"u}neisen parameters.\nCurrently, the values for {\\small AGL}\\ properties available in the {\\small AFLOW}\\ repository are those calculated by numerically fitting the $E_{\\substack{\\scalebox{0.6}{DFT}}}(V)$\ndata and calculating the bulk modulus using Equation~\\ref{eq:art115:bulkmod}.\n{The effect of using different exchange-correlation functionals was investigated for a subset of 16 materials. The results showed that\n{\\small LDA}\\ tended to overestimate thermomechanical properties such as bulk modulus or Debye temperature, compared to {\\small GGA}{}'s tendency\nto underestimate. However, neither functional was consistently better than the other at predicting trends. We therefore use {\\small GGA}-{\\small PBE}\\ for\nthe automated {\\small AEL}-{\\small AGL}\\ calculations in order to maintain consistency with the rest of the {\\small AFLOW}\\ data.}\n\nThe {\\small AEL}-{\\small AGL}\\ evaluation of the Debye temperature provides good\nagreement with experiment for this set of materials, whereas the predictions of the Gr{\\\"u}neisen parameter\nare quite poor. However, since the Gr{\\\"u}neisen parameter is slowly varying for materials sharing crystal structures, the {\\small AEL}-{\\small AGL}\\\nmethodology provides a reliable screening tool for identifying materials with very high or very low thermal conductivity.\nThe correlations between the experimental values of the thermal conductivity and those calculated with {\\small AGL}\\ are summarized in\nTable~\\ref{tab:art115:kappa_correlation}. For the entire set of materials examined we find high values of the Pearson correlation\nbetween $\\kappa^{\\mathrm{exp}}$ and $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$, ranging from $0.880$ to $0.933$. It is particularly high, above $0.9$, for materials\nwith high symmetry (cubic, hexagonal or rhombohedral) structures, but significantly lower for anisotropic materials.\nIn our previous work on {\\small AGL}~\\cite{curtarolo:art96}, we used an approximated the value of $\\sigma = 0.25$ in Equation~\\ref{eq:art115:fpoisson}.\nUsing instead the Poisson ratio calculated in {\\small AEL}, $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$, the overall correlations are improved\nby about 5\\%, from $0.880$ to $0.928$, in the agreement with previous\nwork on metals~\\cite{Liu_Debye_CMS_2015}. The correlations for\nanisotropic materials, such as the body-centered tetragonal set\nexamined here, improved even more, demonstrating the significance of a\ndirect evaluation of the Poisson ratio.\nThis combined algorithm demonstrates the advantage of an integrated high-throughput materials design framework such as {\\small AFLOW},\nwhich enables the calculation of interdependent properties within a single automated workflow.\n\nA direct {\\small AEL}\\ evaluation of the Poisson ratio, instead of assuming a\nsimple approximation, e.g.\\ a Cauchy solid with $\\sigma = 0.25$,\nconsistently improves the correlations of the {\\small AGL}-Debye temperatures\nwith experiments.\nHowever, it has very little effect on the values obtained for the Gr{\\\"u}neisen parameter.\nSimple approximations lead to more numerically-robust and better system-size scaling calculations,\nas they avoid the complications inherent in obtaining the elastic tensor.\n{Therefore, {\\small AGL}\\ could also be used on its own for initial rapid screening,\nwith {\\small AEL}\\ being performed later for potentially interesting materials to increase the accuracy of the results.}\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for the entire set of materials.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.880 & 0.752 & 1.293 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.928 & 0.720 & 2.614 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.879 & 0.735 & 2.673 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.912 & 0.737 & 2.443 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.933 & 0.733 & 2.751 \\\\\n\\end{tabular}\n\\label{tab:art115:kappa_correlation}\n\\end{table}\n\nWith respect to rapid estimation of thermal conductivities,\nthe approximations in the Leibfried-Schl{\\\"o}mann formalism\nmiss some of the details affecting the lattice thermal conductivity, such as the suppression of phonon-phonon scattering due to\nlarge gaps between the branches of the phonon dispersion~\\cite{Lindsay_PRL_2013}.\nNevertheless, the high correlations between $\\kappa^{\\mathrm{exp}}$ and\n$\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ found for most of the structure families in this study demonstrate the utility of the {\\small AEL}-{\\small AGL}\\ approach\nas a screening method for large databases of materials where\nexperimental data is lacking or ambiguous.\nDespite its intrinsic limitations, the synergy presented by the {\\small AEL}-{\\small AGL}\\ approach\nprovides the right balance between accuracy and complexity in identifying materials with\npromising properties for further investigation.\n\n\\subsection{AFLOW AEL-AGL REST-API}\n\\label{subsec:art115:restapi_keywords}\n\nThe {\\small AEL}-{\\small AGL}\\ methodology described in this work is being used to calculate the elastic and thermal properties of materials in a high-throughput\nfashion by the {\\small AFLOW}\\ consortium. The results are now available on the {\\small AFLOW}\\ database~\\cite{aflowlib.org, aflowlibPAPER}\nvia the {\\small AFLOW}\\ {\\small REST-API}~\\cite{aflowAPI}. The following optional materials keywords have now been added to the {\\small AFLOW}\\ {\\small REST-API}\\\nto facilitate accessing this data.\n\n\\def\\item {{\\it Description:}\\ }{\\item {{\\it Description.}\\ }}\n\\def\\item {{\\it Type:}\\ }{\\item {{\\it Type.}\\ }}\n\\def\\item {{\\it Example.}\\ }{\\item {{\\it Example.}\\ }}\n\\def\\item {{\\it Units:}\\ }{\\item {{\\it Units.}\\ }}\n\\def\\item {{\\it Request syntax.}\\ }{\\item {{\\it Request syntax.}\\ }}\n\n\\begin{itemize}\n\n\\item\n\\verb|ael_bulk_modulus_reuss|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ bulk modulus as calculated using the Reuss average.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|ael_bulk_modulus_reuss=105.315|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_bulk_modulus_reuss|.\n\\end{itemize}\n\n\\item\n\\verb|ael_bulk_modulus_voigt|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ bulk modulus as calculated using the Voigt average.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|ael_bulk_modulus_voigt=105.315|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_bulk_modulus_voigt|.\n\\end{itemize}\n\n\\item\n\\verb|ael_bulk_modulus_vrh|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ bulk modulus as calculated using the\nVoigt-Reuss-Hill ({\\small VRH}) average.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|ael_bulk_modulus_vrh=105.315|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_bulk_modulus_vrh|.\n\\end{itemize}\n\n\\item\n\\verb|ael_elastic_anisotropy|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ elastic anisotropy.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } dimensionless.\n\\item {{\\it Example.}\\ } \\verb|ael_elastic_anistropy=0.000816153|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_elastic_anisotropy|.\n\\end{itemize}\n\n\\item\n\\verb|ael_poisson_ratio|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ Poisson ratio.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } dimensionless.\n\\item {{\\it Example.}\\ } \\verb|ael_poisson_ratio=0.21599|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_poisson_ratio|.\n\\end{itemize}\n\n\\item\n\\verb|ael_shear_modulus_reuss|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ shear modulus as calculated using the Reuss average.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|ael_shear_modulus_reuss=73.7868|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_shear_modulus_reuss|.\n\\end{itemize}\n\n\\item\n\\verb|ael_shear_modulus_voigt|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ shear modulus as calculated using the Voigt average.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|ael_shear_modulus_voigt=73.7989|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_shear_modulus_voigt|.\n\\end{itemize}\n\n\\item\n\\verb|ael_shear_modulus_vrh|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ shear modulus as calculated using the\nVoigt-Reuss-Hill ({\\small VRH}) average.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|ael_shear_modulus_vrh=73.7929|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_shear_modulus_vrh|.\n\\end{itemize}\n\n\\item\n\\verb|ael_speed_of_sound_average|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ average speed of sound calculated from the transverse and longitudinal speeds of sound.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } m\/s.\n\\item {{\\it Example.}\\ } \\verb|ael_speed_of_sound_average=500.0|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_speed_of_sound_average|.\n\\end{itemize}\n\n\\item\n\\verb|ael_speed_of_sound_longitudinal|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ speed of sound in the longitudinal direction.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } m\/s.\n\\item {{\\it Example.}\\ } \\verb|ael_speed_of_sound_longitudinal=500.0|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_speed_of_sound_longitudinal|.\n\\end{itemize}\n\n\\item\n\\verb|ael_speed_of_sound_transverse|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ speed of sound in the transverse direction.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } m\/s.\n\\item {{\\it Example.}\\ } \\verb|ael_speed_of_sound_transverse=500.0|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_speed_of_sound_transverse|.\n\\end{itemize}\n\n\\end{itemize}\n\n\\begin{itemize}\n\n\\item\n\\verb|agl_acoustic_debye|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ acoustic Debye temperature.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } K.\n\\item {{\\it Example.}\\ } \\verb|agl_acoustic_debye=492|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_acoustic_debye|.\n\\end{itemize}\n\n\\item\n\\verb|agl_bulk_modulus_isothermal_300K|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ isothermal bulk modulus at 300~K and zero pressure.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|agl_bulk_modulus_isothermal_300K=96.6|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_bulk_modulus_isothermal_300K|.\n\\end{itemize}\n\n\\item\n\\verb|agl_bulk_modulus_static_300K|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ static bulk modulus at 300~K and zero pressure.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|agl_bulk_modulus_static_300K=99.59|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_bulk_modulus_static_300K|.\n\\end{itemize}\n\n\\item\n\\verb|agl_debye|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ Debye temperature.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } K.\n\\item {{\\it Example.}\\ } \\verb|agl_debye=620|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_debye|.\n\\end{itemize}\n\n\\item\n\\verb|agl_gruneisen|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ Gr{\\\"u}neisen parameter.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } dimensionless.\n\\item {{\\it Example.}\\ } \\verb|agl_gruneisen=2.06|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_gruneisen|.\n\\end{itemize}\n\n\\item\n\\verb|agl_heat_capacity_Cv_300K|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ heat capacity at constant volume (C$_V$) at 300~K and zero pressure.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } k$_\\mathrm{B}$\/cell.\n\\item {{\\it Example.}\\ } \\verb|agl_heat_capacity_Cv_300K=4.901|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_heat_capacity_Cv_300K|.\n\\end{itemize}\n\n\\item\n\\verb|agl_heat_capacity_Cp_300K|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ heat capacity at constant pressure (C$_p$) at 300~K and zero pressure.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } k$_\\mathrm{B}$\/cell.\n\\item {{\\it Example.}\\ } \\verb|agl_heat_capacity_Cp_300K=5.502|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_heat_capacity_Cp_300K|.\n\\end{itemize}\n\n\\item\n\\verb|agl_poisson_ratio_source|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns source of Poisson ratio used to calculate Debye temperature in {\\small AGL}. Possible sources include \\verb|ael_poisson_ratio_|, in\nwhich case the Poisson ratio was calculated from first principles using {\\small AEL}; \\verb|empirical_ratio_|, in which case the value was taken\nfrom the literature; and \\verb|Cauchy_ratio_0.25|, in which case the default value of 0.25 of the Poisson ratio of a Cauchy solid\nwas used.\n\\item {{\\it Type:}\\ } \\verb|string|.\n\\item {{\\it Example.}\\ } \\verb|agl_poisson_ratio_source=ael_poisson_ratio_0.193802|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_poisson_ratio_source|.\n\\end{itemize}\n\n\\item\n\\verb|agl_thermal_conductivity_300K|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ thermal conductivity at 300~K.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } W\/m*K.\n\\item {{\\it Example.}\\ } \\verb|agl_thermal_conductivity_300K=24.41|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_thermal_conductivity_300K|.\n\\end{itemize}\n\n\\item\n\\verb|agl_thermal_expansion_300K|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ thermal expansion at 300~K and zero pressure.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } 1\/K.\n\\item {{\\it Example.}\\ } \\verb|agl_thermal_expansion_300K=4.997e-05|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_thermal_expansion_300K|.\n\\end{itemize}\n\n\\end{itemize}\n\\clearpage\n\\chapter{Data-driven Approaches}\n\\section{AFLOW-CHULL: Cloud-Oriented Platform for Autonomous Phase Stability Analysis}\n\\label{sec:art146}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art146}.\n\n\\subsection{Introduction}\nAccelerating the discovery of new functional materials demands an efficient determination of synthesizability.\nIn general, materials synthesis is a multifaceted problem, spanning\n\\textbf{i.} technical challenges, such as experimental apparatus design and growth conditions~\\cite{Jansen_AngChemInt_2002,Potyrailo_ACSCombSci_2011},\nas well as\n\\textbf{ii.} economic and environmental obstacles, including accessibility and handling of necessary components~\\cite{Kuzmin_JPCM_2014,curtarolo:art109}.\nPhase stability is a limiting factor.\nOften, it accounts for the gap between\nmaterials prediction and experimental realization.\nAddressing stability requires an understanding of how phases compete thermodynamically.\nDespite the wealth of available experimental phase diagrams~\\cite{ASMAlloyInternational},\nthe number of systems explored represents a negligible fraction of\nall hypothetical structures~\\cite{Walsh_NChem_2015,curtarolo:art124}.\nLarge materials databases~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux,nomad,APL_Mater_Jain2013,Saal_JOM_2013,cmr_repository,Pizzi_AiiDA_2016}\nenable the construction of calculated phase diagrams,\nwhere aggregate structural and energetic materials data is employed.\nThe analysis delivers many fundamental thermodynamic descriptors,\nincluding stable\/unstable classification,\nphase coexistence, measures of robust stability, and determination of\ndecomposition reactions~\\cite{curtarolo:art109,curtarolo:art113,Bechtel_PRM_2018,Li_CMS_2018,Balachandran_PRM_2018}.\n\nAs with all informatics-based approaches, \\nobreak\\mbox{\\it ab-initio}\\ phase diagrams require an abundance of data:\nwell-converged enthalpies from a variety of different phases.\nMany thermodynamic descriptors computed\nfrom the {\\sf \\AFLOW.org}\\ repository\nhave already demonstrated predictive power in characterizing phase\nstability~\\cite{curtarolo:art49,curtarolo:art51,curtarolo:art53,curtarolo:art57,curtarolo:art63,curtarolo:art67,curtarolo:art70,curtarolo:art74,monsterPGM,curtarolo:art106,curtarolo:art109,curtarolo:art112,curtarolo:art113,curtarolo:art117,curtarolo:art126,curtarolo:art130},\nincluding one investigation that resulted in the synthesis of\ntwo new magnets --- the first ever discovered by computational approaches~\\cite{curtarolo:art109}.\nAs exploration embraces more complex systems, such analyses are expected to\nbecome increasingly critical in confining the search space.\nIn fact, prospects for stable ordered phases diminish with every new component (dimension), despite the growing number of combinations.\nThis is due to increased competition with\n\\textbf{i.} phases of lower dimensionality, \\nobreak\\mbox{\\it e.g.}, ternary phases competing with stable binary phases~\\cite{curtarolo:art130}, and\n\\textbf{ii.} disordered (higher entropy) phases~\\cite{curtarolo:art99,curtarolo:art122,curtarolo:art139}.\n\nTo address the challenge, a new module has been implemented in the autonomous, open-source~\\cite{gnu_license}\n{\\small AFLOW}\\ (\\underline{A}utomatic \\underline{Flow}) framework for \\nobreak\\mbox{\\it ab-initio}\\ calculations~\\cite{curtarolo:art49,curtarolo:art53,curtarolo:art57,aflowBZ,curtarolo:art63,aflowPAPER,curtarolo:art85,curtarolo:art110,monsterPGM,aflowANRL,aflowPI}.\n{\\small \\AFLOWHULLtitle}\\ ({\\small AFLOW}\\ \\underline{c}onvex \\underline{hull}) offers a thermodynamic characterization that can be employed\nlocally from any {\\small UNIX}-like machine, including those running Linux and macOS.\nBuilt-in data curation and validation schemes ensure results are well-converged:\nadhering to proper hull statistics, performing outlier detection, and determining structural equivalence.\n{\\small \\AFLOWHULLtitle}\\ is powered by the {\\small AFLUX}\\ Search-{\\small API}\\ (\\underline{a}pplication \\underline{p}rogramming \\underline{i}nterface)~\\cite{aflux},\nwhich enables access to more than 2 million compounds from the {\\sf \\AFLOW.org}\\ repository.\nWith {\\small AFLUX}\\ integration, data-bindings are flexible enough to serve any materials database,\nincluding large heterogeneous repositories such as {NOMAD}~\\cite{nomad}.\n\nSeveral analysis output types have been created for integration\ninto a variety of design workflows, including plain text and\n{\\small JSON}\\ (\\underline{J}ava\\underline{S}cript \\underline{O}bject \\underline{N}otation) file types.\nA small set of example scripts have been included demonstrating\nhow to employ {\\small \\AFLOWHULLtitle}\\ from within a Python environment, much in the spirit of {\\small AFLOW-SYM}~\\cite{curtarolo:art135}.\nThe {\\small JSON}\\ output also powers an interactive, online web application offering enhanced presentation of thermodynamic descriptors and\nvisualization of 2-\/3-dimensional hulls.\nThe application can be accessed through the {\\sf \\AFLOW.org}\\ portal under ``Apps and Docs'' or directly at {\\sf aflow.org\/aflow-chull}.\n\nAs a test-bed, the module is applied to all 2 million compounds available in the {\\sf \\AFLOW.org}\\ repository.\nAfter enforcing stringent hull convergence criteria, the module resolves a thermodynamic characterization\nfor more than 1,300 binary and ternary systems.\nStable phases are screened for previously explored systems and ranked by their\nrelative stability criterion, a dimensionless quantity capturing the\neffect of the phase on the minimum energy surface~\\cite{curtarolo:art109}.\nSeveral promising candidates are identified, including\n17\\ $C15_{b}$-type structures $\\left(F\\overline{4}3m~\\#216\\right)$ and two half-Heuslers.\nHence, screening criteria based on these thermodynamic descriptors can accelerate the\ndiscovery of new stable phases.\nMore broadly, the design of more challenging materials, including ceramics~\\cite{curtarolo:art80} and metallic glasses~\\cite{curtarolo:art112},\nbenefit from autonomous, integrated platforms such as {\\small \\AFLOWHULLtitle}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig033}\n\\mycaption[Example hull illustrations in 2-\/3-dimensions as generated by {\\small \\AFLOWHULLtitle}.]\n{(\\textbf{a}) Co-Ti and (\\textbf{b}) Mn-Pd-Pt.}\n\\label{fig:art146:hull_examples}\n\\end{figure}\n\n\\subsection{Methods}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig034}\n\\mycaption[Illustration of the convex hull construction for a binary system with {\\small \\AFLOWHULLtitle}.]\n{The approach is inspired by the {\\small Qhull}\\ algorithm~\\cite{qhull}.\nThe points on the plot represent structures from the {\\sf \\AFLOW.org}\\ database~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}.\n(\\textbf{a}) and (\\textbf{g}) denote the beginning and end of the algorithm, respectively.\n(\\textbf{c}-\\textbf{f}) denote the iterative loop that continues until the\ncondition denoted by (\\textbf{b}) is no longer satisfied.\nPoints are marked with crosses if, by that step in the algorithm, they have been determined to be inside the hull,\nand otherwise are marked with circles.\nThe furthest point from the facet in (\\textbf{d}) is marked with a triangle.\nPoints and facets of interest are highlighted in \\textcolor{pranab_red}{{\\bf red}} and \\textcolor{pranab_green}{{\\bf green}}, respectively.}\n\\label{fig:art146:hull_workflow}\n\\end{figure}\n\n\\boldsection{Defining thermodynamic stability.}\nFor a multicomponent system at a fixed temperature ($T$) and pressure ($p$),\nthe minimum Gibbs free energy $G$ (per atom) defines the thermodynamic equilibrium:\n\\begin{equation}\nG(T,p,\\{x_{i}\\})=H-TS\n\\label{eq:art146:gibbs_free_energy}\n\\end{equation}\nwhere $x_{i}$ is the atomic concentration of the $i$-species,\n$H$ is the enthalpy, and $S$ is the entropy.\nA binary phase $A_{x_{A}}B_{x_{B}}$ is stable at equilibrium with respect to its components\n$A$ and $B$ if the corresponding formation reaction releases energy:\n\\begin{equation}\nx_{A} A + x_{B} B \\xrightarrow[]{\\Delta G<0} A_{x_{A}}B_{x_{B}},\n\\label{eq:art146:formation_reaction}\n\\end{equation}\nwhere $\\Delta G$ is the energy difference between the mixed phase\nand the sum of its components.\nConversely, a positive $\\Delta G$ suggests the decomposition of $A_{x_{A}}B_{x_{B}}$ is preferred, and\nis thus unstable.\nIn general, the magnitude of $\\Delta G$ quantifies the propensity for the reaction,\nand the sign determines the direction.\n\nRelative stability can be visualized on a free-energy-concentration diagram\n--- $G$ \\nobreak\\mbox{\\it vs.}\\ $\\left\\{ x_i \\right\\}$ ---\nwhere $\\Delta G$ is depicted as the energetic vertical-distance between $A_{x_{A}}B_{x_{B}}$ and the\ntie-line connecting $A$ and $B$ end-members (elemental phases).\nEnd-members constitute only a single pathway to formation\/decomposition, and\nall feasible reactions should be considered for system-wide stability.\n{Identification of equilibrium phases} is mathematically equivalent to the construction\nof the convex hull --- the set of the most extreme or ``outside'' points (Figure~\\ref{fig:art146:hull_examples}(a)).\n{The convex hull characterizes the phase stability of the system at equilibrium\nand does not include kinetic considerations for synthesis.\nGrowth conditions affect the final outcome leading to formation of polymorphs and\/or metastable phases,\nwhich could differ from the equilibrium phases.\nThis is a formidable task for high-throughput characterization.\nTo help identify kinetic pathways for synthesis, {\\small \\AFLOWHULLtitle}\\\nincludes (more in future releases) potential kinetic descriptors,\n\\nobreak\\mbox{\\it e.g.}, chemical decompositions, distance from stability, entropic temperature~\\cite{curtarolo:art98},\nglass formation ability~\\cite{curtarolo:art112}, and spectral entropy analysis for high-entropy systems.}\n\nIn the zero temperature limit (as is the case for ground-state density functional theory),\nthe entropic term of Equation~\\ref{eq:art146:gibbs_free_energy} vanishes,\nleaving the formation enthalpy term (per atom) as the driving force:\n\\begin{equation}\n H_\\mathrm{f}=H_{A_{x_{A}}B_{x_{B}}}-\\left(x_{A} H_{A} + x_{B} H_{B} \\right).\n\\end{equation}\nBy construction, formation enthalpies of stable elemental phases are zero, restricting\nthe convex hull to the lower hemisphere.\n{Zero-point energies are not yet included in the {\\sf \\AFLOW.org}\\ repository and thus are neglected from the enthalpy calculations.\nEfforts to incorporate vibrational characterizations are underway~\\cite{curtarolo:art96,Nath_QHA_2016}.\nThis contribution could have a large impact on compounds containing light-elements, such as\nhydrogen~\\cite{Majzoub_PRB_2005}, which comprise a small minority (less than 1\\%) of the overall repository.}\n\nBy offsetting the enthalpy with that of the elemental phases,\n$H_\\mathrm{f}$ quantifies the energy gain from forming new bonds between\nunlike components,\\footnote{The formation enthalpy is not to be confused with the cohesive energy, which quantifies\nthe energy difference between the phase and its fully gaseous (single atoms) counterpart, \\nobreak\\mbox{\\it i.e.},\nthe energy in all bonds.} \\nobreak\\mbox{\\it e.g.}, $A-B$.\n{Currently, the {\\small \\AFLOWHULLtitle}\\ framework does not allow the renormalization of chemical potentials to\nimprove the calculation of formation enthalpies when gas phases are involved.\nA new first-principles approach is being developed and tested in {\\small AFLOW},\nand will be implemented in future versions of the {\\small \\AFLOWHULLtitle}\\\nsoftware together with the available approaches}~\\cite{CrUJ,Lany_Zunger_FERE_2012}.\n\nThe tie-lines connecting stable phases in Figure~\\ref{fig:art146:hull_examples}(a)\ndefine regions of phase separation where the two phases coexist at equilibrium.\nThe chemical potentials are equal for each component among coexisting phases,\nimplying the common tangent tie-line construction~\\cite{Ganguly_thermo_2008,Darken_pchemmetals_1953}.\n{Under thermodynamic equilibrium,} phases above a tie-line will decompose into a linear combination of the stable phases that\ndefine the tie-line (Figure~\\ref{fig:art146:hull_analyses}(d)).\nThe Gibbs phase rule~\\cite{McQuarrie} dictates the shape of tie-lines for $N$-ary systems,\nwhich generalizes to $\\left(N-1\\right)$-dimensional triangles (simplexes) and correspond to facets of the convex hull,\n\\nobreak\\mbox{\\it e.g.}, lines in two dimensions (Figure~\\ref{fig:art146:hull_examples}(a)),\ntriangles in three dimensions (Figure~\\ref{fig:art146:hull_examples}(b)),\nand tetrahedra in four.\nThe set of equilibrium facets define the $N$-dimensional minimum energy surface.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig035}\n\\mycaption[Illustration of the {\\small \\AFLOWHULLtitle}\\ iterative hull scheme.]\n{The convex hull and associated properties are first calculated for the binary\nhulls, and then propagated to the ternary hull.\nThis is generalized for $N$-dimensions.}\n\\label{fig:art146:dimensions}\n\\end{figure}\n\n\\boldsection{Hull construction.}\n{\\small \\AFLOWHULLtitle}\\ calculates the $N$-dimensional convex hull corresponding to an $N$-ary system\nwith an algorithm partially inspired by {\\small Qhull}~\\cite{qhull}.\nThe algorithm\nis efficient in identifying the most important points for construction of facets,\nwhich are treated as hyperplanes instead of boundary-defining inequalities.\n{\\small \\AFLOWHULLtitle}\\ uniquely accommodates thermodynamic hulls,\n\\nobreak\\mbox{\\it i.e.}, data occupying the lower half hemisphere and\ndefined by stoichiometric coordinates $\\left(0 \\leq x_{i} \\leq 1 \\right)$.\nPoints corresponding to individual phases are characterized by their stoichiometric and energetic coordinates:\n\\begin{equation}\n\\mathbf{p}=\\left[x_{1}, x_{2}, \\ldots, x_{N-1}, H_\\mathrm{f}\\right] = \\left[\\mathbf{x}, H_\\mathrm{f}\\right],\n\\label{eq:art146:point}\n\\end{equation}\nwhere $x_{N}$ is implicit $\\left(\\sum_{i}x_i=1\\right)$.\nData preparation includes the\n\\textbf{i.} elimination of phases unstable with respect to end-members (points above the zero $H_{\\mathrm{f}}$ tie-line)\nand \\textbf{ii.} organization of phases by stoichiometry and sorted by energy.\nThrough this stoichiometry group structure, all but the minimum energy phases are eliminated from\nthe convex hull calculation.\n\nThe workflow is illustrated in Figure~\\ref{fig:art146:hull_workflow}.\n{\\small \\AFLOWHULLtitle}\\ operates by partitioning space, iteratively defining\n``inside'' \\nobreak\\mbox{\\it vs.}\\ ``outside'' half-spaces until all points are either on the hull or inside of it.\nFirst, a simplex is initialized (Figure~\\ref{fig:art146:hull_workflow}(a)) with the most extreme points:\nstable end-members and the globally stable mixed phase (lowest energy).\nA facet is described as:\n\\begin{equation}\n\\mathbf{n} \\cdot \\mathbf{r} + D = 0,\n\\label{eq:art146:plane_eq}\n\\end{equation}\nwhere $\\mathbf{n}$ is the characteristic normal vector, $\\mathbf{r}$ is the position vector,\nand $D$ is the offset.\nA general hyperplane is defined by $N$ points and $k=\\left(N-1\\right)$ corresponding edges\n$\\mathbf{v}_{k}=\\mathbf{p}_{k}-\\mathbf{p}_{\\mathrm{origin}}$.\nTo construct $\\mathbf{n}$, {\\small \\AFLOWHULLtitle}\\ employs a generalized cross product approach~\\cite{Massey_AMM_1983},\nwhere $n_{i \\in \\{1,\\ldots,N\\}}$ (unnormalized) is the $i$-row cofactor\n$\\left(C_{i,j=0}\\right)$ of the matrix $\\mathbf{V}$ containing $\\mathbf{v}_k$ in its columns:\n\\begin{equation}\n n_{i} = \\left(-1\\right)^{i+1}M_{i,j=0}\\left(\n\\begin{bmatrix}\n | & & | \\\\\n \\mathbf{v}_{1} & \\ldots & \\mathbf{v}_{k} \\\\\n | & & | \\\\\n\\end{bmatrix}\n\\right)\n\\label{eq:art146:hyperplane_normal}\n\\end{equation}\nHere, $M_{i,j=0}\\left(\\mathbf{V}\\right)$ denotes the\n$i$-row minor of $\\mathbf{V}$,\n\\nobreak\\mbox{\\it i.e.}, the determinant of the submatrix formed by removing the $i$-row.\n\nThe algorithm then enters a loop over the facets of the convex hull until no points are declared ``outside'',\ndefined in the hyperplane description by the signed point-plane distance (Figure~\\ref{fig:art146:hull_workflow}(b)).\nEach point outside of the hull is singularly assigned to the outside set of a facet (\\textcolor{pranab_red}{{\\bf red}}\nin Figure~\\ref{fig:art146:hull_workflow}(c)).\nThe furthest point from each facet --- by standard point-plane distance --- is selected from the outside set\n(marked with a triangle in Figure~\\ref{fig:art146:hull_workflow}(d)).\nEach neighboring facet is visited to determine whether the furthest point is also outside of it, defining\nthe set of visible planes (\\textcolor{pranab_green}{{\\bf green}}) and its boundary,\nthe horizon ridges (\\textcolor{pranab_red}{{\\bf red}}) (Figure~\\ref{fig:art146:hull_workflow}(d)).\nThe furthest point is combined with each ridge of the horizon to form new facets (Figure~\\ref{fig:art146:hull_workflow}(e)).\nThe visible planes --- the dotted line in Figure~\\ref{fig:art146:hull_workflow}(e) --- are then removed from the\nconvex hull (Figure~\\ref{fig:art146:hull_workflow}(f)).\nThe fully constructed convex hull --- with all points on the hull or inside of it --- is\nsummarized in Figure~\\ref{fig:art146:hull_workflow}(g).\n\nA challenge arises with lower dimensional data in higher dimensional convex hull constructions.\nFor example, binary phases composed of the same species all exist on the same (vertical) plane in three dimensions.\nA half-space partitioning scheme can make no ``inside'' \\nobreak\\mbox{\\it vs.}\\ ``outside'' differentiation between such points.\nThese ambiguously-defined facets\\nocite{qhull}\\footnote{Ambiguously-defined facets occur when a set of $d+1$ points (or more) define a $(d-1)$-flat~\\cite{qhull}.}\nconstitute a hull outside the scope of the {\\small Qhull}\\ algorithm~\\cite{qhull}.\nIn the case of three dimensions, the creation of ill-defined facets with collinear edges can result.\nHyper-collinearity --- planes defined with collinear edges, tetrahedra defined with coplanar faces, \\nobreak\\mbox{\\it etc.}\\ ---\nis prescribed by the content (hyper-volume) of the facet.\nThe quantity resolves the length of the line ($1$-simplex), the area of a triangle ($2$-simplex),\nthe volume of a tetrahedron ($3$-simplex), \\nobreak\\mbox{\\it etc.},\nand is calculated for a simplex of $N$-dimensions via the Cayley-Menger determinant~\\cite{sommerville_1929_n_dimensional_geometry}.\nBoth vertical and content-less facets are problematic for thermodynamic characterizations,\nparticularly when calculating hull distances, which require facets within finite energetic distances\nand well-defined normals.\n\nA dimensionally-iterative scheme is implemented in {\\small \\AFLOWHULLtitle}\\ to solve the issue.\nIt calculates the convex hull for each dimension consecutively\n(Figure~\\ref{fig:art146:dimensions}).\nIn the case of a ternary hull, the three binary hulls are calculated first, and the relevant\nthermodynamic data is extracted and then propagated forward.\nThough vertical and content-less facets are still created in higher dimensions, no thermodynamic\ndescriptors are extracted from them.\nTo optimize the calculation, only stable binary structures are propagated forward to\nthe ternary hull calculation, and this approach is generalized for $N$-dimensions.\nThe scheme is the default for thermodynamic hulls, resorting back to\nthe general convex hull algorithm otherwise.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig036}\n\\mycaption[Illustrations of various automated convex hull analyses in {\\small \\AFLOWHULLtitle}.]\n{(\\textbf{a}) A plot showing an egregious outlier in the Al-Co convex hull.\n(\\textbf{b}) The corrected Al-Co convex hull with the outlier removed.\n(\\textbf{c}) The Te-Zr convex hull with the traditional compound labels replaced\nwith the corresponding {\\small ICSD}\\ number designations as determined by a structure\ncomparison analysis.\nIf multiple {\\small ICSD}\\ entries are found for the same stoichiometry, the lowest number\n{\\small ICSD}\\ entry is chosen (chronologically reported, usually).\n(\\textbf{d}) The Pd-Pt convex hull. The decomposition energy of Pd$_{2}$Pt$_{3}$ is plotted in\n\\textcolor{pranab_red}{{\\bf red}}, and highlighted in \\textcolor{pranab_green}{{\\bf green}} is\nthe equilibrium facet directly below it.\nThe facet is defined by ground-state phases PdPt$_{3}$ and PdPt.\n(\\textbf{e}) The Pd-Pt convex hull. The stability criterion $\\delta_{\\mathrm{sc}}$ of PdPt is plotted\nin \\textcolor{pranab_green}{{\\bf green}}, with the pseudo-hull plotted with dashed lines.\n(\\textbf{f}) The B-Sm convex hull plotted with the\nideal ``{\\it iso-max-latent-heat}'' lines of the grand-canonical ensemble~\\cite{monsterPGM,curtarolo:art98}\nfor the ground-state structures.}\n\\label{fig:art146:hull_analyses}\n\\end{figure}\n\n\\boldsection{Thermodynamic data.}\nStructural and energetic data employed to construct the convex hull\nis retrieved from the {\\sf \\AFLOW.org}~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}{} repository, which contains more than 2 million compounds and\n200 million calculated properties.\nThe database is generated by the autonomous, \\nobreak\\mbox{\\it ab-initio}\\ framework {\\small AFLOW}~\\cite{curtarolo:art49,curtarolo:art53,curtarolo:art57,aflowBZ,curtarolo:art63,aflowPAPER,curtarolo:art85,curtarolo:art110,monsterPGM,aflowANRL,aflowPI}{}\nfollowing the {\\small AFLOW}\\ Standard for high-throughput materials science\ncalculations~\\cite{curtarolo:art104}.\nIn particular, calculations are performed\nwith {\\small VASP}\\ (\\underline{V}ienna \\textit{\\underline{A}b initio} \\underline{S}imulation \\underline{P}ackage)~\\cite{vasp_prb1996}.\nWavefunctions are represented by a large basis set, including\nall terms with kinetic energy up to a threshold 1.4 times larger than the recommended defaults.\n{\\small AFLOW}\\ also leverages a large $\\mathbf{k}$-point mesh --- as standardized by\na $\\mathbf{k}$-points-per-reciprocal-atom scheme~\\cite{curtarolo:art104} ---\nwhich is critical for convergence and reliability of calculated properties.\nInvestigations show that the {\\small AFLOW}\\ Standard of at least $6,000$ $\\mathbf{k}$-points-per-reciprocal-atom\nfor structural relaxations and $10,000$ for the static calculations ensures\nrobust convergence of the energies to within one meV\/atom in more than 95\\% of systems\n(including metals which suffer from the discontinuity in the occupancy function at zero temperature),\nand within three meV\/atom otherwise~\\cite{Wisesa_Kgrids_PRB_2016}.\n\nSpecial consideration is taken for the calculation of $H_{\\mathrm{f}}$.\nThe reference energies for the elemental phases are calculated and stored in the\n{\\small LIB1}\\ catalog for unary phases in the {\\sf \\AFLOW.org}\\ repository, and include variations for different\nfunctionals and pseudopotentials.\nFor consistency, {\\small \\AFLOWHULLtitle}\\ only employs data calculated with the \\underline{P}erdew-\\underline{B}urke-\\underline{E}rnzerhof\nGeneralized Gradient Approximation functional~\\cite{PBE}\nand pseudopotentials calculated with the\n\\underline{p}rojector \\underline{a}ugmented \\underline{w}ave method~\\cite{PAW} ({\\small PAW}-{\\small PBE}).\n{Calculations employing {\\small DFT}$+U$ corrections to rectify self-interaction errors and energy-gap issues for\nelectronic properties~\\cite{curtarolo:art104} are neglected.\nIn general, these corrections are parameterized\nand material-specific~\\cite{curtarolo:art93}.\nThey artificially augment the energy of the system affecting the reliability of thermodynamic properties.}\nIt is possible to encounter stable (lowest energy) elemental phases with energy differences from the reference\nof order meV\/atom, which is the result of duplicate entries (by relaxation or otherwise)\nas well as reruns with new parameters, \\nobreak\\mbox{\\it e.g.}, a denser $\\mathbf{k}$-point mesh.\nTo avoid any issues with the convex hull calculation, the algorithm fixes\nthe half-space plane at zero.\nHowever, a ``warning'' is prompted in the event that the stable elemental phase differs from\nthe reference energy by more than 15 meV\/atom, yielding a ``skewed'' hull.\n\nData is retrieved via the {\\small AFLUX}\\ Search-{\\small API}~\\cite{aflux}, designed for accessing\nproperty-specific datasets efficiently.\nThe following is an example of a relevant request:\n\\begin{center}\n\\noindent{\\sf http:\/\/aflowlib.duke.edu\/search\/API\/?species(Mn,Pd),nspecies(2),*,paging(0)}\n\\end{center}\nwhere {\\sf http:\/\/aflowlib.duke.edu\/search\/API\/} is the {\\small URL}\\ for the {\\small AFLUX}\\ server and\n{\\sf species(Mn,Pd),nspecies(2),*,paging(0)} is the query.\n{\\sf species(Mn,Pd)} queries for any entry containing the elements\nMn or Pd, {\\sf nspecies(2)} limits the search to binaries only, {\\sf *} returns the data\nfor all available fields, and {\\sf paging(0)} amalgamates all data into a single response\nwithout paginating (warning, this can be a large quantity of data).\nSuch queries are constructed combinatorially for each dimension, \\nobreak\\mbox{\\it e.g.},\na general ternary hull $ABC$ constructs the following seven queries:\n{\\sf species($A$)},\n{\\sf species($B$)}, and\n{\\sf species($C$)} with {\\sf nspecies(1)},\n{\\sf species($A$,$B$)},\n{\\sf species($A$,$C$)}, and\n{\\sf species($B$,$C$)} with {\\sf nspecies(2)}, and\n{\\sf species($A$,$B$,$C$)} with {\\sf nspecies(3)}.\n\n\\boldsection{Validation schemes.}\nVarious statistical analyses and data curation procedures are employed\nby {\\small \\AFLOWHULLtitle}\\ to maximize fidelity.\nAt a minimum, each binary hull must contain 200 structures to ensure\na sufficient sampling size for inference.\nThere is never any guarantee that all stable structures have been identified~\\cite{curtarolo:art54,monsterPGM},\nbut convergence is approached with larger datasets.\nWith continued growth of {\\small LIB3}\\ (ternary phases) and beyond, higher dimensional parameters will be incorporated,\nthough it is expected that the parameters are best defined along tie-lines (\\nobreak\\mbox{\\it vs.}\\ tie-surfaces).\nA comprehensive list of available alloys and structure counts are included in the\nSupporting Information of Reference~\\cite{curtarolo:art146}.\n\n\\boldsection{Outlier detection.} In addition to having been calculated with a standard set of parameters~\\cite{curtarolo:art104},\ndatabase entries\nshould also be well-converged.\nPrior to the injection of new entries into the {\\sf \\AFLOW.org}\\ database,\nvarious verification tests are employed to ensure convergence, including an analysis of the\nrelaxed structure's stress tensor~\\cite{aflux}.\nIssues stemming from poor convergence and failures in the functional parameterization~\\cite{curtarolo:art54,curtarolo:art113}\ncan change the topology of the convex hull,\nresulting in contradictions with experiments.\nHence, an outlier detection algorithm is applied before the hull is constructed:\nstructures are classified as outliers and discarded if\nthey have energies that fall well below the first\nquartile by a multiple of the interquartile range (conservatively set to 3.25 by default)~\\cite{Miller_QJEPSA_1991}.\nOnly points existing in the lower half-space (phases stable against end-members)\nare considered for the outlier analysis, and hence systems need to show\nsome miscibility, \\nobreak\\mbox{\\it i.e.}, at least four points for a proper interquartile range determination.\nDespite its simplicity, the interquartile range is the preferred estimate of scale\nover other measures such as the standard deviation or the median absolute deviation,\nwhich require knowledge of the underlying distribution (normal or otherwise)~\\cite{Leys_JESP_2013}.\nAn example hull (Al-Co) showing an outlier is plotted in Figure~\\ref{fig:art146:hull_analyses}(a)\nand the corrected hull with the outlier removed is presented in Figure~\\ref{fig:art146:hull_analyses}(b).\n\n\\boldsection{Duplicate detection.} A procedure for identifying duplicate entries is also employed.\nBy database construction, near-exact duplicates of elemental phases exist in {\\small LIB2},\nwhich is created spanning the full range of compositions for each alloy system (including elemental phases).\nThese degenerate entries are detected and removed by comparing composition, prototype,\nand formation enthalpy.\nOther structures may have been created distinctly, but converge to duplicates\nvia structural relaxation.\nThese equivalent structures are detected via {\\small AFLOW-XTAL-MATCH}\\\n({\\small AFLOW}\\ crys\\underline{tal} \\underline{match})~\\cite{aflow_compare_2018},\nwhich determines structural\/material uniqueness via the Burzlaff criteria~\\cite{Burzlaff_ActaCrystA_1997}.\nTo compare two crystals, a commensurate representation between structures is resolved by\n\\textbf{i.} identifying common unit cells,\n\\textbf{ii.} exploring cell orientations and origin choices,\nand \\textbf{iii.} matching atomic positions.\nFor each description, the structural similarity is measured by\na composite misfit quantity based on the lattice deviations and mismatch of the mapped atomic positions,\nwith a match occurring for sufficiently small misfit values ($<0.1$).\nDepending on the size of the structures, the procedure can be quite expensive,\nand only applied to find duplicate stable structures.\nCandidates are first screened by composition, space group, and\nformation enthalpies (must be within 15~meV\/atom of the relevant stable configuration).\n{By identifying duplicate stable phases, {\\small \\AFLOWHULLtitle}\\ can\ncross-reference the {\\sf \\AFLOW.org}\\ {\\small ICSD}\\ (\\underline{I}norganic \\underline{C}rystal \\underline{S}tructure \\underline{D}atabase)\ncatalog~\\cite{ICSD,ICSD3}{} to reveal whether the structure has already been observed.}\nThe analysis is depicted in Figure~\\ref{fig:art146:hull_analyses}(c), where\nthe Te-Zr convex hull is plotted with the \\verb|compound| labels replaced with the\ncorresponding {\\small ICSD}\\ number designation.\n\n\\boldsection{Thermodynamic descriptors.}\nA wealth of properties can be extracted from the convex hull construction beyond\na simple determination of stable\/unstable phases.\nFor unstable structures, the energetic vertical-distance to the hull $\\Delta H_{\\mathrm{f}}$,\ndepicted in Figure~\\ref{fig:art146:hull_analyses}(d), serves as a useful metric for quasi-stability.\n$\\Delta H_{\\mathrm{f}}$ is the magnitude of the energy driving the decomposition reaction.\nWithout the temperature and pressure contributions to the energy,\nnear-stable structures should also be considered \\text{(meta-)stable} candidates,\n\\nobreak\\mbox{\\it e.g.}, those within $k_{\\mathrm{B}}T=25$~meV (room temperature) of the hull.\nHighly disordered systems can be realized with even larger distances~\\cite{Sato_Science_2006,curtarolo:art113}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig037}\n\\mycaption[Distance to the hull algorithm.]\n{(\\textbf{a}) The correct distance (shown in \\textcolor{pranab_green}{{\\bf green}}) for $d_1$ is the minimum distance of structure $S_1$\nto all hyperplanes defining the convex hull.\nIn case of structure $S_2$, the minimum distance is not $d_2$ (\\textcolor{pranab_green}{{\\bf green}}) line), an artifact of the hyperplane\ndescription for hull facets.\n(\\textbf{b}) Projecting the points to the zero energy line guarantees that all points will lie within the hull,\nthus enabling the use of minimization algorithm to calculate the correct distance.\nThe distance to the hull $d$ is given as the difference of the projected distance $d_2$ from the distance to the zero energy line $d_1$.\nThe image is adapted from Figure A10 in Reference~\\cite{curtarolo:art113}.}\n\\label{fig:art146:hyperplane_confusion}\n\\end{figure}\n\nTo calculate $\\Delta H_{\\mathrm{f}}$ of phase $\\mathbf{p}$ (Equation~\\ref{eq:art146:point}),\n{\\small \\AFLOWHULLtitle}\\ first resolves the energy of the hull $H_{\\mathrm{hull}}$ at\nstoichiometric coordinates $\\mathbf{x}$, and then\nsubtracts it from\nthe phase's formation enthalpy $H_{\\mathrm{f}}$:\n\\begin{equation}\n\\Delta H_{\\mathrm{f}}[\\mathbf{p}]=\\left|H_{\\mathrm{f}}-H_{\\mathrm{hull}}[\\mathbf{x}]\\right|.\n\\label{eq:art146:dist2hull}\n\\end{equation}\nThe procedure is depicted in Figure~\\ref{fig:art146:hull_analyses}(d), which involves\nidentifying the facet (highlighted in \\textcolor{pranab_green}{{\\bf green}}) that encloses $\\mathbf{x}$ and thus defines\n$H_{\\mathrm{hull}}(\\mathbf{x})$.\nHere, the hyperplane description can be misleading (Equations~\\ref{eq:art146:plane_eq}~and~\\ref{eq:art146:hyperplane_normal}) as\nit lacks information about facet boundaries (Figure~\\ref{fig:art146:hyperplane_confusion}).\nThe enclosing facet is identified as that which\nminimizes the distance to the zero $H_{\\mathrm{f}}$ tie-line at $\\mathbf{x}$:\n\\begin{equation}\nH_{\\mathrm{hull}}[\\mathbf{x}]=-\\min_{\\mathrm{facets}\\in \\mathrm{hull}}\\left|n_N^{-1} \\left(D + \\sum_{i=1}^{N-1} n_i x_i\\right)\\right|.\n\\label{eq:art146:energy_hull}\n\\end{equation}\nVertical facets and those showing hyper-collinearity (having no content) are excluded from the calculation.\n\nWith the appropriate facet identified, the $l$ coefficients of the balanced decomposition reaction\nare derived to yield the full equation.\nThe decomposition of an $N$-ary phase into $l-1$ stable phases\ndefines an $\\left(l \\times N\\right)$-dimensional chemical composition matrix $\\mathbf{C}$,\nwhere $C_{j,i}$ is the atomic concentration of the $i$-species\nof the $j$-phase (the first of which is the unstable mixed phase).\nTake, for example, the decomposition of $\\mathrm{Pd}_{2}\\mathrm{Pt}_{3}$\nto $\\mathrm{PdPt}$ and $\\mathrm{PdPt}_{3}$ as presented in Figure~\\ref{fig:art146:hull_analyses}(d):\n\\begin{equation}\nN_{1}~\\mathrm{Pd}_{0.4}\\mathrm{Pt}_{0.6} \\to N_{2}~\\mathrm{Pd}_{0.5}\\mathrm{Pt}_{0.5} + N_{3}~\\mathrm{Pd}_{0.25}\\mathrm{Pt}_{0.75},\n\\label{eq:art146:decomp_reaction}\n\\end{equation}\nwhere $N_{j}$ is the balanced chemical coefficient for the $j$-phase.\nIn this case, $\\mathbf{C}$ is defined as:\n\\begin{equation}\n\\begin{bmatrix}\nx_{\\mathrm{Pd}} \\in \\mathrm{Pd}_{2}\\mathrm{Pt}_{3} & x_{\\mathrm{Pt}} \\in \\mathrm{Pd}_{2}\\mathrm{Pt}_{3} \\\\\n-x_{\\mathrm{Pd}} \\in \\mathrm{PdPt} & -x_{\\mathrm{Pt}} \\in \\mathrm{PdPt} \\\\\n-x_{\\mathrm{Pd}} \\in \\mathrm{PdPt}_{3} & -x_{\\mathrm{Pt}} \\in \\mathrm{PdPt}_{3} \\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n0.4 & 0.6 \\\\\n-0.5 & -0.5 \\\\\n-0.25 & -0.75 \\\\\n\\end{bmatrix},\n\\end{equation}\nwhere a negative sign differentiates the right hand side of the equation from the left.\nReference~\\onlinecite{Thorne_ARXIV_2011} shows that $N_{j}$ can be extracted from the null space of $\\mathbf{C}$.\n{\\small \\AFLOWHULLtitle}\\ accesses the null space via a full $\\mathbf{QR}$ decomposition of $\\mathbf{C}$, specifically employing a general\nHouseholder algorithm~\\cite{trefethen1997numerical}.\nThe last column of the $\\left(l \\times l\\right)$-dimensional $\\mathbf{Q}$ orthogonal matrix spans the null space $\\mathbf{N}$:\n\\begin{equation}\n \\mathbf{Q} =\n\\begin{bmatrix}\n | & | & 0.8111 \\\\\n \\mathbf{q}_{1} & \\mathbf{q}_{2} & 0.4867 \\\\\n | & | & 0.3244 \\\\\n\\end{bmatrix}.\n\\end{equation}\nBy normalizing $\\mathbf{N}$ such that the first element $N_{1}=1$, the approach yields $N_{2}=0.6$ and $N_{3}=0.4$,\nwhich indeed balances Equation~\\ref{eq:art146:decomp_reaction}.\nThese coefficients can be used to verify\nthe decomposition energy\nobserved in Figure~\\ref{fig:art146:hull_analyses}(d).\nThe formation enthalpies of Pd$_{2}$Pt$_{3}$, PdPt, and PdPt$_{3}$ are\n\\mbox{-286~meV\/(10~atoms)}, \\mbox{-72~meV\/(2~atoms)}, and \\mbox{-104~meV\/(4~atoms)}, respectively.\nThe decomposition energy is calculated as:\n\\begin{equation}\n0.6 H_{\\mathrm{f}}\\left[\\mathrm{PdPt}\\right] + 0.4 H_{\\mathrm{f}}\\left[\\mathrm{PdPt}_{3}\\right] - H_{\\mathrm{f}}\\left[\\mathrm{Pd}_{2}\\mathrm{Pt}_{3}\\right]\n= -3~\\mathrm{meV\/atom},\n\\end{equation}\n\nFor a given stable structure, {\\small \\AFLOWHULLtitle}\\ determines the phases with which it is in equilibrium.\nFor instance, PdPt is in two-phase equilibria with Pd$_{3}$Pt as well as\nwith PdPt$_{3}$ (Figure~\\ref{fig:art146:hull_analyses}(d)).\nPhase coexistence plays a key role in defining a descriptor for precipitate-hardened superalloys.\nCandidates are chosen if a relevant composition is in two-phase equilibrium with the host matrix,\nsuggesting that the formation of coherent precipitates in the matrix is feasible~\\cite{Kirklin_ActaMat_2016,curtarolo:art113}.\n\nAn analysis similar to that quantifying instability $\\left(\\Delta H_{\\mathrm{f}}\\right)$\ndetermines the robustness of stable structures.\nThe stability criterion $\\delta_{\\mathrm{sc}}$ is defined as the distance of a stable\nstructure to the pseudo-hull constructed without it\n(Figure~\\ref{fig:art146:hull_analyses}(e)).\nIts calculation is identical to that of $\\Delta H_{\\mathrm{f}}$ for the pseudo-hull (Equations~\\ref{eq:art146:dist2hull}~and~\\ref{eq:art146:energy_hull}).\nThis descriptor quantifies the effect of the structure on the minimum energy surface, as\nwell as the structure's susceptibility to destabilization by a new phase that has yet to be explored.\nAs with the decomposition analysis, $\\delta_{\\mathrm{sc}}$ also serves to anticipate\nthe effects of temperature and pressure on the minimum energy surface.\nThe descriptor played a pivotal role in screening Heusler structures for new magnetic systems~\\cite{curtarolo:art109}.\n$\\delta_{\\mathrm{sc}}$ calls for the recalculation of facets local to the structure\nand all relevant duplicates as well, thus employing the results of the structure comparison\nprotocol.\n\n{\\small \\AFLOWHULLtitle}\\ can also plot the entropic temperature envelopes characterizing nucleation\nin hyper-thermal synthesis methods for binary systems~\\cite{curtarolo:art98}.\nThe entropic temperature is the ratio of the formation enthalpy to the mixing entropy for an ideal solution ---\na simple quantification for the resilience against disorder~\\cite{monsterPGM}.\nThe ideal ``{\\it iso-max-latent-heat}'' lines\nshown in Figure~\\ref{fig:art146:hull_analyses}(f)\ntry to reproduce the phase's capability to absorb latent heat, which can\npromote its nucleation over more stable phases when starting from large\nQ reservoirs\/feedstock.\nThe descriptor successfully predicts the synthesis of SmB$_{6}$ over SmB$_{4}$\nwith hyper-thermal plasma co-sputtering~\\cite{monsterPGM,curtarolo:art98}.\n\n\\subsection{Results}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig038}\n\\mycaption[Excerpt from the Ag-Au-Cd thermodynamic analysis report.]\n{The document is generated by {\\small \\AFLOWHULLtitle}\\ and showcases\nentry-specific data from the {\\sf \\AFLOW.org}\\ database as well as calculated thermodynamic descriptors.\nStructures highlighted in \\textcolor{pranab_green}{{\\bf green}} are structurally equivalent stable structures,\nand those in \\textcolor{orange}{{\\bf orange}} are structurally similar (same relaxed space group).\nThe working document includes a variety of links,\nincluding hyperlinks to the entry page of each phase (see prototypes)\nand links to relevant parts of the report (see decomposition reaction and\n$N$-phase equilibria).}\n\\label{fig:art146:report}\n\\end{figure}\n\n\\boldsection{Analysis output.}\nFollowing the calculation of the convex hull and relevant thermodynamic descriptors,\n{\\small \\AFLOWHULLtitle}\\ generates a {\\small PDF}\\ file summarizing the results.\nIncluded in the {\\small PDF}\\ are \\textbf{i.} an illustration of the convex hull as shown in\nFigure~\\ref{fig:art146:hull_examples} (for binary and ternary systems)~\\cite{pgfplots_manual} and\n\\textbf{ii.} a report with the aforementioned calculated\nthermodynamic descriptors --- an excerpt is shown in Figure~\\ref{fig:art146:report}.\n\nIn the illustrations, color is used to differentiate points with different enthalpies\nand indicate depth of the facets (3-dimensions).\nThe report includes entry-specific data from the {\\sf \\AFLOW.org}\\ database (prototype, {\\small AUID},\noriginal and relaxed space groups, spin, formation enthalpy $H_{\\mathrm{f}}$, and entropic temperature $T_{\\mathrm{S}}$)\nas well as calculated thermodynamic data (distance to the hull $\\Delta H_{\\mathrm{f}}$,\nthe balanced decomposition reaction for unstable phases, the\nstability criterion $\\delta_{\\mathrm{sc}}$ for stable phases, and\nphases in coexistence).\nStable phases (and those that are structurally equivalent) are highlighted in \\textcolor{pranab_green}{{\\bf green}},\nand similar phases (comparing relaxed space groups) are highlighted in \\textcolor{orange}{{\\bf orange}}.\nLinks are also incorporated in the report, including external\nhyperlinks to entry pages on {\\sf \\AFLOW.org}\\ (see prototypes) and internal\nlinks to relevant parts of the report (see decomposition reaction and $N$-phase equilibria).\nInternal links are also included on the convex hull illustration (see Supporting Information of Reference~\\cite{curtarolo:art146}).\nThe information is provided in the form of plain text and {\\small JSON}\\ files.\nKeys and format are explained in the Supporting Information.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.95\\linewidth]{fig039}\n\\mycaption[The convex hull web application powered by {\\small \\AFLOWHULLtitle}.]\n{(\\textbf{a}) An example 2-dimensional convex hull illustration (Mo-Ti).\n(\\textbf{b}) An example 3-dimensional convex hull illustration (Fe-Rh-Zr).\n(\\textbf{c}) The information component of the hull application.\nPertinent thermodynamic data for selected points is displayed within the grid of cards.\nEach card includes a link to the {\\sf \\AFLOW.org}\\ entry page and the option to remove a point.\nAs points are selected within the visualization, more cards will be added to the grid.\n(\\textbf{d}) The comparison component of the hull application.\nEach hull visualization is displayed as part of a grid of cards.\nFrom this page, new hulls can be added to the store by typing a query in the search box (sidebar).}\n\\label{fig:art146:hull_app}\n\\end{figure}\n\n\\boldsection{Web application.}\nA modern web application has been developed to provide an enhanced, command-line-free platform for {\\small \\AFLOWHULLtitle}.\nThe project includes a rich feature set consisting of binary and ternary convex\nhull visualizations, {\\sf \\AFLOW.org}\\ entry data retrieval, and a convex hull comparison interface.\nThe application is divided into four components: the periodic table, the visualization viewport,\nthe selected entries list, and the comparison page.\n\nThe periodic table component is initially displayed.\nHulls can be queried by selecting\/typing in the elemental combination.\nAs elements are added to the search, the periodic table reacts to the query depending on the\nreliability of the hull:\n\\textcolor{pranab_green}{{\\bf green}} (fully reliable, $N_{\\mathrm{entries}} \\geq 200$),\n\\textcolor{orange}{{\\bf orange}} (potentially reliable, $100 \\leq N_{\\mathrm{entries}} < 200$),\n\\textcolor{pranab_red}{{\\bf red}} (unreliable, $N_{\\mathrm{entries}} < 100$), and\n\\textcolor{gray}{\\bf gray} (unavailable, $N_{\\mathrm{entries}} =0$).\nEach new hull request triggers a fresh data download and analysis,\noffering the most up-to-date results given that new calculations are injected into the\n{\\sf \\AFLOW.org}\\ repository daily.\nOnce the analysis is performed and results are retrieved,\nthe application loads the visualization viewport\nprompting a redirect to the {\\small URL}\\ endpoint of the selected hull, \\nobreak\\mbox{\\it e.g.}, {\\sf \/hull\/AlHfNi}.\nThe {\\small URL}\\ is ubiquitous and can be shared\/cited.\n\nWhen a binary convex hull is selected, the viewport reveals a traditional\n2-dimensional plot (Figure~\\ref{fig:art146:hull_app}(a)),\nwhile a ternary hull yields a 3-dimensional visualization (Figure~\\ref{fig:art146:hull_app}(b)).\nThe scales of both are tunable, and the 3-dimensional visualization offers\nmouse-enabled pan and zoom.\n\nCommon to both types is the ability to select and highlight points.\nWhen a point is selected, its name will appear within the sidebar.\nThe information component is populated with a grid of cards containing properties of each\nselected point (entry), including a link to the {\\sf \\AFLOW.org}\\ entry page (Figure~\\ref{fig:art146:hull_app}(c)).\n\nThe application environment stores all previously selected hulls,\nwhich are retrievable via the hull comparison component (Figure~\\ref{fig:art146:hull_app}(d)).\nOn this page each hull visualization is displayed as a card on a grid.\nThis grid serves as both a history and a means to compare hulls.\n\n\\begin{table}[tp]\\centering\n\\mycaption[The 25 binary phases predicted to be most stable by {\\small \\AFLOWHULLtitle}.]\n{Phases with equivalent structures in the {\\small AFLOW}\\ {\\small ICSD}\\ catalog are excluded.\nThe list is sorted by the absolute value ratio between the stability criterion $\\left(\\delta_{\\mathrm{sc}}\\right)$\nand the formation enthalpy $\\left(H_{\\mathrm{f}}\\right)$ (shown as a percentage).\n${}^{\\dagger}$ indicates no binary phase diagram is available on the\n{\\small ASM}\\ Alloy Phase Diagram database~\\cite{ASMAlloyInternational}.\n{\\small POCC}\\ denotes a \\underline{p}artially-\\underline{occ}upied (disordered) structure~\\cite{curtarolo:art110}.\nComparisons with the {\\small ASM}\\ database include phases that are observed at high temperatures and pressures.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|l|r|r|r|R{4.75in}}\ncompound & {\\small AUID} & relaxed space group & $\\left|\\delta_{\\mathrm{sc}}\/H_{\\mathrm{f}}\\right|$ & Figure & comparison with {\\small ASM}\\ Alloy Phase Diagrams~\\cite{ASMAlloyInternational} \\\\\n\\hline\n\\href{http:\/\/aflow.org\/material.php?id=aflow:38ecc639e4504b9d}{Hf$_{5}$Pb}$^{\\dagger}$ & \\texttt{aflow:38ecc639e4504b9d} & $P4\/mmm~\\#123$ & 78\\% & \\ref{fig:art146:HfPb_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:11ba11a3ee157f2e}{AgIn$_{3}$} & \\texttt{aflow:11ba11a3ee157f2e} & $P6_{3}\/mmc~\\#194$ & 54\\% & \\ref{fig:art146:AgIn_binary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:b60c1f9a1528ba5b}{AgIn$_{2}$} (space group $I4\/mcm$, $\\Delta H_{\\mathrm{f}}$ = 53 meV\/atom) and \\href{http:\/\/aflow.org\/material.php?id=aflow:d30bd203dd3b4049}{In} (space group $I4\/mmm$) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1da75eb5f31b6dd5}{Hf$_{3}$In$_{4}$}$^{\\dagger}$ & \\texttt{aflow:1da75eb5f31b6dd5} & $P4\/mbm~\\#127$ & 45\\% & \\ref{fig:art146:HfIn_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:66dda41a34fe3ad6}{AsTc$_{2}$}$^{\\dagger}$ & \\texttt{aflow:66dda41a34fe3ad6} & $C2\/m~\\#12$ & 41\\% & \\ref{fig:art146:AsTc_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:57e1a1246f813f27}{MoPd$_{8}$} & \\texttt{aflow:57e1a1246f813f27} & $I4\/mmm~\\#139$ & 40\\% & \\ref{fig:art146:MoPd_binary_hull_supp} & composition not found, nearest are Mo$_{0.257}$Pd$_{0.743}$ (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure) and \\href{http:\/\/aflow.org\/material.php?id=aflow:53b1a8ec286d7fe5}{Pd} (space group $Fm\\overline{3}m$) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:32051219452f8e0f}{Ga$_{4}$Tc}$^{\\dagger}$ & \\texttt{aflow:32051219452f8e0f} & $Im\\overline{3}m~\\#229$ & 39\\% & \\ref{fig:art146:GaTc_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:7bd140d7b4c65bc1}{Pd$_{8}$V} & \\texttt{aflow:7bd140d7b4c65bc1} & $I4\/mmm~\\#139$ & 36\\% & \\ref{fig:art146:PdV_binary_hull_supp} & composition not found, nearest are V$_{0.1}$Pd$_{0.9}$ (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure) and \\href{http:\/\/aflow.org\/material.php?id=aflow:4c0207df2fbbd51e}{VPd$_{3}$} (space group $I4\/mmm$, $\\Delta H_{\\mathrm{f}}$ = 5 meV\/atom) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:e7ed70c4711eb718}{InSr$_{3}$} & \\texttt{aflow:e7ed70c4711eb718} & $P4\/mmm~\\#123$ & 35\\% & \\ref{fig:art146:InSr_binary_hull_supp} & composition not found, nearest are Sr$_{28}$In$_{11}$ (space group $Imm2$) and \\href{http:\/\/aflow.org\/material.php?id=aflow:cb9aeb10d6379029}{Sr} (space group $Fm\\overline{3}m$) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:f5cc5eaf65e692a9}{CoNb$_{2}$} & \\texttt{aflow:f5cc5eaf65e692a9} & $I4\/mcm~\\#140$ & 35\\% & \\ref{fig:art146:CoNb_binary_hull_supp} & composition not found, nearest are Nb$_{6.7}$Co$_{6.3}$ (space group $R\\overline{3}m$, {\\small POCC}\\ structure) and Nb$_{0.77}$Co$_{0.23}$ (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:6ee057decaf093d0}{Ag$_{3}$In$_{2}$} & \\texttt{aflow:6ee057decaf093d0} & $Fdd2~\\#43$ & 34\\% & \\ref{fig:art146:AgIn_binary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:89453842555b9d95}{Ag$_{9}$In$_{4}$} (space group $P\\overline{4}3m$, $\\Delta H_{\\mathrm{f}}$ = 21 meV\/atom) and \\href{http:\/\/aflow.org\/material.php?id=aflow:b60c1f9a1528ba5b}{AgIn$_{2}$} (space group $I4\/mcm$, $\\Delta H_{\\mathrm{f}}$ = 53 meV\/atom) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:360240dae753fec6}{AgPt} & \\texttt{aflow:360240dae753fec6} & $P\\overline{6}m2~\\#187$ & 34\\% & \\ref{fig:art146:AgPt_binary_hull_supp} & polymorph found (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:bd3056780447faf0}{OsY$_{3}$} & \\texttt{aflow:bd3056780447faf0} & $Pnma~\\#62$ & 34\\% & \\ref{fig:art146:OsY_binary_hull_supp} & composition found, one-to-one match \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:96142e32718a5ee0}{RuZn$_{6}$} & \\texttt{aflow:96142e32718a5ee0} & $P4_{1}32~\\#213$ & 33\\% & \\ref{fig:art146:RuZn_binary_hull_supp} & composition found, one-to-one match \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1ba6b4b5c0ed9788}{Ag$_{2}$Zn} & \\texttt{aflow:1ba6b4b5c0ed9788} & $P\\overline{6}2m~\\#189$ & 33\\% & \\ref{fig:art146:AgZn_binary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:46dec61deb1ed379}{Ag} (space group $Fm\\overline{3}m$) and Ag$_{4.5}$Zn$_{4.5}$ (space group $P\\overline{3}$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:87d6637b32224f7b}{MnRh} & \\texttt{aflow:87d6637b32224f7b} & $Pm\\overline{3}m~\\#221$ & 32\\% & \\ref{fig:art146:MnRh_binary_hull_supp} & \\href{http:\/\/aflow.org\/material.php?id=aflow:19c39238f5d3feb5}{polymorph} found (space group $P4\/mmm$, $\\Delta H_{\\mathrm{f}}$ = 156 meV\/atom) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:f08f2f61de18aa61}{AgNa$_{2}$} & \\texttt{aflow:f08f2f61de18aa61} & $I4\/mcm~\\#140$ & 32\\% & \\ref{fig:art146:AgNa_binary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:a174f130a5b9b61f}{NaAg$_{2}$} (space group $Fd\\overline{3}m$, $\\Delta H_{\\mathrm{f}}$ = 208 meV\/atom) and \\href{http:\/\/aflow.org\/material.php?id=aflow:95da3ef7fcc58eea}{Na} (space group $R\\overline{3}m$) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:7ce4fcc3660c16cf}{BeRe$_{2}$} & \\texttt{aflow:7ce4fcc3660c16cf} & $I4\/mcm~\\#140$ & 31\\% & \\ref{fig:art146:BeRe_binary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:2bb092148157834d}{Be$_{2}$Re} (space group $P6_{3}\/mmc$) and \\href{http:\/\/aflow.org\/material.php?id=aflow:47d6720be60b12f3}{Re} (space group $P6_{3}\/mmc$) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:e94ab366799a008c}{As$_{2}$Tc}$^{\\dagger}$ & \\texttt{aflow:e94ab366799a008c} & $C2\/m~\\#12$ & 30\\% & \\ref{fig:art146:AsTc_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:eec0d7b6b0d1dfa0}{Be$_{2}$Mn}$^{\\dagger}$ & \\texttt{aflow:eec0d7b6b0d1dfa0} & $P6_{3}\/mmc~\\#194$ & 30\\% & \\ref{fig:art146:BeMn_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:6f3f5b696f5aa391}{AgAu} & \\texttt{aflow:6f3f5b696f5aa391} & $P4\/mmm~\\#123$ & 29\\% & \\ref{fig:art146:AgAu_binary_hull_supp} & polymorph found (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:ca051dbe25c55b92}{Nb$_{5}$Re$_{24}$} & \\texttt{aflow:ca051dbe25c55b92} & $I\\overline{4}3m~\\#217$ & 29\\% & \\ref{fig:art146:NbRe_binary_hull_supp} & composition not found, nearest are Nb$_{0.25}$Re$_{0.75}$ (space group $I\\overline{4}3m$, {\\small POCC}\\ structure) and Nb$_{0.01}$Re$_{0.99}$ (space group $P6_{3}\/mmc$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:a9daa69940d3a59a}{La$_{3}$Os}$^{\\dagger}$ & \\texttt{aflow:a9daa69940d3a59a} & $Pnma~\\#62$ & 28\\% & \\ref{fig:art146:LaOs_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:8ce84acfd6f9ea44}{Be$_{5}$Pt} & \\texttt{aflow:8ce84acfd6f9ea44} & $F\\overline{4}3m~\\#216$ & 28\\% & \\ref{fig:art146:BePt_binary_hull_supp} & composition found, one-to-one match \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:487f7cf6c3fb13f0}{Ir$_{8}$Ru} & \\texttt{aflow:487f7cf6c3fb13f0} & $I4\/mmm~\\#139$ & 27\\% & \\ref{fig:art146:IrRu_binary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:1513b1faeafa2d61}{Ir} (space group $Fm\\overline{3}m$) and Ru$_{0.3}$Ir$_{0.7}$ (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:66af8171e22dc212}{InK} & \\texttt{aflow:66af8171e22dc212} & $R\\overline{3}m~\\#166$ & 27\\% & \\ref{fig:art146:InK_binary_hull_supp} & composition not found, nearest are K$_{8}$In$_{11}$ (space group $R\\overline{3}c$) and \\href{http:\/\/aflow.org\/material.php?id=aflow:a9c9107790b0344c}{K} (space group $Im\\overline{3}m$) \\\\\n\\end{tabular}}\n\\label{tab:art146:stable_binaries}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[The 25 ternary phases predicted to be most stable by {\\small \\AFLOWHULLtitle}.]\n{Phases with equivalent structures in the {\\small AFLOW}\\ {\\small ICSD}\\ catalog are excluded.\nThe list is sorted by the absolute value ratio between the stability criterion $\\left(\\delta_{\\mathrm{sc}}\\right)$\nand the formation enthalpy $\\left(H_{\\mathrm{f}}\\right)$ (shown as a percentage).\n${}^{\\dagger}$ indicates no ternary phase diagram is available on the\n{\\small ASM}\\ Alloy Phase Diagram database~\\cite{ASMAlloyInternational},\nwhile ${}^{\\ddagger}$ indicates all three relevant binaries are available.\n{\\small POCC}\\ denotes a \\underline{p}artially-\\underline{occ}upied (disordered) structure~\\cite{curtarolo:art110}.\nComparisons with the {\\small ASM}\\ database include phases that are observed at high temperatures and pressures.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|l|r|r|r|R{4.75in}}\ncompound & {\\small AUID} & relaxed space group & $\\left|\\delta_{\\mathrm{sc}}\/H_{\\mathrm{f}}\\right|$ & Figure & comparison with {\\small ASM}\\ Alloy Phase Diagrams~\\cite{ASMAlloyInternational} \\\\\n\\hline\n\\href{http:\/\/aflow.org\/material.php?id=aflow:df0cdf0f1ad3110d}{MgSe$_{2}$Zn$_{2}$}$^{\\dagger}$ & \\texttt{aflow:df0cdf0f1ad3110d} & $Fmmm~\\#69$ & 58\\% & \\ref{fig:art146:MgSeZn_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Mg-Se) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:8c51c7ab71f25d11}{Be$_{4}$OsTi}$^{\\dagger}$ & \\texttt{aflow:8c51c7ab71f25d11} & $F\\overline{4}3m~\\#216$ & 38\\% & \\ref{fig:art146:BeOsTi_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Os) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:4e5711451dc4b601}{Be$_{4}$OsV}$^{\\dagger}$ & \\texttt{aflow:4e5711451dc4b601} & $F\\overline{4}3m~\\#216$ & 38\\% & \\ref{fig:art146:BeOsV_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Os) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1684c02e75b0d950}{Ag$_{2}$InZr} & \\texttt{aflow:1684c02e75b0d950} & $Fm\\overline{3}m~\\#225$ & 35\\% & \\ref{fig:art146:AgInZr_ternary_hull_supp} & composition not found, nearest are Ag$_{0.8}$In$_{0.2}$ (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure), Zr$_{0.5}$In$_{0.5}$ (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure), and AgZr$_{5}$In$_{3}$ (space group $P6_{3}\/mcm$) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:b85addbb42c47ae9}{Be$_{4}$RuTi}$^{\\dagger \\ddagger}$ & \\texttt{aflow:b85addbb42c47ae9} & $F\\overline{4}3m~\\#216$ & 32\\% & \\ref{fig:art146:BeRuTi_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:cabd6decf5b6c991}{Be$_{4}$FeTi}$^{\\dagger \\ddagger}$ & \\texttt{aflow:cabd6decf5b6c991} & $F\\overline{4}3m~\\#216$ & 29\\% & \\ref{fig:art146:BeFeTi_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:7010472778d429f7}{Be$_{4}$ReV}$^{\\dagger \\ddagger}$ & \\texttt{aflow:7010472778d429f7} & $F\\overline{4}3m~\\#216$ & 29\\% & \\ref{fig:art146:BeReV_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:e4cc9eea02d9d303}{Ba$_{2}$RhZn}$^{\\dagger}$ & \\texttt{aflow:e4cc9eea02d9d303} & $Cm~\\#8$ & 29\\% & \\ref{fig:art146:BaRhZn_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Ba-Rh) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:2ace5c5383f8ea10}{Be$_{4}$HfOs}$^{\\dagger}$ & \\texttt{aflow:2ace5c5383f8ea10} & $F\\overline{4}3m~\\#216$ & 27\\% & \\ref{fig:art146:BeHfOs_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Os) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:de79192a0c4e751f}{Be$_{4}$ReTi}$^{\\dagger \\ddagger}$ & \\texttt{aflow:de79192a0c4e751f} & $F\\overline{4}3m~\\#216$ & 27\\% & \\ref{fig:art146:BeReTi_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:d484b95ba623f9f7}{Be$_{4}$TcV}$^{\\dagger}$ & \\texttt{aflow:d484b95ba623f9f7} & $F\\overline{4}3m~\\#216$ & 27\\% & \\ref{fig:art146:BeTcV_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Tc) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:c13660b990eb9570}{Be$_{4}$TcTi}$^{\\dagger}$ & \\texttt{aflow:c13660b990eb9570} & $F\\overline{4}3m~\\#216$ & 27\\% & \\ref{fig:art146:BeTcTi_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Tc) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:07840d9e13694f7e}{Be$_{4}$RuV}$^{\\dagger \\ddagger}$ & \\texttt{aflow:07840d9e13694f7e} & $F\\overline{4}3m~\\#216$ & 27\\% & \\ref{fig:art146:BeRuV_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:5778f3b725d5f850}{AsCoTi}$^{\\dagger \\ddagger}$ & \\texttt{aflow:5778f3b725d5f850} & $F\\overline{4}3m~\\#216$ & 26\\% & \\ref{fig:art146:AsCoTi_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:9a10dd8a8224e158}{Be$_{4}$MnTi}$^{\\dagger}$ & \\texttt{aflow:9a10dd8a8224e158} & $F\\overline{4}3m~\\#216$ & 26\\% & \\ref{fig:art146:BeMnTi_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Mn) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:de412213bdefbd14}{Be$_{4}$OsZr}$^{\\dagger}$ & \\texttt{aflow:de412213bdefbd14} & $F\\overline{4}3m~\\#216$ & 26\\% & \\ref{fig:art146:BeOsZr_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Os) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:07bcc161f57da109}{Be$_{4}$IrTi}$^{\\dagger}$ & \\texttt{aflow:07bcc161f57da109} & $F\\overline{4}3m~\\#216$ & 26\\% & \\ref{fig:art146:BeIrTi_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Ir) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:90b98cdcd6eea146}{Mg$_{2}$ScTl}$^{\\dagger}$ & \\texttt{aflow:90b98cdcd6eea146} & $P4\/mmm~\\#123$ & 25\\% & \\ref{fig:art146:MgScTl_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Sc-Tl) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:086b4a89f8d62804}{Be$_{4}$MnV}$^{\\dagger}$ & \\texttt{aflow:086b4a89f8d62804} & $F\\overline{4}3m~\\#216$ & 25\\% & \\ref{fig:art146:BeMnV_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Mn) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:0595e3d45678a85c}{AuBe$_{4}$Cu}$^{\\dagger \\ddagger}$ & \\texttt{aflow:0595e3d45678a85c} & $F\\overline{4}3m~\\#216$ & 25\\% & \\ref{fig:art146:AuBeCu_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:d7fed8d4996290f4}{BiRhZr}$^{\\dagger \\ddagger}$ & \\texttt{aflow:d7fed8d4996290f4} & $F\\overline{4}3m~\\#216$ & 24\\% & \\ref{fig:art146:BiRhZr_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:80bf8ad33a5bb33b}{LiMg$_{2}$Zn} & \\texttt{aflow:80bf8ad33a5bb33b} & $Fm\\overline{3}m~\\#225$ & 21\\% & \\ref{fig:art146:LiMgZn_ternary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:a66c0917c0faf13f}{Li} (space group $Im\\overline{3}m$, $\\Delta H_{\\mathrm{f}}$ = 2 meV\/atom), \\href{http:\/\/aflow.org\/material.php?id=aflow:b83b8ffef10abaa0}{Mg} (space group $P6_{3}\/mmc$), and Li$_{0.77}$MgZn$_{1.23}$ (space group $Fd\\overline{3}m$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:faa814b1222e8aea}{Be$_{4}$RhTi}$^{\\dagger}$ & \\texttt{aflow:faa814b1222e8aea} & $F\\overline{4}3m~\\#216$ & 21\\% & \\ref{fig:art146:BeRhTi_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Rh) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:26cc4fc55644b0d8}{AuCu$_{4}$Hf}$^{\\dagger \\ddagger}$ & \\texttt{aflow:26cc4fc55644b0d8} & $F\\overline{4}3m~\\#216$ & 21\\% & \\ref{fig:art146:AuCuHf_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:ab57b1ae74f4c6d4}{Mg$_{2}$SeZn$_{2}$}$^{\\dagger}$ & \\texttt{aflow:ab57b1ae74f4c6d4} & $Fmmm~\\#69$ & 21\\% & \\ref{fig:art146:MgSeZn_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Mg-Se) \\\\\n\\end{tabular}}\n\\label{tab:art146:stable_ternaries}\n\\end{table}\n\n\\boldsection{Candidates for synthesis.}\nTo demonstrate the capability of {\\small \\AFLOWHULLtitle}, all binary and ternary systems in the {\\sf \\AFLOW.org}\\ repository\nare explored for ones yielding well-converged thermodynamic properties.\nSince reliability constraints are built-in, {no pre-filtering is required and}\nall potential elemental combinations {are attempted.}\nAcross all catalogs present in the database, there exist materials composed of\n86 elements, including:\nH, He, Li, Be, B, C, N, O, F, Ne, Na, Mg, Al, Si, P, S, Cl,\nAr, K, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, Ge, As, Se, Br, Kr, Rb,\nSr, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Cd, In, Sn, Sb, Te, I, Xe, Cs, Ba, La,\nCe, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Hf, Ta, W, Re, Os, Ir,\nPt, Au, Hg, Tl, Pb, Bi, Ac, Th, and Pa.\nHulls are eliminated if systems\n\\textbf{i.} are unreliable based on count (fewer than 200 entries among binary combinations), and\n\\textbf{ii.} show significant immiscibility (fewer than 50 points below the zero $H_{\\mathrm{f}}$ tie-line).\nTernary systems are further screened for those containing ternary ground-state structures.\nThe analysis resulted in the full thermodynamic characterization of 493\\ binary and 873\\ ternary systems.\nThe results are provided in the Supporting Information of Reference~\\cite{curtarolo:art146}.\n\nLeveraging the {\\small JSON}\\ outputs, reliable hulls are further explored for new stable phases.\nPhases are first screened (eliminated) if an equivalent structure exists in the {\\sf \\AFLOW.org}\\ {\\small ICSD}\\\ncatalog, and candidates are sorted by their relative stability criterion,\n\\nobreak\\mbox{\\it i.e.}, $\\left|\\delta_{\\mathrm{sc}}\/H_{\\mathrm{f}}\\right|$.\nThis dimensionless quantity captures the effect of the phase on the minimum energy\nsurface relative to its depth, enabling comparisons across hulls.\n{An example Python script that performs this analysis is provided in the Supporting Information.}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig040}\n\\mycaption[Illustration of the most prevalent stable ternary structures.]\n{(\\textbf{a}) The conventional cubic cell of the ``quaternary-Heusler'' structure, LiMgPdSn~\\cite{Eberz_ZfNaturfB_35_1341_1980,anrl_pt2_2018}.\nEach species occupies a Wyckoff site of space group $F\\overline{4}3m~\\#216$:\nSn (purple) (4a),\nMg (yellow) (4b),\nPd (gray) (4c), and\nLi (blue) (4d).\n(\\textbf{b}) The conventional cubic cell of the Heusler structure, here represented by\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1684c02e75b0d950}{Ag$_{2}$InZr}.\nEach species occupies a Wyckoff site of space group $Fm\\overline{3}m~\\#225$:\nIn (pink) (4a), Zr (green) (4b), Ag (light gray) (8c).\n(\\textbf{c}) The conventional cubic cell of the half-Heusler $C1_{b}$ structure, here represented by\n\\href{http:\/\/aflow.org\/material.php?id=aflow:5778f3b725d5f850}{AsCoTi}.\nEach species occupies a Wyckoff site of space group $F\\overline{4}3m~\\#216$:\nTi (light blue) (4a), As (purple) (4b), Co (dark blue) (4c).\nThe (4d) site is empty.\n(\\textbf{d}) The conventional cubic cell of the $C15_{b}$-type crystal, here represented by\n\\href{http:\/\/aflow.org\/material.php?id=aflow:8c51c7ab71f25d11}{Be$_{4}$OsTi}.\nEach species occupies a Wyckoff site of space group $F\\overline{4}3m~\\#216$:\nTi (light blue) (4a),\nOs (brown) (4c), and\nBe (light green) (8e).\nThe (4d) site is empty, and the Be atoms form a tetrahedron centered around the (4b) site of (\\textbf{a}).}\n\\label{fig:art146:heuslers}\n\\end{figure}\n\nThe top 25 most stable binary and ternary phases are presented in Tables~\\ref{tab:art146:stable_binaries}\nand \\ref{tab:art146:stable_ternaries}, respectively, for which extended analysis is performed\nbased on information stored in the {\\small ASM}\\ (\\underline{A}merican \\underline{S}ociety for \\underline{M}etals)\nAlloy Phase Diagram database~\\cite{ASMAlloyInternational}.\nThe {\\small ASM}\\ database is the largest of its kind, aggregating a wealth of experimental phase diagram information:\n40,300 binary and ternary alloy phase diagrams from over 9,000 systems.\nUpon searching the {\\small ASM}\\ website, many binary systems from Table~\\ref{tab:art146:stable_binaries}\nare unavailable and denoted by the symbol ${}^{\\dagger}$.\nAmong those that are available, some stable phases have already been observed,\nincluding\n\\href{http:\/\/aflow.org\/material.php?id=aflow:bd3056780447faf0}{OsY$_{3}$},\n\\href{http:\/\/aflow.org\/material.php?id=aflow:96142e32718a5ee0}{RuZn$_{6}$},\nand\n\\href{http:\/\/aflow.org\/material.php?id=aflow:8ce84acfd6f9ea44}{Be$_{5}$Pt}.\nFor\n\\href{http:\/\/aflow.org\/material.php?id=aflow:360240dae753fec6}{AgPt},\n\\href{http:\/\/aflow.org\/material.php?id=aflow:87d6637b32224f7b}{MnRh},\nand\n\\href{http:\/\/aflow.org\/material.php?id=aflow:6f3f5b696f5aa391}{AgAu},\nthe composition is successfully predicted,\nbut polymorphs (structurally distinct phases) are observed instead.\nFor all other phases on the list, the composition has not been observed.\nThe discrepancy may be isolated to the phase, or indicative of a more extreme\ncontradiction in the topology of the hull, and thus, nearby phases are also analyzed.\nFor the Be-Re system, though \\href{http:\/\/aflow.org\/material.php?id=aflow:7ce4fcc3660c16cf}{BeRe$_{2}$}\nhas not been observed,\nboth \\href{http:\/\/aflow.org\/material.php?id=aflow:2bb092148157834d}{Be$_{2}$Re}\nand \\href{http:\/\/aflow.org\/material.php?id=aflow:47d6720be60b12f3}{Re}\nare successfully identified.\nMost of the remaining phases show the nearest phase to be a disordered (partially\noccupied) structure, which are excluded from the {\\sf \\AFLOW.org}\\ repository.\nAddressing disorder is a particularly challenging task in \\nobreak\\mbox{\\it ab-initio}\\ studies.\nHowever, recent high-throughput techniques~\\cite{curtarolo:art110} show promise for future investigations\nand will be integrated in future releases of the code.\n\nAmong the most stable ternary phases, only two systems have available\nphase diagrams in the {\\small ASM}\\ database, Ag-In-Zr and Li-Mg-Zn.\nFor the Ag-In-Zr system, the composition of\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1684c02e75b0d950}{Ag$_{2}$InZr}\nis not observed and the nearest stable phases include disordered structures and\nAgZr$_{5}$In$_{3}$, which has not yet been included the {\\sf \\AFLOW.org}\\ repository.\nFor Li-Mg-Zn, the composition of\n\\href{http:\/\/aflow.org\/material.php?id=aflow:80bf8ad33a5bb33b}{LiMg$_{2}$Zn}\nis also not observed and the nearest stable phases include unaries\n\\href{http:\/\/aflow.org\/material.php?id=aflow:a66c0917c0faf13f}{Li},\n\\href{http:\/\/aflow.org\/material.php?id=aflow:b83b8ffef10abaa0}{Mg},\nand a disordered structure.\nAll other ternary systems are entirely unexplored.\nTernary phases with all three binary phase diagrams available\nare denoted with the symbol ${}^{\\ddagger}$, suggesting experimental feasibility.\n\nA striking feature of Table~\\ref{tab:art146:stable_ternaries}\nis that most of the stable structures are\nfound to be in space group $F\\overline{4}3m~\\#216$.\nThis structure has a face-centered cubic lattice with symmetry operations that\ninclude a four-fold rotation about the ${<}001{>}$ axes, a three-fold rotation\nabout the ${<}111{>}$ axes, and no inversion.\nFurther study reveals that these phases, as well as $Fm\\overline{3}m~\\#225$\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1684c02e75b0d950}{Ag$_{2}$InZr}\nand\n\\href{http:\/\/aflow.org\/material.php?id=aflow:80bf8ad33a5bb33b}{LiMg$_{2}$Zn},\ncan be obtained from the ``quaternary-Heusler'' structure,\nLiMgPdSn~\\cite{Eberz_ZfNaturfB_35_1341_1980,anrl_pt2_2018} (Figure~\\ref{fig:art146:heuslers}(a)).\nThe prototype can be considered a $2\\times2\\times2$ supercell of the body-centered cubic structure.\nThe Sn, Mg, Au and Li atoms all occupy different Wyckoff positions of space group\n$F\\overline{4}3m$ and each atom has two sets of nearest neighbors, each four-fold coordinated.\nVarious decorations of these Wyckoff positions generate the other structures:\n\\begin{itemize}\n\\item By decorating two second-neighbor atom sites identically, a Heusler alloy forms\n({\\em Strukturbericht} symbol $L2_{1}$)~\\cite{Bradley_PRSL_A144_340_1934,aflowANRL}.\nFor example, the following substitutions generate\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1684c02e75b0d950}{Ag$_{2}$InZr}\n(Figure~\\ref{fig:art146:heuslers}(b)):\nPd $\\rightarrow$ Ag, Li $\\rightarrow$ Ag,\nSn $\\rightarrow$ In, and Mg $\\rightarrow$ Zr.\nSince the crystal now has an inversion center, the space group becomes\n$Fm\\overline{3}m~\\#225$.\nAs in LiMgPdSn, each atom has two sets of four-fold coordinated nearest neighbors,\neach arranged as a tetrahedron.\nNow, however, one species (Ag) has second-neighbors of the same type.\n\\item By removing the Li atom completely, a half-Heusler forms\n($C1_{b}$)~\\cite{Nowotny_Z_f_Metallk_33_391_1941,aflowANRL}.\nThere are two half-Heusler systems in Table~\\ref{tab:art146:stable_ternaries}:\n\\href{http:\/\/aflow.org\/material.php?id=aflow:5778f3b725d5f850}{AsCoTi}\n(Figure~\\ref{fig:art146:heuslers}(c)) and\n\\href{http:\/\/aflow.org\/material.php?id=aflow:d7fed8d4996290f4}{BiRhZr}.\nThe structure does differ from that of LiMgPdSn and $L2_{1}$,\nas the Ag and Ti atoms are four-fold coordinated, with only Co having the\ncoordination seen in the previous structures.\n\\item The majority of structures in Table~\\ref{tab:art146:stable_ternaries}\nare type $C15_{b}$, prototype AuBe$_{5}$~\\cite{Batchelder_Acta_Crist_11_122_1958,aflowANRL}\n({\\small AFLOW}\\ prototype: \\verb|AB5_cF24_216_a_ce|~\\cite{AB5_cF24_216_a_ce}),\nrepresented by\n\\href{http:\/\/aflow.org\/material.php?id=aflow:8c51c7ab71f25d11}{Be$_{4}$OsTi}\nshown in Figure~\\ref{fig:art146:heuslers}(d).\nCompared to the $C1_{b}$, $C15_{b}$ contains an (8e) Wyckoff position\nforming a tetrahedra centered around the (4b) Wyckoff position.\nReplacing the tetrahedra with a single atom returns the $C1_{b}$ structure.\n\\end{itemize}\nHence, of the 25 most stable ternary structures, 21 are of related structure.\n\nSampling bias likely plays a role in the high prominence of space group $F\\overline{4}3m~\\#216$\nstructures in Table~\\ref{tab:art146:stable_ternaries}, but cannot fully account for the anomaly.\nSpace group $F\\overline{4}3m~\\#216$ constitutes about 17\\% of the {\\small LIB3}\\ catalog,\ncontaining the bulk of the {\\sf \\AFLOW.org}\\ repository (at over 1.5 million ternary systems)\ngenerated largely by small structure prototypes.\nFor context, space group $F\\overline{4}3m~\\#216$ is ranked about twentieth of the\nmost common space groups in the {\\small ICSD}~\\cite{Urusov_JSC_2009},\nappearing in about 1\\% of all entries.\nFurther exploration of larger structure ternary prototypes covering the full range\nof space groups is needed to fully elucidate the nature of this structure's stability.\n\nThe \\mbox{regular-}, inverse-, and half-Heusler prototypes were added to {\\small LIB3}\\\nfor the exploration of new magnets, of which two were discovered~\\cite{curtarolo:art109}.\nThe Heusler set includes more 236,000 structures, most of which remains unexplored.\nThe fully sorted lists of stable binary and ternary phases are presented in the\nSupporting Information of Reference~\\cite{curtarolo:art146}.\n\n\\clearpage\n\n\\subsection{Convex hulls of most stable candidates}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig041}\n\\mycaption{Hf-Pb binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:HfPb_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig042}\n\\mycaption{Ag-In binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AgIn_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig043}\n\\mycaption{Hf-In binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:HfIn_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig044}\n\\mycaption{As-Tc binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AsTc_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig045}\n\\mycaption{Mo-Pd binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:MoPd_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig046}\n\\mycaption{Ga-Tc binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:GaTc_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig047}\n\\mycaption{Pd-V binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:PdV_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig048}\n\\mycaption{In-Sr binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:InSr_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig049}\n\\mycaption{Co-Nb binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:CoNb_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig050}\n\\mycaption{Ag-Pt binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AgPt_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig051}\n\\mycaption{Os-Y binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:OsY_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig052}\n\\mycaption{Ru-Zn binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:RuZn_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig053}\n\\mycaption{Ag-Zn binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AgZn_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig054}\n\\mycaption{Mn-Rh binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:MnRh_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig055}\n\\mycaption{Ag-Na binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AgNa_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig056}\n\\mycaption{Be-Re binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeRe_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig057}\n\\mycaption{Be-Mn binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeMn_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig058}\n\\mycaption{Ag-Au binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AgAu_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig059}\n\\mycaption{Nb-Re binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:NbRe_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig060}\n\\mycaption{La-Os binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:LaOs_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig061}\n\\mycaption{Be-Pt binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BePt_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig062}\n\\mycaption{Ir-Ru binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:IrRu_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig063}\n\\mycaption{In-K binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:InK_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig064}\n\\mycaption{Mg-Se-Zn ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:MgSeZn_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig065}\n\\mycaption{Be-Os-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeOsTi_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig066}\n\\mycaption{Be-Os-V ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeOsV_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig067}\n\\mycaption{Ag-In-Zr ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AgInZr_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig068}\n\\mycaption{Be-Ru-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeRuTi_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig069}\n\\mycaption{Be-Fe-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeFeTi_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig070}\n\\mycaption{Be-Re-V ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeReV_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig071}\n\\mycaption{Ba-Rh-Zn ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BaRhZn_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig072}\n\\mycaption{Be-Hf-Os ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeHfOs_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig073}\n\\mycaption{Be-Re-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeReTi_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig074}\n\\mycaption{Be-Tc-V ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeTcV_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig075}\n\\mycaption{Be-Tc-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeTcTi_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig076}\n\\mycaption{Be-Ru-V ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeRuV_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig077}\n\\mycaption{As-Co-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AsCoTi_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig078}\n\\mycaption{Be-Mn-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeMnTi_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig079}\n\\mycaption{Be-Os-Zr ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeOsZr_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig080}\n\\mycaption{Be-Ir-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeIrTi_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig081}\n\\mycaption{Mg-Sc-Tl ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:MgScTl_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig082}\n\\mycaption{Be-Mn-V ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeMnV_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig083}\n\\mycaption{Au-Be-Cu ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AuBeCu_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig084}\n\\mycaption{Bi-Rh-Zr ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BiRhZr_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig085}\n\\mycaption{Li-Mg-Zn ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:LiMgZn_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig086}\n\\mycaption{Be-Rh-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeRhTi_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig087}\n\\mycaption{Au-Cu-Hf ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AuCuHf_ternary_hull_supp}\n\\end{figure}\n\n\\def\\item {{\\it Description:}\\ }{\\item {{\\it Description:}\\ }}\n\\def\\item {{\\it Type:}\\ }{\\item {{\\it Type:}\\ }}\n\\def\\item {{\\it Units:}\\ }{\\item {{\\it Units:}\\ }}\n\\def\\item {{\\it Similar to:}\\ }{\\item {{\\it Similar to:}\\ }}\n\\def{\\color{blue}{\\item {{\\it Description:}\\ }}}{{\\color{blue}{\\item {{\\it Description:}\\ }}}}\n\\def{\\color{blue}{\\item {{\\it Type:}\\ }}}{{\\color{blue}{\\item {{\\it Type:}\\ }}}}\n\\def{\\color{blue}{\\item {{\\it Similar to:}\\ }}}{{\\color{blue}{\\item {{\\it Similar to:}\\ }}}}\n\n\\subsection{AFLOW-CHULL manual}\n\n\\boldsection{Command-line options.}\n{\\small \\AFLOWHULLtitle}\\ is an integrated module of the {\\small AFLOW}\\ \\nobreak\\mbox{\\it ab-initio}\\ framework\nwhich runs on any {\\small UNIX}-like computer, including those running macOS.\nThe most up-to-date binary can be downloaded from {\\sf aflow.org\/src\/aflow}:\ncurrent version 3.1.207.\n{\\small \\AFLOWHULLtitle}\\ depends on the compiled binary executable and an internet connection,\nas all data is retrieved and analyzed \\textit{in-situ}.\nThe default output option also requires the \\LaTeX\\ package.\nThe results (graphics and {\\small PDF}\\ reports) presented herein are\ncompiled using pdf\\TeX, Version 3.14159265-2.6-1.40.18 (\\TeX\\ Live 2017).\n\n\\vspace{0.5cm}\n\n\\noindent Primary commands:\n\\begin{itemize}\n \\item{\\verb!aflow --chull --alloy=InNiY!}\n \\begin{itemize}\n \\item{Calculates and returns the convex hull for system In-Ni-Y.}\n \\end{itemize}\n \\item{\\verb!aflow --chull --alloy=InNiY! \\\\ \\verb!--distance_to_hull=aflow:375066afdfb5a93f!}\n \\begin{itemize}\n \\item{Calculates and returns the distance to the hull $\\Delta H_{\\mathrm{f}}$ for \\href{http:\/\/aflow.org\/material.php?id=aflow:375066afdfb5a93f}{InNiY$_{4}$}.}\n \\end{itemize}\n \\item{\\verb!aflow --chull --alloy=InNiY! \\\\ \\verb!--stability_criterion=aflow:60a36639191c0af8!}\n \\begin{itemize}\n \\item{Calculates and returns the stability criterion $\\delta_{\\mathrm{sc}}$ for \\href{http:\/\/aflow.org\/material.php?id=aflow:60a36639191c0af8}{InNi$_{4}$Y}.\n The structure and relevant duplicates (if any) are removed to create the pseudo-hull.}\n \\end{itemize}\n \\item{\\verb!aflow --chull --alloy=InNiY --hull_formation_enthalpy=0.25,0.25!}\n \\begin{itemize}\n \\item{Calculates and returns the formation enthalpy of the minimum energy surface at In$_{0.25}$Ni$_{0.25}$Y$_{0.5}$.\n The input composition is specified by implicit coordinates (refer to Equation~\\ref{eq:art146:point}), where the last coordinate\n offers an optional energetic shift.\n }\n \\end{itemize}\n \\item{\\verb!aflow --chull --usage!}\n \\begin{itemize}\n \\item{Prints full set of commands to the screen.}\n \\end{itemize}\n \\item{\\verb!aflow --readme=chull!}\n \\begin{itemize}\n \\item{Prints a verbose manual (commands and descriptions) to the screen.}\n \\end{itemize}\n\\end{itemize}\n\n\\vspace{0.5cm}\n\n\\noindent General options:\n\\begin{myitemize}\n \\item{\\verb!--output=pdf!}\n \\begin{myitemize}\n \\item{Selects the output format. Options include: \\verb|pdf|, \\verb|png|, \\verb|json|, \\verb|txt|, and \\verb|full|. For multiple output, provide a comma-separated value list. A file with the corresponding extension is created, \\nobreak\\mbox{\\it e.g.}, {\\sf aflow\\_InNiY\\_hull.pdf}.}\n \\end{myitemize}\n\\item{\\verb!--destination=$HOME\/!}\n \\begin{myitemize}\n \\item{Sets the output path to {\\sf \\${\\small HOME}}. All output will be redirected to this destination.}\n \\end{myitemize}\n\\item \\verb!--keep=log!\n \\begin{myitemize}\n \\item{Creates a log file with verbose output of the calculation, \\nobreak\\mbox{\\it e.g.}, {\\sf aflow\\_InNiY\\_hull.log}.}\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\noindent Loading options:\n\\begin{myitemize}\n\\item \\verb!--load_library=icsd!\n \\begin{myitemize}\n \\item{Limits the catalogs from which entries are loaded. Options include: \\verb!icsd!, \\verb!lib1!, \\verb!lib2!, and \\verb!lib3!. For multiple catalogs, provide a comma-separated value list.}\n \\end{myitemize}\n\\item \\verb!--load_entries_entry_output!\n \\begin{myitemize}\n \\item{Prints verbose output of the entries loaded. This output is included in the log file by default.}\n \\end{myitemize}\n\\item \\verb!--neglect=aflow:60a36639191c0af8,aflow:3f24d2be765237f1!\n \\begin{myitemize}\n \\item{Excludes individual points from the convex hull calculation.}\n \\end{myitemize}\n\\item \\verb!--see_neglect!\n \\begin{myitemize}\n \\item{Prints verbose output of the entries neglected from the calculation, including ill-calculated entries, duplicates, outliers, and those requested via \\verb!--neglect!.}\n \\end{myitemize}\n\\item \\verb!--remove_extreme_points=-1000!\n \\begin{myitemize}\n \\item{Excludes all points with formation enthalpies below -1000 meV\/atom.}\n \\end{myitemize}\n\\item \\verb!--include_paw_gga!\n \\begin{myitemize}\n \\item{Includes all entries calculated with {\\small PAW}-{\\small GGA}\\ (in addition to those calculated with {\\small PAW}-{\\small PBE}).\n {\\small PAW}-{\\small GGA}\\ refers to the \\underline{G}eneralized \\underline{G}radient \\underline{A}pproximation functional~\\cite{PBE}\n with pseudopotentials calculated with the\n \\underline{p}rojector \\underline{a}ugmented \\underline{w}ave method~\\cite{PAW}.\n This flag is needed to generate Figure~\\ref{fig:art146:hull_analyses}(f).}\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\noindent Analysis options:\n\\begin{myitemize}\n\\item \\verb!--skip_structure_comparison!\n \\begin{myitemize}\n \\item{Avoids determination of structures equivalent to stable phases (speed).}\n \\end{myitemize}\n\\item \\verb!--skip_stability_criterion_analysis!\n \\begin{myitemize}\n \\item{Avoids determination of the stability criterion of stable phases (speed).}\n \\end{myitemize}\n\\item \\verb!--include_skewed_hulls!\n \\begin{myitemize}\n \\item{Proceeds to calculate the hull in the event that it is determined ``skewed'', \\nobreak\\mbox{\\it i.e.},\n the stable elemental phase differs from the reference energy by more than 15~meV\/atom.\n This flag is needed to generate Figure~\\ref{fig:art146:hull_analyses}(f).}\n \\end{myitemize}\n\\item \\verb!--include_unreliable_hulls!\n \\begin{myitemize}\n \\item{Proceeds to calculate the hull in the event that it is determined unreliable (fewer than 200 entries along the binary hulls).}\n \\end{myitemize}\n\\item \\verb!--include_outliers!\n \\begin{myitemize}\n \\item{Includes outliers in the calculation.}\n \\end{myitemize}\n\\item \\verb!--force!\n \\begin{myitemize}\n \\item{Forces an output, ignoring all warnings.}\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\noindent {\\small PDF}\/\\LaTeX\\ options:\n\\begin{myitemize}\n\\item \\verb!--image_only!\n \\begin{myitemize}\n \\item{Creates a {\\small PDF}\\ with the hull illustration only.}\n \\end{myitemize}\n\\item \\verb!--document_only!\n \\begin{myitemize}\n \\item{Creates a {\\small PDF}\\ with the thermodynamic report only. Default for dimensions $N>3$.}\n \\end{myitemize}\n\\item \\verb!--keep=tex!\n \\begin{myitemize}\n \\item{Saves the \\LaTeX\\ input file (deleted by default), allowing for customization of the resulting {\\small PDF}, \\nobreak\\mbox{\\it e.g.}, {\\sf aflow\\_InNiY\\_hull.tex}.}\n \\end{myitemize}\n\\item \\verb!--latex_interactive!\n \\begin{myitemize}\n \\item{Displays the \\LaTeX\\ compilation output and enables interaction with the program.}\n \\end{myitemize}\n\\item \\verb!--plot_iso_max_latent_heat!\n \\begin{myitemize}\n \\item{Plots the entropic temperature envelopes shown in Figure~\\ref{fig:art146:hull_analyses}(f). Limited to binary systems only.}\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\boldsection{{{\\small AFLOW}}rc options.}\nThe {\\sf .aflow.rc} file is a new protocol for specifying {\\small AFLOW}\\ default options.\nThe file emulates the {\\sf .bashrc} script that runs when initializing an interactive environment in\nBash (\\underline{B}ourne \\underline{a}gain \\underline{sh}ell).\nA fresh {\\sf .aflow.rc} file is created in {\\sf \\${\\small HOME}} if one is not already\npresent.\n\n\\noindent Relevant {\\small \\AFLOWHULLtitle}\\ options include:\n\\begin{myitemize}\n\\item \\verb!DEFAULT_CHULL_ALLOWED_DFT_TYPES=\"PAW_PBE\"!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the allowed entries based on \\underline{d}ensity \\underline{f}unctional \\underline{t}heory ({\\small DFT}) calculation type (comma-separated value).\n Options include: \\verb|US|, \\verb|GGA|, \\verb|PAW_LDA|, \\verb|PAW_GGA|, \\verb|PAW_PBE|, \\verb|GW|, and \\verb|HSE06|~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_ALLOW_ALL_FORMATION_ENERGIES=0!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Allows all entries independent of {\\small DFT}\\ calculation type~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_COUNT_THRESHOLD_BINARIES=200!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the minimum number of entries for a reliable binary hull.\n \\item {{\\it Type:}\\ } \\verb|integer|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_PERFORM_OUTLIER_ANALYSIS=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Enables determination of outliers.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_OUTLIER_ANALYSIS_COUNT_THRESHOLD_BINARIES=50!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the minimum number of entries for a reliable outlier analysis.\n Only phases stable with respect to their end-members are considered for the outlier analysis (below the zero $H_{\\mathrm{f}}$ tie-line).\n \\item {{\\it Type:}\\ } \\verb|integer|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_OUTLIER_MULTIPLIER=3.25!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the bounds beyond the interquartile range for which points are considered outliers~\\cite{Miller_QJEPSA_1991}.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_IGNORE_KNOWN_ILL_CONVERGED=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AFLOW}\\ maintains a list of (older) prototypes known to have converged poorly.\n These entries are likely outliers, \\nobreak\\mbox{\\it e.g.}, see prototype $549$ in Figure~\\ref{fig:art146:hull_analyses}(a).\n If this flag is on (\\verb|1|), then the entries are removed before the analysis.\n Turning this flag off (\\verb|0|) is not recommended.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_PLOT_UNARIES=0!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Incorporates the end-members in the convex hull illustration.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_PLOT_OFF_HULL=-1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Incorporates off-hull phases in the convex hull illustration, but excludes phases unstable with respect to their end-members (above the zero $H_{\\mathrm{f}}$ tie-line).\n Only three values are accepted: \\verb|-1| (default: true for 2-dimensional systems, false for 3-dimensional systems), \\verb|0| (false), \\verb|1| (true).\n \\item {{\\it Type:}\\ } \\verb|-1 (default), 0 (false), or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_PLOT_UNSTABLE=0!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Incorporates all unstable phases in the convex hull illustration.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_FILTER_SCHEME=\"energy-axis\"!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the exclusion scheme for the convex hull illustration.\n In contrast to \\verb!--neglect!, this scheme is limited to the illustration, and points are still included in the analysis\/report.\n The following strings are accepted: \\verb!energy-axis!, \\verb!distance!, and an empty string.\n \\verb!energy-axis! refers to a scheme that eliminates structures from the illustration based on their formation enthalpies.\n On the other hand, \\verb!distance! refers to a scheme that eliminates structures from the illustration based on their distances to the hull.\n An empty string signifies no exclusion scheme.\n The criteria (value) for elimination is defined by \\\\ \\verb!DEFAULT_CHULL_LATEX_FILTER_VALUE!.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_FILTER_VALUE=50!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the value beyond which points are excluded per the scheme defined with \\verb!DEFAULT_CHULL_LATEX_FILTER_SCHEME!.\n In this case, \\\\ {\\small \\AFLOWHULLtitle}\\ would filter points with formation enthalpies greater than 50 meV.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_COLOR_BAR=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines whether to show the color bar graphic (3-dimensional illustration only). Colors can still be incorporated without the color bar graphic.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_HEAT_MAP=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines whether to color facets with heat maps illustrating their depth (3-dimensional illustration only).\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_COLOR_GRADIENT=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines whether to incorporate a color scheme at all in the illustration.\n Turning this flag off will also turn off \\verb!DEFAULT_CHULL_LATEX_COLOR_BAR! and \\verb!DEFAULT_CHULL_LATEX_HEAT_MAP!.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_COLOR_MAP=\"\"!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the color map, options are presented in Reference~\\onlinecite{pgfplots_manual}.\n Default is \\\\ \\verb!rgb(0pt)=(0.035,0.270,0.809); rgb(63pt)=(1,0.644,0)!.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_LINKS=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the links scheme. True\/false, \\nobreak\\mbox{\\it i.e.}, \\verb|0|\/\\verb|1|, will toggle all links on\/off.\n \\verb|2| enables external hyperlinks only (no links to other sections of the {\\small PDF}).\n \\verb|3| enables internal links only (no links to external pages).\n \\item {{\\it Type:}\\ } \\verb|0 (false), 1 (true), 2 (external-only), or 3 (internal-only)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_LABEL_NAME=\"\"!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the labeling scheme for phases shown on the convex hull.\n By default, the \\verb|compound| label is shown, while the \\verb|prototype| label can also be specified.\n \\verb|icsd| shows the {\\small ICSD}\\ entry number designation\n (lowest for multiple equivalent ground-state structures reflecting \\verb|icsd_canonical_auid|) if appropriate,\n as shown in Figure~\\ref{fig:art146:hull_analyses}(c).\n Also acceptable: \\verb|both| (\\verb|compound| and \\verb|prototype|) and \\verb|none|.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_META_LABELS=0!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Enables verbose labels, including \\verb|compound|, \\verb|prototype|, $H_{\\mathrm{f}}$, $T_{\\mathrm{S}}$,\n and $\\Delta H_{\\mathrm{f}}$. Warning, significant overlap of labels should be expected.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_LABELS_OFF_HULL=0!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Enables labels for off-hull points.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_HELVETICA_FONT=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Switches the font scheme from Computer Modern (default) to Helvetica.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_FONT_SIZE=\"\"!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the font size of the labels on the convex hull illustration. Warning,\n other settings may override this default. Options include: \\verb|tiny|, \\verb|scriptsize|,\n \\verb|footnotesize|, \\verb|small|, \\verb|normalsize|, \\verb|large| (default), \\verb|Large|,\n \\verb|LARGE|, \\verb|huge|, and \\verb|Huge|.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_ROTATE_LABELS=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Toggles whether labels are rotated.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_BOLD_LABELS=-1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Toggles whether labels are bolded.\n Three values are accepted: \\verb|-1| (default: false unless phase is a ternary), \\verb|0| (false), \\verb|1| (true).\n \\item {{\\it Type:}\\ } \\verb|-1 (default), 0 (false), or 1 (true)|\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\boldsection{Image generation.}\nInstructions for generating the images herein are provided below.\nMany of these images can be generated automatically with {\\small \\AFLOWHULLtitle}. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_examples}(a): run \\verb|aflow --chull --alloy=CoTi --image_only|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_examples}(b): run \\verb|aflow --chull --alloy=MnPdPt --image_only|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_workflow}: \\textbf{i.} the Pd-Pt hull was first generated by running \\\\ \\verb|aflow --chull --alloy=PdPt --image_only --keep=tex|,\n\\textbf{ii.} the resulting \\LaTeX\\ input file ({\\sf aflow\\_PdPt\\_hull.tex}) was modified by hand and compiled to get the various hull illustrations,\n\\textbf{iii.} the overall flowchart was constructed with Microsoft PowerPoint. \\\\\n\\noindent Figure~\\ref{fig:art146:dimensions}: \\textbf{i.} the Al-Ni, Al-Ti, and Ni-Ti binary hulls were first generated by running \\\\ \\verb|aflow --chull --alloy=AlNi,AlTi,NiTi --image_only --keep=tex|,\n\\textbf{ii.} the resulting \\LaTeX\\ input files ({\\sf aflow\\_AlNi\\_hull.tex}, {\\sf aflow\\_AlTi\\_hull.tex}, and {\\sf aflow\\_NiTi\\_hull.tex}) were modified by hand and compiled to get the binary hull images,\n\\textbf{iii.} a snapshot of the Al-Ni-Ti ternary hull was taken from the web application at {\\sf aflow.org\/aflow-chull},\n\\textbf{iv.} the overall illustration was constructed with Adobe Illustrator. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_analyses}(a): set \\verb|DEFAULT_CHULL_IGNORE_KNOWN_ILL_CONVERGED=0| in the {\\sf .aflow.rc} and run \\verb|aflow --chull --alloy=AlCo --image_only|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_analyses}(b): set \\verb|DEFAULT_CHULL_IGNORE_KNOWN_ILL_CONVERGED=1| in the {\\sf .aflow.rc} and run \\verb|aflow --chull --alloy=AlCo --image_only|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_analyses}(c): set \\verb|DEFAULT_CHULL_LATEX_LABEL_NAME=``icsd''| in the {\\sf .aflow.rc} and run \\verb|aflow --chull --alloy=TeZr --image_only|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_analyses}(d): \\textbf{i.} the Pd-Pt hull was first generated by running \\\\ \\verb|aflow --chull --alloy=PdPt --image_only --keep=tex|,\n\\textbf{ii.} the resulting \\LaTeX\\ input file ({\\sf aflow\\_PdPt\\_hull.tex}) was modified by hand and compiled to get the hull illustration.\n$\\Delta H_{\\mathrm{f}}[\\text{aflow:71bc1b15525ffa35}]$ can be calculated individually by running \\verb|aflow --chull --alloy=PdPt --distance_to_hull=aflow:71bc1b15525ffa35|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_analyses}(e): \\textbf{i.} the Pd-Pt hull was first generated by running \\\\ \\verb|aflow --chull --alloy=PdPt --image_only --keep=tex|,\n\\textbf{ii.} the resulting \\LaTeX\\ input file ({\\sf aflow\\_PdPt\\_hull.tex}) was modified by hand and compiled to get the hull illustration.\n$\\delta_{\\mathrm{sc}}[\\text{aflow:f31b0e27897cd162}]$ can be calculated individually by running \\verb|aflow --chull --alloy=PdPt| \\\\ \\verb|--stability_criterion=aflow:f31b0e27897cd162|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_analyses}(f): run \\verb|aflow --chull --alloy=BSm --image_only| \\\\ \\verb|--plot_iso_max_latent_heat --include_paw_gga --include_skewed_hulls|. \\\\\n\\noindent Figure~\\ref{fig:art146:report}: run \\verb|aflow --chull --alloy=AgAuCd|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_app}(a): navigate to {\\sf aflow.org\/aflow-chull} and select the Mo-Ti hull. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_app}(b): navigate to {\\sf aflow.org\/aflow-chull} and select the Fe-Rh-Zr hull. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_app}(c): navigate to {\\sf aflow.org\/aflow-chull}, select the Au-Cu-Zr hull, click on several points in the 3-dimensional illustration to populate the ``Select Points'' table on the left side of the screen, then click on one of the points in the table. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_app}(d): navigate to {\\sf aflow.org\/aflow-chull}, select the Au-Cu-Zr, Au-Cu, and AuZr hulls by clicking ``Periodic Table'' from the navigation bar on the top right corner of the screen between selections, and click ``Hull History'' from the navigation bar on the top right corner of the screen. \\\\\n\\noindent Figures~\\ref{fig:art146:heuslers}(a-d): the structures were visualized with the CrystalMaker X software.\n\n\\vspace{0.5cm}\n\n\\boldsection{Python environment.}\nA module has been created that employs {\\small \\AFLOWHULLtitle}\\\nwithin a Python environment.\nThe module and its description closely follow that of the {\\small AFLOW-SYM}\\ Python module~\\cite{curtarolo:art135}.\nIt connects to a local {\\small AFLOW}\\ installation and imports the {\\small \\AFLOWHULLtitle}\\ results into a\n\\verb|CHull| class.\nA \\verb|CHull| object is initialized with:\n\n\\begin{python}\nfrom aflow_hull import CHull\nfrom pprint import pprint\n\nchull = CHull(aflow_executable = '.\/aflow')\nalloy = 'AlCuZr'\noutput = chull.get_hull(alloy)\npprint(output)\n\\end{python}\n\n\\noindent By default, the \\verb|CHull| object searches for an {\\small AFLOW}\\ executable in\nthe {\\sf \\${\\small PATH}}.\nHowever, the location of an {\\small AFLOW}\\ executable can be specified as\nfollows:\n\n\\noindent \\verb|CHull(aflow_executable=$HOME\/bin\/aflow)|.\n\n\\noindent The \\verb|CHull| object contains built-in methods corresponding to the command line calls mentioned previously:\n\n\\begin{myitemize}\n\\item \\verb|get_hull(`InNiY', options = `--keep=log')|\n\\item \\verb|get_distance_to_hull(`InNiY', `aflow:375066afdfb5a93f',| \\\\ \\verb|options = `--keep=log')|\n\\item \\verb|get_stability_criterion(`InNiY', `aflow:60a36639191c0af8',| \\\\ \\verb|options = `--keep=log')|\n\\item \\verb|get_hull_energy(`InNiY', [0.25,0.25], options = `--keep=log')|\n\\end{myitemize}\nEach method requires an input alloy string and allows an additional parameters\/flags string to be passed via \\verb|options|.\n\\verb|get_distance_to_hull| and \\\\ \\verb|get_stability_criterion| require an additional string input for the\n{\\small AUID}, while \\verb|get_hull_energy| takes an array of doubles as its input\nfor the composition.\n\n\\vspace{0.5cm}\n\n\\boldsection{Python module.}\nThe module to run the aforementioned {\\small \\AFLOWHULLtitle}\\ commands\nis provided below.\nThis module can be modified to incorporate additional\/customized options.\n\\begin{python}\nimport json\nimport subprocess\nimport os\n\nclass CHull:\n\n def __init__(self, aflow_executable='aflow'):\n self.aflow_executable = aflow_executable\n\n def aflow_command(self, cmd):\n try:\n return subprocess.check_output(\n self.aflow_executable + cmd,\n shell=True\n )\n except subprocess.CalledProcessError:\n print('Error aflow executable not found at: ' + self.aflow_executable)\n\n def get_hull(self, alloy, options = None):\n command = ' --chull'\n if options:\n command += ' ' + options\n\n output = ''\n output = self.aflow_command(\n command + ' --print=json --screen_only --alloy=' + alloy\n )\n res_json = json.loads(output)\n return res_json\n\n def get_distance_to_hull(self, alloy, off_hull_point, options = None):\n command = ' --chull --distance_to_hull=' + off_hull_point\n if options:\n command += ' ' + options\n\n output = ''\n output = self.aflow_command(\n command + ' --print=json --screen_only --alloy=' + alloy\n )\n res_json = json.loads(output)\n return res_json\n\n def get_stability_criterion(self, alloy, hull_point, options = None):\n command = ' --chull --stability_criterion=' + hull_point\n if options:\n command += ' ' + options\n\n output = ''\n output = self.aflow_command(\n command + ' --print=json --screen_only --alloy=' + alloy\n )\n res_json = json.loads(output)\n return res_json\n\n def get_hull_energy(self, alloy, composition, options = None):\n command = ' --chull --hull_energy=' + ','.join([ str(comp) for comp in composition ])\n if options:\n command += ' ' + options\n\n output = ''\n output = self.aflow_command(\n command + ' --print=json --screen_only --alloy=' + alloy\n )\n res_json = json.loads(output)\n return res_json\n\\end{python}\n\n\\vspace{0.5cm}\n\n\\noindent{\\textbf{Stability analysis.}\nA Python script is provided below demonstrating\nhow to perform the stability analysis presented in the Results\nsection.\nThe script gathers the most stable binary compounds generated\nfrom 2-element combinations of \\texttt{elements}.\nCompounds are filtered for binary ground-state structures not in the {\\small ICSD}.\nOnly unique compositions are saved.\nThe script writes the results to the {\\small JSON}\\ file {\\sf most\\_stable\\_binaries.json}\nand prints them to screen.\nThe script can be adapted to incorporate the full set of elements and\nfor the calculation of ternary systems.\nConsidering the number of combinations, it is recommended that the script be adapted\nto generate the hulls in parallel.\n}\n\\begin{python}\nfrom aflow_hull import CHull\nimport json\nfrom pprint import pprint\n\nelements = ['Mn', 'Pd', 'Pt'] #extend as needed\nelements.sort()\n\nmost_stable_binaries = [] #final list\nsaved_points_rc = [] #easy way to avoid adding duplicate compositions\n\nchull = CHull(aflow_executable = '.\/aflow') #initialize hull object\nfor i in range(len(elements)): #generate binary alloy combinations\n for j in range(i + 1, len(elements)): #generate binary alloy combinations\n alloy = elements[i]+elements[j] #generate binary alloy combinations\n output = chull.get_hull(alloy) #get hull data\n points_data = output['points_data'] #grab points data\n for point in points_data:\n #filter for binary ground-state structures not in the {\\small ICSD}\\\n if point['ground_state'] and not point['icsd_ground_state'] and point['nspecies'] == 2:\n #easy way to avoid adding duplicate compositions\n if point['reduced_compound'] not in saved_points_rc:\n saved_points_rc.append(point['reduced_compound'])\n #save only what is necessary\n abridged_entry = {}\n abridged_entry['compound'] = point['compound']\n abridged_entry['prototype'] = point['prototype']\n abridged_entry['auid'] = point['auid']\n abridged_entry['aurl'] = point['aurl']\n abridged_entry['relative_stability_criterion'] = point['relative_stability_criterion']\n most_stable_binaries.append(abridged_entry)\n\nmost_stable_binaries = sorted(most_stable_binaries, key=lambda point: -point['relative_stability_criterion']) #sort in descending order\n\n#save data to JSON file\nwith open('most_stable_binaries.json', 'w') as fout:\n json.dump(most_stable_binaries, fout)\n\n#also print output to screen\npprint(most_stable_binaries)\n\\end{python}\n\n\\vspace{0.5cm}\n\n\\boldsection{Output list.}\nThis section details the output fields for the thermodynamic analysis.\nThe lists describe the keywords as they appear in the {\\small JSON}\\ format.\nSimilar keywords are used for the standard text output.\n\n\\boldsection{Points data} (\\verb|points_data|).\n\\begin{myitemize}\n\\item \\verb|auid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small \\underline{A}FLOW} \\underline{u}nique \\underline{ID} ({\\small AUID})~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|aurl|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small \\underline{A}FLOW} \\underline{u}niform \\underline{r}esource \\underline{l}ocator ({\\small AURL})~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|compound|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Compound name~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|enthalpy_formation_atom|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Formation enthalpy per atom $\\left(H_{\\mathrm{f}}\\right)$~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } meV\/atom\n \\end{myitemize}\n\\item \\verb|enthalpy_formation_atom_difference|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The energetic vertical-distance to the hull $\\left(\\Delta H_{\\mathrm{f}}\\right)$, \\nobreak\\mbox{\\it i.e.}, the magnitude of the energy driving the decomposition reaction.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } meV\/atom\n \\end{myitemize}\n\\item \\verb|entropic_temperature|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The ratio of the formation enthalpy and the ideal mixing entropy $\\left(T_{\\mathrm{S}}\\right)$~\\cite{monsterPGM}.\n This term defines the ideal ``{\\it iso-max-latent-heat}'' lines of the grand-canonical ensemble~\\cite{monsterPGM,curtarolo:art98}. Refer to Figure~\\ref{fig:art146:hull_analyses}(f).\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } Kelvin\n \\end{myitemize}\n\\item \\verb|equivalent_structures_auid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AUID}\\ of structurally equivalent entries. This analysis is limited to stable phases only.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|ground_state|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } True for stable phases, and false otherwise.\n \\item {{\\it Type:}\\ } \\verb|boolean|\n \\end{myitemize}\n\\item \\verb|icsd_canonical_auid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AUID}\\ of an equivalent {\\small ICSD}\\ entry. If there are multiple equivalent {\\small ICSD}\\ entries, the one with the lowest number designation is chosen (original usually). This analysis is limited to stable phases only.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|icsd_ground_state|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } True for stable phases with an equivalent {\\small ICSD}\\ entry, and false otherwise.\n \\item {{\\it Type:}\\ } \\verb|boolean|\n \\end{myitemize}\n\\item \\verb|nspecies|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The number of species in the system (\\nobreak\\mbox{\\it e.g.}, binary = 2 and ternary = 3).\n \\item {{\\it Type:}\\ } \\verb|integer|\n \\end{myitemize}\n\\item \\verb|phases_decomposition_auid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AUID}\\ of the products of the decomposition reaction (stable phases). This analysis is limited to unstable phases only.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|phases_decomposition_coefficient|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Coefficients of the decomposition reaction normalized to reactant, \\nobreak\\mbox{\\it i.e.}, $\\textbf{N}$ from Equation~\\ref{eq:art146:decomp_reaction}. Hence, the first entry is always 1. This analysis is limited to unstable phases only.\n \\item {{\\it Type:}\\ } \\verb|array of doubles|\n \\end{myitemize}\n\\item \\verb|phases_decomposition_compound|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } \\verb|compound| of the products of the decomposition reaction (stable phases). This analysis is limited to unstable phases only.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|phases_equilibrium_auid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AUID}\\ of phases in coexistence. This analysis is limited stable phases only.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|phases_equilibrium_compound|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } \\verb|compound| of phases in coexistence. This analysis is limited stable phases only.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|prototype|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AFLOW}\\ prototype designation~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|relative_stability_criterion|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } A dimensionless quantity capturing the effect of the phase on the minimum energy surface relative to its depth, \\nobreak\\mbox{\\it i.e.}, $\\left|\\delta_{\\mathrm{sc}}\/H_{\\mathrm{f}}\\right|$.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\end{myitemize}\n\\item \\verb|space_group_orig|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The space group (symbol and number) of the structure pre-relaxation as determined by {\\small AFLOW-SYM}~\\cite{curtarolo:art135}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|space_group_relax|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The space group (symbol and number) of the structure post-relaxation as determined by {\\small AFLOW-SYM}~\\cite{curtarolo:art135}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|spin_atom|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The magnetization per atom for spin polarized calculations~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } $\\mu_{\\mathrm{B}}$\/atom.\n \\end{myitemize}\n\\item \\verb|stability_criterion|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } A metric for robustness of a stable phase $\\left(\\delta_{\\mathrm{sc}}\\right)$, \\nobreak\\mbox{\\it i.e.},\n the distance of a stable phase from the pseudo-hull constructed without it.\n This analysis is limited to stable phases only.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } meV\/atom\n \\end{myitemize}\n\\item \\verb|url_entry_page|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The {\\small URL}\\ to the entry page: \\\\ {\\sf http:\/\/aflow.org\/material.php?id=aflow:60a36639191c0af8}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\boldsection{Facets data} (\\verb|facets_data|).\n\\begin{myitemize}\n\\item \\verb|artificial|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } True if the facet is artificial, \\nobreak\\mbox{\\it i.e.}, defined solely by artificial end-points, and false otherwise.\n \\item {{\\it Type:}\\ } \\verb|boolean|\n \\end{myitemize}\n\\item \\verb|centroid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The centroid of the facet.\n \\item {{\\it Type:}\\ } \\verb|array of doubles|\n \\item {{\\it Units:}\\ } Stoichiometric-energetic coordinates as defined by Equation~\\ref{eq:art146:point}.\n \\end{myitemize}\n\\item \\verb|content|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The content (hyper-volume) of the facet.\n \\item {{\\it Type:}\\ } \\verb|array of doubles|\n \\item {{\\it Units:}\\ } Stoichiometric-energetic coordinates as defined by Equation~\\ref{eq:art146:point}.\n \\end{myitemize}\n\\item \\verb|hypercollinearity|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } True if the facet has no content, \\nobreak\\mbox{\\it i.e.}, exhibits hyper-collinearity, and false otherwise.\n \\item {{\\it Type:}\\ } \\verb|boolean|\n \\end{myitemize}\n\\item \\verb|normal|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The normal vector characterizing the facet, \\nobreak\\mbox{\\it i.e.}, $\\mathbf{n}$ in Equation~\\ref{eq:art146:plane_eq}.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } Stoichiometric-energetic coordinates as defined by Equation~\\ref{eq:art146:point}.\n \\end{myitemize}\n\\item \\verb|offset|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The offset characterizing the facet, \\nobreak\\mbox{\\it i.e.}, $D$ in Equation~\\ref{eq:art146:plane_eq}.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } Stoichiometric-energetic coordinates as defined by Equation~\\ref{eq:art146:point}.\n \\end{myitemize}\n\\item \\verb|vertical|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } True if the facet is vertical along the energetic axis, and false otherwise.\n \\item {{\\it Type:}\\ } \\verb|boolean|\n \\end{myitemize}\n\\item \\verb|vertices_auid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AUID}\\ of the phases that define the vertices of the facet.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|vertices_compound|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } \\verb|compound| of the phases that define the vertices of the facet.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|vertices_position|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Coordinates that define the vertices of the facet.\n \\item {{\\it Type:}\\ } \\verb|array of arrays of doubles|\n \\item {{\\it Units:}\\ } Stoichiometric-energetic coordinates as defined by Equation~\\ref{eq:art146:point}.\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\boldsection{{\\small AFLOW}\\ forum.}\nUpdates about {\\small \\AFLOWHULLtitle}\\ are discussed in the {\\small AFLOW}\\ forum ({\\sf aflow.org\/forum}):\n``Thermodynamic analysis''.\n\n\\subsection{Conclusions}\nThermodynamic analysis is a critical step for any effective materials design workflow.\nBeing a collective characterization, thermodynamics requires comparisons between many configurations of the system.\nThe availability of large databases \\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux,nomad,APL_Mater_Jain2013,Saal_JOM_2013,cmr_repository}\nallows the construction of computationally-based phase diagrams.\n{\\small \\AFLOWHULLtitle}\\ presents a complete software infrastructure, including\nflexible protocols for data retrieval, analysis, and verification~\\cite{aflowPI,nomad}.\nThe module is exhaustively applied to the {\\sf \\AFLOW.org}\\ repository and identified\nseveral new candidate phases: 17\\ promising $C15_{b}$-type structures and two half-Heuslers.\nThe extension of {\\small \\AFLOWHULLtitle}\\ to repositories beyond {\\sf \\AFLOW.org}\\ {can}\nbe performed {by adapting} the open-source \\texttt{C++} code and\/or Python module.\nComputational platforms such as {\\small \\AFLOWHULLtitle}\\ are valuable tools for guiding synthesis, including high-throughput and\neven autonomous approaches~\\cite{Xiang06231995,Takeuchi:2003fe,koinuma_nmat_review2004,nmatHT}.\n\n\\clearpage\n\\section{Modeling Off-Stoichiometry Materials with a High Throughput \\nobreak\\mbox{\\it Ab-initio}\\ Approach}\n\\label{sec:art110}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art110}.\nAuthor contributions are as follows:\nStefano Curtarolo designed the study.\nKesong Yang and Corey Oses implemented the {\\small AFLOW-POCC}\\ framework and performed proof of concept studies.\nAll authors discussed the results and their implications and contributed to the paper.\n\n\\subsection{Introduction}\n\nCrystals are characterized by their regular, repeating structures.\nSuch a description allows us to reduce our focus from the macroscopic material to a microscopic subset of\nunique atoms and positions.\nA full depiction of material properties, including mechanical, electronic, and magnetic features,\nfollows from an analysis of the primitive lattice.\nFirst principles quantum mechanical calculations have been largely successful in reproducing ground state properties of\nperfectly ordered crystals~\\cite{DFT,Hohenberg_PR_1964,nmatHT}.\nHowever, such perfection does not exist in nature.\nInstead, crystals display a degree of randomness, or disorder, in their lattices.\nThere are several types of disorder; including topological, spin, substitutional, and vibrational~\\cite{Elliott_PoAM_1990}.\nThis work focuses on substitutional disorder, in which equivalent sites of a crystal are not uniquely or fully occupied.\nRather, each site is characterized by a statistical, or partial, occupation.\nSuch disorder is intrinsic in many technologically significant systems, including those used in fuel cells~\\cite{Xie_ACatB_2015},\nsolar cells~\\cite{Kurian_JPCC_2013}, high-temperature superconductors~\\cite{Bednorz_ZPBCM_1986,Maeno_Nature_1994}, low thermal conductivity\nthermoelectrics~\\cite{Winter_JACerS_2007}, imaging and communications devices~\\cite{Patra_JAP_2012}, as well as promising\nrare-earth free materials for use in sensors, actuators, energy-harvesters, and spintronic devices~\\cite{Wang_SR_2013}.\nHence, a comprehensive computational study of substitutionally disordered materials at the atomic scale is of paramount importance for\noptimizing key physical properties of materials in technological applications.\n\nUnfortunately, structural parameters with partial occupancy cannot be used directly in first principles\ncalculations --- a significant hindrance for computational studies of disordered systems.\nTherefore, additional efforts must be made to model disorder or aperiodic systems~\\cite{curtarolo:art25,Mihalkovic_PHM_2006,\nNordheim_1931_AP_VCA,Vanderbilt_2000_PRB_VCA,Soven_PhysRev_1967,Korringa1947392,\nKohn_1954_PhysRev,Stocks_PRL_1978,zunger_sqs,Shan_PRL_1999,Popescu_PRL_2010,Faulkner_PMS_1982}.\nA rigorous statistical treatment of substitutional disorder at the atomic scale requires utility of large ordered supercells\ncontaining a composition consistent with the compound's stoichiometry~\\cite{sod,Habgood_PCCP_2011,Haverkort_ArXiv_2011}.\nHowever, the computational cost of such large supercell calculations has traditionally inhibited their use.\nFortunately, the emergence of high-throughput (HT) computational techniques \\cite{nmatHT}\ncoupled with the exponential growth of computational power is\nnow allowing the study of disordered systems from first principles~\\cite{MGI}.\n\nHerein, we present an approach to perform such a treatment working within the HT computational framework\n{\\small AFLOW}~\\cite{curtarolo:art104,aflowPAPER}.\nWe highlight three novel and attractive features central to this method: complete implementation into an automatic high throughput framework (optimizing speed without\nmitigating accuracy), utility of a novel occupancy optimization algorithm, and use of the Universal Force Field method \\cite{Rappe_1992_JCAS_UFF}\nto reduce the number of {\\small DFT}\\ calculations needed per system.\nTo illustrate the effectiveness of the approach, {\\small AFLOW-POCC}\\ is applied to three disordered systems,\na zinc chalcogenide (ZnS$_{1-x}$Se$_x$), a wide-gap oxide semiconductor (Mg$_{x}$Zn$_{1-x}$O), and an iron alloy (Fe$_{1-x}$Cu$_{x}$).\nExperimental observations are successfully reproduced and new phenomena are predicted:\n\\begin{itemize}\n\\item ZnS$_{1-x}$Se$_x$ shows a small, yet smooth optical bowing over the complete compositional space.\nAdditionally, the stoichiometrically-evolving ensemble average DOS demonstrates that\nthis system is of the amalgamation type and not of the persistence type.\n\\item Mg$_{x}$Zn$_{1-x}$O exhibits an abrupt transition in optical bowing consistent with a phase transition\nover its compositional range.\n\\item The ferromagnetic behavior of Fe$_{1-x}$Cu$_{x}$ is predicted to be smoothly stifled as more\ncopper is introduced into the structure, even through a phase transition.\n\\end{itemize}\nOverall, these systems exhibit highly-tunable properties already exploited in many technologies.\nThrough the approach, these features are not only recovered, but additional insight into the underlying physical mechanisms is\nalso revealed.\n\n\\subsection{Methodology}\n\nThis section details the technicalities of representing a partially occupied disordered system as a series of unique supercells.\nHere is an outline of the approach:\n\\begin{enumerate}\n\\item For a given disordered material, optimize its partial occupancy values and determine the size of the derivative superlattice $n$.\n\\item\n\\begin{enumerate}\n\\item Use the superlattice size $n$ to generate a set of unique derivative superlattices and corresponding sets of\nunique supercells with the required stoichiometry.\n\\item Import these non-equivalent supercells into the automatic computational framework {\\small AFLOW}\\ for HT\nfirst principles electronic structure calculations.\n\\end{enumerate}\n\\item Obtain and use the relative formation enthalpy to calculate the equilibrium probability of each\nsupercell as a function of temperature $T$ according to the Boltzmann distribution.\n\\item Determine the disordered system's material properties through ensemble averages of those calculated for each supercell.\nSpecifically, the following properties are resolved: the density of states (DOS), band gap energy $E_{\\mathrm{gap}}$, and magnetic moment $M$.\n\\end{enumerate}\n\nIn the following sections, a model disordered system, Ag$_{8.733}$Cd$_{3.8}$Zr$_{3.267}$, is presented\nto illustrate the technical procedures mentioned above.\nThis disordered system has two partially occupied sites: one shared between silver and zirconium, and another shared between\ncadmium and a vacancy.\nWorking within the {\\small AFLOW}\\ framework~\\cite{aflowBZ}, a simple structure file has been designed for partially occupied systems.\nAdapted from {\\small VASP}{}'s {\\small POSCAR}~\\cite{vasp_cms1996,vasp_prb1996}, the {\\small PARTCAR}\\ contains within it a description of lattice parameters and\nsite coordinates\/occupants, along\nwith a concentration tolerance (explained in the next section), and (partial) occupancy values for each site.\nTo see more details about this structure or its {\\small PARTCAR}, see Section~\\ref{subsec:art110:PARCAR}.\n\n\\subsubsection{Determining superlattice size}\nIn order to fully account for the partial occupancy of the disordered system, the set of superlattices of\na size corresponding to the lowest common denominator of the fractional partial occupancy values should be generated.\nWith partial occupancy values of 0.733 (733\/1000) and 0.267 (267\/1000) in the disordered system Ag$_{8.733}$Cd$_{3.8}$Zr$_{3.267}$,\nsuperlattices of size 1000 would need to be constructed.\nNot only would this require working with correspondingly large supercells (16,000 atoms per supercell in this example),\nbut the number of unique supercells in the set would be substantial.\nThis extends well beyond the capabilities of first principles calculations, and thus, is not practical.\nIt is therefore necessary to optimize the partial occupancy values to produce an appropriate superlattice size.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Evolution of the algorithm used to optimize the partial occupancy values and superlattice size for the disordered system\nAg$_{8.733}$Cd$_{3.8}$Zr$_{3.267}$.]\n{$f_i$ indicates the iteration's choice fraction for each partially occupied site, ($i$ = 1, 2, 3, \\ldots);\n$e_i$ indicates the error between the iteration's choice fraction and the actual partial occupancy value.\n$e_{\\mathrm{max}}$ is the maximum error of the system.}\n\\vspace{3mm}\n\\begin{tabular}{llrlrlrrr}\n\\multirow{2}{*}{$n^{\\prime}$} & \\multicolumn{2}{c}{occup. 1 (Ag)} & \\multicolumn{2}{c}{occup. 2 (Zr)} & \\multicolumn{2}{c}{occup. 3 (Cd)} & \\multirow{2}{*}{\\textit{$e_{\\mathrm{max}}$} } & \\multirow{2}{*}{$n$} \\\\\n\\cline{2-7}\n & \\textit{$f_{1}$} & \\textit{$e_{1}$} & \\textit{$f_{2}$} & \\textit{$e_{2}$} & \\textit{$f_{3}$} & \\textit{$e_{3}$} & & \\\\\n\\hline\n1\t& 1\/1 & 0.267 & 0\/1 & 0.267 & 1\/1 & 0.2 & 0.267 & 1 \\\\\n\\hline\n2\t& 1\/2 & 0.233 & 1\/2 & 0.233 & 2\/2 & 0.2 & 0.233 & 2 \\\\\n\\hline\n3\t& 2\/3 & 0.067 & 1\/3 & 0.067 & 2\/3 & 0.133 & 0.133 & 3 \\\\\n\\hline\n4\t& 3\/4 & 0.017 & 1\/4 & 0.017 & 3\/4 & 0.05 & 0.05 & 4 \\\\\n\\hline\n5\t& 4\/5 & 0.067 & 1\/5 & 0.067 & 4\/5 & 0 & 0.067 & 5 \\\\\n\\hline\n6\t& 4\/6 & 0.067 & 2\/6 & 0.067 & 5\/6 & 0.033 & 0.067 & 6 \\\\\n\\hline\n7\t& 5\/7 & 0.019 & 2\/7 & 0.019 & 6\/7 & 0.057 & 0.057 & 7 \\\\\n\\hline\n8\t& 6\/8 & 0.017 & 2\/8 & 0.017 & 6\/8 & 0.05 & 0.05 & 4 \\\\\n\\hline\n9\t& 7\/9 & 0.044 & 2\/9 & 0.044 & 7\/9 & 0.022 & 0.044 & 9 \\\\\n\\hline\n10\t& 7\/10 & 0.033 & 3\/10 & 0.033 & 8\/10 & 0 & 0.033 & 10 \\\\\n\\hline\n11\t& 8\/11 & 0.006 & 3\/11 & 0.006 & 9\/11 & 0.018 & 0.018 & 11 \\\\\n\\hline\n12\t& 9\/12 & 0.017 & 3\/12 & 0.017 & 10\/12 & 0.033 & 0.033 & 12 \\\\\n\\hline\n13\t& 10\/13 & 0.036 & 3\/13 & 0.036 & 10\/13 & 0.031 & 0.036 & 13 \\\\\n\\hline\n14\t& 10\/14 & 0.019 & 4\/14 & 0.019 & 11\/14 & 0.014 & 0.019 & 14 \\\\\n\\hline\n15\t& 11\/15 & 0.00003 & 4\/15 & 0.00003 & 12\/15 & 0 & 0.00003 & 15 \\\\\n\\end{tabular}\n\\label{tab:art110:pocc_algo}\n\\end{table}\n\nAn efficient algorithm is presented to calculate the optimized partial occupancy values and corresponding superlattice size\nwith the example disordered system Ag$_{8.733}$Cd$_{3.8}$Zr$_{3.267}$ in Table~\\ref{tab:art110:pocc_algo}.\nFor convenience, the algorithm's iteration step is referred as $n^{\\prime}$,\nthe superlattice index, and $n$ as the superlattice size.\nQuite simply, the algorithm iterates, increasing the superlattice index from 1 to $n^{\\prime}$ until the optimized partial occupancy values reach the required accuracy.\nAt each iteration, a fraction is generated for each partially occupied site, all of which have the common denominator $n^{\\prime}$.\nThe numerator is determined to be the integer that reduces the overall fraction's error relative to the actual site's fractional partial occupancy value.\nThe superlattice size corresponds to the lowest common denominator of the irreducible fractions (\\nobreak\\mbox{\\it e.g.}, see iteration step 8).\nThe maximum error among all of the sites is chosen to be the accuracy metric for the system.\n\nFor the disordered system Ag$_{8.733}$Cd$_{3.8}$Zr$_{3.267}$, given a tolerance of 0.01, the calculated superlattice size is 15\n(240 atoms per supercell).\nBy choosing a superlattice with a nearly equivalent stoichiometry as the disordered system, the supercell size has been\nreduced by over a factor of 60 and entered the realm of feasibility with this calculation.\nNotice that the errors in partial occupancy values calculated for\nsilver and zirconium are the same, as they share the same site.\nThe same holds true for cadmium and its vacant counterpart (not shown).\nTherefore, the algorithm only needs to determine one choice fraction per site, instead of per occupant (as shown).\nSuch an approach reduces computational costs by guaranteeing that only the smallest supercells (both in number and size)\nwith the lowest tolerable error in composition are funneled into the HT first principles calculation framework.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig088}\n\\mycaption[Structure enumeration for off-stoichiometric materials modeling.]\n{For the off-stoichiometric material ZnS$_{0.25}$Se$_{0.75}$, a superlattice of size $n=4$ accommodates the stoichiometry exactly.\nBy considering all possibilities of decorated supercells and eliminating duplicates by UFF energies, seven structures are identified as unique.\nThese representative structures are fully characterized by {\\small AFLOW}\\ and {\\small VASP}, and are ensemble-averaged to resolve the system-wide properties.}\n\\label{fig:art110:pocc}\n\\end{figure}\n\n\\subsubsection{Unique supercells generation}\nWith the optimal superlattice size $n$, the unique derivative superlattices of the disordered system can be generated using\nHermite Normal Form (HNF) matrices~\\cite{enum1} as depicted in Figure~\\ref{fig:art110:pocc}.\nEach HNF matrix generates a superlattice of a size corresponding to its determinant, $n$.\nThere exists many HNF matrices with the same determinant, each creating a variant superlattice.\nFor each unique superlattice, a complete set of possible supercells is generated with the required stoichiometry by exploring all\npossible occupations of partially occupied sites.\nHowever, not all of these combinations are unique --- nominally warranting an involved structure comparison analysis that becomes\nextremely time consuming for large supercells~\\cite{enum1}.\nInstead, duplicates are identified by estimating the total energy of each supercell in a HT manner based on the Universal Force Field (UFF)\nmethod~\\cite{Rappe_1992_JCAS_UFF}.\nThis classical molecular mechanics force field approximates the energy of a structure by considering its composition,\nconnectivity, and geometry, for which parameters have been tabulated.\nOnly supercells with the same total energy are structurally compared and potentially treated as duplicate structures to be discarded, if necessary.\nThe count of duplicate structures determines the degeneracy of the structure.\nOnly non-equivalent supercells are imported into the automatic computational framework {\\small AFLOW}\\ for HT\nquantum mechanics.\n\n\\subsubsection{Supercell equilibrium probability calculation}\nThe unique supercells representing a partially occupied disordered material are labeled as {$S_1$, $S_2$, $S_3$, \\ldots,\n$S_n$}.\nTheir formation enthalpies (per atom) are labeled as {$H_{\\mathrm{f},1}$, $H_{\\mathrm{f},2}$, $H_{\\mathrm{f},3}$, \\ldots, $H_{\\mathrm{f},n}$}, respectively.\nThe formation enthalpy of each supercell is automatically calculated from HT first principles calculations using the {\\small AFLOW}\\\nframework~\\cite{curtarolo:art104,aflowPAPER}.\nThe supercell with the lowest formation enthalpy is selected as a reference (ground state structure), and its formation enthalpy is denoted as $H_{\\mathrm{f},0}$.\nThe relative formation enthalpy of the \\emph{i}th supercell is calculated as $\\Delta {H_{\\mathrm{f},i}} = {H_{\\mathrm{f},i}} - {H_{\\mathrm{f},0}}$\nand characterizes its disorder relative to the ground state.\nThe probability $P_i$ of the \\emph{i}th supercell is determined by the Boltzmann factor:\n\\begin{equation}\n{P_i} = \\frac{{{g_ie^{ - \\Delta {H_{\\mathrm{f},i}}\/{k_{\\mathrm{B}}}T}}}}{{\\sum\\limits_{i = 1}^n {{g_ie^{ - \\Delta {H_{\\mathrm{f},i}}\/{k_{\\mathrm{B}}}T}}} }},\n\\end{equation}\nwhere $g_i$ is the degeneracy of the \\emph{i}th supercell,\n$\\Delta {H_{\\mathrm{f},i}}$ is the relative formation enthalpy of the \\emph{i}th supercell,\n$k_{\\mathrm{B}}$ is the Boltzmann constant,\nand $T$ is a virtual ``roughness'' temperature.\n$T$ is not a true temperature \\textit{per se}, but instead a parameter describing how much disorder has been\nstatistically explored during synthesis.\nTo elaborate further, consider two extremes in the ensemble average (ignoring structural degeneracy):\n\\begin{enumerate}\n\\item $k_{\\mathrm{B}} T \\lesssim \\max\\left(\\Delta {H_{\\mathrm{f},i}}\\right)$\nneglecting highly disordered structures $(\\Delta {H_{\\mathrm{f},i}} \\ggg 0)$\nas $T\\to 0$, and\n\\item $k_{\\mathrm{B}} T\\ggg\\max\\left(\\Delta {H_{\\mathrm{f},i}}\\right)$\nrepresenting the annealed limit ($T\\to \\infty$) in which all structures are equiprobable.\n\\end{enumerate}\nThe probability $P_i$ describes the weight of the \\emph{i}th supercell among the thermodynamically equivalent states of the disordered material\nat equilibrium.\n\n\\subsubsection{Ensemble average density of states, band gap energy, and magnetic moment}\nWith the calculated material properties of each supercell and its equilibrium probability in hand, the overall system properties\ncan be determined by ensemble averages of those calculated for each supercell.\nThis work focuses on the calculation of the ensemble average density of states (DOS), band gap energy $E_{\\mathrm{gap}}$, and magnetic moment $M$.\nThe DOS of the \\emph{i}th supercell is labeled as $N_i(E)$ and indicates the number of electronic states per energy interval.\nThe ensemble average DOS of the system is then determined by the following formula:\n\\begin{equation}\nN(E) = \\sum\\limits_{i = 1}^n {{P_i} \\times {N_i}(E)}.\n\\end{equation}\nAdditionally, a band gap $E_{\\mathrm{gap},i}$ can be extracted from the DOS of each supercell.\nIn this fashion, an ensemble average band gap $E_{\\mathrm{gap}}$ can be calculated for the system.\nIt is important to note that standard density functional theory ({\\small DFT}) calculations are limited to a description of the ground\nstate~\\cite{DFT,Hohenberg_PR_1964,nmatHT}.\nAs such, calculated excited state properties may contain substantial errors.\nIn particular, {\\small DFT}\\ tends to underestimate the band gap~\\cite{Perdew_IJQC_1985}.\nDespite these known hindrances in the theory, the framework is capable of predicting significant trends\nspecific to the disordered systems.\nAs a bonus, the calculation of these results are performed in a high-throughput fashion.\nIt is expected that a more accurate, fine-grained description of the electronic structure in such systems will be obtained through a combination of\nthis software framework and more advanced first principles approaches~\\cite{GW,Hedin_GW_1965,Heyd2003,Liechtenstein1995,curtarolo:art86,curtarolo:art93,curtarolo:art103}.\n\nIn the same spirit as the $N(E)$ and $E_{\\mathrm{gap}}$, {\\small AFLOW-POCC}\\ calculates the ensemble average magnetic moment $M$ of the system.\nThe magnetic moment of the \\emph{i}th supercell is labeled as $M_i$.\nIf the ground state of the \\emph{i}th structure is non-spin-polarized, then its magnetic moment is set to zero, \\nobreak\\mbox{\\it i.e.}, $M_i=0$.\nTaking into account the impact of signed spins on the ensemble average, this approach is limited only to ferromagnetic solutions.\nAdditionally, as an initialization for the self-consistent run, the same ferromagnetic alignment is assumed among all of the spins in the system\n(an {\\small AFLOW}\\ calculation standard)~\\cite{curtarolo:art104}.\nFinally, the ensemble average magnetic moment of the system is calculated with the following formula:\n\\begin{equation}\nM = \\sum\\limits_{i = 1}^n {{P_i} \\times } |{M_i}|.\n\\end{equation}\n\n\\subsection{Example applications}\nThree disordered systems of technological importance are analyzed using {\\small AFLOW-POCC}:\na zinc chalcogenide, a wide-gap oxide semiconductor, and an iron alloy.\nUnless otherwise stated, the supercells used in these calculations were generated with the lowest superlattice size $n_{\\mathrm{xct}}$ needed to represent\nthe composition exactly.\n\n\\subsubsection{Zinc chalcogenides}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig089}\n\\mycaption[Disordered ZnS$_{1-x}$Se$_x$.]\n{(\\textbf{a}) A comparison of the experimental \\cite{Larach_PR_1957,Ebina_PRB_1974,El-Shazly_APA_1985} \\nobreak\\mbox{\\it vs.}\\\ncalculated compositional dependence of the band gap energy $E_{\\mathrm{gap}}$ at room temperature.\nA rigid shift in the $E_{\\mathrm{gap}}$ axis relative to the experimental results of ZnSe (second ordinate axis) accounts for the expected systematic\ndeviation in {\\small DFT}\\ calculations~\\cite{Perdew_IJQC_1985}.\nOnly the lowest empirical $E_{\\mathrm{gap}}$ trends are shown.\nError bars indicate the weighted standard deviation of the ensemble average $E_{\\mathrm{gap}}$.\n(\\textbf{b}) Calculated density of states plots for various compositions:\n$x_{\\mathrm{Se}}=0.00$ ($n=1$),\n$0.33$ ($n=3$),\n$0.67$ ($n=3$), and\n$1.00$ ($n=1$).\nThe straight black line indicates the position of the valence band maximum,\nwhile the straight magenta and cyan lines indicate the positions of the valence band minimum at $x_{\\mathrm{Se}}=0.33$\nand the conduction band minimum at $x_{\\mathrm{Se}}=0.00$, respectively. }\n\\label{fig:art110:ZnSSe}\n\\end{figure}\n\nOver the years, zinc chalcogenides have garnered interest for a dynamic range of applications --- beginning with the creation\nof the first blue-light emitting laser diodes~\\cite{Haase_APL_1991}, and recently have been studied\nas inorganic graphene analogues (IGAs) with potential applications in flexible and transparent nanodevices~\\cite{Sun_NComm_2012}.\nThese wide-gap II-VI semiconductors have demonstrated a smoothly tunable band gap energy $E_{\\mathrm{gap}}$ with respect to\ncomposition~\\cite{Larach_PR_1957,Ebina_PRB_1974,El-Shazly_APA_1985}.\nBoth linear and quadratic dependencies have been observed, with the latter phenomenon referred to as\n\\textit{optical bowing}~\\cite{Bernard_PRB_1986}.\nSpecifically, given the pseudo-ternary system $A_{x}B_{1-x}C$,\n\\begin{equation}\nE_{\\mathrm{gap}}(x)=\\left[x \\epsilon_{AC}+(1-x)\\epsilon_{BC}\\right] - b x(1-x),\n\\end{equation}\nwith $b$ characterizing the bowing.\nWhile Larach \\nobreak\\mbox{\\it et al.}\\ reported a linear dependence ($b=0$)~\\cite{Larach_PR_1957},\nEbina \\nobreak\\mbox{\\it et al.}\\ \\cite{Ebina_PRB_1974} and\nEl-Shazly \\nobreak\\mbox{\\it et al.}\\ \\cite{El-Shazly_APA_1985} reported similar bowing parameters of\n$b=0.613\\pm0.027$~eV and $b=0.457\\pm0.044$~eV, respectively, averaged over the two observed direct transitions.\n\nAs a proof of concept, {\\small AFLOW-POCC}\\ is employed to calculate the compositional dependence of the $E_{\\mathrm{gap}}$ and\nDOS for ZnS$_{1-x}$Se$_x$ at room temperature (annealed limit).\nOverall, this system shows relatively low disorder ($\\max\\left(\\Delta {H_{\\mathrm{f},i}}\\right)\\sim 0.005$~eV),\nexhibiting negligible variations in the ensemble average properties at higher temperatures.\nThese results are compared to experimental measurements~\\cite{Larach_PR_1957,Ebina_PRB_1974,El-Shazly_APA_1985} in Figure~\\ref{fig:art110:ZnSSe}.\nCommon among all three trends (Figure~\\ref{fig:art110:ZnSSe}(a)) is the $E_{\\mathrm{gap}}$ shrinkage with increasing $x_{\\mathrm{Se}}$,\nas well as a near 1~eV tunable $E_{\\mathrm{gap}}$ range.\nThe calculated trend demonstrates a non-zero bowing similar to that observed by both Ebina \\nobreak\\mbox{\\it et al.}~\\cite{Ebina_PRB_1974} and\nEl-Shazly \\nobreak\\mbox{\\it et al.}~\\cite{El-Shazly_APA_1985}.\nA fit shows a bowing parameter of $b=0.585\\pm0.078$~eV, lying in the range between the two experimental bowing parameters.\n\nThe ensemble average DOS plots at room temperature are illustrated in Figure~\\ref{fig:art110:ZnSSe}(b) for $x_{\\mathrm{Se}}=0.00$ ($n=1$), $0.33$ ($n=3$), $0.67$ ($n=3$),\nand $1.00$ ($n=1$).\nThe plots echo the negatively correlated band gap relationship illustrated in Figure~\\ref{fig:art110:ZnSSe}(a), highlighting\nthat the replacement of sulfur with selenium atoms reduces the band gap.\nSpecifically, two phenomena are observed as the concentration of selenium increases: (\\textcolor{red}{\\bf red arrows})\nthe reduction of the valence band width\n(with the exception of $x_{\\mathrm{Se}} = 0.00$ (ZnS) concentration), and (\\textcolor{blue}{\\bf blue arrows})\na shift of the conduction band peak back towards the Fermi energy.\nThe valence band of ZnS more closely resembles that of its extreme concentration counterpart at $x_{\\mathrm{Se}} = 1.00$\n(ZnSe) than the others.\nThe extreme concentration conduction peaks appear more defined than their intermediate concentration counterparts, which is likely an artifact\nof the ensemble averaging calculation.\n\nFinally, a partial-DOS analysis is performed in both species and orbitals (not shown).\nIn the valence band, sulfur and selenium account for the majority of the states, in agreement with their relative concentrations.\nMeanwhile, zinc accounts for the majority of the states in the conduction band at all concentrations.\nCorrespondingly, at all concentrations, the $p$-orbitals make up the majority of the valence band,\nwhereas the conduction band consists primarily of $s$- and $p$-orbitals.\nThese observations are consistent with conclusions drawn from previous optical reflectivity measurements that optical transitions\nare possible from sulfur or selenium valence bands to zinc conduction bands~\\cite{Kirschfeld_PRL_1972}.\n\nOverall, the concentration-evolving $E_{\\mathrm{gap}}$ trend and DOS plots support a continuing line of\nwork~\\cite{Larach_PR_1957,Ebina_PRB_1974,El-Shazly_APA_1985} corroborating that this system\nis of the amalgamation type~\\cite{Onodera_JPSJ_1968}\nand not of the persistence type~\\cite{Kirschfeld_PRL_1972}.\nNotably, however, reflectivity spectra shows that the peak position in the $E_{\\mathrm{gap}}$ for ZnS rich alloys may remain\nstationary~\\cite{Ebina_PRB_1974},\nwhich may have manifested itself in the aforementioned anomaly observed in this structure's valence band width.\n\n\\subsubsection{Wide-gap oxide semiconductor alloys}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig090}\n\\mycaption[Disordered Mg$_{x}$Zn$_{1-x}$O.]\n{(\\textbf{a}) A comparison of the experimental~\\cite{Ohtomo_SST_2005,Takeuchi_JAP_2003,\nChen_JAPCM_2003,Takagi_JJAP_2003,Choopun_APL_2002,Minemoto_TSF_2000,Sharma_APL_1999,Ohtomo_APL_1998} \\nobreak\\mbox{\\it vs.}\\ calculated compositional\ndependence of the band gap energy $E_{\\mathrm{gap}}$ at room temperature.\nA rigid shift in the $E_{\\mathrm{gap}}$ axis relative to the experimental results of MgO (second ordinate axis) accounts for the expected\nsystematic deviation in {\\small DFT}\\ calculations~\\cite{Perdew_IJQC_1985}.\nThe \\textcolor{blue}{\\bf wurtzite} and \\textcolor{red}{\\bf rocksalt} structures are highlighted in blue and red, respectively,\nwhile the mixed phase structures are shown in black.\nError bars indicate the weighted standard deviation of the ensemble average $E_{\\mathrm{gap}}$.\n(\\textbf{b}) Calculated density of states plots for various compositions:\n$x_{\\mathrm{Mg}}=0.00$ ($n=1$),\n$0.33$ ($n=3$),\n$0.67$ ($n=3$), and\n$1.00$ ($n=1$).\nThe straight black line indicates the position of the valence band maximum,\nwhile the straight cyan line indicates the position of the conduction band minimum at $x_{\\mathrm{Mg}}=0.00$.}\n\\label{fig:art110:MgZnO}\n\\end{figure}\n\nZinc oxide (ZnO) has proven to be a pervasive material, with far reaching applications such as paints, catalysts,\npharmaceuticals (sun creams), and optoelectronics~\\cite{Takeuchi_MgZnO_Patent}.\nIt has long been investigated for its electronic properties, and falls into the class of transparent conducting\noxides~\\cite{Ellmer_ZnO_2007}.\nJust as with the previous zinc chalcogenide example, ZnO is a wide-gap II-VI semiconductor that has demonstrated\na tunable band gap energy $E_{\\mathrm{gap}}$ with composition.\nIn particular, ZnO has been engineered to have an $E_{\\mathrm{gap}}$ range as large as 5~eV by synthesizing it with magnesium.\nThis pairing has been intensively studied because of the likeness in ionic radius between zinc and magnesium\nwhich results in mitigated misfit strain in the heterostructure~\\cite{Yoo_TSF_2015}.\nWhile the solubility of MgO and ZnO is small, synthesis has been made possible throughout the full compositional\nspectrum~\\cite{Ohtomo_SST_2005,Takeuchi_JAP_2003,Chen_JAPCM_2003,Takagi_JJAP_2003,Choopun_APL_2002,\nMinemoto_TSF_2000,Sharma_APL_1999,Ohtomo_APL_1998}.\n\nAs another proof of concept, the compositional dependence of the $E_{\\mathrm{gap}}$ and DOS for Mg$_{x}$Zn$_{1-x}$O\nare modeled at room temperature (annealed limit).\nIn particular, this disordered system is chosen to illustrate the breath of materials which this framework can model.\nSimilar to ZnS$_{1-x}$Se$_x$, this system shows relatively low disorder ($\\max\\left(\\Delta {H_{\\mathrm{f},i}}\\right)\\sim 0.007$~eV),\nexhibiting negligible variations in the ensemble average properties at higher temperatures.\nThe results are compared to that observed empirically~\\cite{Ohtomo_SST_2005,Takeuchi_JAP_2003,Chen_JAPCM_2003,Takagi_JJAP_2003,Choopun_APL_2002,\nMinemoto_TSF_2000,Sharma_APL_1999,Ohtomo_APL_1998} in Figure~\\ref{fig:art110:MgZnO}.\nAs illustrated in Figure~\\ref{fig:art110:MgZnO}(a), Ohtomo \\nobreak\\mbox{\\it et al.}\\ observed a composition dependent phase transition\nfrom a wurtzite to a rocksalt structure with increasing $x_{\\mathrm{Mg}}$; the transition occurring around the mid concentrations.\nThis transition is enforced in the calculations.\nEmpirically, the overall trend in the wurtzite phase shows a negligible bowing in the $E_{\\mathrm{gap}}$ trend,\ncontrasting the significant bowing observed in the rocksalt phase.\nThe wurtzite phase $E_{\\mathrm{gap}}$ trend shows a slope of $2.160\\pm0.080$~eV, while the rocksalt phase shows a bowing\nparameter of $3.591\\pm0.856$~eV.\nCalculated trends are shown in Figure~\\ref{fig:art110:MgZnO}(a).\nQualitatively, linear and non-linear $E_{\\mathrm{gap}}$ trends are also observed in the wurtzite and rocksalt phases, respectively.\nThe fits are as follows: a slope of $2.147\\pm0.030$~eV in the wurtzite phase and a bowing parameter of\n$5.971\\pm1.835$~eV in the rocksalt phase.\nThese trends match experiment well within the margins of error.\nA larger margin of error is detected in the rocksalt phase, particular in the phase separated region\n($0.4\\lesssim x_{\\mathrm{Mg}} \\lesssim 0.6$).\nThis may be indicative of the significant shear strain and complex nucleation behavior characterizing the region~\\cite{Takeuchi_JAP_2003}.\n\nThe ensemble average DOS plots at room temperature are illustrated in Figure~\\ref{fig:art110:MgZnO}(b) for $x_{\\mathrm{Mg}}=0.00$ ($n=1$), $0.33$ ($n=3$), $0.67$ ($n=3$),\nand $1.00$ ($n=1$)\nThe plots not only echo the positively correlated band gap relationship illustrated in Figure~\\ref{fig:art110:MgZnO}(a),\nbut also exhibit the aforementioned change from a linear to non-linear trend.\nThis is most easily seen by observing the shift in the conduction band away from the Fermi energy,\nhighlighted by the \\textcolor{blue}{\\bf blue arrows}.\nContrasting ZnS$_{1-x}$Se$_x$, a significant change in width of the valence band is not observed over the range of the stoichiometry.\n\nFinally, a partial-DOS analysis is performed in both species and orbitals (not shown).\nOverall, the constant oxygen backbone plays a major role in defining the shape of both the valence and conduction bands,\nparticularly as $x_{\\mathrm{Mg}}$ increases.\nThis resonates with the strong $p$-orbital presence in both bands throughout all concentrations.\nZinc and its $d$-orbitals play a particularly dominant role in the valence band in magnesium-poor structures.\n\n\\subsubsection{Iron alloys}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig091}\n\\mycaption[Disordered Fe$_{1-x}$Cu$_{x}$.]\n{(\\textbf{a}) A comparison of the experimental~\\cite{Sumiyama_JPSJ_1984} \\nobreak\\mbox{\\it vs.}\\ calculated compositional\ndependence of the magnetic moment $M$.\nThe calculations mimic the following phases observed at 4.2~K:\n$x_{\\mathrm{Cu}}\\leq0.42$ \\textcolor{blue}{\\bf bcc} phase shown in blue,\n$0.42\\!0.95\\right)$ features,\nthe final feature vector captures 2,494 total descriptors.\n\nDescriptor construction is inspired by the topological charge indices~\\cite{Galvez_JCICS_1995}\nand the Kier-Hall\nelectro-topological state indices~\\cite{Kier_Electrotopological_1999}.\nLet $\\mathbf{M}$ be the matrix obtained by multiplying the adjacency\nmatrix $\\mathbf{A}$ by the reciprocal square distance matrix $\\mathbf{D}$ $\\left(D_{ij}=1\/r_{i,j}^{2}\\right)$:\n\\begin{equation}\n\\mathbf{M}=\\mathbf{A} \\cdot \\mathbf{D}.\n\\end{equation}\nThe matrix $\\mathbf{M}$, called the Galvez matrix, is a square $n \\times n$ matrix,\nwhere $n$ is the number of atoms in the unit cell.\nFrom $\\mathbf{M}$, descriptors of reference property $\\mathbf{q}$ are calculated as\n\\begin{equation}\nT^{\\mathrm{E}}=\\sum_{i=1}^{n-1}\\sum_{j=i+1}^{n}\\left|q_{i}-q_{j}\\right|M_{ij}\n\\end{equation}\nand\n\\begin{equation}\nT_{{\\substack{\\scalebox{0.6}{bond}}}}^{\\mathrm{E}}=\\sum_{\\{i,j\\}\\in\\mathrm{bonds}}\\left|q_{i}-q_{j}\\right|M_{ij},\n\\end{equation}\nwhere the first set of indices count over all pairs of atoms and the second\nis restricted to all pairs $i,j$ of bonded atoms.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig095}\n\\mycaption[Outline of the modeling work-flow.]\n{{\\small ML}\\ models are represented by orange diamonds. Target properties predicted by these models are highlighted in green.}\n\\label{fig:art124:figure2}\n\\end{figure}\n\n\\boldsection{Quantitative materials structure-property relationship modeling.}\nIn training the models, the same {\\small ML}\\ method and descriptors are employed without any hand tuning or variable selection.\nSpecifically, models are constructed using gradient boosting decision tree\n({\\small GBDT}) technique~\\cite{Friedman_AnnStat_2001}.\nAll models were validated through $y$-randomization (label scrambling).\nFive-fold cross validation is used to assess how well each model will generalize to an independent dataset.\nHyperparameters are determined with grid searches on the training set and 10-fold cross validation.\n\nThe gradient boosting decision trees ({\\small GBDT}) method~\\cite{Friedman_AnnStat_2001}\nevolved from the application of boosting\nmethods~\\cite{gbm} to regression trees~\\cite{Loh_ISR_2014}.\nThe boosting method is based on the observation that finding many weakly accurate\nprediction rules can be a lot easier than finding a single, highly accurate rule~\\cite{Schapire_ML_1990}.\nThe boosting algorithm calls this ``weak'' learner repeatedly, at each stage feeding it\na different subset of the training examples.\nEach time it is called, the weak learner generates a new weak prediction rule.\nAfter many iterations, the boosting algorithm combines these weak rules into\na single prediction rule aiming to be much more accurate than any single weak rule.\n\nThe {\\small GBDT}\\ approach is an additive model of the following form:\n\\begin{equation}\nF(\\mathbf{x};\\{\\gamma_{m},\\mathbf{a}\\}_{1}^{M})=\\sum_{m=1}^{M}\\gamma_{m} h_{m}(\\mathbf{x};\\mathbf{a}_{m}),\n\\end{equation}\nwhere $h_{m}(\\mathbf{x};\\mathbf{a}_{m})$ are the weak learners (decision trees in this case)\ncharacterized by parameters\n$\\mathbf{a}_{m}$, and $M$ is the total\ncount of decision trees obtained through boosting.\n\nIt builds the additive model in a forward stage-wise fashion:\n\\begin{equation}\nF_m(\\mathbf{x})=F_{m-1}(\\mathbf{x})+\\gamma_{m} h_{m}(\\mathbf{x};\\mathbf{a}_{m}).\n\\end{equation}\nAt each stage $\\left(m=1,2,\\ldots,M\\right)$, $\\gamma_{m}$ and $\\mathbf{a}_{m}$ are chosen to minimize the loss function\n$f_L$ given the current model $F_{m-1}(x_{i})$ for all data points (count $N$),\n\\begin{equation}\n\\left(\\gamma_{m},\\mathbf{a}_{m}\\right)=\\argmin_{\\gamma,\\mathbf{a}} \\sum_{i=1}^{N}\nf_{L} \\left[y_{i},F_{m-1}\\left(\\mathbf{x_{i}}\\right)+\\gamma h\\left(\\mathbf{x}_{i};\\mathbf{a}\\right)\\right].\n\\end{equation}\nGradient boosting attempts to solve this minimization problem numerically via steepest descent.\nThe steepest descent direction is the negative gradient of the loss function\nevaluated at the current model $F_{m-1}$, where the step length is chosen using line search.\n\nAn important practical task is to quantify variable importance.\nFeature selection in decision tree ensembles cannot differentiate between primary\neffects and effects caused by interactions between variables.\nTherefore, unlike regression coefficients, a direct comparison of captured effects is prohibited.\nFor this purpose, variable influence is quantified in the\nfollowing way~\\cite{Friedman_AnnStat_2001}.\nLet us define the influence of variable $j$ in a single tree $h$.\nConsider that the tree has $l$ splits and therefore $l-1$ levels.\nThis gives rise to the definition of the variable influence,\n\\begin{equation}\nK_{j}^2(h)=\\sum_{i=1}^{l-1} I_{i}^{2} \\mathbbm{1}\\left(x_{i}=j\\right),\n\\end{equation}\nwhere $I_{i}^{2}$ is the empirical squared improvement resulting from this split,\nand $\\mathbbm{1}$ is the indicator function.\nHere, $\\mathbbm{1}$ has a value of one if the split at node $x_{i}$ is on variable $j$, and\nzero otherwise,\n\\nobreak\\mbox{\\it i.e.}, it measures the number of times a variable $j$ is selected for splitting.\nTo obtain the overall influence of variable $j$ in the ensemble of decision trees (count $M$),\nit is averaged over all trees,\n\\begin{equation}\nK_{j}^{2}={M}^{-1} \\sum_{m=1}^{M} K_{j}^{2} (h_{m}).\n\\end{equation}\nThe influences $K_{j}^{2}$ are normalized so that they add to one.\nInfluences capture the importance of the variable, but\nnot the direction of the response (positive or negative).\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig096}\n\\mycaption[Five-fold cross validation plots for the eight {\\small ML}\\ models predicting electronic and thermomechanical properties.]\n{\\textbf{(a)} Receiver operating characteristic ({\\small ROC}) curve for the classification {\\small ML}\\ model.\n\\textbf{(b)}-\\textbf{(h)} Predicted \\nobreak\\mbox{\\it vs.}\\ calculated values\nfor the regression {\\small ML}\\ models:\n\\textbf{(b)} band gap energy $\\left(E_{\\scriptstyle \\mathrm{BG}}\\right)$,\n\\textbf{(c)} bulk modulus $\\left(B_{\\scriptstyle \\mathrm{VRH}}\\right)$,\n\\textbf{(d)} shear modulus $\\left(G_{\\scriptstyle \\mathrm{VRH}}\\right)$,\n\\textbf{(e)} Debye temperature $\\left(\\theta_{\\scriptstyle \\mathrm{D}}\\right)$,\n\\textbf{(f)} heat capacity at constant pressure $\\left(C_{\\scriptstyle \\mathrm{P}}\\right)$,\n\\textbf{(g)} heat capacity at constant volume $\\left(C_{\\scriptstyle \\mathrm{V}}\\right)$, and\n\\textbf{(h)} thermal expansion coefficient $\\left(\\alpha_{\\scriptstyle \\mathrm{V}}\\right)$.\n}\n\\label{fig:art124:figure3}\n\\end{figure}\n\n\\boldsection{Integrated modeling work-flow.}\nEight predictive models are developed in this work, including:\na binary classification model that predicts if a material is a metal or an insulator\nand seven regression models that predict:\nthe band gap energy $\\left(E_{\\substack{\\scalebox{0.6}{BG}}}\\right)$ for insulators,\nbulk modulus $\\left(B_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$,\nshear modulus $\\left(G_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$,\nDebye temperature $\\left(\\theta_{\\substack{\\scalebox{0.6}{D}}}\\right)$,\nheat capacity at constant pressure $\\left(C_{\\substack{\\scalebox{0.6}{p}}}\\right)$,\nheat capacity at constant volume $\\left(C_{\\substack{\\scalebox{0.6}{V}}}\\right)$, and\nthermal expansion coefficient $\\left(\\alpha_{\\substack{\\scalebox{0.6}{V}}}\\right)$.\n\nFigure~\\ref{fig:art124:figure2} shows the overall application work-flow.\nA novel candidate material is first classified as a metal or an insulator.\nIf the material is classified as an insulator, $E_{\\substack{\\scalebox{0.6}{BG}}}$ is predicted,\nwhile classification as a metal implies that the material has no $E_{\\substack{\\scalebox{0.6}{BG}}}$.\nThe six thermomechanical properties are then predicted independent of the material's metal\/insulator classification.\nThe integrated modeling work-flow has been implemented as a web application at\n\\href{http:\/\/aflow.org\/aflow-ml}{aflow.org\/aflow-ml},\nrequiring only the atomic species and positions as input for predictions.\n\nWhile all three models were trained independently, the accuracy of the\n$E_{\\substack{\\scalebox{0.6}{BG}}}$ regression model is inherently dependent on the accuracy of the metal\/insulator classification model\nin this work-flow.\nHowever, the high accuracy of the metal\/insulator classification model suggests this not to be a practical concern.\n\n\\boldsection{Model generalizability.}\nOne technique for assessing model quality is five-fold cross validation, which gauges how well\nthe model is expected to generalize to an independent dataset.\nFor each model, the scheme involves randomly partitioning the set into five groups and predicting the value of\neach material in one subset while training the model on the other four subsets.\nHence, each subset has the opportunity to play the role of the ``test set''.\nFurthermore, any observed deviations in the predictions are addressed.\nFor further analysis, all predicted and calculated results are available in\nSupplementary Note 2 of Reference~\\cite{curtarolo:art124}.\n\nThe accuracy of the metal\/insulator classifier is reported as the\narea under the curve ({\\small AUC})\nof the receiver operating characteristic ({\\small ROC}) plot (Figure~\\ref{fig:art124:figure3}(a)).\nThe {\\small ROC}\\ curve illustrates the model's ability to differentiate between metallic and insulating input materials.\nIt plots the prediction rate for insulators (correctly \\nobreak\\mbox{\\it vs.}\\ incorrectly predicted) throughout the\nfull spectrum of possible prediction thresholds.\nAn area of 1.0 represents a perfect test, while an area of 0.5 characterizes a random guess (the dashed line).\nThe model shows excellent external predictive power with the {\\small AUC}\\ at 0.98,\nan insulator-prediction success rate (sensitivity) of 0.95,\na metal-prediction success rate (specificity) of 0.92,\nand an overall classification rate ({\\small CCR}) of 0.93.\nFor the complete set of \\PLMFelectronicTotal\\ materials, this corresponds to\n2,103 misclassified materials, including 1,359 misclassified metals and 744 misclassified insulators.\nEvidently, the model exhibits positive bias toward predicting insulators, where bias refers to whether a\n{\\small ML}\\ model tends to over- or under-estimate the predicted property.\nThis low false-metal rate is fortunate as the model is unlikely to\nmisclassify a novel, potentially interesting semiconductor as a metal.\nOverall, the metal classification model is robust enough to handle the full complexity of the periodic table.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Statistical summary of the five-fold cross-validated predictions for the seven regression models.]\n{The summary corresponds with Figure~\\ref{fig:art124:figure3}.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & {\\small RMSE}\\ & {\\small MAE}\\ & $r^{2}$ \\\\\n\\hline\n$E_{\\substack{\\scalebox{0.6}{BG}}}$ & 0.51~eV & 0.35~eV & 0.90 \\\\\n$B_{\\substack{\\scalebox{0.6}{VRH}}}$ & 14.25~GPa & 8.68~GPa & 0.97 \\\\\n$G_{\\substack{\\scalebox{0.6}{VRH}}}$ & 18.43~GPa & 10.62~GPa & 0.88 \\\\\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}$ & 56.97~K & 35.86~K & 0.95 \\\\\n$C_{\\substack{\\scalebox{0.6}{p}}}$ & 0.09~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.05~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.95 \\\\\n$C_{\\substack{\\scalebox{0.6}{V}}}$ & 0.07~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.04~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.95 \\\\\n$\\alpha_{\\substack{\\scalebox{0.6}{V}}}$ & $1.47 \\times 10^{-5}$~K$^{-1}$ & $5.69 \\times 10^{-6}$~K$^{-1}$ & 0.91 \\\\\n\\end{tabular}\n\\label{tab:art124:table1}\n\\end{table}\n\nThe results of the five-fold cross validation analysis for the band gap energy $\\left(E_{\\substack{\\scalebox{0.6}{BG}}}\\right)$ regression model\nare plotted in Figure~\\ref{fig:art124:figure3}(b).\nAdditionally, a statistical profile of these predictions, along with that of the six thermomechanical regression models,\nis provided in Table~\\ref{tab:art124:table1}, which includes metrics such as\nthe root-mean-square error ({\\small RMSE}),\nmean absolute error ({\\small MAE}), and coefficient of determination $\\left(r^2\\right)$.\nSimilar to the classification model, the $E_{\\substack{\\scalebox{0.6}{BG}}}$ model exhibits a positive predictive bias.\nThe biggest errors come from materials with narrow band gaps,\n\\nobreak\\mbox{\\it i.e.}, the scatter in the lower left corner in Figure~\\ref{fig:art124:figure3}(b).\nThese materials predominantly include complex fluorides and nitrides.\nN$_{2}$H$_{6}$Cl$_{2}$ ({\\small ICSD}\\ \\#23145)\nexhibits the worst prediction accuracy with signed error SE = 3.78 eV~\\cite{Donohue_JCP_1947}.\nThe most underestimated materials are HCN ({\\small ICSD}\\ \\#76419) and, respectively\nN$_{2}$H$_{6}$Cl$_{2}$ ({\\small ICSD}\\ \\#240903) with SE = -2.67 and -3.19 eV~\\cite{Dulmage_ActaCrist_1951,Kruszynski_ActaCristE_2007}, respectively.\nThis is not surprising considering that all three are molecular crystals.\nSuch systems are anomalies in the {\\small ICSD}, and fit better in other databases, such as\nthe Cambridge Structural Database~\\cite{Groom_CSD_2016}.\nOverall, 10,762 materials are predicted within 25\\% accuracy of calculated values,\nwhereas 824 systems have errors over 1 eV.\n\nFigures~\\ref{fig:art124:figure3}(c-h) and Table~\\ref{tab:art124:table1} showcase the results of the five-fold cross validation analysis\nfor the six thermomechanical regression models.\nFor both bulk $\\left(B_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$ and shear $\\left(G_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$ moduli,\nover 85\\% of materials are predicted within 20~GPa of their calculated values.\nThe remaining models also demonstrate high accuracy, with\nat least 90\\% of the full training set $\\left(>2,546~\\mathrm{systems}\\right)$\npredicted to within 25\\% of the calculated values.\nSignificant outliers in predictions of the bulk modulus include\ngraphite ({\\small ICSD}\\ \\#187640, SE = 100 GPa, likely\ndue to extreme anisotropy) and two theoretical high-pressure boron nitrides ({\\small ICSD}\\ \\#162873 and \\#162874,\nunder-predicted by over 110 GPa)~\\cite{Lian_JCP_2013,Doll_PRB_2008}.\nOther theoretical systems are ill-predicted throughout the six properties, including\nZN ({\\small ICSD}\\ \\#161885), CN$_{2}$ ({\\small ICSD}\\ \\#247676), C$_{3}$N$_{4}$ ({\\small ICSD}\\ \\#151782),\nand CH ({\\small ICSD}\\ \\#187642)~\\cite{EscorciaSalas_MJ_2008,Li_PCCP_2012,Marques_PRB_2004,Lian_JCP_2013}.\nPredictions for the $G_{\\substack{\\scalebox{0.6}{VRH}}}$, Debye temperature $\\left(\\theta_{\\substack{\\scalebox{0.6}{D}}}\\right)$, and thermal expansion coefficient\n$\\left(\\alpha_{\\substack{\\scalebox{0.6}{V}}}\\right)$\ntend to be slightly underestimated, particularly for higher calculated values.\nAdditionally, mild scattering can be seen for $\\theta_{\\substack{\\scalebox{0.6}{D}}}$ and\n$\\alpha_{\\substack{\\scalebox{0.6}{V}}}$, but not enough to have a significant\nimpact on the error or correlation metrics.\n\nDespite minimal deviations, both {\\small RMSE}\\ and {\\small MAE}\\ are within 4\\% of the ranges covered for each property,\nand the predictions demonstrate excellent correlation with the calculated properties.\nNote the tight clustering of points just below 3 $k_{\\substack{\\scalebox{0.6}{B}}}$\/atom for the heat\ncapacity at constant volume $\\left(C_{\\substack{\\scalebox{0.6}{V}}}\\right)$.\nThis is due to $C_{\\substack{\\scalebox{0.6}{V}}}$ saturation in accordance with the Dulong-Petit law occurring at or below\n300 K for many compounds.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.65\\linewidth]{fig097}\n\\mycaption[Semi-log scatter plot of the full dataset (\\PLMFelectronicTotal\\ unique materials) in a dual-descriptor space.]\n{$\\avg\\left(\\Delta H_{\\scriptstyle \\mathrm{fusion}}\\lambda^{-1}\\right)$ \\nobreak\\mbox{\\it vs.}\\\n$\\avg\\left(V_{\\scriptstyle \\mathrm{molar}}r_{\\scriptstyle \\mathrm{cov}}^{-1}\\right)$.\nInsulators and metals are colored in red and blue, respectively.}\n\\label{fig:art124:figure4}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig098}\n\\mycaption[Partial dependence plots of the $E_{\\scriptstyle \\mathrm{BG}}$, $B_{\\scriptstyle \\mathrm{VRH}}$, and\n$\\theta_{\\scriptstyle \\mathrm{D}}$ models.]\n{\\textbf{(a)} Partial dependence of $E_{\\scriptstyle \\mathrm{BG}}$ on the $\\avg\\left(\\Delta IP_{\\mathrm{bond}}\\right)$\ndescriptor.\nFor $E_{\\scriptstyle \\mathrm{BG}}$,\nthe 2D interaction between $\\std\\left(\\Delta IP_{\\mathrm{bond}}\\right)$ and $\\avg\\left(\\Delta IP_{\\mathrm{bond}}\\right)$\nand between $\\rho$ (density) and $\\avg\\left(\\Delta IP_{\\mathrm{bond}}\\right)$ are illustrated in panels\n\\textbf{(b)} and \\textbf{(c)}, respectively.\n\\textbf{(d)} Partial dependence of the $B_{\\scriptstyle \\mathrm{VRH}}$ on the crystal volume per atom descriptor.\nFor $\\theta_{\\scriptstyle \\mathrm{D}}$,\nthe 2D interaction between\n$\\avg\\left(\\Delta EA_{\\scriptstyle \\mathrm{bond}}\\right)$ and\n$\\std\\left(\\Delta H_{\\scriptstyle \\mathrm{vapor}} \\Delta H_{\\scriptstyle \\mathrm{atom}}^{-1}\\right)$\nand between\ncrystal lattice parameters $b$ and $c$ are illustrated\nin panels \\textbf{(e)} and \\textbf{(f)}, respectively.}\n\\label{fig:art124:figure5}\n\\end{figure}\n\n\\boldsection{Model interpretation.}\nModel interpretation is of paramount importance in any {\\small ML}\\ study.\nThe significance of each descriptor is determined in order to gain insight into\nstructural features that impact molecular properties of interest.\nInterpretability is a strong advantage of decision tree methods, particularly with the {\\small GBDT}\\ approach.\nOne can quantify the predictive power of a specific descriptor by analyzing the reduction\nof the {\\small RMSE}\\ at each node of the tree.\n\nPartial dependence plots offer yet another opportunity for {\\small GBDT}\\ model interpretation.\nSimilar to the descriptor significance analysis, partial dependence resolves the\neffect of a variable (descriptor) on a property, but only after marginalizing over all other\nexplanatory variables~\\cite{Hastie_StatLearn_2001}.\nThe effect is quantified by the change of that property as relevant descriptors are varied.\nThe plots themselves highlight the most important interactions among relevant descriptors\nas well as between properties and their corresponding descriptors.\nWhile only the most important descriptors are highlighted and discussed,\nan exhaustive list of relevant descriptors and their relative contributions\ncan be found in\nSupplementary Note 1 of Reference~\\cite{curtarolo:art124}.\n\nFor the metal\/insulator classification model, the descriptor significance analysis\nshows that two descriptors have the highest importance (equally), namely\n$\\avg\\left(\\Delta H_{\\substack{\\scalebox{0.6}{fusion}}}\\lambda^{-1}\\right)$ and\n$\\avg\\left(V_{\\substack{\\scalebox{0.6}{molar}}} r_{\\substack{\\scalebox{0.6}{cov}}}^{-1}\\right)$.\n$\\avg\\left(\\Delta H_{\\substack{\\scalebox{0.6}{fusion}}}\\lambda^{-1}\\right)$ is the ratio between the\nfusion enthalpy $\\left(\\Delta H_{\\substack{\\scalebox{0.6}{fusion}}}\\right)$\nand the thermal conductivity $\\left(\\lambda\\right)$ averaged over all atoms in the material, and\n$\\avg\\left(V_{\\substack{\\scalebox{0.6}{molar}}} r_{\\substack{\\scalebox{0.6}{cov}}}^{-1}\\right)$ is the ratio between the\nmolar volume $\\left(V_{\\substack{\\scalebox{0.6}{molar}}}\\right)$\nand the covalent radius $\\left(r_{\\substack{\\scalebox{0.6}{cov}}}\\right)$ averaged over all atoms in the material.\nBoth descriptors are simple node-specific features.\nThe presence of these two prominent descriptors accounts for the high accuracy of the classification model.\n\nFigure~\\ref{fig:art124:figure4} shows the projection of the full dataset onto the dual-descriptor space of\n$\\avg\\left(\\Delta H_{\\substack{\\scalebox{0.6}{fusion}}}\\lambda^{-1}\\right)$ and $\\avg\\left(V_{\\substack{\\scalebox{0.6}{molar}}} r_{\\substack{\\scalebox{0.6}{cov}}}^{-1}\\right)$.\nIn this 2D space, metals and insulators are substantially partitioned.\nTo further resolve this separation, the plot is split into four quadrants\n(see dashed lines) with an origin approximately at\n$\\avg\\left(V_{\\substack{\\scalebox{0.6}{molar}}} r_{\\substack{\\scalebox{0.6}{cov}}}^{-1}\\right)=11$,~$\\avg\\left(\\Delta H_{\\substack{\\scalebox{0.6}{fusion}}}\\lambda^{-1}\\right)=2$.\nInsulators are predominately located in quadrant I.\nThere are several clusters (one large and several small) parallel to the $x$-axis.\nMetals occupy a compact square block in quadrant III within intervals\n$5<\\avg\\left(V_{\\substack{\\scalebox{0.6}{molar}}} r_{\\substack{\\scalebox{0.6}{cov}}}^{-1}\\right)<12$ and $0.02<\\avg\\left(\\Delta H_{\\substack{\\scalebox{0.6}{fusion}}}\\lambda^{-1}\\right)<2$.\nQuadrant II is mostly empty with a few materials scattered about the origin.\nIn the remaining quadrant (IV), materials have mixed character.\n\nAnalysis of the projection shown in Figure~\\ref{fig:art124:figure4} suggests a simple heuristic rule:\nall materials within quadrant I are classified as insulators $\\left(E_{\\substack{\\scalebox{0.6}{BG}}}\\!>\\!0\\right)$,\nand all materials outside of this quadrant are metals.\nRemarkably, this unsupervised projection approach achieves a very high\nclassification accuracy of 86\\% for the entire dataset of \\PLMFelectronicTotal\\ materials.\nThe model misclassifies only 3,621 materials:\n2,414 are incorrectly predicted as insulators and 1,207 are incorrectly predicted as metals.\nThis example illustrates how careful model analysis of the most significant descriptors\ncan yield simple heuristic rules for materials design.\n\nThe regression model for the band gap energy $\\left(E_{\\substack{\\scalebox{0.6}{BG}}}\\right)$ is more complex.\nThere are a number of descriptors in the model with comparable contributions,\nand thus, all individual contributions are small.\nThis is expected as a number of conditions can affect $E_{\\substack{\\scalebox{0.6}{BG}}}$.\nThe most important are $\\avg\\left(\\chi Z_{\\mathrm{eff}}^{-1}\\right)$ and $\\avg\\left(C \\lambda^{-1}\\right)$ with\nsignificance scores of 0.075 and 0.071,\nrespectively, where $\\chi$ is the electronegativity, $Z_{\\mathrm{eff}}$ is the effective nuclear charge,\n$C$ is the specific heat capacity, and $\\lambda$ is the thermal conductivity of each atom.\n\nFigure~\\ref{fig:art124:figure5} shows partial dependence plots focusing on $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$ as an example.\nIt is derived from edge fragments of bonded atoms $\\left(l=1\\right)$ and defined as an absolute difference in\nionization potentials averaged over the material.\nIn other words, it is a measure of bond polarity, similar to electronegativity.\nFigure~\\ref{fig:art124:figure5}(a) shows a steady monotonic increase in $\\Delta E_{\\substack{\\scalebox{0.6}{BG}}}$ for larger\nvalues of $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$.\nThe effect is small, but captures an expected physical principle:\npolar inorganic materials (\\nobreak\\mbox{\\it e.g.}, oxides, fluorides) tend to have larger $E_{\\substack{\\scalebox{0.6}{BG}}}$.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig099}\n\\mycaption[Model performance evaluation for the six {\\small ML}\\ models predicting thermomechanical properties\nof \\PLMFthermoTestTotal\\ newly characterized materials.]\n{Predicted \\nobreak\\mbox{\\it vs.}\\ calculated values for the regression {\\small ML}\\ models:\n\\textbf{(a)} bulk modulus $\\left(B_{\\scriptstyle \\mathrm{VRH}}\\right)$,\n\\textbf{(b)} shear modulus $\\left(G_{\\scriptstyle \\mathrm{VRH}}\\right)$,\n\\textbf{(c)} Debye temperature $\\left(\\theta_{\\scriptstyle \\mathrm{D}}\\right)$,\n\\textbf{(d)} heat capacity at constant pressure $\\left(C_{\\scriptstyle \\mathrm{P}}\\right)$,\n\\textbf{(e)} heat capacity at constant volume $\\left(C_{\\scriptstyle \\mathrm{V}}\\right)$, and\n\\textbf{(f)} thermal expansion coefficient $\\left(\\alpha_{\\scriptstyle \\mathrm{V}}\\right)$.}\n\\label{fig:art124:figure6}\n\\end{figure}\n\nGiven the number of significant interactions involved with this phenomenon,\ntailoring $E_{\\substack{\\scalebox{0.6}{BG}}}$ involves the\noptimization of a highly non-convex, multidimensional object.\nFigure~\\ref{fig:art124:figure5}(b) illustrates a 2D slice of this object as\n$\\std\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$ and $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$ vary simultaneously.\nLike $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$,\n$\\std\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$ is the standard deviation of the set of absolute differences in $IP$ among\nall bonded atoms.\nIn the context of these two variables, $E_{\\substack{\\scalebox{0.6}{BG}}}$ responds to deviations in $\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}$\namong the set of bonded atoms, but remains constant across shifts in $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$.\nThis suggests an opportunity to tune $E_{\\substack{\\scalebox{0.6}{BG}}}$ by considering another composition that varies the deviations among bond polarities.\nAlternatively, a desired $E_{\\substack{\\scalebox{0.6}{BG}}}$ can be maintained\nby considering another composition that preserves the deviations among bond polarities, even as the overall average\nshifts.\nSimilarly, Figure~\\ref{fig:art124:figure5}(c) shows the partial dependence on both\nthe density $\\left(\\rho\\right)$ and $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$.\nContrary to the previous trend, larger $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$\nvalues correlate with smaller $E_{\\substack{\\scalebox{0.6}{BG}}}$, particularly for low density structures.\nMaterials with higher density and lower $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$ tend to have higher $E_{\\substack{\\scalebox{0.6}{BG}}}$.\nConsidering the elevated response (compared to Figure~\\ref{fig:art124:figure5}(b)), the inverse correlation of $E_{\\substack{\\scalebox{0.6}{BG}}}$ with the average\nbond polarity in the context of density suggests an even more effective means of tuning $E_{\\substack{\\scalebox{0.6}{BG}}}$.\n\nA descriptor analysis of the thermomechanical property models reveals the importance of\none descriptor in particular, the volume per atom of the crystal.\nThis conclusion certainly resonates with the nature of these properties, as they generally correlate\nwith bond strength~\\cite{curtarolo:art115}.\nFigure~\\ref{fig:art124:figure5}(d) exemplifies such a relationship, which shows\nthe partial dependence plot of the bulk modulus $\\left(B_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$ on the volume per atom.\nTightly bound atoms are generally indicative of stronger bonds.\nAs the interatomic distance increases, properties like $B_{\\substack{\\scalebox{0.6}{VRH}}}$ generally reduce.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Statistical summary of the new predictions for the six thermomechanical regression models.]\n{The summary corresponds with Figure~\\ref{fig:art124:figure6}.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & {\\small RMSE}\\ & {\\small MAE}\\ & $r^{2}$ \\\\\n\\hline\n$B_{\\substack{\\scalebox{0.6}{VRH}}}$ & 21.13~GPa & 12.00~GPa & 0.93 \\\\\n$G_{\\substack{\\scalebox{0.6}{VRH}}}$ & 18.94~GPa & 13.31~GPa & 0.90 \\\\\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}$ & 64.04~K & 42.92~K & 0.93 \\\\\n$C_{\\substack{\\scalebox{0.6}{p}}}$ & 0.10~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.06~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.92 \\\\\n$C_{\\substack{\\scalebox{0.6}{V}}}$ & 0.07~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.05~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.95 \\\\\n$\\alpha_{\\substack{\\scalebox{0.6}{V}}}$ & $1.95 \\times 10^{-5}$~K$^{-1}$ & $5.77 \\times 10^{-6}$~K$^{-1}$ & 0.76 \\\\\n\\end{tabular}\n\\label{tab:art124:table2}\n\\end{table}\n\nTwo of the more interesting dependence plots are also shown in Figure~\\ref{fig:art124:figure5}(e-f),\nboth of which offer opportunities for tuning the Debye temperature ($\\theta_{\\substack{\\scalebox{0.6}{D}}}$).\nFigure~\\ref{fig:art124:figure5}(e) illustrates the interactions among two descriptors,\nthe absolute difference in electron affinities among bonded atoms\naveraged over the material\n$\\left(\\avg\\left(\\Delta EA_{\\substack{\\scalebox{0.6}{bond}}}\\right)\\right)$, and\nthe standard deviation of the set of ratios of the enthalpies of vaporization $\\left(\\Delta H_{\\substack{\\scalebox{0.6}{vapor}}}\\right)$\nand atomization $\\left(\\Delta H_{\\substack{\\scalebox{0.6}{atom}}}\\right)$ for all atoms in the material\n$\\left(\\std\\left(\\Delta H_{\\substack{\\scalebox{0.6}{vapor}}} \\Delta H_{\\substack{\\scalebox{0.6}{atom}}}^{-1}\\right)\\right)$.\nWithin these dimensions, two distinct regions emerge of increasing\/decreasing $\\theta_{\\substack{\\scalebox{0.6}{D}}}$ separated by a\nsharp division\nat about $\\avg\\left(\\Delta EA_{\\substack{\\scalebox{0.6}{atom}}}\\right) = 3$.\nWithin these partitions, there are clusters of maximum gradient in $\\theta_{\\substack{\\scalebox{0.6}{D}}}$---peaks within the left\npartition and troughs within the right.\nThe peaks and troughs alternate with varying $\\std\\left(\\Delta H_{\\substack{\\scalebox{0.6}{vapor}}} \\Delta H_{\\substack{\\scalebox{0.6}{atom}}}^{-1}\\right)$.\nAlthough $\\std\\left(\\Delta H_{\\substack{\\scalebox{0.6}{vapor}}} \\Delta H_{\\substack{\\scalebox{0.6}{atom}}}^{-1}\\right)$\nis not an immediately intuitive descriptor, the alternating clusters may be a manifestation\nof the periodic nature of $\\Delta H_{\\substack{\\scalebox{0.6}{vapor}}}$ and $\\Delta H_{\\substack{\\scalebox{0.6}{atom}}}$~\\cite{webelements_periodicity}.\nAs for the partitions themselves,\nthe extremes of $\\avg\\left(\\Delta EA_{\\substack{\\scalebox{0.6}{atom}}}\\right)$ characterize covalent and ionic materials, as\nbonded atoms with similar $EA$ are likely to share electrons, while those\nwith varying $EA$ prefer to donate\/accept electrons.\nConsidering that $EA$ is also periodic, various opportunities for carefully tuning $\\theta_{\\substack{\\scalebox{0.6}{D}}}$\nshould be available.\n\nFinally, Figure~\\ref{fig:art124:figure5}(f) shows the partial dependence of $\\theta_{\\substack{\\scalebox{0.6}{D}}}$ on the lattice parameters $b$ and $c$.\nIt resolves two notable correlations:\n\\textit{(i)} uniformly increasing the cell size of the system decreases $\\theta_{\\substack{\\scalebox{0.6}{D}}}$, but\n\\textit{(ii)} elongating the cell ($c\/b \\gg 1$) increases it.\nAgain, \\textit{(i)} can be attributed to the\ninverse relationship between volume per atom and bond strength,\nbut does little to address \\textit{(ii)}.\nNevertheless, the connection between elongated, or layered, systems and the Debye temperature is certainly not\nsurprising---anisotropy can be leveraged to enhance phonon-related interactions associated with\nthermal conductivity~\\cite{Minnich_PRB_2015}\nand superconductivity~\\cite{Shimahara_PRB_2002,Jha_PT_1989,Klein_SSC_1980}.\nWhile the domain of interest is quite narrow,\nthe impact is substantial, particularly in comparison to that shown in Figure~\\ref{fig:art124:figure5}(e).\n\n\\boldsection{Model validation.}\nWhile the expected performances of the {\\small ML}\\ models can be projected through five-fold cross validation,\nthere is no substitute for validation against an independent dataset.\nThe {\\small ML}\\ models for the thermomechanical properties are leveraged to make predictions\nfor materials previously uncharacterized, and subsequently validated\nthese predictions via the {\\small AEL}-{\\small AGL}\\ integrated framework~\\cite{curtarolo:art96, curtarolo:art115}.\nFigure~\\ref{fig:art124:figure6} illustrates the models' performance on the set of \\PLMFthermoTestTotal\\ additional materials,\nwith relevant statistics displayed in Table~\\ref{tab:art124:table2}.\nFor further analysis, all predicted and calculated results are available in\nSupplementary Note 3 of Reference~\\cite{curtarolo:art124}.\n\nComparing with the results of the generalizability analysis shown in Figure~\\ref{fig:art124:figure3} and Table~\\ref{tab:art124:table1},\nthe overall errors are consistent with five-fold cross validation.\nFive out of six models have $r^2$ of 0.9 or higher.\nHowever, the $r^2$ value for the thermal expansion coefficient\n$\\left(\\alpha_{\\substack{\\scalebox{0.6}{V}}}\\right)$ is lower than forecasted.\nThe presence of scattering suggests the need for a larger training set---as new,\nmuch more diverse materials were likely introduced in the test set.\nThis is not surprising considering the number of variables that can affect thermal expansion~\\cite{Figge_APL_2009}.\nOtherwise, the accuracy of these predictions confirm the effectiveness of the {\\small PLMF}\\ representation,\nwhich is particularly compelling considering:\n\\textit{(i)} the limited diversity training dataset (only about 11\\% as large as that available for\npredicting the electronic properties), and\n\\textit{(ii)} the relative size of the test set (over a quarter the size of the training set).\n\nIn the case of the bulk modulus $\\left(B_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$, 665 systems (86\\% of test set) are predicted within 25\\%\nof calculated values.\nOnly the predictions of four materials, Bi ({\\small ICSD}\\ \\#51674), PrN ({\\small ICSD}\\ \\#168643),\nMg$_{3}$Sm ({\\small ICSD}\\ \\#104868), and ZrN ({\\small ICSD}\\ \\#161885), deviate beyond 100~GPa from calculated values.\nBi is a high-pressure phase (Bi-III) with a caged, zeolite-like structure~\\cite{McMahon_BiIII_2001}.\nThe structures of zirconium nitride (wurtzite phase) and praseodymium nitride (B3 phase) were hypothesized and\ninvestigated via {\\small DFT}\\ calculations~\\cite{EscorciaSalas_MJ_2008,Kocak_PSCB_2010} and have yet to be observed\nexperimentally.\n\nFor the shear modulus $\\left(G_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$, 482 materials (63\\% of the test set) are predicted within 25\\%\nof calculated values.\nJust one system, C$_{3}$N$_{4}$ ({\\small ICSD}\\ \\#151781), deviates beyond 100~GPa from its calculated value.\nThe Debye temperature $\\left(\\theta_{\\substack{\\scalebox{0.6}{D}}}\\right)$ is predicted to within 50 K accuracy for 540 systems (70\\% of the test set).\nBeF$_{2}$ ({\\small ICSD}\\ \\#173557), yet another cage (sodalite) structure~\\cite{Zwijnenburg_JACS_2008}, has among the largest errors\nin three models including $\\theta_{\\substack{\\scalebox{0.6}{D}}}$ (SE = -423 K) and both heat capacities\n($C_{\\substack{\\scalebox{0.6}{p}}}$: SE = 0.65 $k_{\\substack{\\scalebox{0.6}{B}}}$\/atom; $C_{\\substack{\\scalebox{0.6}{V}}}$: SE = 0.61 $k_{\\substack{\\scalebox{0.6}{B}}}$\/atom).\nSimilar to other ill-predicted structures, this polymorph is theoretical, and has yet to be synthesized.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig100}\n\\mycaption[Comparison of the {\\small AEL}-{\\small AGL}\\ calculations and {\\small ML}\\ predictions with experimental values for three thermomechanical properties.]\n{\\textbf{(a)} bulk modulus $\\left(B\\right)$,\n\\textbf{(b)} shear modulus $\\left(G\\right)$,\nand\n\\textbf{(c)} Debye temperature $\\left(\\theta_{\\scriptstyle \\mathrm{D}}\\right)$.\n}\n\\label{fig:art124:figure7}\n\\end{figure}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Statistical summary of the {\\small AEL}-{\\small AGL}\\ calculations and\n{\\small ML}\\ predictions \\nobreak\\mbox{\\it vs.}\\ experimental values for three thermomechanical properties.]\n{The summary corresponds with Figure~\\ref{fig:art124:figure7}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r}\n\\multirow{2}{*}{property} & \\multicolumn{2}{c|}{{\\small RMSE}} & \\multicolumn{2}{c|}{{\\small MAE}} & \\multicolumn{2}{c}{$r^{2}$} \\\\\n\\cline{2-7}\n & exp. \\nobreak\\mbox{\\it vs.}\\ calc. & exp. \\nobreak\\mbox{\\it vs.}\\ pred. & exp. \\nobreak\\mbox{\\it vs.}\\ calc. & exp. \\nobreak\\mbox{\\it vs.}\\ pred. & exp. \\nobreak\\mbox{\\it vs.}\\ calc. & exp. \\nobreak\\mbox{\\it vs.}\\ pred. \\\\\n\\hline\n$B$ & 8.90~GPa & 10.77~GPa & 6.36~GPa & 8.12~GPa & 0.99 & 0.99 \\\\\n$G$ & 7.29~GPa & 9.15~GPa & 4.76~GPa & 6.09~GPa & 0.99 & 0.99 \\\\\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}$ & 76.13~K & 65.38~K & 49.63~K & 42.92~K & 0.97 & 0.97 \\\\\n\\end{tabular}}\n\\label{tab:art124:table3}\n\\end{table}\n\n\\boldsection{Comparison with experiments.}\nA comparison between calculated, predicted, and experimental results is presented in\nFigure~\\ref{fig:art124:figure7}, with relevant statistics summarized in Table~\\ref{tab:art124:table3}.\nData is considered for the bulk modulus $B$, shear modulus $G$, and (acoustic) Debye temperature $\\theta_{\\substack{\\scalebox{0.6}{a}}}$\nfor 45 well-characterized materials with\ndiamond (SG\\# 227, {\\small AFLOW}\\ prototype \\texttt{A\\_cF8\\_227\\_a}),\nzincblende (SG\\# 216, \\texttt{AB\\_cF8\\_216\\_c\\_a}),\nrocksalt (SG\\# 225, \\texttt{AB\\_cF8\\_225\\_a\\_b}),\nand wurtzite (SG\\# 186, \\texttt{AB\\_hP4\\_186\\_b\\_b})\nstructures~\\cite{Morelli_Slack_2006,Semiconductors_BasicData_Springer}.\nExperimental $B$ and $G$ are compared to the $B_{\\substack{\\scalebox{0.6}{VRH}}}$ and $G_{\\substack{\\scalebox{0.6}{VRH}}}$ values predicted here, and\n$\\theta_{\\substack{\\scalebox{0.6}{a}}}$ is converted to the traditional Debye temperature $\\theta_{\\substack{\\scalebox{0.6}{D}}}=\\theta_{\\substack{\\scalebox{0.6}{a}}} n^{1\/3}$,\nwhere $n$ is the number of atoms in the unit cell.\nAll relevant values are listed in\nSupplementary Note 4 of Reference~\\cite{curtarolo:art124}.\n\nExcellent agreement is found between experimental and calculated values,\nbut more importantly, between experimental and predicted results.\nWith error metrics close to or under expected tolerances from the generalizability analysis,\nthe comparison highlights effective experimental confidence in the approach.\nThe experiments\/prediction validation is clearly the ultimate objective of the research presented here.\n\n\\subsection{Discussion}\nTraditional trial-and-error approaches have proven ineffective in discovering practical materials.\nComputational models developed with {\\small ML}\\ techniques may provide\na truly rational approach to materials design.\nTypical high-throughput {\\small DFT}\\ screenings involve exhaustive\ncalculations of all materials in the database, often without\nconsideration of previously calculated results.\nEven at high-throughput rates, an average {\\small DFT}\\ calculation of a medium\nsize structure (about 50 atoms per unit cell) takes about 1,170 CPU-hours of\ncalculations or about 37 hours on a 32-CPU cores node.\nHowever, in many cases, the desired range of values for the target property is known.\nFor instance,\nthe optimal band gap energy and thermal conductivity for optoelectronic applications\nwill depend on the power and voltage conditions of the device~\\cite{Figge_APL_2009,Zhou_JACerS_2016}.\nSuch cases offer an opportunity to leverage previous results and savvy {\\small ML}\\ models,\nsuch as those developed in this work, for rapid pre-screening of potential materials.\nResearchers can quickly narrow the list of candidate materials and avoid many extraneous\n{\\small DFT}\\ calculations---saving money, time, and computational resources.\nThis approach takes full advantage of previously calculated results,\ncontinuously accelerating materials discovery.\nWith prediction rates of about 0.1 seconds per material, the same 32-CPU cores node can screen\nover 28 million material candidates per day with this framework.\n\nFurthermore, interaction diagrams as depicted in Figure~\\ref{fig:art124:figure5} offer a pathway to design\nmaterials that meet certain constraints and requirements.\nFor example, substantial differences in thermal expansion coefficients among the materials used\nin high-power, high-frequency optoelectronic applications leads to bending and cracking of the structure\nduring the growth process~\\cite{Figge_APL_2009,Zhou_JACerS_2016}.\nNot only would this work-flow facilitate the search for semiconductors with large band gap energies,\nhigh Debye temperatures (thermal conductivity),\nbut also materials with similar thermal expansion coefficients.\n\nWhile the models themselves demonstrate excellent predictive power with minor deviations, outlier analysis reveals\ntheoretical structures to be among the worst offenders.\nThis is not surprising, as the true stability conditions (\\nobreak\\mbox{\\it e.g.}, high-pressure\/high-temperature) have yet\nto be determined, if they exist at all.\nThe {\\small ICSD}\\ estimates that structures for over 7,000 materials (or roughly 4\\%) come\nfrom calculations rather than actual experiment.\nSuch discoveries exemplify yet another application for {\\small ML}\\ modeling, rapid\/robust curation of large datasets.\n\nTo improve large-scale high-throughput computational screening for the identification\nof materials with desired properties, fast and accurate data mining approaches\nshould be incorporated into the standard work-flow.\nIn this work, we developed a universal {\\small QMSPR}\\ framework for predicting electronic\nproperties of inorganic materials.\nIts effectiveness is validated through the prediction of eight key materials properties\nfor stoichiometric inorganic crystalline materials, including\nthe metal\/insulator classification,\nband gap energy, bulk and shear moduli, Debye temperature, heat capacity (at constant\npressure and volume), and thermal expansion coefficient.\nIts applicability extends to all 230 space groups and the vast majority of\nelements in the periodic table.\nAll models are freely available at \\href{http:\/\/aflow.org\/aflow-ml}{aflow.org\/aflow-ml}.\n\n\\subsection{Methods}\n\\boldsection{Data preparation.}\nTwo independent datasets were prepared for the creation and validation of the {\\small ML}\\ models.\nThe training set includes\nelectronic~\\cite{aflowlibPAPER,aflowPAPER,aflowBZ,curtarolo:art67,monsterPGM,curtarolo:art49}\nand thermomechanical properties~\\cite{curtarolo:art96, curtarolo:art115} for a broad diversity of\ncompounds already characterized in the {\\small AFLOW}\\ database.\nThis set is used to build and analyze the {\\small ML}\\ models, one model per property.\nThe constructed thermomechanical models are then employed to make predictions of previously uncharacterized compounds in the {\\small AFLOW}\\ database.\nBased on these predictions and consideration of computational cost, several compounds are selected to validate the models' predictive\npower.\nThese compounds and their newly computed properties define the test set.\nThe compounds used in both datasets are specified in\nSupplementary Notes 2 and 3 of Reference~\\cite{curtarolo:art124}, respectively.\n\n\\boldsection{Training set.}\n{\\bf I.}\nBand gap energy data for 49,934 materials were extracted from the {\\small AFLOW}\\\nrepository~\\cite{aflowlibPAPER,aflowPAPER,aflowBZ,curtarolo:art67,monsterPGM,curtarolo:art49}, representing approximately\n60\\% of the known stoichiometric inorganic crystalline materials listed in the\nInorganic Crystal Structure Database ({\\small ICSD})~\\cite{ICSD,ICSD3}.\nWhile these band gap energies are generally underestimated with respect to experimental\nvalues~\\cite{Perdew_IJQC_1985}, {\\small DFT}+$U$ is robust enough to\ndifferentiate between metallic (no $E_{\\substack{\\scalebox{0.6}{BG}}}$) and insulating $\\left(E_{\\substack{\\scalebox{0.6}{BG}}}\\!>\\!0\\right)$ systems~\\cite{curtarolo:art104}.\nAdditionally, errors in band gap energy prediction are typically systematic.\nTherefore, the band gap energy values can be corrected \\textit{ad-hoc} with fitting\nschemes~\\cite{Yazyev_PRB_2012,Zheng_PRL_2011}.\nPrior to model development, both {\\small ICSD}\\ and {\\small AFLOW}\\ data were curated:\nduplicate entries, erroneous structures, and ill-converged calculations were corrected or removed.\nNoble gases crystals are not considered.\nThe final dataset consists of \\PLMFelectronicTotal\\ unique materials (\\PLMFmetalTotal\\ with no $E_{\\substack{\\scalebox{0.6}{BG}}}$\nand \\PLMFinsulatorTotal\\ with $E_{\\substack{\\scalebox{0.6}{BG}}}\\!>\\!0$),\ncovering the seven lattice systems, 230 space groups, and 83 elements\n(H-Pu, excluding noble gases, Fr, Ra, Np, At, and Po).\nAll referenced {\\small DFT}\\ calculations were performed with the Generalized Gradient Approximation\n({\\small GGA}) {\\small PBE}~\\cite{PBE}\nexchange-correlation functional and projector-augmented wavefunction ({\\small PAW})\npotentials~\\cite{PAW,kresse_vasp_paw} according to the\n{\\small AFLOW}\\ Standard for High-Throughput (HT) Computing~\\cite{curtarolo:art104}.\nThe Standard ensures reproducibility of the data, and provides visibility\/reasoning for any parameters\nset in the calculation, such as accuracy thresholds, calculation\npathways, and mesh dimensions.\n{\\bf II.}\nThermomechanical properties data for just over 3,000 materials were extracted from the {\\small AFLOW}\\\nrepository~\\cite{curtarolo:art115}.\nThese properties include the bulk modulus, shear modulus, Debye temperature, heat capacity at constant pressure,\nheat capacity at constant volume, and thermal expansion coefficient, and were\ncalculated using the {\\small AEL}-{\\small AGL}\\ integrated framework~\\cite{curtarolo:art96, curtarolo:art115}.\nThe {\\small AEL}\\ ({\\small AFLOW}\\ Elasticity Library)\nmethod~\\cite{curtarolo:art115} applies a set of independent normal and shear strains to the structure, and then fits the calculated stress\ntensors to obtain the elastic constants~\\cite{curtarolo:art100}.\nThese can then be used to calculate the elastic moduli in\nthe Voigt and Reuss approximations, as well as the Voigt-Reuss-Hill ({\\small VRH}) averages which are the values of the bulk and\nshear moduli modeled in this work.\nThe {\\small AGL}\\ ({\\small AFLOW}\\ {\\small GIBBS}\\ Library) method~\\cite{curtarolo:art96}\nfits the energies from a set of isotropically\ncompressed and expanded volumes of a structure to a quasiharmonic Debye-Gr{\\\"u}neisen model~\\cite{Blanco_CPC_GIBBS_2004}\nto obtain thermomechanical\nproperties, including the bulk modulus, Debye temperature, heat capacity, and thermal expansion coefficient.\n{\\small AGL}\\ has been\ncombined with {\\small AEL}\\ in a single workflow, so that it can utilize the Poisson ratios obtained from {\\small AEL}\\ to improve the\naccuracy of the thermal properties predictions~\\cite{curtarolo:art115}.\nAfter a similar curation of ill-converged calculations, the final dataset consists of\n\\PLMFthermoTrainingTotal\\ materials.\nIt covers the seven lattice systems, includes unary, binary, and ternary compounds, and\nspans broad ranges of each thermomechanical property, including\nhigh thermal conductivity systems such as C ({\\small ICSD}\\ \\#182729), BN ({\\small ICSD}\\ \\#162874), BC$_{5}$ ({\\small ICSD}\\ \\#166554),\nCN$_{2}$ ({\\small ICSD}\\ \\#247678), MnB$_{2}$ ({\\small ICSD}\\ \\#187733), and SiC ({\\small ICSD}\\ \\#164973), as well as\nlow thermal conductivity systems such as Hg$_{33}$(Rb,K)$_{3}$ ({\\small ICSD}\\ \\#410567 and \\#410566),\nCs$_{6}$Hg$_{40}$ ({\\small ICSD}\\ \\#240038), Ca$_{16}$Hg$_{36}$ ({\\small ICSD}\\ \\#107690), CrTe ({\\small ICSD}\\ \\#181056),\nand Cs ({\\small ICSD}\\ \\#426937).\nMany of these systems additionally exhibit extreme values of the bulk and shear moduli,\nsuch as C (high bulk and shear moduli) and Cs (low bulk and shear moduli).\nInteresting systems such as\nRuC ({\\small ICSD}\\ \\#183169) and NbC ({\\small ICSD}\\ \\#189090)\nwith a high bulk modulus ($B_{\\substack{\\scalebox{0.6}{VRH}}}$ = 317.92 GPa, 263.75 GPa) but\nlow shear modulus ($G_{\\substack{\\scalebox{0.6}{VRH}}}$ = 16.11 GPa, 31.86 GPa)\nalso populate the set.\n\n\\boldsection{Test set.}\nWhile nearly all {\\small ICSD}\\ compounds are characterized electronically within the {\\small AFLOW}\\ database,\nmost have not been characterized thermomechanically due to the added computational cost.\nThis presented an opportunity to validate the {\\small ML}\\ models.\nOf the remaining compounds, several were prioritized for immediate characterization via\nthe {\\small AEL}-{\\small AGL}\\ integrated framework~\\cite{curtarolo:art96, curtarolo:art115}.\nIn particular, focus was placed on systems predicted to have a large bulk modulus, as this property\nis expected to scale well with the other aforementioned thermomechanical\nproperties~\\cite{curtarolo:art96, curtarolo:art115}.\nThe set also includes various other small cell, high symmetry systems expected to span the full\napplicability domains of the models.\nThis effort resulted in the characterization of \\PLMFthermoTestTotal\\ additional compounds.\n\n\\boldsection{Data availability.}\nAll the \\nobreak\\mbox{\\it ab-initio}\\ data are freely available to the public as\npart of the {\\small AFLOW}\\ online repository and can be accessed through {\\sf \\AFLOW.org}\\\nfollowing the {\\small REST-API}\\ interface~\\cite{aflowPAPER}.\n\\clearpage\n\\chapter{Applications}\n\\section{Materials Cartography: Representing and Mining Materials Space Using Structural and Electronic Fingerprints}\n\\label{sec:art094}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art94},\nwhich was awarded with ACS Editors' Choice.\nAuthor contributions are as follows:\nStefano Curtarolo and Alexander Tropsha designed the study.\nOlexandr Isayev and Denis Fourches developed the fingerprinting and cartography methods.\nEugene N. Muratov adapted the SiRMS method for materials.\nCorey Oses and Kevin M. Rasch prepared the data and worked with the {\\sf \\AFLOW.org}\\ database.\nAll authors discussed the results and their implications and contributed to the paper.\n\n\\subsection{Introduction}\nDesigning materials with desired physical and chemical properties is recognized as an\noutstanding challenge in materials research~\\cite{Rajan_materialstoday_2005,nmatHT,Potyrailo_ACSCombSci_2011}.\nMaterial properties directly depend on a large number of key variables, often making the property prediction complex.\nThese variables include constitutive elements, crystal forms, and geometrical and electronic characteristics; among others.\nThe rapid growth of materials research has led to the accumulation of vast amounts of data.\nFor example, the Inorganic Crystal Structure Database ({\\small ICSD}) includes more than 170,000 entries~\\cite{ICSD}.\nExperimental data are also included in other databases, such as MatWeb~\\cite{MatWeb} and MatBase~\\cite{Matbase}.\nIn addition, there are several large databases such as the {\\sf \\AFLOW.org}\\ repository~\\cite{aflowBZ,aflowSCINT},\nthe Materials Project~\\cite{APL_Mater_Jain2013},\nand the Harvard Clean Energy Project~\\cite{Hachmann_JPCL_2011,Hachmann_EES_2014}\nthat contain thousands of unique materials and their theoretically calculated properties.\nThese properties include electronic structure profiles estimated with quantum mechanical methods.\nThe latter databases have great potential to serve as a source of novel functional materials.\nPromising candidates from these databases may in turn be selected for experimental\nconfirmation using rational design approaches~\\cite{MGI}.\n\nThe rapidly growing compendium of experimental and theoretical materials data offers\na unique opportunity for scientific discovery.\nSpecialized data mining and data visualization methods are being developed within\nthe nascent field of materials\ninformatics~\\cite{Rajan_materialstoday_2005,Suh_MST_2009,Olivares-Amaya_EES_2011,Potyrailo_ACSCombSci_2011,nmatHT,Schuett_PRB_2014,Seko_PRB_2014}.\nSimilar approaches have been used extensively in cheminformatics with resounding success.\nFor example, in many cases, these approaches have served to help identify and design\nsmall organic molecules with desired biological activity and acceptable\nenvironmental\/human-health safety profiles~\\cite{Laggner_NCB_2012,Besnard_Nature_2012,Cherkasov_JMC_2013,Lusci_JCIM_2013}.\nApplication of cheminformatics approaches to materials science would allow researchers to\n{\\bf i.} define, visualize, and navigate through materials space,\n{\\bf ii.} analyze and model structural and electronic characteristics of materials\nwith regard to a particular physical or chemical property, and\n{\\bf iii.} employ predictive materials informatics models to forecast the experimental properties of\n{\\it de novo} designed or untested materials.\nSuch rational design approaches in materials science constitute a rapidly growing\nfield~\\cite{Olivares-Amaya_EES_2011,Balachandran_PRSA_2011,Kong_JCIM_2012,Balachandran_ActaCristB_2012,Srinivasan_MAT_2013,Schuett_PRB_2014,Seko_PRB_2014,Broderick_APL_2014,Dey_CMS_2014}.\n\nHerein, we introduce a novel materials fingerprinting approach.\nWe combine this with graph theory, similarity searches, and machine learning algorithms.\nThis enables the unique characterization, comparison, visualization, and design of materials.\nWe introduce the concept and describe the development of materials fingerprints that encode\nmaterials' band structures, density of states ({\\small DOS}), crystallographic, and constitutional information.\nWe employ materials fingerprints to visualize this territory via advancing the new concept of ``{\\it materials cartography}''.\nWe show this technology identifies clusters of materials with similar properties.\nFinally, we develop Quantitative Materials Structure-Property Relationship ({\\small QMSPR}) models\nthat rely on these materials fingerprints.\nWe then employ these models to discover novel materials with desired properties that\nlurk within the materials databases.\n\n\\subsection{Methods}\n\\label{subsec:art094:methods}\n\n\\subsubsection{{\\sf \\AFLOW.org}\\ repository and data}\nThe {\\sf \\AFLOW.org}\\ repository of density functional theory ({\\small DFT}) calculations is managed\nby the software package {\\small AFLOW}~\\cite{aflowPAPER,aflowlibPAPER}.\nAt the time of the study, the {\\sf \\AFLOW.org}\\ database included the results of calculations\ncharacterizing over 20,000 crystals, but has since grown to include 50,000 entries ---\nrepresenting about a third of the contents of the {\\small ICSD}~\\cite{ICSD}.\nOf the characterized systems, roughly half are metallic and half are insulating.\n{\\small AFLOW}\\ leverages the {\\small VASP}\\ Package~\\cite{vasp_cms1996} to calculate the total energy\nof a given crystal structure with {\\small PAW}\\ pseudopotentials~\\cite{PAW} and the {\\small PBE}~\\cite{PBE} exchange-correlation functional.\nThe entries of the repositories have been described previously~\\cite{aflowBZ,aflowlibPAPER,aflowAPI}.\n\n\\subsubsection{Data set of superconducting materials}\nWe have compiled experimental data for superconductivity critical temperatures,\n$T_{\\mathrm{c}}$, for more than 700 records from the Handbook of Superconductivity~\\cite{Poole_Superconductivity_2000} and the\nCRC Handbook of Chemistry and Physics~\\cite{Lide_CRC_2004}, as well as the SuperCon Database~\\cite{SuperCon}.\nAs we have shown recently~\\cite{Fourches_JCIM_2010}, data curation is a necessary\nstep for any Quantitative Structure-Property Relationship ({\\small QSAR}) modeling.\nIn the compiled data set, several $T_{\\mathrm{c}}$ values have been measured under strained conditions,\nsuch as different pressures and magnetic fields.\nWe have only kept records taken under standard pressure and with no external magnetic fields.\nFor materials with variations in reported $T_{\\mathrm{c}}$ values in excess of 4~K,\noriginal references were revisited and records have been discarded when no reliable information was available.\n$T_{\\mathrm{c}}$ values with a variation of less than 3~K have been averaged.\nOf the remaining 465 materials ($T_{\\mathrm{c}}$ range of 0.1-133~K), most records show\na variability in $T_{\\mathrm{c}}$ of $\\pm$1~K between different sources.\nSuch a level of variability would be extremely influential in materials with\nlow $T_{\\mathrm{c}}$ ($T_{\\mathrm{c}}\\!<\\!1$~K) because we have used the decimal\nlogarithm of the experimentally measured critical temperature ($\\log(T_{\\mathrm{c}})$) as our target property.\n\nTo appropriately capture information inherent to materials over the full range of\n$T_{\\mathrm{c}}$, we have constructed two data sets for the development of three models.\nThe {\\bf continuous model} serves to predict $T_{\\mathrm{c}}$ and utilizes\nrecords excluding materials with $T_{\\mathrm{c}}$ values less than 2~K.\nThis data set consists of 295 unique materials with a $\\log(T_{\\mathrm{c}})$ range of 0.30-2.12.\nThe {\\bf classification model} serves to predict the position of $T_{\\mathrm{c}}$\n(above\/below) with respect to the threshold $T_{\\mathrm{thr}}$\n(unbiasedly set to 20~K as observed in Figure~\\ref{fig:art094:bands}(e), see the \\nameref{subsec:art094:results} section).\nIt utilizes records incorporating the aforementioned excluded materials,\nas well as lanthanum cuprate (La$_2$CuO$_4$, {\\small ICSD}\\ \\#19003).\nLanthanum cuprate had been previously discarded for high variability\n($T_{\\mathrm{c}}$ = 21-39~K), but now satisfies the classification criteria.\nThis data set consists of 464 materials (29 with $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$\nand 435 with $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$).\nFinally, the {\\bf structural model} serves to identify geometrical components that most\ninfluence $T_{\\mathrm{c}}$. It utilizes the same data set as the continuous model.\n\n\\subsubsection{Materials fingerprints}\nFollowing the central paradigms of structure-property relationships, we assume that\n{\\bf i.} properties of materials are a direct function of their structure and\n{\\bf ii.} materials with similar structures (as determined by constitutional,\ntopological, spatial, and electronic characteristics) are likely to have similar physical and chemical properties.\n\nThus, encoding material characteristics in the form of numerical arrays,\nnamely descriptors~\\cite{nmatHT,Schuett_PRB_2014} or\n``{\\it fingerprints}''~\\cite{Valle_ActaCristA_2010}, enables the use of classical cheminformatics and\nmachine-learning approaches to mine, visualize, and model any set of materials.\nWe have encoded the electronic structure diagram for each material as two distinct types of arrays\n(Figure~\\ref{fig:art094:fingerprints_construction}):\na {\\it symmetry-dependent fingerprint} (band structure based ``B-fingerprint'') and a\n{\\it symmetry-independent fingerprint} ({\\small DOS}\\ based ``D-fingerprint'').\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig101}\n\\mycaption[Construction of materials fingerprints from the band structure and {\\small DOS}.]\n{For simplicity, we illustrate the idea of B-fingerprints with only 8 bins.}\n\\label{fig:art094:fingerprints_construction}\n\\end{figure}\n\n\\boldsection{B-fingerprint.} Along every special high-symmetry point of the Brillouin zone ({\\small BZ}),\nthe energy diagram has been discretized into 32 bins to serve as our fingerprint array.\nEach {\\small BZ}\\ has a unique set of high-symmetry points~\\cite{aflowBZ}.\nThe comparison set of high-symmetry points belonging to a single {\\small BZ}\\ type is considered symmetry-dependent.\nTo name a few examples, the Brillouin zone path of a cubic lattice\n($\\Gamma\u2013X\u2013M\u2013\\Gamma\u2013R\u2013X\\!\\!\\mid\\!\\! M\u2013R$) is encoded with just four points ($\\Gamma, M, R, X$),\ngiving rise to a fingerprint array of length 128.\nThe body-centered orthorhombic lattice is more complex~\\cite{aflowBZ,aflowSCINT}\n($\\Gamma\u2013X\u2013L\u2013T\u2013W\u2013R\u2013X_1\u2013Z\u2013\\Gamma\u2013Y\u2013S\u2013W\\!\\mid\\!L_1\u2013Y\\!\\mid\\!Y_1\u2013Z$)\nand is represented by 13 points ($\\Gamma, L, L_1, L_2, R, S, T, W, X, X_1, Y, Y_1, Z)$,\ngiving a fingerprint array of length 416.\nConversely, the comparison of identical {\\bf k}-points not specifically belonging to any {\\small BZ}\\\nis always possible when only restricted to $\\Gamma$.\nConsequently, we limit our models to the $\\Gamma$ point B-fingerprint in the present work.\n\n\\boldsection{D-fingerprint.} A similar approach can be taken for the {\\small DOS}\\ diagrams,\nwhich are sampled in 256 bins (from min to max) and the magnitude of each bin is discretized in 32 bits.\nTherefore, the D-fingerprint is a total of 1024 bytes.\nOwing to the complexity and limitations of the symmetry-dependent B-fingerprints,\nwe have only generated symmetry-independent D-fingerprints.\nThe length of these fingerprints is tunable depending on the objects, applications, and other factors.\nWe have carefully designed the domain space and length of these fingerprints to avoid\nthe issues of enhancing boundary effects or discarding important features.\n\n\\boldsection{SiRMS descriptors for materials.}\nTo characterize the structure of materials from several different perspectives,\nwe have developed descriptors similar to those used for small organic molecules\nthat can reflect their compositional, topological, and spatial (stereochemical) characteristics.\nClassical cheminformatics tools can only handle small organic molecules.\nTherefore, we have modified the Simplex (SiRMS) approach~\\cite{Kuzmin_JCAMD_2008}\nbased on our experience with mixtures~\\cite{Muratov_SC_2013,Muratov_MI_2012}\nin order to make this method suitable for computing descriptors for materials.\n\nThe SiRMS approach~\\cite{Kuzmin_JCAMD_2008} characterizes small organic molecules\nby splitting them into multiple molecular fragments called simplexes.\nSimplexes are tetratomic fragments of fixed composition (1D), topology (2D), and chirality and symmetry (3D).\nThe occurrences of each of these fragments in a given compound are then counted.\nAs a result, each molecule of a given data set can be characterized by its SiRMS fragment profiles.\nThese profiles take into account atom types, connectivity, \\nobreak\\mbox{\\it etc.}~\\cite{Kuzmin_JCAMD_2008}.\nHere, we have adapted the SiRMS approach to describe materials with their fragmental compositions.\n\nEvery material is represented according to the structure of its crystal unit cell (Figure~\\ref{fig:art094:sirms_generation}).\nComputing SiRMS descriptors for materials is equivalent to the computation of\nSiRMS fragments for nonbonded molecular mixtures.\nBounded simplexes describe only a single component of the mixture.\nUnbounded simplexes could either belong to a single component, or could span up to four components of the unit cell.\nA special label is used during descriptor generation to distinguish ``mixture''\nsimplexes (belonging to different molecular moieties) from those incorporating elements from a single compound~\\cite{Muratov_MI_2012}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig102}\n\\mycaption{Generation of SiRMS descriptors for materials.}\n\\label{fig:art094:sirms_generation}\n\\end{figure}\n\nThus, the structure of every material is characterized by both bounded and unbounded\nSiRMS descriptors as illustrated in Figure~\\ref{fig:art094:sirms_generation}.\nThe descriptor value of a given simplex fragment is equal to the number of its occurrences in the system.\nIn the case of materials, this value has been summed throughout all the constituents of a system;\ntaking into account their stoichiometric ratios and crystal lattices (see Figure~\\ref{fig:art094:sirms_generation}).\n``Mixture'' descriptors are weighted according to the smallest stoichiometric\nratio of constituents within this mixture, and added throughout all the mixtures in a system.\nAtoms in simplexes are differentiated according to their type (element) and partial charge.\nFor the latter, atoms are divided into six groups corresponding to their partial charge:\n$A\\!\\leq\\!-2\\!<\\!B\\!\\leq\\!-1\\!<\\!C\\!\\leq\\!0\\!<\\!D\\!\\leq\\!1\\!<\\!E\\!\\leq\\!2\\!<\\!F$.\nIn addition, we have developed a special differentiation of atoms in simplexes to account for their groups on the periodic table.\nThat is, all elements belonging to the same group are encoded by the same symbol.\n\n\\subsubsection{Network representation (materials cartograms)}\nTo represent the library of materials as a network, we considered each material, encoded by its fingerprints, as a node.\nEdges exist between nodes with similarities greater than or equal to certain thresholds.\nIn this study, we use fingerprint-based Tanimoto similarity and a threshold $S=0.7$.\nThis network representation of materials is defined as the graph $G(V,E)$, where $V=\\left\\{\\nu_1|\\nu_2\\in L\\right\\}$ and\n$E\\!=\\!\\left\\{(\\nu_1,\\nu_2)\\mid\\mathrm{sim}(\\nu_1,\\nu_2)\\geq T\\right\\}$.\nHere, $L$ denotes a materials library, $\\mathrm{sim}(\\nu_1,\\nu_2)$\ndenotes a similarity between materials $\\nu_1$ and $\\nu_2$, and $T$ denotes a similarity threshold.\n\nTo examine if the materials networks are scale-free, we analyzed the degree distributions of the networks.\nNetworks are considered scale-free if the distribution of vertex degrees of the nodes follows the power law:\n$p(x)=kx^{-\\alpha}$ where $k$ is the normalization constant, and $\\alpha$ is the exponent.\nThe materials networks have been visualized using the Gephi package~\\cite{Bastian_ICWSM_2009}.\nThe ForceAtlas 2 algorithm~\\cite{Jacomy_PLoS_2014}, a type of force-directed layout\nalgorithm, has been used for the graph layout.\nA force-directed layout algorithm considers a force between any two nodes,\nand minimizes the ``energy'' of the system by moving the nodes and changing the forces between them.\nThe algorithm guarantees that the topological similarity among nodes determines their vicinity, leading to accurate and\nvisually-informative representations of materials space.\n\n\\subsection{Results and discussion}\n\\label{subsec:art094:results}\n\n\\subsubsection{Similarity search in materials space}\nIn the first phase of this study, the optimized geometries, symmetries,\nband structures, and {\\small DOS}{}s available in the {\\sf \\AFLOW.org}\\ repository were converted\ninto fingerprints, or arrays of numbers.\n\nWe encoded the electronic structure diagram for each material as two distinct types of\nfingerprints (Figure~\\ref{fig:art094:fingerprints_construction}):\nband structure symmetry-dependent fingerprints (B-fingerprints) and\n{\\small DOS}\\ symmetry-independent fingerprints (D-fingerprints).\nThe B-fingerprint is defined as a collated digitalized histogram of energy eigenvalues\nsampled at the high-symmetry reciprocal points with 32 bins.\nThe D-fingerprint is a string containing 256 4-byte real numbers,\neach characterizing the strength of the {\\small DOS}\\ in one of the 256 bins dividing the [-10, 10]~eV interval.\nMore details are in the \\nameref{subsec:art094:methods} section.\n\nThis unique, condensed representation of materials enabled the use of cheminformatics methods,\nsuch as similarity searches, to retrieve materials with similar properties but different compositions from the {\\sf \\AFLOW.org}\\ database.\nAs an added benefit, our similarity search can also quickly find duplicate records.\nFor example, we have identified several barium titanate (BaTiO$_3$) records with identical fingerprints\n({\\small ICSD}\\ \\#15453, \\#27970, \\#6102, and \\#27965 in the {\\sf \\AFLOW.org}\\ database).\nThus, fingerprint representation afforded rapid identification of duplicates,\nwhich is the standard first step in our cheminformatics data curation workflow~\\cite{Fourches_JCIM_2010}.\nIt is well known that standard {\\small DFT}\\ has severe limitations in the description of excited states, and needs to be substituted\nwith more advanced approaches to characterize semiconductors and\ninsulators~\\cite{Hedin_GW_1965,GW,Heyd2003,Liechtenstein1995,Cococcioni_reviewLDAU_2014}.\nHowever, there is a general trend of {\\small DFT}\\ errors being comparable in similar classes of systems.\nThese errors may thus be considered ``systematic'', and are irrelevant when one seeks only similarities between materials.\n\nThe first test case is gallium arsenide, GaAs ({\\small ICSD}\\ \\#41674),\na very important material for electronics~\\cite{INSPEC_PGA_1986} in the {\\sf \\AFLOW.org}\\ database.\nGaAs is taken as the reference material, and the remaining 20,000+ materials from the\n{\\sf \\AFLOW.org}\\ database are taken as the virtual screening library.\nThe pairwise similarity between GaAs and any of the materials represented by our D-fingerprints\nis computed using the Tanimoto similarity coefficient ($S$)~\\cite{Maggiora_JMC_2014}.\nThe top five materials (GaP, Si, SnP, GeAs, InTe) retrieved show very high similarity ($S\\!>\\!0.8$)\nto GaAs, and all five are known to be semiconductor materials~\\cite{Lide_CRC_2004,Littlewood_CRSSMS_1983,Madelung_Semiconductors_2004}.\n\nIn addition, we have searched the {\\sf \\AFLOW.org}\\ database for materials similar to BaTiO$_3$\nwith the perovskite structure ({\\small ICSD}\\ \\#15453) using B-fingerprints.\nBaTiO$_3$ is widely used as a ferroelectric ceramic or piezoelectric~\\cite{Bhalla_MRI_2000}.\nOut of the six most similar materials with $S>0.8$, five (BiOBr, SrZrO$_3$, BaZrO$_3$, KTaO$_3$ and KNbO$_3$)\nare well known for their optical properties~\\cite{Rabe_Ferroelectrics_2010}.\nThe remaining material, cubic YbSe ({\\small ICSD}\\ \\#33675), is largely unexplored.\nOne can therefore formulate a testable hypothesis suggesting that this material may be ferroelectric or piezoelectric.\n\nWe also investigated the challenging case of topological insulators.\nThey form a rare group of insulating materials with conducting surface-segregated states (or interfaces)~\\cite{nmatTI}\narising from a combination of spin-orbit coupling and time-reversal symmetry~\\cite{RevModPhys.82.3045}.\nAlthough {\\small DFT}\\ calculations conducted for materials in the {\\sf \\AFLOW.org}\\ repository do not\nincorporate spin-orbit coupling for the most part~\\cite{nmatTI}, various topological insulators show exceptionally\nhigh band-structure similarities --- validating the B-fingerprints scheme.\nThe two materials most similar to Sb$_2$Te$_3$~\\cite{RevModPhys.82.3045} (based on B-fingerprints)\nwith $S\\!>\\!0.9$ are Bi$_2$Te$_3$~\\cite{Chen09science,zhang_PRL_2009} and Sb$_2$Te$_2$Se~\\cite{Xu2010arxiv1007}.\nFive out of six materials most similar to Bi$_2$Te$_2$Se~\\cite{Xu2010arxiv1007,Arakane2010NC}\nare also known topological insulators: Bi$_2$Te$_2$S, Bi$_2$Te$_3$, Sb$_2$Te$_2$Se,\nGeBi$_2$Te$_4$~\\cite{Xu2010arxiv1007}, and Sb$_2$Se$_2$Te~\\cite{nmatTI,Zhang_Nat.Phys._2009}.\n\nThese examples demonstrate proof of concept and illustrate the power of simple yet uncommon\nfingerprint-based similarity searches for rapid and effective identification of\nmaterials with similar properties in large databases.\nThey also illuminate the intricate link between structures and properties of materials by demonstrating\nthat similar materials (as defined by their fingerprint similarity) have similar\nproperties (such as being ferroelectric or insulating).\nThis observation sets the stage for building and exploring {\\small QMSPR}\\ models; as discussed in the following sections.\n\n\\subsubsection{Visualizing and exploring materials space}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig103}\n\\mycaption[Materials cartograms with D- (top) and B-fingerprint network representations (bottom).]\n{({\\bf a}) D-fingerprint network representation of materials. Materials are color-coded\naccording to the number of atoms per unit cell.\nRegions corresponding to pure elements, binary, ternary and quaternary compounds are outlined.\n({\\bf b}) Distribution of connectivity within the network.\n({\\bf c}) Mapping band gaps of materials. Points colored in deep blue are metals;\ninsulators are color-coded according to the band gap value. Four large communities are outlined.\n({\\bf d}) Mapping the superconductivity critical temperature, $T_{\\mathrm{c}}$, with relevant regions outlined.}\n\\label{fig:art094:cartograms}\n\\end{figure}\n\nThe use of fingerprint representation and similarity concepts led us to develop the materials network.\nCompounds are mapped as nodes.\nWe use the ``{\\it force directed graph drawing}'' algorithm~\\cite{Herman_IEEEtvcg_2000}\nin which positions of the compounds are initially taken randomly.\nThere is a force between the nodes: a repulsive Coulomb component and an optional\nattractive contribution with a spring constant equal to the Tanimoto coefficient\nbetween D-fingerprints (effective when $S\\ge0.7$).\nTwo nodes are connected only when the coefficient is greater than or equal to the threshold.\nThe model is equilibrated through a series of heating and quenching steps.\nFigure~\\ref{fig:art094:cartograms}(a) shows the result in which we add\nBezier-curved lines depicting regions of accumulation.\nWe shall refer to this approach to visualizing and analyzing materials and their properties as ``{\\it materials cartography}''.\n\nThe network shown in Figure~\\ref{fig:art094:cartograms}(a) is color-coded according to overall complexity.\nPure systems, 79\\% of the total 246 unary nodes, are confined in a small, enclosed region.\nBinary nodes cover more configurational space, with 82\\% of the 3700+ binaries lying in a compact region.\nTernaries are scattered. They mostly populate the center of the space (91\\% of the 5300+ ternaries).\nQuaternaries and beyond are located at the top part of the network (92\\% of the 1080 nodes).\nThis region is the most distant from that of the unary nodes, which tends to be disconnected from the others.\nIndeed, overlap between binaries and ternaries is substantial.\nThe diversification of electronic properties and thickness of the compact envelope grows with structural complexity.\nOrphans are defined as nodes with a very low degree of connectivity: only the vertices (materials)\nconnected by edges are shown ($\\sim$39\\% of the database).\nInterestingly, of the 200 materials with connectivity smaller than 12,\nmost are La-based (36 bimetallic and 126 polymetallic) or Ce-based (10 nodes).\n\n\\begin{table}[tp]\\centering\n\\mycaption[Topological properties for constructed materials cartograms.]\n{In network theory, a ``component'' is a group of nodes that are all connected to each other.\nA ``giant component'' is a connected component of a given random graph that contains a constant\nfraction of the entire graph's vertices~\\cite{Chung_Complex_2006}.\nFigures in parenthesis are calculated by fitting only the asymptotic portion of the curve in Figure~\\ref{fig:art094:cartograms}(b).\n}\n\\vspace{3mm}\n\\begin{tabular}{l | r r}\n & D-fingerprints network & B-fingerprints network \\\\\n\\hline\ntotal number of cases & 17420 & 17420 \\\\\ngiant component & 10521 (60.4\\%) & 15535 (89.2\\%)\\\\\nedges & 466,000 & 564,000 \\\\\naverage degree & 88.60 & 72.59 \\\\\nnetwork diameter (edges) & 27 & 23 \\\\\npower law $\\gamma$ & 2.745 & 0.916 (2.04) \\\\\n\\end{tabular}\n\\label{tab:art094:cartograms}\n\\end{table}\n\nThe degree of connectivity is illustrated in Figure~\\ref{fig:art094:cartograms}(b).\nThe panel indicates the log-log distribution of connectivity across the sample set.\nThe red and blue points measure the D-fingerprints (Figure~\\ref{fig:art094:cartograms}(a))\nand B-fingerprints connectivity (Figure~\\ref{fig:art094:cartograms}(c)), respectively.\nTable~\\ref{tab:art094:cartograms} contains relevant statistical information about the cartograms.\nAlthough the power law distribution of Figure~\\ref{fig:art094:cartograms}(b) is typical of\nscale-free networks and similar to many networks examined in cheminformatics and\nbioinformatics~\\cite{Girvan_PNAS_2002,Newman_SiRev_2003,Yildirim_NB_2007}, in our case, connectivity differs.\nIn previous examples~\\cite{Girvan_PNAS_2002,Newman_SiRev_2003,Yildirim_NB_2007},\nmost of the nodes have only a few connections; with a small minority being highly\nconnected to a small set of ``hubs''~\\cite{Jeong_Nature_2000,Barabasi_Science_1999}.\nIn contrast, the {\\sf \\AFLOW.org}\\ database is highly heterogeneous:\nmost of the hubs' materials are concentrated along the long, narrow belt along the middle of the network.\nThe top 200 nodes (ranked by connectivity) are represented by 83 polymetallics\n(CoCrSi, Al$_2$Fe$_3$Si$_3$, Al$_8$Cr$_4$Y, \\nobreak\\mbox{\\it etc.}),\n102 bimetallics (Al$_3$Mo, As$_3$W$_2$, FeZn$_{13}$, \\nobreak\\mbox{\\it etc.}),\n14 common binary compounds (GeS, AsIn, \\nobreak\\mbox{\\it etc.}), and boron ({\\small ICSD}\\ \\#165132).\nThis is not entirely surprising, since these materials are well studied\nand represent the lion's share of the {\\small ICSD}\\ database.\nAl$_3$FeSi$_2$ ({\\small ICSD}\\ \\#79710), an uncommonly used material, has the highest connectivity of 946.\nMeanwhile, complex ceramics and exotic materials are relatively disconnected.\n\nA second network, built with B-fingerprints, is illustrated in Figure~\\ref{fig:art094:cartograms}(c).\nWhile this network preserves most of the topological features described\nin the D-fingerprint case (Figure~\\ref{fig:art094:cartograms}(a)), critical distinctions appear.\nThe B-fingerprint network separates metals from insulators.\nClustering and subsequent community analyses show four large groups of materials.\nGroup-A ($\\sim$3000 materials) consists predominately of insulating compounds (63\\%) and semiconductors (10\\%).\nGroup-B distinctly consists of compounds with polymetallic character (70\\% of $\\sim$2500 materials).\nIn contrast, Group-C includes $\\sim$500 zero band gap materials with nonmetal atoms,\nincluding halogenides, carbides, silicides, \\nobreak\\mbox{\\it etc.}\\\nLastly, Group-D has a mixed character with $\\sim$300 small band gap materials (below 1.5~eV);\nand $\\sim$500 semimetals and semiconductors.\n\nLithium scandium diphosphate, LiScP$_2$O$_7$ ({\\small ICSD}\\ \\#91496), has the highest connectivity\nof 746 in the B-fingerprint network.\nVery highly connected materials are nearly evenly distributed between Groups-A and -B,\nforming dense clusters within their centers.\nAs in the case of the D-fingerprint network, the connectivity distribution follows a power law\n(Figure~\\ref{fig:art094:cartograms}(b), see Table~\\ref{tab:art094:cartograms} for\nadditional statistics); indicating that this is a scale-free network.\n\nTo illustrate one possible application of the materials networks, we chose superconductivity ---\none of the most elusive challenges in solid-state physics.\nWe have compiled experimental data for 295 stoichiometric superconductors that\nare also available in the {\\sf \\AFLOW.org}\\ repository.\nAll materials in the data set are characterized with the fingerprints specified in\nthe \\nameref{subsec:art094:methods} section.\nThe data set includes both prominently high temperature superconducting materials\nsuch as layered cuprates, ferropnictides, iron arsenides-122, MgB$_2$; as well as more\nconventional compounds such as A15, ternary pnictides, \\nobreak\\mbox{\\it etc.}\\\nOur model does not consider the effect of phonons, which play a\ndominant role in many superconductors~\\cite{tinkham_superconductivity}.\nHigh-throughput parameterization of phonon spectra is still in its infancy~\\cite{curtarolo:Ru},\nand only recently have vibrational descriptors been\nadapted to large databases~\\cite{curtarolo:art96}.\nWe envision that future development of vibrational fingerprints\nfollowing these guidelines will capture similarities between\nknown, predicted, and verified superconductors (\\nobreak\\mbox{\\it i.e.},\nMgB$_2$ \\nobreak\\mbox{\\it vs.}\\ LiB$_2$~\\cite{curtarolo:art21,curtarolo:art26} and MgB$_2$ \\nobreak\\mbox{\\it vs.}\\\nFe-B compounds~\\cite{Kolmogorov_FeB_PRL2010,Gou_PRL_2013_FeB_superconductor}).\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig104}\n\\mycaption[Comparison high-low $T_{\\mathrm{c}}$ aligned band structures and $T_{\\mathrm{c}}$ predictions.]\n{({\\bf a}) Band structure of Ba$_2$Ca$_2$Cu$_3$HgO$_8$ ($T_{\\mathrm{c}}=$133~K).\n({\\bf b}) Band structure of SrCuO$_2$ ({\\small ICSD}\\ \\#16217, $T_{\\mathrm{c}}=$91~K~\\cite{Takahashi_PSCC_1994_SrCuO2_superconductor}).\n({\\bf c}) Aligned B-fingerprints for the 15 materials with the highest and lowest $T_{\\mathrm{c}}$.\n({\\bf d}) Band structure of Nb$_2$Se$_3$ ({\\small ICSD}\\ \\#42981, $T_{\\mathrm{c}}=$0.4~K).\n({\\bf e}) Plot of the predicted \\nobreak\\mbox{\\it vs.}\\ experimental critical temperatures for the continuous model.\nMaterials are color-coded according to the classification model: solid\/open green (red) circles indicate\ncorrect\/incorrect predictions in $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$\n($T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$), respectively.}\n\\label{fig:art094:bands}\n\\end{figure}\n\nAll materials are identified and marked on the B-fingerprint network, and are\ncolor-coded according to their critical temperature, $T_{\\mathrm{c}}$ (Figure~\\ref{fig:art094:cartograms}(d)).\nAll high-$T_{\\mathrm{c}}$ superconductors are localized in a relatively compact region.\nThe distribution is centered on a tight group of Ba$_2$Cu$_3X$O$_7$ compounds\n(the so-called Y123, where $X$= lanthanides).\nThe materials with the two highest $T_{\\mathrm{c}}$ values in our set are\nBa$_2$Ca$_2$Cu$_3$HgO$_8$ ({\\small ICSD}\\ \\#75730, $T_{\\mathrm{c}}=$133~K) and\nBa$_2$CaCu$_2$HgO$_6$ ({\\small ICSD}\\ \\#75725, $T_{\\mathrm{c}}=$125~K).\nTheir close grouping manifests a significant superconductivity hot-spot of materials with similar fingerprints.\nWe aligned the B-fingerprints for the 15 superconductors with the highest $T_{\\mathrm{c}}$\nvalues in Figure~\\ref{fig:art094:bands}(c).\n\nAll the top 15 high $T_{\\mathrm{c}}$ superconductors are layered cuprates,\nwhich have dominated high $T_{\\mathrm{c}}$ superconductor research since 1986~\\cite{Bednorz_ZPBCM_1986}.\nThese compounds are categorized as Charge-Transfer Mott Insulators (CTMI)~\\cite{Zaanen_PRL_1985}.\nThere are three distinct bands that are conserved for these structures around -6, -1, and 4~eV\nrelative to the Fermi energy at $\\Gamma$ (within the simple {\\small DFT}+$U$ description available in the {\\sf \\AFLOW.org}\\ repository,\nFigure~\\ref{fig:art094:bands}(c)).\nThese features are consistent with the three-band Hubbard-like picture characteristic of\nCTMIs~\\cite{Manske_Superconductors_2004,Emery_PRL_1987}.\n\nMeanwhile, the fingerprint distribution for the 15 materials with the lowest\n$T_{\\mathrm{c}}$ is random (Figure~\\ref{fig:art094:bands}(c)).\nThe importance of band structure features in superconductivity has long\nbeen recognized~\\cite{Zaanen_NPhys_2006,Micnas_RMP_1990,Orenstein_Science_2000}.\nThus, materials cartography based on the B-fingerprint network allows us to visualize this phenomenon concisely.\n\n\\subsubsection{Predictive QMSPR modeling}\nWe developed {\\small QMSPR}\\ models (continuous~\\cite{Bramer_PDM_2007}, classification, and structural)\nto compute superconducting properties of materials from their structural characteristics.\nTo achieve this objective, we compiled two superconductivity data sets consisting of\n{\\bf i.} 295 materials with continuous $T_{\\mathrm{c}}$ values ranging from 2~K to 133~K; and\n{\\bf ii.} 464 materials with binary $T_{\\mathrm{c}}$ values.\nThe models were generated with Random Forest (RF)~\\cite{Breiman_ML_2001} and\nPartial Least Squares (PLS)~\\cite{Wold_CILS_2001} techniques.\nThese used both B- and D-fingerprints, as well as Simplex (SiRMS)~\\cite{Kuzmin_JCAMD_2008} descriptors.\nThese fingerprints were adapted for materials modeling for the first time in this study\n(see the \\nameref{subsec:art094:methods} section).\nAdditionally, we incorporated atomic descriptors that differentiate\nby element, charge, and group within the periodic table.\nStatistical characteristics for all 464 materials used for the {\\small QMSPR}\\ analysis\nare reported in Tables~\\ref{tab:art094:continuous}-\\ref{tab:art094:fragments}.\n\nAttempts to develop {\\small QMSPR}\\ models using B- and D-fingerprints for both data sets were not satisfactory,\nindicating that our fingerprints, while effective in qualitative clustering,\ndo not contain enough information for quantitatively predicting target properties\n({\\small QMSPR}\\ model acceptance criteria has been discussed previously~\\cite{Tropsha_MI_2010}).\nThus, we employed more sophisticated chemical fragment descriptors,\nsuch as SiRMS~\\cite{Kuzmin_JCAMD_2008}, and adapted them for\nmaterials modeling (see the \\nameref{subsec:art094:methods} section).\n\n\\boldsection{Continuous model.}\nWe constructed a continuous model which serves to predict the value of\n$T_{\\mathrm{c}}$ with a consensus RF- and PLS-SiRMS approach.\nIt has a cross-validation determination coefficient of $Q^2=0.66$ (five-fold external CV;\nsee Table~\\ref{tab:art094:continuous}).\nFigure~\\ref{fig:art094:bands}(e) shows predicted \\nobreak\\mbox{\\it vs.}\\ experimental $T_{\\mathrm{c}}$\nvalues for the continuous model: all materials having\n$\\log(T_{\\mathrm{c}})$$\\leq$1.3 are scattered, but within the correct range.\nInterestingly, we notice that systems with\n$\\log(T_{\\mathrm{c}})$$\\geq$1.3 received higher accuracy, with the exceptions of\nMgB$_{2}$ ({\\small ICSD}\\ \\#26675), Nb$_{3}$Ge ({\\small ICSD}\\ \\#26573), Cu$_{1}$Nd$_{2}$O$_{4}$ ({\\small ICSD}\\ \\#4203),\nAs$_{2}$Fe$_{2}$Sr ({\\small ICSD}\\ \\#163208), Ba$_{2}$CuHgO$_{4}$ ({\\small ICSD}\\ \\#75720), and\nClHfN ({\\small ICSD}\\ \\#87795) (all highly underestimated).\nNot surprisingly MgB$_2$~\\cite{Buzea_SST_2001_MgB2} is an outlier in our statistics.\nThis is in agreement with the fact that to date\nno superconductor with an electronic structure similar to MgB$_2$ has been found.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Statistical characteristics of the continuous {\\small QMSPR}\\ models for superconductivity.]\n{$Q^{2}$(ext) refers to the leave-one-out five-fold external cross-validation coefficient,\nRMSE refers to root-mean-square error,\nMAE refers to the mean absolute error,\nRF-SiRMS refers to the application of the Random Forest technique with Simplex descriptors,\nPLS-SiRMS refers to the application of the Partial Least Squares regression technique with Simplex descriptors,\nand consensus refers to the average of the RF-SiRMS and PLS-SiRMS results.}\n\\vspace{3mm}\n\\begin{tabular}{l | r r r r}\nmodel & $N$ & $Q^{2}$(ext) & RMSE & MAE \\\\\n\\hline\nRF-SiRMS & 295 & 0.64 & 0.24 & 0.18\\% \\\\\nPLS-SiRMS & 295 & 0.61 & 0.25 & 0.20\\% \\\\\nconsensus & 295 & 0.66 & 0.23 & 0.18\\% \\\\\n\\end{tabular}\n\\label{tab:art094:continuous}\n\\end{table}\n\n\\boldsection{Classification model.}\nBy observing the existence of the threshold $T_{\\mathrm{thr}}$=20~K ($\\log(T_{\\mathrm{thr}})$=1.3),\nwe developed a classification model.\nIt is based on the same RF-SiRMS technique, but it is strictly used to predict the position of\n$T_{\\mathrm{c}}$ with respect to the threshold, above or below.\nThe classification model has a balanced accuracy of 0.97 with five-fold external CV analysis.\nThe type of points in Figure~\\ref{fig:art094:bands}(e) illustrates the classification model outcome:\nsolid\/open green (red) circles for correct\/incorrect\npredictions in $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ ($T_{\\mathrm{c}}\\leq T_{\\mathrm{thr}}$), respectively.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Statistical characteristics of the classification {\\small QMSPR}\\ models for superconductivity.]\n{AD refers to applicability domain~\\cite{Tropsha_CPD_2007}.\nAccuracy is determined by the ratio of correct predictions to the total number of predictions,\nsensitivity is determined by the ratio of correctly predicted $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$\nto the number of empirical $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$,\nspecificity is determined by the ratio of correctly predicted $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$\nto the number of empirical $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$,\nCCR (correct classification rate) is the average of the sensitivity and the specificity,\nand coverage is determined by the ratio of the total number of predictions to the total number of cases.}\n\\vspace{3mm}\n\\begin{tabular}{l | r r}\n & no AD & with AD \\\\\n\\hline\ntotal number of cases & 464 & 464\\\\\ntotal number of predictions & 464 & 451\\\\\nnumber of correct predictions & 452 & 446\\\\\nnumber of wrong predictions & 12 & 5\\\\\nnumber of empirical $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ & 29 & 22\\\\\nnumber of empirical $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$ & 435 & 429\\\\\nnumber of correctly predicted $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ & 19 & 17\\\\\nnumber of correctly predicted $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$ & 433 & 429\\\\\nnumber of incorrectly predicted $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ & 2 & 0\\\\\nnumber of incorrectly predicted $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$ & 10 & 5\\\\\n$T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ prediction value & 0.90 & 1.00\\\\\n$T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$ prediction value & 0.98 & 0.99\\\\\naccuracy & 0.97 & 0.99\\\\\nsensitivity & 0.66 & 0.77\\\\\nspecificity & 1.00 & 1.00\\\\\nCCR & 0.83 & 0.89\\\\\ncoverage & 1.00 & 0.97\\\\\n\\end{tabular}\n\\label{tab:art094:classification}\n\\end{table}\n\nFor $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$ and $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$,\naccuracies of prediction are 98\\% and 90\\% (cumulative 94\\%).\n(Figure~\\ref{fig:art094:bands}(e), see Table~\\ref{tab:art094:classification} for additional statistics).\nAmong the 464 materials, ten systems with experimental $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ are predicted to have\n$T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$)\n[FeLaAsO ({\\small ICSD}\\ \\#163496), AsFeO$_{3}$Sr$_{2}$V ({\\small ICSD}\\ \\#165984), As$_{2}$EuFe$_{2}$ ({\\small ICSD}\\ \\#163210),\nAs$_{2}$Fe$_{2}$Sr, CuNd$_{2}$O$_{4}$ ({\\small ICSD}\\ \\#86754), As$_{2}$BaFe$_{2}$\n({\\small ICSD}\\ \\#166018), MgB$_{2}$, ClHfN, La$_{2}$CuO$_{4}$, and Nb$_{3}$Ge].\nOnly two with experimental $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$\nare predicted with $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$\n(AsFeLi ({\\small ICSD}\\ \\#168206), As$_{2}$CaFe$_{2}$ ({\\small ICSD}\\ \\#166016)).\nOwing to the spread around the threshold, additional information about\nborates and Fe-As compounds is required for proper training of the learning algorithm.\n\nIn the past, it has been shown that {\\small QSAR}\\ approaches can be used for the\ndetection of mis-annotated chemical compounds, a critical step in data curation~\\cite{Fourches_JCIM_2010}.\nWe have employed a similar approach here.\nIn our models, three materials, ReB$_2$ ({\\small ICSD}\\ \\#23871),\nLi$_2$Pd$_3$B ({\\small ICSD}\\ \\#84931), and La$_2$CuO$_4$, were significantly mis-predicted.\nMore careful examination of the data revealed that the $T_{\\mathrm{c}}$'s of\nReB$_2$ and Li$_2$Pd$_3$B were incorrectly extracted from literature.\nWe also found that La$_2$CuO$_4$ has the largest variation of reported values within the data set.\nTherefore, it was excluded from the regression.\nThis approach illustrates that {\\small QMSPR}\\ modeling should be automatically implemented to reduce and correct erroneous entries.\n\n\\boldsection{Structural model.}\nWe also developed a structural model meant to capture the geometrical\nfeatures that most influence $T_{\\mathrm{c}}$.\nIt employs SiRMS descriptors, PLS approaches, and five-fold external cross-validation.\nThe predictive performance of this model ($Q^2=0.61$) is comparable to that\nof the SiRMS-based RF model (see Table~\\ref{tab:art094:continuous} for additional statistics).\nThe top 10 statistically significant geometrical fragments and their contributions to\n$T_{\\mathrm{c}}$ variations are shown in Table~\\ref{tab:art094:classification}.\nAll descriptor contributions were converted to atomic contributions\n(details discussed previously~\\cite{Muratov_FMC_2010}) and related to material structures.\nExamples of unit cell structures for pairs of similar materials\nwith different $T_{\\mathrm{c}}$ values were color-coded according to\natomic contributions to $T_{\\mathrm{c}}$, and are shown in Figure~\\ref{fig:art094:structure_fragments}\n(green for $T_{\\mathrm{c}}\\!\\uparrow$, red for $T_{\\mathrm{c}}\\!\\downarrow$, and gray for neutral).\n\n\\begin{table}[tp]\\centering\n\\mycaption[Top statistically significant fragments and their contributions to $T_{\\mathrm{c}}$ variation.]\n{``-'' demonstrates that the collection is bonded, while ``and'' demonstrates that the collection is not bonded.}\n\\vspace{3mm}\n\\begin{tabular}{l | r}\nfragment name & contribution to $log(T_{\\mathrm{c}})$ score\\\\\n\\hline\nO-Cu-O & 18\\%\\\\\nperiodic groups IB-IIB-IVA & 14\\%\\\\\nperiodic groups IIA and IB & 12\\%\\\\\nAs, As, Fe fragment count & 5\\%\\\\\nperiodic groups IIB-IVA & 5\\%\\\\\nperiodic groups IIA and IVA & 5\\%\\\\\ncharges~\\cite{bader_atoms_1994} (-1.5)(-1.5)(+2.5) & 3\\%\\\\\nO element count & 2\\%\\\\\nCu element count & 2\\%\\\\\nO, O, O fragment count & 2\\%\\\\\ncharge~\\cite{bader_atoms_1994} (+2.5) & 2\\%\\\\\nNb element count & 2\\%\\\\\ncharge~\\cite{bader_atoms_1994} (-1.5) & 2\\%\\\\\n\\end{tabular}\n\\label{tab:art094:fragments}\n\\end{table}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig105}\n\\mycaption[Materials color-coded according to atom contributions to $\\log(T_{\\mathrm{c}})$.]\n{Atoms and structural fragments that decrease superconductivity critical temperatures\nare colored in red and those enhancing $T_{\\mathrm{c}}$ are shown in green.\nNon-influential fragments are in gray.\n({\\bf a}) Ba$_2$Ca$_2$Cu$_3$HgO$_8$,\n({\\bf b}) As$_2$Ni$_2$O$_6$Sc$_2$Sr$_4$,\n({\\bf c}) Mo$_6$PbS$_8$,\n({\\bf d}) Mo$_6$NdS$_8$,\n({\\bf e}) Li$_2$Pd$_3$B,\n({\\bf f}) Li$_2$Pt$_3$B,\n({\\bf g}) FeLaAsO, and\n({\\bf h}) FeLaPO. }\n\\label{fig:art094:structure_fragments}\n\\end{figure}\n\nExamples of fragments for materials having\n$T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ [Ba$_2$Ca$_2$Cu$_3$HgO$_8$, {\\small ICSD}\\ \\#75730,\n$\\log(T_{\\mathrm{c}})$=2.12]\nand $T_{\\mathrm{c}}\\!\\!\\leq T_{\\mathrm{thr}}$\n[As$_2$Ni$_2$O$_6$Sc$_2$Sr$_4$, {\\small ICSD}\\ \\#180270, $\\log(T_{\\mathrm{c}})$=0.44]\nare shown in Figures~\\ref{fig:art094:structure_fragments}(a) and~\\ref{fig:art094:structure_fragments}(b), respectively.\nThey indicate that individual atom contributions are nonlocal as they strongly\ndepend upon the atomic environment (Figures~\\ref{fig:art094:structure_fragments}(c)-\\ref{fig:art094:structure_fragments}(h)).\nFor example, Mo$_6$PbS$_8$ [{\\small ICSD}\\ \\#644102, $\\log(T_{\\mathrm{c}})$=1.13] and\nMo$_6$NdS$_8$ [{\\small ICSD}\\ \\#603458, $\\log(T_{\\mathrm{c}})$=0.54]\ndiffer by a substitution --- yet the difference in $T_{\\mathrm{c}}$ is substantial.\nFurthermore, substitution of Nd for Pb affects contributions to the\ntarget property from all the remaining atoms in the unit cell\n(Figure~\\ref{fig:art094:structure_fragments}(c) and~\\ref{fig:art094:structure_fragments}(d)).\nThe same observation holds for Li$_2$Pd$_3$B [{\\small ICSD}\\ \\#84931,\n$\\log(T_{\\mathrm{c}})$=0.89] and Li$_2$Pt$_3$B [{\\small ICSD}\\ \\#84932,\n$\\log(T_{\\mathrm{c}})$=0.49] Figure~\\ref{fig:art094:structure_fragments}(e) and\n~\\ref{fig:art094:structure_fragments}(f); as well as FeLaAsO [{\\small ICSD}\\ \\#163496, $\\log(T_{\\mathrm{c}})$=1.32]\nand FeLaPO [{\\small ICSD}\\ \\#162724, $\\log(T_{\\mathrm{c}})$=0.82]\nFigure~\\ref{fig:art094:structure_fragments}(g) and~\\ref{fig:art094:structure_fragments}(h).\n\n\\subsection{Conclusion}\nWith high-throughput approaches in materials science\nincreasing the data-driven content of the field,\nthe gap between accumulated-information and derived knowledge widens.\nThe issue can be overcome by adapting the data-analysis approaches\ndeveloped during the past decade for chem- and bioinformatics.\n\nOur study gives an example of this.\nWe introduce novel materials fingerprint descriptors that lead to the generation of\nnetworks called ``{\\it materials cartograms}'': nodes represent compounds;\nconnections represent similarities.\nThe representation can identify regions with distinct physical and chemical properties,\nthe key step in searching for interesting, yet unknown compounds.\n\nStarting from atomic-compositions, bond-topologies, structure-geometries,\nand electronic properties of materials publicly available in the {\\sf \\AFLOW.org}\\ repository,\nwe have introduced cheminformatics models leveraging novel materials fingerprints.\nWithin our formalism, simple band-structure and {\\small DOS}\\ fingerprints are adequate to\nlocate metals, semiconductors, topological insulators, piezoelectrics, and superconductors.\nMore complex {\\small QMSPR}\\ modeling~\\cite{Kuzmin_JCAMD_2008} are used to tackle\nqualitative and quantitative values of superconducting critical temperature\nand geometrical features helping\/hindering criticality~\\cite{Kuzmin_JCAMD_2008}.\n\nIn summary, the fingerprinting cartography introduced in this work\nhas demonstrated its utility in an initial set of problems.\nThis shows the possibility of designing new materials and gaining\ninsight into the relationship between the structure\nand physical properties of materials.\nFurther advances in the analysis and exploration of databases may become the\nfoundation for rationally designing novel compounds with desired properties.\n\\clearpage\n\\section{Machine Learning Modeling of Superconducting Critical Temperature}\n\\label{sec:art137}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art137}.\nAuthor contributions are as follows:\nValentin Stanev, Ichiro Takeuchi and Aaron Gilad Kusne designed the research.\nValentin Stanev worked on the model.\nCorey Oses and Stefano Curtarolo performed the {\\small AFLOW}\\ calculations.\nValentin Stanev, Ichiro Takeuchi, Efrain Rodriguez and Johnpierre Paglione analyzed the results.\nValentin Stanev, Corey Oses, Ichiro Takeuchi and Efrain Rodriguez wrote the text of the manuscript.\nAll authors discussed the results and commented on the manuscript.\n\n\\subsection{Introduction}\n\nSuperconductivity, despite being the subject of intense physics,\nchemistry and materials science research for more than a century,\nremains among one of the most puzzling scientific topics~\\cite{SCSpecial_PSCC_2015}.\nIt is an intrinsically quantum phenomenon caused by a finite attraction between paired electrons,\nwith unique properties including zero DC resistivity, Meissner and Josephson effects, and\nwith an ever-growing list of current and potential applications.\nThere is even a profound connection between phenomena in the\nsuperconducting state and the Higgs mechanism in particle physics~\\cite{PWAnderson_PR_1963}.\nHowever, understanding the relationship between superconductivity and materials'\nchemistry and structure presents significant theoretical and experimental challenges.\nIn particular, despite focused research efforts in the last 30 years,\nthe mechanisms responsible for high-temperature superconductivity in\ncuprate and iron-based families remain elusive~\\cite{Chu_PSCC_2015,Paglione_NatPhys_2010}.\n\nRecent developments, however, allow a different approach to investigate what ultimately determines the superconducting\ncritical temperatures $\\left(T_{\\mathrm{c}}\\right)$ of materials.\nExtensive databases covering various measured and calculated materials properties have been created over\nthe years~\\cite{ICSD,aflowPAPER,cmr_repository,Saal_JOM_2013,APL_Mater_Jain2013}.\nThe shear quantity of accessible information also makes possible,\nand even necessary,\nthe use of data-driven approaches, \\nobreak\\mbox{\\it e.g.}, statistical and machine learning (ML)\nmethods~\\cite{Agrawal_APLM_2016,Lookman_MatInf_2016,Jain_JMR_2016,Mueller_MLMS_2016}.\nSuch algorithms can be\ndeveloped\/trained on the variables collected in these databases,\nand employed to predict macroscopic properties\nsuch as the melting temperatures of binary compounds~\\cite{Seko_PRB_2014},\nthe likely crystal structure at a given composition~\\cite{Balachandran_SR_2015},\nband gap energies~\\cite{Pilania_SR_2016,curtarolo:art124} and\ndensity of states~\\cite{Pilania_SR_2016} of certain classes of materials.\n\nTaking advantage of this immense increase of readily accessible and potentially relevant information, we develop several\nML methods modeling $T_{\\mathrm{c}}$ from\nthe complete list of reported (inorganic) superconductors~\\cite{SuperCon}.\nIn their simplest form, these methods take as input a number of predictors\ngenerated from the elemental composition of each material.\nModels developed with these basic features are surprisingly accurate, despite\nlacking information of relevant properties, such as space group, electronic structure,\nand phonon energies.\nTo further improve the predictive power of the models,\nas well as the ability to extract useful information out of them,\nanother set of features are constructed based on crystallographic and electronic information\ntaken from the {\\small AFLOW}\\ Online Repositories~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}.\n\nApplication of statistical methods in the context of superconductivity began in the early\neighties with simple clustering methods~\\cite{Villars_PRB_1988,Rabe_PRB_1992}.\nIn particular, three ``golden'' descriptors confine the sixty known (at the time) superconductors with\n$T_{\\mathrm{c}} > 10$~K to three small islands in space:\nthe averaged valence-electron numbers, orbital radii differences, and metallic\nelectronegativity differences.\nConversely, about $600$ other superconductors with $T_{\\mathrm{c}} < 10$~K appear randomly dispersed\nin the same space.\nThese descriptors were selected heuristically due to their\nsuccess in classifying binary\/ternary structures and predicting stable\/metastable ternary quasicrystals.\nRecently, an investigation stumbled on this clustering problem again by observing a\nthreshold $T_{\\mathrm{c}}$ closer to $\\log\\left(T_{\\mathrm{c}}^{\\mathrm{thres}}\\right)\\approx1.3$\n$\\left(T_{\\mathrm{c}}^{\\mathrm{thres}}=20~\\text{K}\\right)$~\\cite{curtarolo:art94}.\nInstead of a heuristic approach, random forests and simplex fragments were\nleveraged on the structural\/electronic properties\ndata from the {\\small AFLOW}\\ Online Repositories to find the optimum clustering descriptors.\nA classification model was developed showing good performance.\nSeparately, a sequential learning framework was evaluated on superconducting materials,\nexposing the limitations of relying on random-guess (trial-and-error) approaches for\nbreakthrough discoveries~\\cite{Ling_IMMI_2017}.\nSubsequently, this study also highlights the impact machine learning can have\non this particular field.\nIn another early work,\nstatistical methods were used to find correlations between normal state properties and $T_{\\mathrm{c}}$\nof the metallic elements in the first six rows of the periodic table~\\cite{Hirsch_PRB_1997}.\nOther contemporary works hone in on specific materials~\\cite{Owolabi_JSNM_2015,Ziatdinov_NanoTech_2016}\nand families of superconductors~\\cite{Klintenberg_CMS_2013,Owolabi_APTA_2014} (see also Reference~\\cite{Norman_RPP_2016}).\n\nWhereas previous investigations explored several hundred compounds at most,\nthis work considers more than $16,000$ different compositions.\nThese are extracted from the SuperCon database, which contains an exhaustive\nlist of superconductors, including many closely-related materials varying only by small changes in stoichiometry (doping plays a significant role in optimizing $T_{\\mathrm{c}}$).\nThe order-of-magnitude increase in training data\n\\textbf{i.} presents crucial subtleties in chemical composition among related compounds,\n\\textbf{ii.} affords family-specific modeling exposing different superconducting mechanisms, and\n\\textbf{iii.} enhances model performance overall.\nIt also enables the optimization of several model construction procedures.\nLarge sets of independent variables can be constructed and rigorously filtered\nby predictive power (rather than selecting them by intuition alone).\nThese advances are crucial to uncovering insights into the\nemergence\/suppression of superconductivity with composition.\n\nAs a demonstration of the potential of ML methods in looking for novel superconductors, we combined and\napplied several models to search for candidates among the roughly\n$110,000$ different compositions contained in the Inorganic Crystallographic Structure Database ({\\small ICSD}),\na large fraction of which have not been tested for superconductivity.\nThe framework highlights 35 compounds with predicted $T_{\\mathrm{c}}$'s\nabove 20~K for experimental validation.\nOf these, some exhibit interesting chemical and structural similarities to cuprate superconductors, demonstrating the ability of the ML models to identify meaningful patterns in the data.\nIn addition, most materials from the list share a peculiar feature in their electronic band structure:\none (or more) flat\/nearly-flat bands just below the energy of the highest occupied electronic state.\nThe associated large peak in the density of states (infinitely large in the limit of truly flat bands)\ncan lead to strong electronic instability, and has been discussed recently as one possible way to\nhigh-temperature superconductivity~\\cite{Kopnin_PRB_2011,Peotta_NComm_2015}.\n\n\\subsection{Results}\n\\boldsection{Data and predictors.}\nThe success of any ML method ultimately depends on access to reliable and plentiful data.\nSuperconductivity data used in this work is extracted from the SuperCon database~\\cite{SuperCon},\ncreated and maintained by the Japanese National Institute for Materials Science.\nIt houses information such as the $T_{\\mathrm{c}}$\nand reporting journal publication for superconducting materials known from experiment.\nAssembled within it is a uniquely exhaustive list of all reported superconductors,\nas well as related non-superconducting compounds.\nAs such, SuperCon is the largest database of its kind, and has never before been employed\n{\\it en masse} for machine learning modeling.\n\nFrom SuperCon, we have extracted a list of approximately $16,400$ compounds,\nof which $4,000$ have no $T_{\\mathrm{c}}$ reported (see Methods for details).\nOf these, roughly $5,700$ compounds are cuprates and $1,500$ are iron-based\n(about 35\\% and 9\\%, respectively), reflecting the significant research efforts invested in these two families.\nThe remaining set of about $8,000$ is a mix of various materials, including conventional phonon-driven superconductors\n(\\nobreak\\mbox{\\it e.g.}, elemental superconductors, A15 compounds), known unconventional superconductors like the\nlayered nitrides and heavy fermions, and many materials for which the mechanism of superconductivity\nis still under debate (such as bismuthates and borocarbides).\nThe distribution of materials by $T_{\\mathrm{c}}$ for the three groups is shown in Figure~\\ref{fig:art137:Class_score}(a).\n\nUse of this data for the purpose of creating ML models can be problematic.\nML models have an intrinsic applicability domain, \\nobreak\\mbox{\\it i.e.}, predictions\nare limited to the patterns\/trends encountered in the training set.\nAs such, training a model only on superconductors can lead to significant selection bias\nthat may render it ineffective when applied to new\nmaterials\\footnote{\\textit{N.B.}, a model suffering from selection bias\ncan still provide valuable statistical information about known superconductors.}.\nEven if the model learns to correctly recognize factors promoting superconductivity,\nit may miss effects that strongly inhibit it.\nTo mitigate the effect, we incorporate about $300$ materials found by H.\nHosono's group not to display superconductivity~\\cite{Hosono_STAM_2015}.\nHowever, the presence of non-superconducting materials,\nalong with those without $T_{\\mathrm{c}}$ reported in SuperCon, leads to a conceptual problem.\nSome of these compounds emerge as non-superconducting ``end-members'' from\ndoping\/pressure studies, indicating no superconducting transition was observed despite some efforts to find one.\nHowever, the transition may still exist,\nalbeit at experimentally difficult to reach or altogether inaccessible temperatures\n(for most practical purposes below $10$~mK)\\footnote{There are theoretical arguments for this --- according\nto the Kohn-Luttinger theorem, a\nsuperconducting instability should be present as $T \\rightarrow 0$ in any fermionic metallic system\nwith Coulomb interactions~\\cite{Kohn_PRL_1965}.}.~\\nocite{Kohn_PRL_1965}\nThis presents a conundrum:\nignoring compounds with no reported $T_{\\mathrm{c}}$ disregards a potentially important\npart of the dataset, while assuming $T_{\\mathrm{c}} = 0$~K prescribes an inadequate description\nfor (at least some of) these compounds.\nTo circumvent the problem,\nmaterials are first partitioned in two groups by their $T_{\\mathrm{c}}$,\nabove and below a threshold temperature $\\left(T_{\\mathrm{sep}}\\right)$,\nfor the creation of a classification model.\nCompounds with no reported critical temperature can be classified in the ``below-$T_{\\mathrm{sep}}$'' group\nwithout the need to specify a $T_{\\mathrm{c}}$ value (or assume it is zero).\nThe ``above-$T_{\\mathrm{sep}}$'' bin also enables the development of a regression model\nfor $\\ln{(T_{\\mathrm{c}})}$, without problems arising in the $T_{\\mathrm{c}}\\to0$ limit.\n\nFor most materials, the SuperCon database provides\nonly the chemical composition and $T_{\\mathrm{c}}$.\nTo convert this information into meaningful features\/predictors (used interchangeably),\nwe employ the Materials Agnostic Platform for Informatics and Exploration (Magpie)~\\cite{Ward_ML_GFA_NPGCompMat_2016}.\nMagpie computes a set of attributes for each material, including elemental property\nstatistics like the mean and the standard deviation of 22 different elemental properties\n(\\nobreak\\mbox{\\it e.g.}, period\/group on the periodic table,\natomic number, atomic radii, melting temperature), as well as electronic structure attributes, such as the average\nfraction of electrons from the $s$, $p$, $d$ and $f$ valence shells among all\nelements present.\n\nThe application of Magpie predictors, though appearing to lack \\textit{a priori} justification,\nexpands upon past clustering approaches by Villars and Rabe~\\cite{Villars_PRB_1988,Rabe_PRB_1992}.\nThey show that, in the space of a few judiciously chosen\nheuristic predictors, materials separate and cluster according to their\ncrystal structure and even complex properties such as high-temperature\nferroelectricity and superconductivity.\nSimilar to these features, Magpie predictors capture significant chemical information, which\nplays a decisive role in determining\nstructural and physical properties of materials.\n\nDespite the success of Magpie predictors in modeling materials properties~\\cite{Ward_ML_GFA_NPGCompMat_2016},\ninterpreting their connection to superconductivity presents a serious challenge.\nThey do not encode (at least directly) many important properties, particularly those\npertinent to superconductivity.\nIncorporating features\nlike lattice type and density of states would undoubtedly lead to significantly more powerful and interpretable models.\nSince such information is not generally available in SuperCon,\nwe employ data from the {\\small AFLOW}\\ Online Repositories~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}.\nThe materials database houses more than 200 million properties calculated with\nthe software package {\\small AFLOW}~\\cite{curtarolo:art49,curtarolo:art53,curtarolo:art57,aflowBZ,curtarolo:art63,aflowPAPER,curtarolo:art85,curtarolo:art110,monsterPGM,aflowANRL,aflowPI}.\nIt contains information for the vast majority of compounds in the {\\small ICSD}~\\cite{ICSD}.\nAlthough the {\\small AFLOW}\\ Online Repositories contain calculated properties,\nthe density functional theory ({\\small DFT}) results have been extensively validated with\n{\\small ICSD}\\ records~\\cite{curtarolo:art94,curtarolo:art96,curtarolo:art112,curtarolo:art115,curtarolo:art120,curtarolo:art124}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig106}\n\\mycaption[Schematic of the random forest ML approach.]\n{Example of a single decision tree used to classify materials depending on whether\n$T_{\\mathrm{c}}$ is above or below $10$~K.\nA tree can have many levels, but only the three top are shown.\nThe decision rules leading to each subset are written inside individual rectangles.\nThe subset population percentage\nis given by ``samples'', and the node color\/shade\nrepresents the degree of separation,\n\\nobreak\\mbox{\\it i.e.}, dark blue\/orange illustrates a high proportion of\n$T_{\\mathrm{c}} >10$~K\/$T_{\\mathrm{c}} < 10$~K materials\n(the exact value is given by ``proportion'').\nA random forest consists of a large number --- could be hundreds or thousands --- of such individual trees.}\n\\label{fig:art137:tree_example}\n\\end{figure}\n\nUnfortunately, only a small subset of materials in SuperCon overlaps with those in the {\\small ICSD}:\nabout $800$ with finite $T_{\\mathrm{c}}$ and less than $600$ are contained within {\\small AFLOW}.\nFor these, a set of 26 predictors are incorporated\nfrom the {\\small AFLOW}\\ Online Repositories, including structural\/chemical information like the lattice type, space group,\nvolume of the unit cell, density, ratios of the lattice parameters,\nBader charges and volumes, and formation energy (see Methods for details).\nIn addition, electronic properties are considered, including the\ndensity of states near the Fermi level as calculated by {\\small AFLOW}.\nPrevious investigations exposed limitations in applying ML methods to a similar dataset\nin isolation~\\cite{curtarolo:art94}.\nInstead, a framework is presented here for combining models built on Magpie descriptors\n(large sampling, but features limited to compositional data) and {\\small AFLOW}\\ features\n(small sampling, but diverse and pertinent features).\n\nOnce we have a list of relevant predictors, various ML models can be applied to the\ndata~\\cite{Bishop_ML_2006,Hastie_StatLearn_2001}.\nAll ML algorithms in this work are\nvariants of the random forest method~\\cite{randomforests}.\nFundamentally, this approach combines many individual decision trees, where\neach tree is a non-parametric supervised learning method used for\nmodeling either categorical or numerical variables (\\nobreak\\mbox{\\it i.e.},\nclassification or regression modeling).\nA tree predicts the value of a target variable by learning simple decision rules\ninferred from the available features (see Figure~\\ref{fig:art137:tree_example} for an example).\n\nRandom forest is one of the most powerful, versatile, and widely-used ML methods~\\cite{Caruana_2006}.\nThere are several advantages that make it especially suitable for this problem.\nFirst, it can learn complicated non-linear dependencies from the data.\nUnlike many other methods (\\nobreak\\mbox{\\it e.g.}, linear regression),\nit does not make assumptions about the functional form of the relationship between the predictors\nand the target variable (\\nobreak\\mbox{\\it e.g.}, linear, exponential or some other {\\it a priori} fixed function).\nSecond, random forests are quite tolerant to heterogeneity in the training data.\nIt can handle both numerical and categorical data which, furthermore, does not\nneed extensive and potentially dangerous preprocessing, such as scaling or normalization.\nEven the presence of strongly correlated predictors is not a problem for model\nconstruction (unlike many other ML algorithms).\nAnother significant advantage of this method is that, by combining information from\nindividual trees, it can estimate the importance of each predictor, thus making the model more interpretable.\nHowever, unlike model construction, determination of predictor importance is complicated by the presence of\ncorrelated features.\nTo avoid this, standard feature selection procedures are employed along with\na rigorous predictor elimination scheme (based on their strength and correlation with others).\nOverall, these methods\nreduce the complexity of the models and improve our\nability to interpret them.\n\n\\boldsection{Classification models.}\nAs a first step in applying ML methods to the dataset, a sequence of classification models\nare created, each designed to separate materials into two distinct groups depending on whether\n$T_{\\mathrm{c}}$ is above or below some predetermined value.\nThe temperature that separates the two groups ($T_{\\mathrm{sep}}$)\nis treated as an adjustable parameter of the model, though some physical\nconsiderations should guide its choice as well.\nClassification ultimately allows compounds with no reported $T_{\\mathrm{c}}$ to be used\nin the training set by including them in the below-$T_{\\mathrm{sep}}$ bin.\nAlthough discretizing continuous variables is not generally recommended, in this case\nthe benefits of including compounds without $T_{\\mathrm{c}}$ outweigh\nthe potential information loss.\n\nIn order to choose the optimal value of $T_{\\mathrm{sep}}$, a series of random forest models\nare trained with different threshold temperatures separating the two classes.\nSince setting $T_{\\mathrm{sep}}$ too low or too high creates strongly imbalanced classes\n(with many more instances in one group), it is important to compare the models using several different metrics.\nFocusing only on the accuracy (count of correctly-classified instances)\ncan lead to deceptive results.\nHypothetically, if $95\\%$ of the observations in the dataset are in the below-$T_{\\mathrm{sep}}$ group,\nsimply classifying all materials as such would\nyield a high accuracy ($95\\%$), while being trivial in any other sense\\footnote{There are more sophisticated techniques to deal with severely imbalanced datasets, like undersampling the majority class or generating synthetic data points for the minority class (see, for example, Reference~\\cite{SMOTE}.}~\\nocite{SMOTE}.\nTo avoid this potential pitfall, three other standard metrics\nfor classification are considered: precision, recall, and $F_{\\mathrm{1}}$ score.\nThey are defined using the values $tp$, $tn$, $fp$, and $fn$ for\nthe count of true\/false positive\/negative\npredictions of the model:\n\\begin{eqnarray}\n\\text{accuracy} \\equiv \\frac{tp+tn}{tp+tn+fp+fn},\n\\label{eq:art137:accur}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\text{precision}\\equiv\\frac{tp}{tp+fp},\n\\label{eq:art137:precision}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\text{recall} \\equiv\\frac{tp}{tp+fn},\n\\label{eq:art137:recall}\n\\end{eqnarray}\n\\begin{eqnarray}\nF_{\\mathrm{1}}\\equiv2*\\frac{\\text{precision}*\\text{recall}}{\\text{precision}+\\text{recall}},\n\\label{eq:art137:f1}\n\\end{eqnarray}\nwhere positive\/negative refers to above-$T_{\\mathrm{sep}}$\/below-$T_{\\mathrm{sep}}$.\nThe accuracy of a classifier is the total proportion of correctly-classified materials,\nwhile precision measures the proportion of correctly-classified\nabove-$T_{\\mathrm{sep}}$ superconductors out of all predicted above-$T_{\\mathrm{sep}}$.\nThe recall is the proportion of correctly-classified above-$T_{\\mathrm{sep}}$\nmaterials out of all truly above-$T_{\\mathrm{sep}}$ compounds.\nWhile the precision measures the probability that a\nmaterial selected by the model actually has $T_{\\mathrm{c}} > T_{\\mathrm{sep}}$,\nthe recall reports how sensitive the model is to above-$T_{\\mathrm{sep}}$ materials.\nMaximizing the precision or recall would require some compromise with\nthe other, \\nobreak\\mbox{\\it i.e.}, a model that labels all materials as above-$T_{\\mathrm{sep}}$ would have perfect recall but dismal precision.\nTo quantify the trade-off between recall and precision, their harmonic mean ($F_{\\mathrm{1}}$ score) is\nwidely used to measure the performance of a classification model.\nWith the exception of accuracy, these metrics are not symmetric with respect to the exchange of positive and negative labels.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.90\\linewidth]{fig107}\n\\mycaption[SuperCon dataset and classification model performance.]\n{(\\textbf{a}) Histogram of materials categorized by\n$T_{\\mathrm{c}}$ (bin size is $2$~K, only those with finite $T_{\\mathrm{c}}$ are counted).\nBlue, green, and red denote low-$T_{\\mathrm{c}}$, iron-based, and cuprate superconductors, respectively.\nIn the inset: histogram of materials categorized by $\\ln{(T_{\\mathrm{c}})}$\nrestricted to those with $T_{\\mathrm{c}} >10$~K.\n(\\textbf{b}) Performance of different classification models as a function of the threshold temperature\n$\\left(T_{\\mathrm{sep}}\\right)$ that separates materials in two classes by $T_{\\mathrm{c}}$.\nPerformance is measured by accuracy (gray), precision (red), recall (blue), and $F_{\\mathrm{1}}$ score (purple).\nThe scores are calculated from predictions on an independent test set, \\nobreak\\mbox{\\it i.e.}, one separate\nfrom the dataset used to train the model.\nIn the inset: the dashed red curve gives the proportion of materials in the above-$T_{\\mathrm{sep}}$ set.\n(\\textbf{c}) Accuracy, precision, recall, and $F_{\\mathrm{1}}$\nscore as a function of the size of the training set with a fixed test set.\n(\\textbf{d}) Accuracy, precision, recall, and $F_{\\mathrm{1}}$ as a function of the number of\npredictors.}\n\\label{fig:art137:Class_score}\n\\end{figure}\n\nFor a realistic estimate of the performance of each model,\nthe dataset is randomly split ($85\\%\/15\\%$) into training and test subsets.\nThe training set is employed to fit the model, which is then applied to the test set for subsequent benchmarking.\nThe aforementioned metrics (Equations~\\ref{eq:art137:accur}-\\ref{eq:art137:f1}) calculated on the test set provide\nan unbiased estimate of how well the model is expected to generalize to a new (but similar) dataset.\nWith the random forest method, similar estimates can be obtained intrinsically at the training stage.\nSince each tree is trained only on a bootstrapped subset of the data,\nthe remaining subset can be used as an internal test set.\nThese two methods for quantifying model performance usually yield very similar results.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig108}\n\\mycaption[Scatter plots of $3,000$ superconductors in the space of the four most important classification predictors.]\n{Blue\/red represent below-$T_{\\mathrm{sep}}$\/above-$T_{\\mathrm{sep}}$ materials, where $T_{\\mathrm{sep}} = 10$~K.\n(\\textbf{a}) Feature space of the first and second most important predictors:\nstandard deviations of the column numbers and electronegativities (calculated over the values for the constituent elements in each compound).\n(\\textbf{b}) Feature space of the third and fourth most important predictors:\nstandard deviation of the elemental melting temperatures and average of the\natomic weights.}\n\\label{fig:art137:Class_features}\n\\end{figure}\n\nWith the procedure in place, the models' metrics are evaluated for a range of $T_{\\mathrm{sep}}$ and illustrated\nin Figure~\\ref{fig:art137:Class_score}(b).\nThe accuracy increases as $T_{\\mathrm{sep}}$ goes from $1$~K to $40$~K,\nand the proportion of above-$T_{\\mathrm{sep}}$ compounds drops from above $70\\%$ to about $15\\%$,\nwhile the recall and $F_{\\mathrm{1}}$ score generally decrease.\nThe region between $5-15$~K is especially appealing in (nearly) maximizing\nall benchmarking metrics while balancing the sizes of the bins.\nIn fact, setting $T_{\\mathrm{sep}}=10$~K is a particularly convenient choice.\nIt is also the temperature used in References~\\cite{Villars_PRB_1988,Rabe_PRB_1992}\nto separate the two classes, as it is just above the\nhighest $T_{\\mathrm{c}}$ of all elements and pseudoelemental\nmaterials (solid solution whose range of composition includes a pure element).\nHere, the proportion of above-$T_{\\mathrm{sep}}$ materials is approximately $38\\%$ and\nthe accuracy is about $92\\%$, \\nobreak\\mbox{\\it i.e.}, the model can\ncorrectly classify nine out of ten materials --- much better than random guessing.\nThe recall --- quantifying how well all above-$T_{\\mathrm{sep}}$ compounds are labeled and,\nthus, the most important metric when searching for new superconducting materials --- is even higher.\n(Note that the models' metrics also depend on random factors such as the composition of the\ntraining and test sets, and their exact values can vary.)\n\n\\begin{table}[tp]\\centering\n\\mycaption[The most relevant predictors and their importances for the classification and general regression models.]\n{$\\avg(x)$ and $\\std(x)$ denote the composition-weighted average and\nstandard deviation, respectively,\ncalculated over the vector of elemental values for each compound~\\cite{Ward_ML_GFA_NPGCompMat_2016}.\nFor the classification model, all predictor importances are quite close.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l | l r| l r}\npredictor & \t\\multicolumn{4}{c}{model}\n\\\\\n\\cline{2-5}\nrank& \\multicolumn{2}{l|}{classification} & \\multicolumn{2}{l}{regression (general; $T_{\\mathrm{c}}>10$ K)} \\\\\n\\hline\n1 & $\\std ($column number$)$ & \\setlength{\\tabcolsep}{4pt} 0.26 & $\\avg ($number of unfilled orbitals$)$ & \\setlength{\\tabcolsep}{4pt} 0.26\\\\\n2 & $\\std ($electronegativity$)$ & \\setlength{\\tabcolsep}{4pt} 0.26 & $\\std ($ground state volume$)$ & \\setlength{\\tabcolsep}{4pt} 0.18\\\\\n3 & $\\std ($melting temperature$)$ & \\setlength{\\tabcolsep}{4pt} 0.23 & $\\std ($space group number$)$ & \\setlength{\\tabcolsep}{4pt} 0.17\\\\\n4 & $\\avg ($atomic weight$)$ & \\setlength{\\tabcolsep}{4pt} 0.24 & $\\avg ($number of $d$ unfilled orbitals$)$ & \\setlength{\\tabcolsep}{4pt} 0.17\\\\\n5 & & \\setlength{\\tabcolsep}{4pt} - & $\\std ($number of $d$ valence electrons$)$ & \\setlength{\\tabcolsep}{4pt} 0.12\\\\\n6 & & \\setlength{\\tabcolsep}{4pt} - & $\\avg ($melting temperature$)$ & \\setlength{\\tabcolsep}{4pt} 0.1\\\\\n\\end{tabular}}\n\\label{tab:art137:Table1}\n\\end{table}\n\nThe most important factors that determine the model's performance are the size of the available\ndataset and the number of meaningful predictors.\nAs can be seen in Figure~\\ref{fig:art137:Class_score}(c), all\nmetrics improve significantly with the increase of the training set size. The effect is most dramatic for sizes between several hundred and few thousands instances, but there is no obvious saturation even for the largest available datasets. This validates efforts herein to incorporate as much relevant data as possible into model training.\nThe number of predictors is another very important model parameter.\nIn Figure~\\ref{fig:art137:Class_score}(d),\nthe accuracy is calculated at each step of the backward feature elimination process.\nIt quickly saturates when the number of predictors reaches $10$.\nIn fact, a model using only the five most informative predictors, selected out of the full list of 145 ones,\nachieves almost $90\\%$ accuracy.\n\nFor an understanding of what the model has learned, an analysis of the chosen predictors is needed.\nIn the random forest method, features can be ordered by their importance quantified via\nthe so-called Gini importance or\n``mean decrease in impurity''~\\cite{Bishop_ML_2006,Hastie_StatLearn_2001}.\nFor a given feature, it is the sum of the Gini impurity\\footnote{Gini impurity is calculated as\n\\unexpanded{$\\sum_i p_i \\left(1-p_i\\right)$},\nwhere \\unexpanded{$p_i$} is the probability of randomly chosen data point\nfrom a given decision tree leaf to be in class\n\\unexpanded{$i$}~\\cite{Bishop_ML_2006,Hastie_StatLearn_2001}.}~\\nocite{Bishop_ML_2006,Hastie_StatLearn_2001}\nover the number of splits that include the feature, weighted by the number of samples\nit splits, and averaged over the entire forest.\nDue to the nature of the algorithm, the closer to the top of the tree a predictor is used,\nthe greater number of predictions it impacts.\n\nAlthough correlations between predictors do not affect the model's ability to learn, it can distort importance estimates.\nFor example, a material property with a strong effect on $T_{\\mathrm{c}}$ can be shared\namong several correlated predictors.\nSince the model can access the same information through any of these variables,\ntheir relative importances are diluted across the group.\nTo reduce the effect and limit the list of predictors to a manageable size,\nthe backward feature elimination method is employed.\nThe process begins with a model constructed with the full list of predictors,\nand iteratively removes the least significant one, rebuilding the model and recalculating importances\nwith every iteration.\n(This iterative procedure is necessary since the ordering of the predictors by importance can change at each step.)\nPredictors are removed until the overall accuracy of the model drops by $2\\%$, at which point there are only five left.\nFurthermore, two of these predictors are strongly correlated with each other, and we remove the less important one. This\nhas a negligible impact on the model performance,\nyielding four predictors total (see Table~\\ref{tab:art137:Table1})\nwith an above $90\\%$ accuracy score --- only slightly worse than the full model.\nScatter plots of the pairs of the most important predictors are shown in Figure~\\ref{fig:art137:Class_features}, where\nblue\/red denotes whether the material is in the below-$T_{\\mathrm{sep}}$\/above-$T_{\\mathrm{sep}}$ class.\nFigure~\\ref{fig:art137:Class_features}(a) shows a scatter plot of $3,000$ compounds\nin the space spanned by the standard deviations of the column numbers and electronegativities\ncalculated over the elemental values.\nSuperconductors with $T_{\\mathrm{c}} > 10$~K tend to\ncluster in the upper-right corner of the plot and in a relatively thin elongated region extending to the left of it.\nIn fact, the points in the upper-right corner represent mostly cuprate materials,\nwhich with their complicated compositions and large number of elements are likely\nto have high standard deviations in these variables.\nFigure~\\ref{fig:art137:Class_features}(b) shows\nthe same compounds projected in the space of the standard deviations of\nthe melting temperatures and the averages of the atomic weights of the elements forming each compound.\nThe above-$T_{\\mathrm{sep}}$ materials tend to cluster in areas with lower mean atomic weights --- not\na surprising result given the role of phonons in conventional superconductivity.\n\nFor comparison, we create another classifier based on the average number of valence electrons,\nmetallic electronegativity differences, and orbital radii differences, \\nobreak\\mbox{\\it i.e.}, the predictors used\nin References~\\cite{Villars_PRB_1988,Rabe_PRB_1992} to cluster materials with $T_{\\mathrm{c}} > 10$ K.\nA classifier built only with these three predictors\nis less accurate than both the full and the truncated models presented herein,\nbut comes quite close: the full model has about $3\\%$ higher\naccuracy and $F_{\\mathrm{1}}$ score, while the truncated model with four predictors is less that $2\\%$ more accurate.\nThe rather small (albeit not insignificant) differences demonstrates that even on the scale of the\nentire SuperCon dataset, the predictors used by Villars and Rabe~\\cite{Villars_PRB_1988,Rabe_PRB_1992}\ncapture much of the relevant chemical information for superconductivity.\n\n\\begin{table}[tp]\\centering\n\\mycaption[The most significant predictors and their importances for the three material-specific regression models.]\n{$\\avg(x)$, $\\std(x)$, $\\max(x)$ and $\\fraction(x)$ denote the composition-weighted average,\nstandard deviation, maximum, and fraction, respectively,\ntaken over the elemental values for each compound.\n$l^2$-norm of a composition is calculated by $||x||_{2} = \\sqrt{\\sum_i x_i^2}$, where $x_i$ is the proportion of each element $i$ in the compound.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l | l r| l r |l r}\npred. & \t\\multicolumn{6}{c}{model}\n\\\\\n\\cline{2-7}\nrank & \\multicolumn{2}{l|}{regression (low-$T_{\\mathrm{c}}$)} & \\multicolumn{2}{l}{regression (cuprates)}& \\multicolumn{2}{|l} {regression (Fe-based)} \\\\\n\\hline\n1 & $\\fraction (d$ valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.18 & $\\avg ($number of unfilled orbitals$)$ &\\setlength{\\tabcolsep}{4pt} 0.22 & $\\std ($column number$)$ &\\setlength{\\tabcolsep}{4pt} 0.17\\\\\n2 & $\\avg ($number of $d$ unfilled orbitals$)$ &\\setlength{\\tabcolsep}{4pt} 0.14 & $\\std ($number of $d$ valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.13 & $\\avg ($ionic character$)$ &\\setlength{\\tabcolsep}{4pt} 0.15\\\\\n3 & $\\avg ($number of valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.13 & $\\fraction (d$ valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.13 & $\\std ($Mendeleev number$)$ &\\setlength{\\tabcolsep}{4pt} 0.14\\\\\n4 & $\\fraction (s$ valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.11 & $\\std ($ground state volume$)$ &\\setlength{\\tabcolsep}{4pt} 0.13 & $\\std ($covalent radius$)$ &\\setlength{\\tabcolsep}{4pt} 0.14\\\\\n5 & $\\avg ($number of $d$ valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.09 & $\\std ($number of valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.1 & $\\max ($melting temperature$)$ &\\setlength{\\tabcolsep}{4pt} 0.14\\\\\n6 & $\\avg ($covalent radius$)$ &\\setlength{\\tabcolsep}{4pt} 0.09 & $\\std ($row number$)$ &\\setlength{\\tabcolsep}{4pt} 0.08 & $\\avg ($Mendeleev number$)$ &\\setlength{\\tabcolsep}{4pt} 0.14\\\\\n7 & $\\avg ($atomic weight$)$ &\\setlength{\\tabcolsep}{4pt} 0.08 & $||$composition$||_{2}$ &\\setlength{\\tabcolsep}{4pt} 0.07 & $||$composition$||_{2}$ &\\setlength{\\tabcolsep}{4pt} 0.11\\\\\n8 & $\\avg ($Mendeleev number$)$ &\\setlength{\\tabcolsep}{4pt} 0.07 & $\\std ($number of $s$ valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.07 & &\\setlength{\\tabcolsep}{4pt} -\\\\\n9 & $\\avg ($space group number$)$ &\\setlength{\\tabcolsep}{4pt} 0.07 & $\\std ($melting temperature$)$ &\\setlength{\\tabcolsep}{4pt} 0.07 & &\\setlength{\\tabcolsep}{4pt} -\\\\\n10 & $\\avg ($number of unfilled orbitals$)$ &\\setlength{\\tabcolsep}{4pt} 0.06 & &\\setlength{\\tabcolsep}{4pt} - & &\\setlength{\\tabcolsep}{4pt} -\\\\\n\\end{tabular}}\n\\label{tab:art137:Table2}\n\\end{table}\n\n\\boldsection{Regression models.}\nAfter constructing a successful classification model, we now move to the more difficult challenge of predicting $T_{\\mathrm{c}}$.\nCreating a regression model may enable better understanding of the factors controlling\n$T_{\\mathrm{c}}$ of known superconductors,\nwhile also serving as an organic part of a system for identifying potential new ones.\nLeveraging the same set of elemental predictors as the classification model, several regression models are presented\nfocusing on materials with $T_{\\mathrm{c}} > 10$~K.\nThis approach avoids the problem of materials with no reported $T_{\\mathrm{c}}$ with the assumption that,\nif they were to exhibit superconductivity at all, their critical temperature would be below $10$~K.\nIt also enables the substitution of $T_{\\mathrm{c}}$ with $\\ln{(T_{\\mathrm{c}})}$ as the target variable\n(which is problematic as $T_{\\mathrm{c}}\\to0$), and thus addresses the problem of the uneven distribution\nof materials along the $T_{\\mathrm{c}}$ axis (see Figure~\\ref{fig:art137:Class_score}(a)).\nUsing $\\ln{(T_{\\mathrm{c}})}$ creates a more uniform distribution (Figure~\\ref{fig:art137:Class_score}(a) inset),\nand is also considered a best practice when the range of a target variable covers more than one\norder of magnitude (as in the case of $T_{\\mathrm{c}}$).\nFollowing this transformation, the dataset is parsed randomly ($85\\%$\/$15\\%$) into training\nand test subsets (similarly performed for the classification model).\n\nPresent within the dataset are distinct families of superconductors with different driving\nmechanisms for superconductivity, including cuprate and iron-based high-temperature superconductors,\nwith all others denoted ``low-$T_{\\mathrm{c}}$'' for brevity (no specific mechanism in this group).\nSurprisingly, a single regression model does reasonably well among the\ndifferent families -- benchmarked on the test set,\nthe model achieves $R^2 \\approx 0.88$ (Figure~\\ref{fig:art137:Rerg_r2}(a)).\nIt suggests that the random forest algorithm is flexible and powerful enough\nto automatically separate the compounds into groups\nand create group-specific branches with distinct predictors (no explicit group labels were used during training and testing).\nAs validation, three separate models are constructed, each trained only on a specific family, namely the\nlow-$T_{\\mathrm{c}}$, cuprate, and iron-based superconductors, respectively.\nBenchmarking on mixed-family test sets, the models performed well on compounds belonging\nto their training set family while demonstrating no predictive power on the others.\nFigures~\\ref{fig:art137:Rerg_r2}(b)-(d) illustrate a cross-section of this comparison.\nSpecifically, the model trained on low-$T_{\\mathrm{c}}$ compounds dramatically underestimates\nthe $T_{\\mathrm{c}}$ of both high-temperature superconducting families (Figures~\\ref{fig:art137:Rerg_r2}(b) and (c)),\neven though this test set only contains compounds with $T_{\\mathrm{c}} < 40$~K.\nConversely, the model trained on the cuprates tends to overestimate the $T_{\\mathrm{c}}$\nof low-$T_{\\mathrm{c}}$ (Figure~\\ref{fig:art137:Rerg_r2}(d)) and iron-based (Figure~\\ref{fig:art137:Rerg_r2}(e)) superconductors.\nThis is a clear indication that superconductors from these groups have different factors determining their $T_{\\mathrm{c}}$.\nInterestingly, the family-specific models do not perform better than the general regression containing\nall the data points: $R^2$ for the low-$T_{\\mathrm{c}}$ materials is about $0.85$, for cuprates is just below $0.8$,\nand for iron-based compounds is about $0.74$.\nIn fact, it is a purely geometric effect that\nthe combined model has the highest $R^2$.\nEach group of superconductors contributes mostly to a distinct $T_{\\mathrm{c}}$ range, and, as a result, the combined regression is better determined over longer temperature interval.\n\nIn order to reduce the number of predictors and increase the interpretability of these models without\nsignificant detriment to their performance, a backward feature elimination process is again employed.\nThe procedure is very similar to the one described previously for the classification model,\nwith the only difference being that the reduction is guided by $R^2$ of the model, rather than the accuracy\n(the procedure stops when $R^2$ drops by $3\\%$).\n\nThe most important predictors for the four models (one general and three family-specific) together with\ntheir importances are shown in Tables~\\ref{tab:art137:Table1} and \\ref{tab:art137:Table2}.\nDifferences in important predictors across the family-specific models reflect the fact that\ndistinct mechanisms are responsible for driving superconductivity among these groups.\nThe list is longest for the low-$T_{\\mathrm{c}}$ superconductors, reflecting the eclectic nature of\nthis group.\nSimilar to the general regression model,\ndifferent branches are likely created for distinct sub-groups.\nNevertheless, some important predictors have straightforward interpretation.\nAs illustrated in Figure~\\ref{fig:art137:Tc_atomWeigth}(a),\nlow average atomic weight is a necessary (albeit not sufficient) condition for\nachieving high $T_{\\mathrm{c}}$ among the low-$T_{\\mathrm{c}}$ group.\nIn fact, the maximum $T_{\\mathrm{c}}$ for a given weight roughly follows $1\/\\sqrt{m_A}$.\nMass plays a significant role in conventional superconductors\nthrough the Debye frequency of phonons, leading to the well-known formula $T_{\\mathrm{c}} \\sim 1\/\\sqrt{m}$,\nwhere $m$ is the ionic mass\\footnote{See, for example, References~\\cite{Maxwell_PR_1950,Reynolds_PR_1950,Reynolds_PR_1951}.}~\\nocite{Maxwell_PR_1950,Reynolds_PR_1950,Reynolds_PR_1951}.\nOther factors like density of states are also important,\nwhich explains the spread in $T_{\\mathrm{c}}$ for a given $m_A$.\nOutlier materials clearly lying above the $\\sim 1\/\\sqrt{m_A}$ line include\nbismuthates and chloronitrates, suggesting the conventional electron-phonon mechanism is not driving\nsuperconductivity in these materials.\nIndeed, chloronitrates exhibit a very weak isotope effect~\\cite{Kasahara_PSCC_2015}, though\nsome unconventional electron-phonon coupling could still be relevant for superconductivity~\\cite{Yin_PRX_2013}.\nAnother important feature for low-$T_{\\mathrm{c}}$ materials\nis the average number of valence electrons.\nThis recovers the empirical relation first discovered by Matthias more than sixty years ago~\\cite{Matthias_PR_1955}.\nSuch findings validate the ability of ML approaches\nto discover meaningful patterns that encode true physical phenomena.\n\nSimilar $T_{\\mathrm{c}}$-\\nobreak\\mbox{\\it vs.}-predictor plots reveal more interesting and subtle features.\nA narrow cluster of materials with $T_{\\mathrm{c}} > 20$~K emerges in the context of the mean covalent radii of compounds\n(Figure ~\\ref{fig:art137:Tc_atomWeigth}(b)) --- another\nimportant predictor for low-$T_{\\mathrm{c}}$ superconductors.\nThe cluster includes (left-to-right) alkali-doped C$_{60}$, MgB$_2$-related compounds, and bismuthates.\nThe sector likely characterizes a region of strong covalent bonding and corresponding high-frequency phonon modes\nthat enhance $T_{\\mathrm{c}}$ (however, frequencies that are too high become irrelevant for superconductivity).\nAnother interesting relation appears in the context of the average number of $d$ valence electrons.\nFigure~\\ref{fig:art137:Tc_atomWeigth}(c) illustrates a fundamental bound on\n$T_{\\mathrm{c}}$ of all non-cuprate and non-iron-based superconductors.\n\nA similar limit exists for cuprates based on the average number of unfilled orbitals (Figure ~\\ref{fig:art137:Tc_atomWeigth}(d)).\nIt appears to be quite rigid --- several data points found above it on inspection are actually\nincorrectly recorded entries in the database and were subsequently removed.\nThe connection between $T_{\\mathrm{c}}$ and the average number of unfilled orbitals\\footnote{The\nnumber of unfilled orbitals refers to the\nelectron configuration of the substituent elements before combining to form oxides.\nFor example, Cu has one unfilled orbital ([Ar]$4s^23d^9$) and Bi has\nthree ([Xe]$4f^{14}6s^25d^{10}6p^3$).\nThese values are averaged per formula unit.}\nmay offer new insight into the mechanism for superconductivity in this family.\nKnown trends include higher $T_{\\mathrm{c}}$'s for structures that\n\\textbf{i.} stabilize more than one superconducting Cu-O plane per unit cell\nand \\textbf{ii.} add more polarizable cations such as Tl$^{3+}$ and Hg$^{2+}$ between these planes.\nThe connection reflects these observations,\nsince more copper and oxygen per formula unit\nleads to lower average number of unfilled orbitals (one for copper, two for oxygen).\nFurther, the lower-$T_{\\mathrm{c}}$ cuprates typically consist of Cu$^{2-}$\/Cu$^{3-}$-containing\nlayers stabilized by the addition\/substitution of hard cations,\nsuch as Ba$^{2+}$ and La$^{3+}$, respectively.\nThese cations have a large number of unfilled orbitals, thus increasing the compound's average.\nTherefore, the ability of between-sheet cations\nto contribute charge to the Cu-O planes may be indeed quite important.\nThe more polarizable the $A$ cation, the more electron density it can contribute\nto the already strongly covalent Cu$^{2+}$--O bond.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig109}\n\\mycaption[Benchmarking of regression models predicting $\\ln(T_{\\mathrm{c}})$.]\n{(\\textbf{a}) Predicted\n\\nobreak\\mbox{\\it vs.}\\ measured $\\ln(T_{\\mathrm{c}})$ for the general regression model.\nThe test set comprises of a mix of low-$T_{\\mathrm{c}}$, iron-based, and cuprate superconductors\nwith $T_{\\mathrm{c}}>10$~K.\nWith an $R^2$ of about $0.88$, this one model can accurately predict\n$T_{\\mathrm{c}}$ for materials in different superconducting groups.\n(\\textbf{b} and \\textbf{c}) Predictions of the regression model\ntrained solely on low-$T_{\\mathrm{c}}$ compounds\nfor test sets containing cuprate and iron-based materials.\n(\\textbf{d} and \\textbf{e}) Predictions of the regression model\ntrained solely on cuprates for test sets containing low-$T_{\\mathrm{c}}$ and iron-based superconductors.\nModels trained on a single group have no predictive power for materials from other groups.}\n\\label{fig:art137:Rerg_r2}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig110}\n\\mycaption[Scatter plots of $T_{\\mathrm{c}}$ for superconducting materials in the space of significant,\nfamily-specific regression predictors.]\n{For $4,000$ ``low-$T_{\\mathrm{c}}$'' superconductors (\\nobreak\\mbox{\\it i.e.}, non-cuprate and non-iron-based),\n$T_{\\mathrm{c}}$ is plotted\n\\nobreak\\mbox{\\it vs.}\\ the\n(\\textbf{a}) average atomic weight,\n(\\textbf{b}) average covalent radius, and\n(\\textbf{c}) average number of $d$ valence electrons.\nThe dashed red line in (\\textbf{a}) is $\\sim 1\/\\sqrt{m_A}$.\nHaving low average atomic weight and low average number of $d$ valence\nelectrons are necessary (but not sufficient) conditions for achieving high $T_{\\mathrm{c}}$\nin this group.\n(\\textbf{d}) Scatter plot of $T_{\\mathrm{c}}$ for all known superconducting cuprates \\nobreak\\mbox{\\it vs.}\\ the mean number of unfilled orbitals.\n(\\textbf{c} and \\textbf{d}) suggest that the values of these predictors lead to\nhard limits on the maximum achievable $T_{\\mathrm{c}}$.}\n\\label{fig:art137:Tc_atomWeigth}\n\\end{figure}\n\n\\boldsection{Including AFLOW.}\nThe models described previously demonstrate\nsurprising accuracy and predictive power, especially considering the difference between the\nrelevant energy scales of most Magpie predictors (typically in the range of eV) and superconductivity (meV scale).\nThis disparity, however, hinders the interpretability of the models,\n\\nobreak\\mbox{\\it i.e.}, the ability to extract meaningful physical correlations.\nThus, it is highly desirable to create accurate ML models with features based on\nmeasurable macroscopic properties of the actual compounds\n(\\nobreak\\mbox{\\it e.g.}, crystallographic and electronic properties)\nrather than composite elemental predictors.\nUnfortunately, only a small subset of materials in SuperCon\nis also included in the {\\small ICSD}:\nabout $1,500$ compounds in total, only about $800$ with finite $T_{\\mathrm{c}}$,\nand even fewer are characterized with \\nobreak\\mbox{\\it ab initio}\\ calculations\\footnote{Most of the superconductors in\n\\protect{\\small ICSD}\\ but not in \\protect{\\small AFLOW}\\ are non-stoichiometric\/doped compounds, and thus not amenable to conventional {\\small DFT}\\ methods.\nFor the others, \\protect{\\small AFLOW}\\ calculations were attempted but did not converge to a reasonable solution.}.\nIn fact, a good portion of known superconductors are disordered (off-stoichiometric) materials and\nnotoriously challenging to address with {\\small DFT}\\ calculations.\nCurrently, much faster and efficient methods are becoming available~\\cite{curtarolo:art110}\nfor future applications.\n\nTo extract suitable features, data is incorporated from\nthe {\\small AFLOW}\\ Online Repositories --- a database of\n{\\small DFT}\\ calculations managed by the software package {\\small AFLOW}.\nIt contains information for the vast majority of compounds\nin the {\\small ICSD}\\ and about 550 superconducting materials.\nIn Reference~\\onlinecite{curtarolo:art94}, several\nML models using a similar set of materials are presented.\nThough a classifier shows good accuracy, attempts to create a\nregression model for $T_{\\mathrm{c}}$ led to disappointing results.\nWe verify that using Magpie predictors for the superconducting compounds in the {\\small ICSD}\\\nalso yields an unsatisfactory regression model.\nThe issue is not the lack of compounds \\textit{per se}, as\nmodels created with randomly drawn subsets from SuperCon with\nsimilar counts of compounds perform much better.\nIn fact, the\nproblem is the chemical sparsity of superconductors in the {\\small ICSD}, \\nobreak\\mbox{\\it i.e.},\nthe dearth of closely-related compounds (usually created by chemical substitution).\nThis translates to compound scatter in predictor space --- a challenging learning environment for the model.\n\nThe chemical sparsity in {\\small ICSD}\\ superconductors is a significant hurdle, even when both sets of predictors\n(\\nobreak\\mbox{\\it i.e.}, Magpie and {\\small AFLOW}\\ features) are combined via feature fusion.\nAdditionally, this approach alone neglects the majority of the $16,000$ compounds available via SuperCon.\nInstead, we constructed separate models employing\nMagpie and {\\small AFLOW}\\ features, and then judiciously combined the results\nto improve model metrics --- known as late or decision-level fusion.\nSpecifically, two independent classification models are developed,\none using the full SuperCon dataset and Magpie predictors, and another based on\nsuperconductors in the {\\small ICSD}\\ and {\\small AFLOW}\\ predictors.\nSuch an approach can improve the recall, for example, in the case where we classify ``high-$T_{\\mathrm{c}}$''\nsuperconductors as those predicted by either model to be above-$T_{\\mathrm{sep}}$.\nIndeed, this is the case here where, separately, the models obtain a recall of $40\\%$ and $ 66\\%$, respectively, and\ntogether achieve a recall of about $76\\%$\\footnote{These numbers are based on (a relatively small) test set benchmarking and their uncertainty is roughly $3\\%$.}.\nIn this way, the models' predictions complement each other in a constructive way such that\nabove-$T_{\\mathrm{sep}}$ materials missed by one model (but not the other) are now accurately classified.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig111}\n\\mycaption[DOS of four compounds identified by the ML algorithm as potential materials with $T_{\\mathrm{c}} > 20$~K.]\n{The partial DOS contributions from $s$, $p$ and $d$ electrons and total DOS are shown in blue, green, red, and black, respectively.\nThe large peak just below $E_F$ is a direct consequence of the flat band(s) present in all these materials.\nThese images were generated automatically via {\\small AFLOW}~\\cite{curtarolo:art53}.\nIn the case of substantial overlap among \\textbf{k}-point labels, the right-most label is offset below.}\n\\label{fig:art137:flat_bands}\n\\end{figure}\n\n\\boldsection{Searching for new superconductors in the ICSD.}\nAs a final proof of concept demonstration,\nthe classification and regression models\ndescribed previously are integrated in one pipeline\nand employed to screen the entire {\\small ICSD}\\ database for candidate ``high-$T_{\\mathrm{c}}$'' superconductors.\n(Note that ``high-$T_{\\mathrm{c}}$'' is a simple label,\nthe precise meaning of which can be adjusted.)\nSimilar tools power high-throughput screening workflows for materials with desired\nthermal conductivity and magnetocaloric properties~\\cite{curtarolo:art120,Bocarsly_ChemMat_2017}.\nAs a first step, the full set of Magpie predictors are generated for all\ncompounds in SuperCon.\nA classification model similar to the one presented above is constructed,\nbut trained only on materials in SuperCon and not in the {\\small ICSD}\\ (used\nas an independent test set).\nThe model is then applied on the {\\small ICSD}\\ set\nto create a list of materials with predicted $T_{\\mathrm{c}}$ above $10$~K.\nOpportunities for model benchmarking are limited to those\nmaterials both in the SuperCon and {\\small ICSD}\\ datasets, though this test\nset is shown to be problematic.\nThe set includes about 1,500 compounds, with $T_{\\mathrm{c}}$ reported for only about half of them.\nThe model achieves an impressive accuracy of $0.98$, which is overshadowed by the fact that\n$96.6\\%$ of these compounds belong to the $T_{\\mathrm{c}} < 10$~K class.\nThe precision, recall, and $F_{\\mathrm{1}}$ scores are about $0.74$,\n$0.66$, and $0.70$, respectively.\nThese metrics are lower than the estimates\ncalculated for the general classification model,\nwhich is expected given that this set cannot\nbe considered randomly selected.\nNevertheless, the performance suggests a good opportunity to identify new candidate superconductors.\n\n\\begin{table}[tp]\\centering\n\\mycaption[List of potential superconductors identified by the pipeline.]\n{Also shown are their {\\small ICSD}\\ numbers and symmetries.\nNote that for some compounds there are several entries.\nAll of the materials contain oxygen.}\n\\vspace{3mm}\n{\\small\n\\begin{tabular}{l|r|r}\ncompound & {\\small ICSD}\\ & SYM \\\\\n\\hline\nCsBe(AsO$_4$) & 074027 & orthorhombic \\\\\nRbAsO$_2$ & 413150 & orthorhombic \\\\\nKSbO$_2$ & 411214 & monoclinic \\\\\nRbSbO$_2$ & 411216 & monoclinic \\\\\nCsSbO$_2$ & 059329 & monoclinic \\\\\n\\hline\nAgCrO$_2$ & 004149\/025624 & hexagonal \\\\\nK$_{0.8}$(Li$_{0.2}$Sn$_{0.76}$)O$_2$ & 262638 & hexagonal \\\\\n\\hline\nCs(MoZn)(O$_3$F$_3$)& 018082 & cubic \\\\\n\\hline\nNa$_3$Cd$_2$(IrO$_6$) & 404507 & monoclinic \\\\\nSr$_3$Cd(PtO$_6$) & 280518 & hexagonal \\\\\nSr$_3$Zn(PtO$_6$) & 280519 & hexagonal \\\\\n\\hline\n(Ba$_5$Br$_2)$Ru$_2$O$_9$ & 245668 & hexagonal \\\\\n\\hline\nBa$_4$(AgO$_2$)(AuO$_4)$ & 072329 & orthorhombic \\\\\nSr$_5$(AuO$_4$)$_2$ & 071965 & orthorhombic \\\\\n\\hline\nRbSeO$_2$F & 078399 & cubic \\\\\nCsSeO$_2$F & 078400 & cubic \\\\\nKTeO$_2$F & 411068 & monoclinic \\\\\n\\hline\nNa$_2$K$_4$(Tl$_2$O$_6$) & 074956 & monoclinic \\\\\n\\hline\nNa$_3$Ni$_2$BiO$_6$ & 237391 & monoclinic \\\\\nNa$_3$Ca$_2$BiO$_6$ & 240975 & orthorhombic\\\\\n\\hline\n\nCsCd(BO$_3$) & 189199 & cubic \\\\\n\\hline\nK$_2$Cd(SiO$_4)$ & 083229\/086917 & orthorhombic \\\\\nRb$_2$Cd(SiO$_4$) & 093879 & orthorhombic \\\\\nK$_2$Zn(SiO$_4$) & 083227 & orthorhombic \\\\\nK$_2$Zn(Si$_2$O$_6$) & 079705 & orthorhombic \\\\\n\\hline\n\nK$_2$Zn(GeO$_4$) & 069018\/085006\/085007 & orthorhombic \\\\\n(K$_{0.6}$Na$_{1.4})$Zn(GeO$_4)$ & 069166 & orthorhombic \\\\\nK$_2$Zn(Ge$_2$O$_6$) & 065740 & orthorhombic \\\\\nNa$_6$Ca$_3$(Ge$_2$O$_6$)$_3$ & 067315 & hexagonal \\\\\nCs$_3$(AlGe$_2$O$_7$) & 412140 & monoclinic \\\\\nK$_4$Ba(Ge$_3$O$_9$) & 100203 & monoclinic \\\\\nK$_{16}$Sr$_4$(Ge$_3$O$_9$)$_{4}$ & 100202 & cubic \\\\\nK$_3$Tb[Ge$_3$O$_8$(OH)$_2$] & 193585 & orthorhombic \\\\\nK$_3$Eu[Ge$_3$O$_8$(OH)$_2$] & 262677 & orthorhombic \\\\\n\\hline\nKBa$_6$Zn$_4$(Ga$_7$O$_{21}$) & 040856 & trigonal \\\\\n\\end{tabular}}\n\\label{tab:art137:Table3}\n\\end{table}\n\nNext in the pipeline, the list is fed into a random forest regression\nmodel (trained on the entire SuperCon database)\nto predict $T_{\\mathrm{c}}$.\nFiltering on the materials with $T_{\\mathrm{c}} > 20$~K,\nthe list is further reduced to about 2,000 compounds.\nThis count may appear daunting, but should\nbe compared with the total number of compounds in the database --- about 110,000.\nThus, the method selects less than two percent of all materials,\nwhich in the context of the training set (containing more than $20\\%$ with ``high-$T_{\\mathrm{c}}$''),\nsuggests that the model is not overly biased toward predicting high critical temperatures.\n\nThe vast majority of the compounds identified as\ncandidate superconductors are cuprates,\nor at least compounds that contain copper and oxygen.\nThere are also some materials clearly related to the iron-based superconductors.\nThe remaining set has 35 members, and is composed of materials that are not obviously\nconnected to any high-temperature superconducting families (see Table~\\ref{tab:art137:Table3})\\footnote{For at least one compound\nfrom the list --- Na$_3$Ni$_2$BiO$_6$ --- low-temperature measurements have been performed and no signs\nof superconductivity were observed~\\cite{Seibel_InChem_2013}.}~\\nocite{Seibel_InChem_2013}.\nNone of them is predicted to have\n$T_{\\mathrm{c}}$ in excess of $40$~K, which is not surprising, given that no such instances exist in the training dataset. All contain oxygen --- also not a surprising result, since the group of\nknown superconductors with $T_{\\mathrm{c}} > 20$~K is dominated by oxides.\n\nThe list comprises several distinct groups.\nMost of the materials are insulators, similar to stoichiometric (and underdoped) cuprates that\ngenerally require charge doping and\/or pressure to drive these materials into a superconducting state.\nEspecially interesting are the compounds containing heavy metals (such as Au, Ir, Ru), metalloids (Se, Te),\nand heavier post-transition metals (Bi, Tl), which are or could be pushed into interesting\/unstable oxidation states.\nThe most surprising and non-intuitive of the compounds in the list are the silicates and the germanates.\nThese materials form corner-sharing SiO$_4$ or GeO$_4$ polyhedra, similar to quartz glass,\nand also have counter cations with full or empty shells such as Cd$_2$$^+$ or K$^+$.\nConverting these insulators to metals (and possibly superconductors) likely requires\nsignificant charge doping. However, the similarity between these compounds and cuprates is meaningful.\nIn compounds like K$_2$CdSiO$_4$ or K$_2$ZnSiO$_4$, K$_2$Cd (or K$_2$Zn) unit carries\na 4+ charge that offsets the (SiO$_4$)$^{4-}$ (or (GeO$_4$)$^{4-}$) charges.\nThis is reminiscent of the way Sr$_2$ balances the (CuO$_4$)$^{4-}$ unit in Sr$_2$CuO$_4$.\nSuch chemical similarities based on charge balancing and stoichiometry were likely identified and exploited by the ML algorithms.\n\nThe electronic properties calculated by {\\small AFLOW}\\ offer additional insight into the results of the search, and suggest a possible connection among these candidate.\nPlotting the electronic structure of the potential superconductors exposes a rather unusual feature shared\nby almost all --- one or several (nearly) flat bands just below the energy of the highest occupied electronic state\\footnote{The\nflat band attribute is unusual for a superconducting material: the average DOS of the known superconductors in the \\protect{\\small ICSD}\\\n(at least those available in the \\protect{\\small AFLOW}\\ Online Repositories) has no distinct features, demonstrating roughly uniform distribution of electronic states.\nIn contrast, the average DOS of the potential superconductors in Table~\\ref{tab:art137:Table3} shows a sharp peak just below $E_{\\mathrm{F}}$.\nAlso, most of the flat bands in the potential superconductors we discuss have a notable contribution from the oxygen $p$-orbitals.\nAccessing\/exploiting the potential strong instability this electronic structure feature creates can require significant charge doping.}.\nSuch bands lead to a large peak in the DOS (see Figure~\\ref{fig:art137:flat_bands}) and\ncan cause a significant enhancement in $T_{\\mathrm{c}}$.\nPeaks in the DOS elicited by van Hove singularities can enhance $T_{\\mathrm{c}}$\nif sufficiently close to $E_{\\mathrm{F}}$~\\cite{Labbe_PRL_1967,Hirsch_PRL_1986,Dzyaloshinskii_JETPLett_1987}.\nHowever, note that unlike typical van Hove points, a true flat band creates divergence\nin the DOS (as opposed to its derivatives), which in turn leads to a critical temperature\ndependence that is linear in the pairing interaction strength, rather than the usual exponential relationship\nyielding lower $T_{\\mathrm{c}}$~\\cite{Kopnin_PRB_2011}.\nAdditionally, there is significant similarity\nwith the band structure and DOS of layered\nBiS$_2$-based superconductors~\\cite{Yazici_PSCC_2015}.\n\nThis band structure feature came as the surprising\nresult of applying the ML model.\nIt was not sought for, and, moreover,\nno explicit information about the electronic band structure has been\nincluded in these predictors.\nThis is in contrast to the algorithm presented in Reference~\\onlinecite{Klintenberg_CMS_2013},\nwhich was specifically designed to filter {\\small ICSD}\\ compounds based on several preselected electronic structure features.\n\nWhile at the moment it is not clear if some (or indeed any) of these compounds are really superconducting,\nlet alone with $T_{\\mathrm{c}}$'s above 20~K,\nthe presence of this highly unusual electronic structure feature is encouraging.\nAttempts to synthesize several of these compounds are already underway.\n\n\\subsection{Discussion}\nHerein, several machine learning tools are developed to study the critical temperature of superconductors.\nBased on information from the SuperCon database, initial coarse-grained\nchemical features are generated using the Magpie software.\nAs a first application of ML methods, materials are divided into two classes depending on\nwhether $T_{\\mathrm{c}}$ is above or below $10$~K.\nA non-parametric random forest classification model is constructed\nto predict the class of superconductors.\nThe classifier shows excellent performance, with out-of-sample accuracy and $F_{\\mathrm{1}}$\nscore of about $92\\%$.\nNext,\nseveral successful random forest regression models are created to predict the value of $T_{\\mathrm{c}}$,\nincluding separate models for three material sub-groups, \\nobreak\\mbox{\\it i.e.},\ncuprate, iron-based, and low-$T_{\\mathrm{c}}$ compounds.\nBy studying the importance of predictors for each family of superconductors,\ninsights are obtained about the\nphysical mechanisms driving superconductivity among the different groups.\nWith the incorporation of crystallographic-\/electronic-based features\nfrom the {\\small AFLOW}\\ Online Repositories, the ML models are further improved.\nFinally, we combined these models into one integrated pipeline, which is employed to search the entire\n{\\small ICSD}\\ database for new inorganic superconductors.\nThe model identified 35 oxides as candidate materials.\nSome of these are chemically and structurally similar to cuprates (even though no explicit structural information was provided during training of the model). Another feature that unites almost all of these materials is the presence of flat or nearly-flat bands just below the energy of the highest occupied electronic state.\n\nIn conclusion, this work demonstrates the important role\nML models can play in superconductivity research.\nRecords collected over several decades in SuperCon and other relevant databases can be consumed by ML models,\ngenerating insights and promoting better understanding of the connection\nbetween materials' chemistry\/structure and superconductivity.\nApplication of sophisticated ML algorithms has the potential to dramatically accelerate\nthe search for candidate high-temperature superconductors.\n\n\\subsection{Methods}\n\n\\boldsection{Superconductivity data.}\nThe SuperCon database consists of two separate subsets: ``Oxide \\& Metallic''\n(inorganic materials containing metals, alloys, cuprate high-temperature superconductors, \\nobreak\\mbox{\\it etc.})\nand ``Organic'' (organic superconductors).\nDownloading the entire inorganic materials dataset and removing compounds with\nincompletely-specified chemical compositions leaves about $22,000$ entries.\nIf a single $T_{\\mathrm{c}}$ record exists for a given material, it is taken to accurately reflect the critical temperature of this material.\nIn the case of multiple records for the same compound,\nthe reported material's $T_{\\mathrm{c}}$'s are averaged, but only if\ntheir standard deviation is less than $5$~K, and discarded otherwise.\nThis brings the total down to about $16,400$ compounds,\nof which around $4,000$ have no critical temperature reported. Each entry in the set contains fields for the chemical composition,\n$T_{\\mathrm{c}}$, structure, and a journal reference to the information source.\nHere, structural information is ignored as it is not always available.\n\nThere are occasional problems with the validity and consistency of some of the data.\nFor example, the database includes some reports based on tenuous experimental evidence and\nonly indirect signatures of superconductivity, as well as reports of inhomogeneous (surface, interfacial)\nand nonequilibrium phases.\nEven in cases of \\textit{bona fide} bulk superconducting phases, important relevant variables\nlike pressure are not recorded.\nThough some of the obviously erroneous records were removed from the data,\nthese issues were largely ignored\nassuming their effect on the entire dataset to be relatively modest. The data cleaning and processing is carried out using the Python Pandas package for data analysis~\\cite{Mckinney_Pandas_2012}.\n\n\\boldsection{Chemical and structural features.}\nThe predictors are calculated using the Magpie software \\cite{magpie_software}.\nIt computes a set of 145 attributes\nfor each material, including:\n\\textbf{i.} stoichiometric features (depends only on the ratio of elements and\nnot the specific species);\n\\textbf{ii.} elemental property statistics: the mean, mean absolute deviation, range, minimum,\nmaximum, and mode of 22 different elemental properties\n(\\nobreak\\mbox{\\it e.g.}, period\/group on the periodic table,\natomic number, atomic radii, melting temperature);\n\\textbf{iii.} electronic structure attributes: the average\nfraction of electrons from the $s$, $p$, $d$ and $f$ valence shells among all\nelements present; and\n\\textbf{iv.} ionic compound features that include whether it is possible to form an ionic\ncompound assuming all elements exhibit a single oxidation state.\n\nML models are also constructed with the\nsuperconducting materials in the {\\small AFLOW}\\ Online Repositories.\n{\\small AFLOW}\\ is a high-throughput \\nobreak\\mbox{\\it ab initio}\\ framework that manages density functional theory ({\\small DFT})\ncalculations in accordance with the {\\small AFLOW}\\ Standard~\\cite{curtarolo:art104}.\nThe Standard ensures that the calculations and derived properties are empirical (reproducible), reasonably\nwell-converged, and above all, consistent (fixed set of parameters), a particularly attractive feature for ML modeling.\nMany materials properties important for superconductivity have been calculated within the {\\small AFLOW}\\ framework,\nand are easily accessible through the {\\small AFLOW}\\ Online Repositories.\nThe features are built with the following properties:\nnumber of atoms, space group, density, volume, energy per atom, electronic entropy per atom, valence of the cell,\nscintillation attenuation length, the ratios of the unit cell's dimensions, and Bader charges and volumes.\nFor the Bader charges and volumes (vectors), the following statistics\nare calculated and incorporated:\nthe maximum, minimum, average, standard deviation, and range.\n\n\\boldsection{Machine learning algorithms.}\nOnce we have a list of relevant predictors, various ML models can be applied to the\ndata~\\cite{Bishop_ML_2006,Hastie_StatLearn_2001}.\nAll ML algorithms in this work are\nvariants of the random forest method~\\cite{randomforests}.\nIt is based on creating a set of individual decision trees (hence the ``forest''),\neach built to solve the same classification\/regression problem.\nThe model then combines their results, either by voting or averaging depending on the problem.\nThe deeper individual tree are, the more complex the relationships the model can learn,\nbut also the greater the danger of overfitting, \\nobreak\\mbox{\\it i.e.}, learning\nsome irrelevant information or just ``noise''.\nTo make the forest more robust to overfitting, individual trees in the ensemble are\nbuilt from samples drawn with replacement (a bootstrap sample) from the training set.\nIn addition, when splitting a node during the construction of a tree, the model chooses the best split\nof the data only considering a random subset of the features.\n\nThe random forest models above are developed using scikit-learn --- a powerful and efficient machine\nlearning Python library \\cite{Pedregosa_JMLR_2011}.\nHyperparameters of these models include the number of trees in the forest,\nthe maximum depth of each tree, the minimum number of samples required to split an internal node,\nand the number of features to consider when looking for the best split.\nTo optimize the classifier and the combined\/family-specific regressors, the\nGridSearch function in scikit-learn is employed, which generates and compares candidate models from a grid of parameter values.\nTo reduce computational expense, models are not optimized at each step of the backward feature selection process.\n\nTo test the influence of using log-transformed target variable $\\ln(T_{\\mathrm{c}})$,\na general regression model is trained and tested on raw $T_{\\mathrm{c}}$ data.\nThis model is very similar to the one described in section ``Results'', and its $R^2$ value is fairly similar as well\n(although comparing $R^2$ scores of models built using different target data can be misleading).\nHowever, note the relative sparsity of data points in some $T_{\\mathrm{c}}$ ranges, which makes the model susceptible to outliers.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.6\\linewidth]{fig112}\n\\mycaption[Regression model predictions of $T_{\\mathrm{c}}$.]\n{Predicted\n\\nobreak\\mbox{\\it vs.}\\ measured $T_{\\mathrm{c}}$ for general regression model.\n$R^2$ score is comparable to the one obtained testing regression modeling $\\ln(T_{\\mathrm{c}})$.}\n\\label{fig:art137:Regr_non_loc}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig113}\n\\mycaption[Histograms of $\\Delta\\ln(T_{\\mathrm{c}}) * \\ln(T_{\\mathrm{c}})^{-1}$ for the four regression models.]\n{$\\Delta\\ln(T_{\\mathrm{c}}) \\equiv (\\ln(T^{\\mathrm{meas}}_{\\mathrm{c}}) - \\ln(T^{\\mathrm{pred}}_{\\mathrm{c}}))$\nand $\\ln(T_{\\mathrm{c}}) \\equiv \\ln(T^{\\mathrm{meas}}_{\\mathrm{c}})$.}\n\\label{fig:art137:Regr_err}\n\\end{figure}\n\n\\boldsection{Prediction errors of the regression models.}\nPreviously, several regression models were described,\neach one designed to predict the critical temperatures of materials from different superconducting groups.\nThese models achieved an impressive $R^{2}$ score, demonstrating\ngood predictive power for each group.\nHowever, it is also important to consider the accuracy of the predictions\nfor individual compounds (rather than on the aggregate set),\nespecially in the context of searching for new materials.\nTo do this, we calculate the prediction errors for about 300 materials from a test set.\nSpecifically, we consider the difference between the logarithm of the predicted and measured\ncritical temperature $[\\ln(T^{\\mathrm{meas}}_{\\mathrm{c}})- \\ln(T^{\\mathrm{pred}}_{\\mathrm{c}})]$\nnormalized by the value of $\\ln(T^{\\mathrm{meas}}_{\\mathrm{c}})$\n(normalization compensates the different $T_{\\mathrm{c}}$ ranges of different groups).\nThe models show comparable spread of errors.\nThe histograms of errors for the four models\n(combined and three group-specific) are shown in Fig.~\\ref{fig:art137:Regr_err}.\nThe errors approximately follow a normal distribution,\ncentered not at zero but at a small negative value.\nThis suggests the models are marginally biased, and on average tend to slightly underestimate $T_{\\mathrm{c}}$.\nThe variance is comparable for all models, but largest for the model trained\nand tested on iron-based materials, which also shows the smallest $R^2$.\nPerformance of this model is expected to benefit from a larger training set.\n\n\\boldsection{Data availability.} The superconductivity data used to generate the results\nin this work can be downloaded from \\url{https:\/\/github.com\/vstanev1\/Supercon}.\n\\clearpage\n\\section{High Throughput Thermal Conductivity of High Temperature Solid Phases: The Case of Oxide and Fluoride Perovskites}\n\\label{sec:art120}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art120}.\n\n\\subsection{Introduction}\n\\label{subsec:art120:Introduction}\n\nHigh throughput \\nobreak\\mbox{\\it ab-initio}\\ screening of materials is a new and rapidly\ngrowing discipline~\\cite{nmatHT}. Amongst the basic properties\nof materials, thermal conductivity is a particularly relevant one.\nThermal management is a crucial factor to a vast range of technologies,\nincluding power electronics, CMOS interconnects, thermoelectric energy\nconversion, phase change memories, turbine thermal coatings and many\nothers~\\cite{Cahill_APR_2014}. Thus, rapid determination\nof thermal conductivity for large pools of compounds is a desirable\ngoal in itself, which may enable the identification of suitable compounds\nfor targeted applications. A few recent works have investigated thermal\nconductivity in a high throughput fashion~\\cite{aflowKAPPA,Seko_PRL_2015}.\nA drawback of these studies is that they were restricted to use the\nzero Kelvin phonon dispersions. This is often fine when the room temperature\nphase is mechanically stable at 0~K. It however poses a problem\nfor materials whose room or high temperature phase is not the 0~K\nstructure: when dealing with structures exhibiting displacive distortions,\nincluding temperature effects in the phonon spectrum is a crucial\nnecessity.\n\nSuch a phenomenon often happens for perovskites. Indeed, the perovskite\nstructure can exhibit several distortions from the ideal cubic lattice,\nwhich is often responsible for rich phase diagrams. When the structure\nis not stable at low temperatures, a simple computation of the phonon\nspectrum using forces obtained from density functional theory and\nthe finite displacement method yields imaginary eigenvalues. This\nprevents us from assessing the mechanical stability of those compounds\nat high temperatures or calculating their thermal conductivity. Moreover,\ntaking into account finite-temperature effects in phonon calculations\nis currently a very demanding task, especially for a high-throughput\ninvestigation.\n\nIn this study, we are interested in the \\textit{high-temperature}\nproperties of perovskites, notably for thermoelectric applications.\nFor this reason, we focus on perovskites with the highest symmetry\ncubic structure, which are most likely to exist at high temperatures\n\\cite{Landau_CTP5_SP_1969,Howard_ActaCrisA_2005,Thomas_PRL_1968,Cochran_PSSB_1968,Angel_PRL_2005}.\nWe include the effects of anharmonicity in our \\nobreak\\mbox{\\it ab-initio}\\ calculations\nof mechanical and thermal properties.\n\n\\subsection{Finite-temperature calculations of mechanical stability and thermal properties}\n\\label{subsec:art120:Finite-T_calculations}\n\nRecently, several methods have been developed to deal with anharmonic\neffects at finite temperatures in solids~\\cite{Souvatzis_PRL_2008,Hellman_PRB_2011,Hellman_PRB_2013,Errea_PRL_2013,Tadano_PRB_2015,VanRoekeghem_ARXIV_2016}.\nIn this study, we use the method presented in Reference~\\cite{VanRoekeghem_ARXIV_2016}\nto compute the temperature-dependent interatomic force constants,\nwhich uses a regression analysis of forces from density functional\ntheory coupled with a harmonic model of the quantum canonical ensemble.\nThis is done in an iterative way to achieve self-consistency of the\nphonon spectrum.\nThe workflow is summarized in Figure~\\ref{fig:art120:finite-T-phonon}.\nIn the following (in particular Section~\\ref{subsec:art120:PCA-regression}),\nit will be referred as ``SCFCS'' -- standing for self-consistent\nforce constants. As a trade-off between accuracy and throughput, we\nchoose a 3x3x3 supercell and a cutoff of 5~\\AA\\ for the third order\nforce constants. Special attention is paid to the computation of the\nthermal displacement matrix~\\cite{VanRoekeghem_ARXIV_2016}, due to the imaginary\nfrequencies that can appear during the convergence process, as well\nas the size of the supercell that normally prevents us from sampling\nthe usual soft modes at the corners of the Brillouin zone (see Supplementary Material of Reference~\\cite{curtarolo:art120}).\nThis allows us to assess the stability at 1000~K of the\n391 hypothetical compounds mentioned in Section~\\ref{subsec:art120:Introduction}.\nAmong this set, we identify 92 mechanically stable compounds, for\nwhich we also check the stability at 300~K. The phonon spectra of\nthe stable compounds are provided in the Supplementary Material of Reference~\\cite{curtarolo:art120}.\nFurthermore, we compute the thermal conductivity using the finite temperature force\nconstants and the full solution of the Boltzmann transport equation\nas implemented in the ShengBTE code~\\cite{Li_ShengBTE_CPC_2014}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.6\\linewidth]{fig114}\n\\mycaption{Workflow of the method used to calculate the phonon spectrum and thermal\nconductivity including finite-temperature anharmonic effects.}\n\\label{fig:art120:finite-T-phonon}\n\\end{figure}\n\nWe list the stable compounds and their thermal conductivities in Table\n\\ref{tab:art120:List-of-perovskites}.\nRemarkably, this list contains 37\nperovskites that have been reported experimentally in the ideal cubic\nstructure (see References in Table~\\ref{tab:art120:List-of-perovskites}),\nwhich lends support to our screening method.\nOn the other hand, we\nalso find that 11 compounds are reported only in a non-perovskite\nform. This is not necessarily indicative of mechanical instability,\nbut instead suggests thermodynamical stability may be an issue for\nthese compounds, at least near this temperature and pressure. 36 compounds\nremain unreported experimentally in the literature to our knowledge.\nThus, by screening only for mechanical stability at high-temperatures,\nwe reduce the number of potential new perovskites by a factor of 10.\nFurthermore, we find that 50 of them are mechanically stable in the\ncubic form close to room temperature.\n\nOf the full list of perovskites, only a few measurements of thermal\nconductivity are available in the literature. They are displayed in\nparentheses in Table~\\ref{tab:art120:List-of-perovskites} along with their\ncalculated values. Our method tends to slightly underestimate the\nvalue of the thermal conductivity, due to the compromises we made\nto limit the computational cost of the study (see Supplementary Material of Reference~\\cite{curtarolo:art120}).\nThis discrepancy could also be partially related to the electronic\nthermal conductivity, which was not subtracted in the measurements.\nStill, we expect the order of magnitude of the thermal conductivity\nand the relative classification of different materials to be consistent.\nMore importantly, this large dataset allows us to analyze the global\ntrends driving thermal conductivity. These trends are discussed in\nSection~\\ref{subsec:art120:Descriptors}.\n\n\\newcommand{\\fluoridestabfootone}{\nAuMgF$_{3}$ was mentioned theoretically in Reference~\\cite{Uetsuji_TJSME_2006}.}\n\\newcommand{\\fluoridestabfoottwo}{\nThe thermal diffusivity of BaLiF$_{3}$ was measured at 300~K in\nReference~\\cite{Duarte_MSEB_1994} as $\\alpha$=0.037~cm$^{2}$s$^{-1}$.}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[List of cubic perovskites found to be mechanically stable at 1000~K\nand their corresponding computed lattice thermal conductivity (in\nW\/m\/K).]\n{We also report the computed lattice thermal conductivity at\n300~K (in W\/m\/K) when we obtain stability at that temperature. We\nhighlight in blue the compounds that are experimentally reported in\nthe ideal cubic perovskite structure, and in red those that are reported\nonly in non-perovskite structures (references provided in the table).\nWhen no reference is provided, no mention of the compound in this\nstoichiometry has been found in the experimental literature. Experimental\nmeasurements of the thermal conductivity are reported in parentheses,\nand in italics when the structure is not cubic.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|rcl|r|r|r|r|rcl|r|r|r|r|r}\n & $\\kappa_{1000}$ & & $\\kappa_{300}$ & & References & & & $\\kappa_{1000}$ & & $\\kappa_{300}$ & & References & & & $\\kappa_{1000}$ & & $\\kappa_{300}$ & & References\\tabularnewline\n\\cline{1-6} \\cline{8-13} \\cline{15-20}\n\\textcolor{blue}{CaSiO$_{3}$} & 4.89 & & & & \\cite{Komabayashi_EPSL_2007} & & CdYF$_{3}$ & 1.29 & & 3.51 & & & & TlOsF$_{3}$ & 0.62 & & 0.95 & & \\tabularnewline\n\\textcolor{blue}{RbTaO$_{3}$} & 3.61 & & & & \\cite{Lebedev_PhysSolStat_2015} & & \\textcolor{blue}{RbCaF$_{3}$} & 1.15 & & 2.46 & (3.2) & \\cite{Ludekens_ActaCrist_1952,Ridou_Ferroelectrics_1976,Martin_Phonons_1976} & & InZnF$_{3}$ & 0.61 & & 1.86 & & \\tabularnewline\n\\textcolor{blue}{NaTaO$_{3}$} & 3.45 & & & & \\cite{Kennedy_JPCM_1999} & & HgInF$_{3}$ & 1.15 & & 3.85 & & & & \\textcolor{blue}{CsCdF$_{3}$} & 0.59 & & 1.73 & & \\cite{Rousseau_PRB_1975}\\tabularnewline\n\\textcolor{red}{CuCF$_{3}$} & 3.32 & & 8.79 & & \\cite{Zanardi_JACS_2011} & & AlFeF$_{3}$ & 1.14 & & & & & & AlMgF$_{3}$ & 0.56 & & & & \\tabularnewline\n\\textcolor{blue}{SrSiO$_{3}$} & 3.23 & & 10.10 & & \\cite{Xiao_AM_2013} & & \\textcolor{blue}{PbHfO$_{3}$} & 1.12 & & & & \\cite{Kwapulinski_JPCM_1994} & & AuZnF$_{3}$ & 0.53 & & & & \\tabularnewline\n\\textcolor{blue}{NaNbO$_{3}$} & 3.05 & & & (\\textit{1.5}) & \\cite{Shirane_PR_1954,Mishra_PRB_2011,Tachibana_APL_2008} & & \\textcolor{blue}{AgMgF$_{3}$} & 1.11 & & & & \\cite{Portier_CRASC_1970} & & InOsF$_{3}$ & 0.52 & & & & \\tabularnewline\n\\textcolor{blue}{BaHfO$_{3}$} & 3.04 & (4.5) & 8.26 & (10.4) & \\cite{Maekawa_BaHfO3_SrHfO3_JAC_2006} & & ZnScF$_{3}$ & 1.10 & & 3.66 & & & & \\textcolor{blue}{RbSrF$_{3}$} & 0.51 & & & & \\cite{Pies_Landolt_Bornstein_1973}\\tabularnewline\n\\textcolor{blue}{KNbO$_{3}$} & 2.94 & & & (\\textit{10}) & \\cite{Shirane_PR_1954,Tachibana_APL_2008} & & \\textcolor{blue}{RbFeF$_{3}$} & 1.09 & & 4.62 & & \\cite{Kestigian_IC_1966} & & \\textcolor{blue}{CsSrF$_{3}$} & 0.50 & & 1.13 & & \\cite{Pies_Landolt_Bornstein_1973}\\tabularnewline\n\\textcolor{red}{TlTaO$_{3}$} & 2.86 & & & & \\cite{Ramadass_SSC_1975} & & \\textcolor{black}{TlMgF$_{3}$} & 1.06 & & 3.42 & & \\cite{Arakawa_JPCM_2006} & & BeYF$_{3}$ & 0.48 & & 2.34 & & \\tabularnewline\n\\textcolor{blue}{AgTaO$_{3}$} & 2.77 & & & & \\cite{Kania_PT_1981,Pawelczyk_PT_1987} & & \\textcolor{blue}{KCaF$_{3}$} & 1.06 & & & & \\cite{Demetriou_SSI_2005} & & BeScF$_{3}$ & 0.48 & & 1.59 & & \\tabularnewline\n\\textcolor{blue}{KMgF$_{3}$} & 2.74 & & 8.25 & (10) & \\cite{Wood_JACryst_2002,Martin_Phonons_1976} & & HgScF$_{3}$ & 1.01 & & 5.42 & & & & \\textcolor{blue}{TlCdF$_{3}$} & 0.44 & & & & \\cite{Rousseau_PRB_1975}\\tabularnewline\n\\textcolor{red}{GaTaO$_{3}$} & 2.63 & & & & \\cite{Xu_Thesis_2000,Armiento_PRB_2011,Castelli_EES_2012} & & \\textcolor{blue}{CsCaF$_{3}$} & 0.98 & & 3.03 & & \\cite{Rousseau_SSC_1981} & & \\textcolor{blue}{RbHgF$_{3}$} & 0.43 & & & & \\cite{Hoppe_ZAAC_1969}\\tabularnewline\n\\textcolor{blue}{BaTiO$_{3}$} & 2.51 & & 4.99 & (\\textit{4-5}) & \\cite{Tachibana_APL_2008,Strukov_JPCM_2003} & & AuMgF$_{3}$ & 0.96 & & & & \\tablefootnote{\\fluoridestabfootone} & & PdYF$_{3}$ & 0.43 & & 0.99 & & \\tabularnewline\n\\textcolor{blue}{PbTiO$_{3}$} & 2.42 & & & (\\textit{5}) & \\cite{Tachibana_APL_2008} & & InMgF$_{3}$ & 0.96 & & 3.53 & & & & AlZnF$_{3}$ & 0.39 & & & & \\tabularnewline\n\\textcolor{blue}{SrTiO$_{3}$} & 2.36 & (4) & 6.44 & (10.5) & \\cite{Muta_JAC_2005,Popuri_RSCA_2014,Yamanaka_JSSC_2004} & & \\textcolor{blue}{RbZnF$_{3}$} & 0.91 & & 2.64 & & \\cite{Daniel_PRB_1995} & & \\textcolor{black}{KHgF$_{3}$} & 0.37 & & & & \\cite{Hoppe_ZAAC_1969}\\tabularnewline\n\\textcolor{blue}{SrHfO$_{3}$} & 2.20 & (\\textit{2.7}) & & (\\textit{5.2}) & \\cite{Kennedy_PRB_1999,Yamanaka_JSSC_2004} & & ZnInF$_{3}$ & 0.88 & & 1.89 & & & & \\textcolor{red}{RbSnF$_{3}$} & 0.37 & & 0.82 & & \\cite{Tran_JSSC_2014}\\tabularnewline\n\\textcolor{blue}{BaZrO$_{3}$} & 2.13 & (2.9) & 5.61 & (5.2) & \\cite{Yamanaka_JAC_2003} & & \\textcolor{black}{BaSiO$_{3}$} & 0.87 & & & & \\cite{Yusa_AM_2007} & & ZnBiF$_{3}$ & 0.37 & & 1.29 & & \\tabularnewline\nXeScF$_{3}$ & 1.87 & & 4.40 & & & & TlCaF$_{3}$ & 0.86 & & & & & & \\textcolor{blue}{CsHgF$_{3}$} & 0.37 & & 1.00 & & \\cite{Hoppe_ZAAC_1969}\\tabularnewline\nHgYF$_{3}$ & 1.84 & & 5.37 & & & & CdScF$_{3}$ & 0.85 & & 2.37 & & & & \\textcolor{red}{KSnF$_{3}$} & 0.35 & & & & \\cite{Tran_JSSC_2014}\\tabularnewline\n\\textcolor{blue}{AgNbO$_{3}$} & 1.79 & & & & \\cite{Lukaszewski_PT_1983,Sciau_JPCM_2004} & & XeBiF$_{3}$ & 0.82 & & 2.13 & & & & CdBiF$_{3}$ & 0.33 & & 0.98 & & \\tabularnewline\n\\textcolor{red}{TlNbO$_{3}$} & 1.75 & & & & \\cite{Ramadass_SSC_1975} & & \\textcolor{blue}{AgZnF$_{3}$} & 0.80 & & & & \\cite{Portier_CRASC_1970} & & \\textcolor{black}{RbPbF$_{3}$} & 0.32 & & & & \\cite{Yamane_SSI_2008}\\tabularnewline\n\\textcolor{blue}{KFeF$_{3}$} & 1.72 & & 6.37 & (3.0) & \\cite{Okazaki_JPSJ_1961,Suemune_kappa_JPSJ_1964} & & PdScF$_{3}$ & 0.79 & & 1.63 & & & & BeAlF$_{3}$ & 0.30 & & 1.70 & & \\tabularnewline\nSnSiO$_{3}$ & 1.66 & & 4.22 & & \\cite{Clark_IC_2001,Armiento_PRB_2014} & & \\textcolor{blue}{KCdF$_{3}$} & 0.75 & & & & \\cite{Hidaka_SSC_1977,Hidaka_PT_1990} & & \\textcolor{red}{KPbF$_{3}$} & 0.30 & & & & \\cite{Hull_JPCM_1999}\\tabularnewline\n\\textcolor{red}{PbSiO$_{3}$} & 1.66 & & 3.69 & & \\cite{Mackay_MM_1952,Xiao_AM_2012} & & \\textcolor{blue}{BaLiF$_{3}$} & 0.73 & & 2.21 & \\tablefootnote{\\fluoridestabfoottwo} & \\cite{Mortier_SSC_1994,Duarte_MSEB_1994} & & CsBaF$_{3}$ & 0.29 & & & & \\tabularnewline\n\\textcolor{black}{AuNbO$_{3}$} & 1.56 & & & & \\cite{Wu_AngChemInt_2013} & & HgBiF$_{3}$ & 0.72 & & 2.37 & & & & InCdF$_{3}$ & 0.29 & & & & \\tabularnewline\n\\textcolor{red}{CaSeO$_{3}$} & 1.42 & & & & \\cite{Wildner_NJMA_2007} & & ZnAlF$_{3}$ & 0.72 & & 1.92 & & & & BaCuF$_{3}$ & 0.28 & & & & \\tabularnewline\n\\textcolor{red}{NaBeF$_{3}$} & 1.40 & & 2.53 & & \\cite{ODaniel_NJMMAA_1945,Roy_JACerS_1953} & & GaZnF$_{3}$ & 0.69 & & & & & & \\textcolor{red}{TlSnF$_{3}$} & 0.27 & & 0.63 & & \\cite{Foulon_EJSSIC_1993}\\tabularnewline\n\\textcolor{blue}{RbMgF$_{3}$} & 1.37 & & 4.54 & & \\cite{Shafer_JoPACoS_1969} & & \\textcolor{blue}{RbCdF$_{3}$} & 0.68 & & 1.46 & & \\cite{Rousseau_PRB_1975} & & \\textcolor{blue}{TlHgF$_{3}$} & 0.26 & & & & \\cite{Hebecker_Naturwissenschaften_1973}\\tabularnewline\nGaMgF$_{3}$ & 1.34 & & 2.11 & & & & GaRuF$_{3}$ & 0.67 & & & & & & CdSbF$_{3}$ & 0.26 & & & & \\tabularnewline\n\\textcolor{blue}{KZnF$_{3}$} & 1.33 & & 4.15 & (5.5) & \\cite{Suemune_JPSJ_1964,Martin_Phonons_1976} & & \\textcolor{black}{CsZnF$_{3}$} & 0.67 & & 1.12 & & \\cite{Longo_JSSC_1969} & & \\textcolor{blue}{TlPbF$_{3}$} & 0.22 & & & & \\cite{Buchinskaya_RCR_2004}\\tabularnewline\nZnYF$_{3}$ & 1.32 & & 3.72 & & & & \\textcolor{black}{TlZnF$_{3}$} & 0.64 & & 1.96 & & \\cite{Babel_TlZnF3_1967} & & & & & & & \\tabularnewline\n\\end{tabular}}\n\\label{tab:art120:List-of-perovskites}\n\\end{table}\n\n\\clearpage\n\nWe also investigate the (potentially) negative thermal expansion of\nthese compounds. Indeed, the sign of the coefficient of thermal expansion\n$\\alpha_{\\text{V}}$ is the same as the sign of the weighted Gr\\\"{u}neisen\nparameter $\\gamma$, following $\\alpha_{\\text{V}}=\\frac{\\gamma c_{\\text{V}}\\rho}{K_{\\text{T}}}$,\nwhere $K_{\\text{T}}$ is the isothermal bulk modulus, $c_{\\text{V}}$ is the isochoric\nheat capacity and $\\rho$ is the density~\\cite{Gruneisen_AnnPhys_1912,ashcroft_mermin}.\nThe weighted Gr\\\"{u}neisen parameter is obtained by summing the contributions\nof the mode-dependent Gr\\\"{u}neisen parameters: $\\gamma=\\sum\\gamma_{i}c_{Vi}\/\\sum c_{Vi}$.\nFinally the mode-dependent parameters are related to the volume variation\nof the mode frequency $\\omega_{i}$ via $\\gamma_{i}=-(V\/\\omega_{i})(\\partial\\omega_{i}\/\\partial V)$.\nIn our case, we calculate those parameters directly using the second\nand third order force constants at a given temperature~\\cite{Fabian_PRL_1997,Broido_PRB_2005,Hellman_PRB_2013}:\n\\begin{equation}\n\\gamma_{m}=-\\frac{1}{6\\omega_{m}^{2}}\\sum_{ijk\\alpha\\beta\\gamma}\\frac{\\epsilon_{mi\\alpha}^{*}\\epsilon_{mj\\beta}}{\\sqrt{M_{i}M_{j}}}r_{k}^{\\gamma}\\Psi_{ijk}^{\\alpha\\beta\\gamma}e^{i\\mathbf{q}\\cdot\\mathbf{r}_{j}}\n\\end{equation}\n\nThis approach has been very successful in predicting the thermal expansion\nbehavior in the empty perovskite ScF$_{3}$~\\cite{VanRoekeghem_ARXIV_2016},\nwhich switches from negative to positive around 1100~K~\\cite{Greve_JACS_2010}.\nIn our list of filled perovskites, we have found only two candidates\nwith negative thermal expansion around room temperature: TlOsF$_{3}$\nand BeYF$_{3}$, and none at 1000~K.\nThis shows that filling the perovskite structure is probably detrimental to the negative thermal\nexpansion.\n\nWe also examine the evolution of the thermal conductivity as a function\nof temperature, for the compounds that are mechanically stable at\n300~K and 1000~K. There is substantial evidence that the thermal\nconductivity in cubic perovskites generally decreases more slowly\nthan the model $\\kappa\\propto T^{-1}$ behavior~\\cite{Peierls_AnnPhys_1929,Roufosse_JGPR_1974}\nat high temperatures, in contrast to the thermal conductivity of \\nobreak\\mbox{\\it e.g.}\\\nSi or Ge that decreases faster than $\\kappa\\propto T^{-1}$~\\cite{Glassbrenner_PR_1964}.\nThis happens for instance in SrTiO$_{3}$~\\cite{Muta_JAC_2005,Popuri_RSCA_2014},\nKZnF$_{3}$~\\cite{Suemune_JPSJ_1964,Martin_Phonons_1976},\nKMgF$_{3}$~\\cite{Martin_Phonons_1976}, KFeF$_{3}$~\\cite{Suemune_kappa_JPSJ_1964},\nRbCaF$_{3}$~\\cite{Martin_Phonons_1976}, BaHfO$_{3}$\n\\cite{Maekawa_BaHfO3_SrHfO3_JAC_2006}, BaSnO$_{3}$~\\cite{Maekawa_BaSnO3_JAC_2006}\nand BaZrO$_{3}$~\\cite{Yamanaka_JAC_2003}.\nWe also predicted an anomalous behavior in ScF$_{3}$ using \\nobreak\\mbox{\\it ab-initio}\\ calculations,\ntracing its origin to the important anharmonicity of the soft modes\n\\cite{VanRoekeghem_ARXIV_2016}.\nFigure~\\ref{fig:art120:Kappa} displays several experimentally\nmeasured thermal conductivities from the literature on a logarithmic\nscale, along with the results of our high-throughput calculations.\nAs discussed above, the absolute values of the calculated thermal\nconductivities are generally underestimated, but their relative magnitude\nand the overall temperature dependence are generally consistent. Although\nthe behavior of the thermal conductivity $\\kappa(T)$ is in general\nmore complex than a simple power-law behavior, we model the deviation\nto the $\\kappa\\propto T^{-1}$ law by using a parameter $\\alpha$\nthat describes approximately the temperature-dependence of $\\kappa$\nbetween 300~K and 1000~K as $\\kappa\\propto T^{-\\alpha}$. For\ninstance, in Figure~\\ref{fig:art120:Kappa}, KMgF$_{3}$ appears to have\nthe fastest decreasing thermal conductivity with $\\alpha=0.9$ both\nfrom experiment and calculations, while SrTiO$_{3}$ is closer to\n$\\alpha=0.6$. At present, there are too few experimental measurements\nof the thermal conductivities in cubic perovskites to state that the\n$\\kappa\\propto T^{-\\alpha}$ behavior with $\\alpha<1$ is the general\nrule in this family. However, the large number of theoretical predictions\nprovides a way to assess this trend. Of the 50 compounds that we found\nto be mechanically stable at room temperature, we find a mean $\\alpha\\simeq0.85$,\nsuggesting that this behavior is likely general and correlated to\nstructural characteristics of the perovskites.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=8.5cm]{fig115}\n\\mycaption{A comparison between total thermal conductivities from References~\\cite{Suemune_JPSJ_1964,Martin_Phonons_1976,Muta_JAC_2005,Popuri_RSCA_2014,Maekawa_BaHfO3_SrHfO3_JAC_2006,Yamanaka_JAC_2003},\nhigh-throughput calculations of the lattice thermal conductivity at\n300~K and 1000~K, and model behaviors in $\\kappa\\propto T^{-1}$\nand $\\kappa\\propto T^{-0.7}$.}\n\\label{fig:art120:Kappa}\n\\end{figure}\n\n\\subsection{Accelerating the discovery of stable compounds at high temperature}\n\\label{subsec:art120:PCA-regression}\n\nThrough brute-force calculations of the initial list of 391 compounds,\nwe extracted 92 that are mechanically stable at 1000~K. However,\nthis type of calculation is computationally expensive. Thus, it is\ndesirable for future high-throughput searches of other material classes\nto define a strategy for exploring specific parts of the full combinatorial\nspace. In this section, we propose and test such a strategy based\non an iterative machine-learning scheme using principal component\nanalysis and regression.\n\nWe begin by calculating the second order force constants $\\Phi_{0\\text{~K}}$\nof all compounds using the finite displacement method, which is more\nthan an order of magnitude faster than finite-temperature calculations.\nThis gives us a list of 29 perovskites that are mechanically stable\nin the cubic phase at 0~K. Since this is the highest symmetry phase,\nthey are likely also mechanically stable at high-temperatures\\footnote{However, we note that transitions to other structures can take place,\nin particular with one of hexagonal symmetry, such as in BaTiO$_{3}$\n\\cite{Glaister_PPS_1960}, RbZnF$_{3}$\\cite{Daniel_PRB_1995}\nor RbMgF$_{3}$~\\cite{Shafer_JoPACoS_1969}.\nThis phase transition is of\nfirst order, in contrast to displacive transitions that are of second\norder.}.\nWe calculate their self-consistent finite-temperature force constants\n$\\Phi{}_{1000\\text{~K}}^{\\text{SCFCS}}$ as described in Section~\\ref{subsec:art120:Finite-T_calculations}.\nThis initial set allows us to perform principal component analysis\nof the 0~K force constants so that we obtain a transformation that\nretains the 10 most important components. In a second step, we use\nregression analysis to find a relation between the principal components\nat 0~K and at 1000~K. This finally gives us a model that extracts\nthe principal components of the force constants at 0~K, interpolate\ntheir values at 1000~K, and reconstruct the full force constants\nmatrix at 1000~K: $\\Phi_{1000\\text{~K}}^{\\text{model}}$. We say that this\nmodel has been ``trained'' on the particular set of compounds described\nabove. Applying it to the previously calculated $\\Phi_{0\\text{~K}}$\nfor all compounds, we can efficiently span the full combinatorial\nspace to search for new perovskites with a phonon spectrum that is\nunstable at 0~K but stable at 1000~K. For materials determined\nmechanically stable with $\\Phi_{1000\\text{~K}}^{\\text{model}}$, we calculate\n$\\Phi_{1000\\text{~K}}^{\\text{SCFCS}}$. If the mechanical stability is confirmed,\nwe add the new compound to the initial set and subsequently train\nthe model again with the enlarged set. When no new compounds with\nconfirmed mechanical stability at high temperatures are found, we\nstop the search. This process is summarized in Figure~\\ref{fig:art120:PCA-regression}.\nFollowing this strategy, we find 79 perovskites that are stable according\nto the model, 68 of which are confirmed to be stable by the full calculation.\nThis means that we have reduced the total number of finite-temperature\ncalculations by a factor of 5, and that we have retrieved mechanically\nstable compounds with a precision of 86\\% and a recall of 74\\% \\footnote{Precision is defined as the fraction of true positives in all positives\nreported by the model and recall as the fraction of true positives\nfound using the model with respect to all true positives.}. It allows us to obtain approximate phonon spectra for unstable compounds,\nwhich is not possible with our finite-temperature calculations scheme\n(see Supplementary Material of Reference~\\cite{curtarolo:art120}). It also allows us to find compounds\nthat had not been identified as mechanically stable by the first exhaustive\nsearch due to failures in the workflow. Considering the generality\nof the approach, we expect this method to be applicable to other families\nof compounds as well. Most importantly, it reduces the computational\nrequirements, particularly if the total combinatorial space is much\nlarger than the space of interest.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.6\\linewidth]{fig116}\n\\mycaption{Depiction of strategy for exploring the relevant combinatorial space\nof compounds that are mechanically stable at high temperature.}\n\\label{fig:art120:PCA-regression}\n\\end{figure}\n\n\\subsection{Simple descriptors of the thermal conductivity}\n\\label{subsec:art120:Descriptors}\n\nWe now focus on the analysis of the thermal conductivity data provided\nin Table~\\ref{tab:art120:List-of-perovskites}. We note that this set contains\nabout two times more fluorides than oxides. This was already the case\nafter the first screening in which we kept only the semiconductors,\nand it can be explained by the strong electronegativity of fluorine,\nwhich generally forms ionic solids with the alkali and alkaline earth\nmetals easily, as well as with elements from groups 12, 13 and 14.\nThis is shown on Figure~\\ref{fig:art120:Columns}, in which we display histograms\nof the columns of elements at sites \\textit{A} and \\textit{B} of the\nperovskite in our initial list of paramagnetic semiconductors and\nafter screening for mechanical stability.\n\nWe can also see that the oxides tend to display a higher thermal conductivity\nthan the fluorides, as shown on the density plot of Figure~\\ref{fig:art120:Fluorides_vs_oxides}.\nThis is once again due to the charge of the fluorine ion, which is\nhalf that of the oxygen ion. In a model of a purely ionic solid, this\nwould cause the interatomic forces created by electrostatic interactions\nto be divided by two in fluorides as compared to oxides. This is roughly\nwhat we observe in our calculations of the second order force constants.\nIt translates into smaller phonon frequencies and mean group velocities\nin fluorides as compared to oxides. Fluorides also have smaller heat\ncapacities, due to their larger lattice parameters (see Supplementary material of Reference~\\cite{curtarolo:art120}).\nThose two factors mainly drive the important discrepancy\nof the thermal conductivity between fluorides and oxides. Following\nthe same reasoning, it means that halide perovskites in general should\nhave a very low thermal conductivity.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=8.5cm]{fig117}\n\\mycaption[Column number of the element at site (\\textbf{a}) \\textit{A} and (\\textbf{b}) \\textit{B}\nof the perovskite \\textit{ABX}$_{3}$.]\n{Counts in the initial list of fluorides (red) and oxides (blue) paramagnetic semiconductors and\nafter screening for mechanical stability are shown in violet and cyan, respectively.}\n\\label{fig:art120:Columns}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=8.5cm]{fig118}\n\\mycaption[Distribution of compounds as a function of the lattice thermal conductivity\nat 1000~K.]\n{The red curve corresponds to the distribution for all\nmechanically stable compounds. The blue curve corresponds to the distribution\nfor fluorides only. The green curve corresponds to the distribution\nfor oxides only.}\n\\label{fig:art120:Fluorides_vs_oxides}\n\\end{figure}\n\nFinally, we analyze the correlations between the thermal conductivity\nand different simple structural descriptors. Figure~\\ref{fig:art120:Correlograms}\ndisplays the correlograms for fluorides and oxides between the following\nvariables: the thermal conductivity $\\kappa$, the thermal conductivity\nin the small grain limit $\\kappa_{\\text{sg}}$~\\cite{curtarolo:art85,aflowKAPPA},\nthe mean phonon group velocity v$_{\\text{g}}$, the heat capacity\nc$_{\\text{V}}$, the root mean square Gr\\\"{u}neisen parameter $\\gamma_{\\text{rms}}$\n\\cite{Madsen_PRB_2014,Madsen_PSSA_2016}, the masses of atoms\nat sites \\textit{A} and \\textit{B} of the perovskite \\textit{ABX}$_{3}$,\ntheir electronegativity, their Pettifor number~\\cite{pettifor:1984},\ntheir ionic radius, the lattice parameter of the compound and its\nelectronic gap. Remarkably, sites \\textit{A} and \\textit{B} play very\ndifferent roles in fluorides and oxides. In particular, the thermal\nconductivity of fluorides is mostly influenced by substitutions of\nthe atom inside the fluorine octahedron (site \\textit{B}), while the\ninterstitial atom at site \\textit{A} has a negligible impact. The\nopposite is true for the oxides. This means that when searching for\nnew compounds with a low lattice thermal conductivity, substitutions\nat the \\textit{A} site of fluorides can be performed to optimize cost\nor other considerations without impacting thermal transport. It is\nalso interesting to note that the gap is largely correlated with the\nelectronegativity of atom \\textit{B}, suggesting the first electronic\nexcitations likely involve electron transfer from the anion to the\n\\textit{B} atom.\n\nCommon to both fluorides and oxides, the lattice parameter is mostly\ncorrelated with the ionic radius of atom \\textit{B} rather than atom\n\\textit{A}. Interestingly, the lattice parameter is larger for fluorides,\nalthough the ionic radius of fluorine is smaller than for oxygen.\nThis is presumably due to partially covalent bonding in oxides (see\n\\nobreak\\mbox{\\it e.g.}, Reference~\\onlinecite{Kolezynski_Ferroelectrics_2005}). In contrast, fluorides\nare more ionic: the mean degree of ionicity of the \\textit{X-B} bond\ncalculated from Pauling's electronegativities~\\cite{Pauling_JACS_1932}\n$e_{X}$ and $e_{B}$ as $I{}_{XB}=100\\left(1-e^{\\left(e_{X}-e_{B}\\right)\/4}\\right)$\nyields a value of 56\\% for oxides \\nobreak\\mbox{\\it vs.}\\ 74\\% for fluorides. Ionicity\nis also reflected by the band structure, as can be seen from the weak\ndispersion and hybridization of the F-2$p$ bands \\footnote{See for instance the band structure of SrTiO$_{3}$~\\cite{vanBenthem_JAP_2001}\ncompared to the one of KCaF$_{3}$~\\cite{Ghebouli_SSS_2015}. In those\ntwo compounds, the degree of ionicity of the \\textit{X}-\\textit{B}\nbond calculated from Pauling's electronegativity is 59\\% and 89\\%,\nrespectively.}. This may explain why the role of atoms at site \\textit{A} and \\textit{B}\nis so different between the two types of perovskites. We think that\nthe more ionic character combined to the small nominal charge in fluorides\nmakes the octahedron cage enclosing the atom \\textit{B} less rigid,\nsuch that the influence of the atom \\textit{B} on the thermal conductivity\nbecomes more significant.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig119}\n\\mycaption[Correlograms among properties of\nmechanically stable (\\textbf{a}) fluorides and\n(\\textbf{b}) oxides at 1000~K.]\n{Properties compared include the thermal conductivity $\\kappa$, the thermal\nconductivity in the small grain limit $\\kappa_{\\text{sg}}$, the mean\nphonon group velocity v$_{\\text{g}}$, the heat capacity c$_{\\text{V}}$,\nthe root mean square Gr\\\"{u}neisen parameter $\\gamma_{\\text{rms}}$,\nthe masses m$_{A}$ and m$_{B}$\nof atoms at sites \\textit{A} and \\textit{B} of the perovskite \\textit{ABX}$_{3}$,\ntheir electronegativity e$_{A}$, e$_{B}$,\ntheir Pettifor scale $\\chi_{A}$, $\\chi_{B}$,\ntheir ionic radius\nr$_{A}$, r$_{B}$,\nthe lattice parameter of the compound a$_{\\text{latt}}$\nand its electronic gap.}\n\\label{fig:art120:Correlograms}\n\\end{figure}\n\n\\subsection{Conclusion}\n\nEmploying finite-temperature \\nobreak\\mbox{\\it ab-initio}\\ calculations of force\nconstants in combination with machine learning techniques, we have\nassessed the mechanical stability and thermal conductivity of hundreds\nof oxides and fluorides with cubic perovskite structures at high temperatures.\nWe have shown that the thermal conductivities of fluorides are generally\nmuch smaller than those of oxides, and we found new potentially stable\nperovskite compounds. We have also shown that the thermal conductivity\nof cubic perovskites generally decreases more slowly than the inverse\nof temperature. Finally, we provide simple ways of tuning the thermal\nproperties of oxides and fluorides by contrasting the effects of substitutions\nat the \\textit{A} and \\textit{B} sites. We hope that this work will\ntrigger further interest in halide perovskites for applications that\nrequire a low thermal conductivity.\n\\clearpage\n\\section{Accelerated Discovery of New Magnets in the Heusler Alloy Family}\n\\label{sec:art109}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art109}.\nAuthor contributions are as follows:\nThe initial idea for the project was developed by Stefano Sanvito and Stefano Curtarolo.\nJunkai Xue and Thomas Archer constructed the Heusler database.\nCorey Oses and Mario \\v{Z}ic performed additional {\\small DFT}\\ calculations for tetragonally distorted Heusler alloys.\nAnurag Tiwari performed the regression analysis for the $T_\\mathrm{C}$.\nCorey Oses also performed the convex hull calculations.\nCrystal growth and experimental characterization has been performed by Pelin Tozman under the supervision of\nMunuswamy Venkatesan and J. Michael D. Coey.\nThe project was supervised by Stefano Sanvito, Stefano Curtarolo and J. Michael D. Coey, who also produced the manuscript.\n\n\\subsection{Introduction}\nVery few types of macroscopic order in condensed matter are as sensitive to details as magnetism. The magnetic\ninteraction is usually based on the $m$-$J$ paradigm, where localized magnetic moments, $m$, are magnetically\ncoupled through the exchange interaction, $J$. Only a few elements in the periodic table can provide localized moments\nin the solid state, namely 3$d$ transition metals, 4$f$ rare earths and some 4$d$ ions. Lighter 2$p$\nelements are prone to form close shells, while in heavier ones the Hund's coupling is not strong enough to\nsustain a high-spin configuration~\\cite{Janak_PRB_1977}. The magnetic coupling then depends on how the wave-functions\nof the magnetic ions overlap with each other, either directly, through other ions or via delocalized electrons.\nThis generates a multitude of mechanisms for magnetic coupling, operating at both sides of the\nmetal\/insulator transition boundary, and specific to the details of the chemical environment. In general $J$ is sensitive\nto the bond length, the bond angle, the magnetic ion valence. It is then not surprising\nthat among the $\\sim$100,000 unique inorganic compounds known to mankind~\\cite{ICSD4}, only about 2,000\nshow magnetic order of any kind~\\cite{CoeyBook}.\n\nWhen one focuses on the magnets that are useful for consumer applications, then the choice becomes even more restricted\nwith no more than two dozen compounds taking practically the entire global market. A useful magnet, regardless of the particular\ntechnology, should operate in the -50$^\\circ$C to +120$^\\circ$C range, imposing the ordering temperature, $T_\\mathrm{C}$,\nto be at least 300$^\\circ$C. Specific technologies then impose additional constraints. Permanent magnets should display a\nlarge magnetization and hysteresis~\\cite{CoeyBook}. Magnetic electrodes in high-performance magnetic tunnel junctions should\ngrow epitaxially on a convenient insulator and have a band-structure suitable for spin-filtering~\\cite{Handbook_Spin_Mag_2011}. If the same tunnel\njunction is used as spin-transfer torque magnetic random access memory element, the magnet should also have a low Gilbert\ndamping coefficient and a high Fermi-level spin polarization~\\cite{Handbook_Spin_Mag_2011}. Indeed, there are not many magnets matching all the\ncriteria, hence the design of a new one suitable for a target application is a complex and multifaceted task.\n\nThe search for a new magnet usually proceeds by trial and error, but the path may hide surprises. For instance,\nchemical intuition suggests that SrTcO$_3$ should be a poor magnet, since all Sr$X$O$_3$ perovskites with $X$ in\nthe chemical neighborhood of Tc are either low-temperature magnetic ($X$ = Ru, Cr, Mn, Fe) or do not present any\nmagnetic order ($X$ = Mo). Yet, SrTcO$_3$ is a G-type antiferromagnet~\\cite{Rodriguez_STO2} with a remarkably\nhigh N\\'{e}el temperature, 750$^\\circ$C, originating from a subtle interplay between $p$-$d$ hybridization and Jahn-Teller\ndistortion~\\cite{Franchini_STOus}. This illustrates that often a high-performance magnet may represent a singularity\nin physical\/chemical trends and that its search can defy intuition. For this reason we take a completely different\napproach to the discovery process and demonstrate that a combination of advanced electronic structure theory and\nmassive database creation and search, the high-throughput computational materials design approach~\\cite{nmatHT},\ncan provide a formidable tool for finding new magnetic materials.\n\nOur computational strategy consists of three main steps. Firstly, we construct an extensive database containing the\ncomputed electronic structures of potential novel magnetic materials. Here we consider Heusler alloys (HAs), a prototypical\nfamily of ternary compounds populated with several high-performance magnets~\\cite{Graf_PSSC_2011}. A rough stability analysis,\nbased on evaluating the enthalpy of formation against reference single-phase compounds provides a first\nscreening of the database. This, however, is not a precise measure of the thermodynamic stability of a material, since\nit does not consider decomposition into competing phases (single-element, binary, and ternary compounds). Such analysis\nrequires the computation of the electronic structure of all possible decomposition members associated with the given Heusler compounds.\nThis is our second step and it is carried out here only for intermetallic HAs, for which an extensive binary database is\navailable~\\cite{aflowlibPAPER}. Finally, we analyze the magnetic order of the predicted stable magnetic intermetallic HAs and, via\na regression trained on available magnetic data, estimate their $T_\\mathrm{C}$. The theoretical screening is then validated by\nexperimental synthesis of a few of the predicted compounds.\n\n\\subsection{Construction of the database}\n\nThe prototypical HA, $X_2YZ$ (Cu$_2$MnAl-type), crystallizes in the {\\it Fm$\\overline{3}$m} cubic\nspace group, with the $X$ atoms occupying the 8$c$ Wyckoff position (1\/4, 1\/4, 1\/4) and the $Y$ and $Z$ atoms\nbeing respectively at 4$a$ (0, 0, 0) and 4$b$ (1\/2, 1\/2, 1\/2). The crystal can be described as four interpenetrating\n{\\it fcc} lattices with $Y$ and $Z$ forming an octahedral-coordinated rock-salt structure, while the $X$ atoms occupy the\ntetrahedral voids [see Figure~\\ref{fig:art109:Fig1}(a)]. Two alternative structures also exist. In the inverse Heusler $(XY)XZ$\n(Hg$_2$CuTi-type), now $X$ and $Z$ form the rock-salt lattice, while the remaining $X$ and the $Y$ atoms fill the\ntetrahedral sites [Figure~\\ref{fig:art109:Fig1}(b)], so that one $X$ atom presents sixfold octahedral coordination, while the other\nfourfold tetrahedral coordination. The second structure, the half-Heusler $XYZ$ (MgCuSb-type), is obtained by removing one of\nthe $X$ atoms, thus leaving a vacancy at one of the tetrahedral site [Figure~\\ref{fig:art109:Fig1}(c)].\nThe minimal unit cell describing all three types can be constructed as a tetrahedral {\\it F$\\overline{4}$3m} cell, containing\n4 (3 for the case of the half Heusler) atoms [Figure~\\ref{fig:art109:Fig1}(d)]. Such a cell allows for a ferromagnetic spin configuration\nand for a limited number of antiferromagnetic ones.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig120}\n\\mycaption[Heusler structures.]\n{(\\textbf{a}) regular Heusler,\n(\\textbf{b}) inverse Heusler, and\n(\\textbf{c}) half Heusler.\n(\\textbf{d}) the tetrahedral {\\it F$\\overline{4}$3m} cell used to construct the electronic structure database.\n(\\textbf{e}) Ternary convex hull diagram for Al-Mn-Ni.\nNote the presence of the stable HA, Ni$_2$MnAl.}\n\\label{fig:art109:Fig1}\n\\end{figure}\n\n\\subsection{Results}\n\nWe construct the HAs database by considering all possible three-element combinations made of atoms from\nthe 3$d$, 4$d$ and 5$d$ periods and some elements from group III, IV, V and VI. In particular we use Ag, Al, As, Au,\nB, Ba, Be, Bi, Br, Ca, Cd, Cl, Co, Cr, Cu, Fe, Ga, Ge, Hf, Hg, In, Ir, K, La, Li, Mg, Mn, Mo, Na, Nb, Ni, Os, P, Pb, Pd,\nPt, Re, Rh, Ru, Sb, Sc, Se, Si, Sn, Sr, Ta, Tc, Te, Ti, Tl, V, W, Y, Zn and Zr. Note that we have deliberately excluded\nrare earths, responding to the global need to design new magnets with a reduced rare earth content.\nFurthermore, we have not imposed constraints on the total number of valence electrons~\\cite{zhang_sorting_2012,Yan_NComm_2015},\nsince magnetism is found for a broad range of electron counts.\nFor each combination of three elements ($X$, $Y$, $Z$) all the possible regular, inverse and half HAs are constructed.\nThese total to 236,115 decorations. The electronic structure of all the structures is computed by density functional theory\n({\\small DFT}) in the generalized gradient approximation ({\\small GGA}) of the exchange correlation functional as parameterized by\nPerdew-Burke-Ernzerhof~\\cite{PBE}. Our {\\small DFT}\\ platform is the {\\small VASP}\\ code~\\cite{vasp_cms1996} and each structure is fully relaxed.\nThe typical convergence tolerance is 1~meV\/atom and this is usually achieved by sampling the Brillouin zone\nover a dense grid of 3000-4000 $k$-points per reciprocal atom. A much denser grid of 10,000 $k$-points is employed\nfor the static run to obtain accurate charge densities and density of states. The large volume of data is managed by the\nAFLOW code~\\cite{aflowPAPER}, which creates the appropriate entries for the AFLOW database~\\cite{aflowlibPAPER}.\nMore details about the computational method are in Reference~\\cite{curtarolo:art104}.\n\nLet us begin our analysis by providing a broad overview of the database. Among the 236,115 decorations only 104,940\nare unique, meaning that only a single structure is likely to form for a given stoichiometry. Strictly speaking, this is not true\nsince there are many examples of HAs presenting various degrees of site occupation disorder, and the\nestimate gives an initial idea on how many compounds one may expect. Then a minimal criterion of stability is that the\nenthalpy of formation of the $X_2YZ$ structure, $H_{X_2YZ}$, is lower than the sum of the enthalpies of formation\nof its elementary constituents, namely $H_{\\mathrm{f}}=H_{X_2YZ}-(2H_{X}+H_{Y}+H_{Z})<0$.\nSuch criterion returns us 35,602 compounds, with 6,778 presenting a magnetic moment. Note that this number can be\nslightly underestimated as our unit cell can describe only a handful of possible anti-ferromagnetic configurations, meaning that\ncompounds where the magnetic cell is larger than the unit cell may then converge to a diamagnetic solution.\nIn any case, such a number is certainly significantly larger than the actual number of stable magnetic HAs. This can\nonly be established by computing the entire phase diagram of each ternary compound, \\nobreak\\mbox{\\it i.e.}, by assessing the stability of\nany given $X_2YZ$ structure against decomposition over all the possible alternative binary and ternary prototypes (for example\n$X_2YZ$ can decompose into $XY$+$XZ$, $X_2Y$+$Z$, $XYZ$+$X$, \\nobreak\\mbox{\\it etc.}). Such a calculation is extremely intensive. An\ninformative phase diagram for a binary alloy needs to be constructed over approximately 10,000 prototypes~\\cite{monsterPGM}, which\nmeans that at least 30,000 calculations are needed for every ternary. As a consequence mapping the stability of every calculated HA\nwill require the calculation of approximately 15,000,000 prototypes, quite a challenging task.\n\nWhen the electronic structure and the enthalpy of formation of the relevant binaries are available, then one can\nconstruct the convex hull diagram for the associated ternary compounds~\\cite{Lukas_CALPHAD_2007}. An example of such convex\nhull diagram for Al-Mn-Ni is presented in Figure~\\ref{fig:art109:Fig1}(e). The figure shows that there is a stable phase, Ni$_2$MnAl,\nwith a formation energy of -404~meV\/atom. In this case, there are also three other unstable ternary structures with\n$H_{\\mathrm{f}}<0$, namely Mn$_2$NiAl, NiMnAl and Al$_2$MnNi. The enthalpy of formation of Mn$_2$NiAl is\n$H_{\\mathrm{f}}=-209$~meV\/atom and it is 121~meV\/atom higher than the tie-plane, that of NiMnAl is\n-39~meV\/atom (400~meV\/atom above the tie plane), and that of Al$_2$MnNi is\n-379~meV\/atom (100~meV above the tie plane). This illustrates that $H_{\\mathrm{f}}<0$\nalone is not a stringent criterion for stability and that a full analysis needs to be performed before making the call on\na given ternary. Notably, Ni$_2$MnAl has been synthesized in a mixture of B2 and L2$_1$\nphases~\\cite{Ziebeck_JPFMP_1975} and it is a well-established magnetic shape memory alloy.\n\nGiven the enormous computational effort of mapping the stability of the entire database we have limited\nfurther analysis to intermetallic HAs made only with elements of the 3$d$, 4$d$ and 5$d$\nperiods. These define 36,540 structures, for which the corresponding binaries are available in the {\\sf \\AFLOW.org}\\\ndatabase~\\cite{aflowlibPAPER}. Our convex hull analysis then returns 248 thermodynamically stable compounds (full\nlist provided in Tables~\\ref{fig:art109:magnetic_heusler_1}-\\ref{fig:art109:magnetic_heusler_8}), of which only 22 possess a magnetic ground state in the tetrahedral\n{\\it F$\\overline{4}$3m} unit cell. The details of their electronic structure are presented in Table~\\ref{tab:art109:BLtab}.\nNote that in the last column of the table we include an estimate of the robustness of a particular compound against\ndecomposition, $\\delta_{\\mathrm{sc}}^{30}$. A material is deemed as decomposable (`Y' in the table) if its enthalpy of formation\nis negative but less than 30~meV\/atom lower than the most stable balanced decomposition. In contrast a material is\ndeemed robust (`N' in the table) when $H_{\\mathrm{f}}$ is more than 30~meV\/atom away from that of the closest balanced\ndecomposition. When such a criterion is applied we find that 14 of the predicted HAs can potentially decompose,\nwhile the other 8 are robust.\n\nWe have further checked whether such magnetic ground states are stable against tetragonal distortion, which may\noccur in HAs in particular with the Mn$_2YZ$ composition. Indeed we find that the ground state of five structures, namely\nCo$_2$NbZn, Co$_2$TaZn, Pd$_2$MnAu, Pd$_2$MnZn and Pt$_2$MnZn, is tetragonally distorted. Furthermore for\ntwo of them, Co$_2$NbZn and Co$_2$TaZn, the tetragonal distortion suppresses the magnetic order indicating that\nthe competition between the Stoner and band Jahn-Teller instability~\\cite{Labbe_JPFrance_1966} favors a distorted non-magnetic ground\nstate. The analysis so far tells us that the incidence of stable magnetic HAs among the possible intermetallics is\nabout 0.057\\%. When this is extrapolated to the entire database we can forecast a total of about 140 stable magnetic\nalloys, of which about 60 are already known. In the same way we can estimate approximately 1,450 stable non-magnetic\nHAs, although this is just a crude forecast, since regions of strong chemical stability may be present in the complete\ndatabase and absent in the intermetallic subset.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Electronic structure parameters of the 22 magnetic HAs found among all possible\nintermetallics.]\n{The table lists the unit cell volume of the {\\it F$\\overline{4}$3m} cell, the $c\/a$ ratio for tetragonal\ncells, $a$, the Mn-Mn distance for Mn-containing alloys, $d_\\mathrm{Mn-Mn}$, the magnetic moment per formula\nunit, $m$, the spin polarization at the Fermi level, $P_\\mathrm{F}$, the enthalpy of formation $H_{\\mathrm{f}}$, the entropic\ntemperature, $T_\\mathrm{S}$, and the magnetic ordering temperature, $T_\\mathrm{C}$. Compounds labeled with\n$*$ are not stable against tetragonal distortion (Co$_2$NbZn and Co$_2$TaZn become diamagnetic after distortion).\nNote that $T_\\mathrm{C}$ is evaluated only for Co$_2YZ$ and $X_2$Mn$Z$ compounds for which a sufficiently large\nnumber of experimental data are available for other chemical compositions. In the case of Mn$_2YZ$ compounds we\nreport the magnetic moment of the ground state and in brackets that of the ferromagnetic solution. The last column\nprovides a more stringent criterion of stability. $\\delta_{\\mathrm{sc}}^{30}=$~Y if the given compound has an enthalpy within 30\nmeV\/atom from that of its most favorable balanced decomposition (potentially decomposable), and $\\delta_{\\mathrm{sc}}^{30}=$~N if\nsuch enthalpy is more than 30~meV\/atom lower (robust).}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\nalloy & volume (\\AA$^3$) & $c\/a$ & $a$ (\\AA) & $d_\\mathrm{Mn-Mn}$ (\\AA) & $m$ ($\\mu_\\mathrm{B}$\/f.u.) & $P_\\mathrm{F}$\n& $H_{\\mathrm{f}}$ (eV\/atom) & $T_\\mathrm{S}$~{\\small (K)}\\ & $T_\\mathrm{C}$~{\\small (K)}\\ & $\\delta_{\\mathrm{sc}}^{30}$\\\\\n\\hline\nMn$_2$PtRh & 58.56 & & 6.16 & 3.08 & 0.00 (9.05) & 0.00 (0.86) & -0.29 & 3247 & -- & N \\\\\nMn$_2$PtCo & 54.28 && 6.00 & 3.00 & 1.13 (9.04) & 0.00 (0.86) & -0.17 & 1918 & -- & Y \\\\\nMn$_2$PtPd & 60.75 && 6.24 & 3.12 & 0.00 (8.86) & 0.00 (0.38) & -0.29 & 3218 & -- & N \\\\\nMn$_2$PtV & 55.73 && 6.06 & 3.03 & 4.87 (4.87) & 0.67 & -0.30 & 3353 & -- & Y \\\\\nMn$_2$CoCr & 47.19 && 5.73 & 2.87 & 4.84 (4.84) & 0.016 & -0.05 & 529 & -- & N \\\\\nCo$_2$MnTi & 49.68 && 5.84 & & 4.92 & 0.58 & -0.28 & 3122 & 940 & N \\\\\nCo$_2$VZn & 46.87 && 5.73 & & 1.01 & 0.93 & -0.15 & 1653 & 228 & Y \\\\\nCo$_2$NbZn$^*$ & 51.87 &1.0& 5.9 &&1.00 & 0.95 & -0.18 & 2034 & 212 & Y \\\\\nCo$_2$NbZn & 51.52 & 1.15 & 5.63 && 0.0 & 0.0 & -0.20& 2034 & 0 & Y \\\\\nCo$_2$TaZn$^*$ & 51.80&1.0& 5.92 && 0.98& 0.63 & -0.22 & 2502 & 125 & N \\\\\nCo$_2$TaZn & 51.55 & 1.12 & 5.70 && 0.0 & 0.0 & -0.23& 2502 & 0 & N \\\\\nRh$_2$MnTi & 58.08 && 6.15 & 4.35 & 4.80 & 0.51 & -0.58 & 6500 & 417 & Y\\\\\nRh$_2$MnZr & 64.50 && 6.37 &4.50&4.75 & 0.34 & -0.58 & 6518 & 338 & Y \\\\\nRh$_2$MnHf & 63.22 && 6.32 & 4.47&4.74 & 0.34 & -0.67 & 7474 & 364 & Y \\\\\nRh$_2$MnSc & 61.62 && 6.27& 4.43&4.31 & 0.77 & -0.63 & 7031 & 429 & N \\\\\nRh$_2$MnZn & 54.95 && 6.03&4.27&3.37 & 0.63 & -0.31 & 3444 & 372 & Y \\\\\nPd$_2$MnAu$^*$ & 64.21 &1.0& 6.36& 4.49&4.60 & 0.06 & -0.20 & 2203 & 853 & Y \\\\\nPd$_2$MnAu & 63.50 & 1.35 & 5.75&4.07 & 4.28 & 0.28 & -0.33 & 2203 & 331 & Y \\\\\nPd$_2$MnCu & 57.63 && 6.13&4.34&4.53 & 0.06 & -0.22 & 2492 & 415 & Y \\\\\nPd$_2$MnZn$^*$ & 58.88 & 1.0 &6.17&4.37& 4.33 & 0.38 & -0.39& 4399 & 894 & Y \\\\\nPd$_2$MnZn & 58.74 & 1.18 &5.84&4.13& 4.22 & 0.16 & -0.47& 4399 & 402 & Y \\\\\nPt$_2$MnZn$^*$ & 59.23 &1.0&6.19&4.37&4.34 & 0.34 & -0.45 & 5035 & 694 & Y \\\\\nPt$_2$MnZn & 58.95 & 1.22 &5.79&4.10& 4.13 & 0.017 & -0.65& 5035 & 381 & Y \\\\\nRu$_2$MnNb & 59.64 &&6.20&4.39& 4.07 & 0.85 & -0.19 & 2068 & 276 & Y \\\\\nRu$_2$MnTa & 59.72 &&6.20&4.39& 4.06 & 0.86 & -0.26 & 2912 & 305 & N \\\\\nRu$_2$MnV & 54.38 &&6.01&4.25& 4.00 & 0.707 & -0.16 & 1832 & 342 & Y \\\\\nRh$_2$FeZn & 54.60 &&6.02&& 4.24 & 0.49 & -0.28 & 3150 & -- & N\\\\\n\\end{tabular}}\n\\label{tab:art109:BLtab}\n\\end{table}\n\nIn Table~\\ref{tab:art109:BLtab}, together with structural details, the magnetic moment per formula unit, $m$, and the enthalpy\nof formation we report a few additional quantities that help us in understanding the potential of a given alloy as\nhigh-performance magnet. The spin polarization of the density of states at the Fermi level, $n_\\mathrm{F}^\\sigma$\n($\\sigma=\\uparrow, \\downarrow$) is calculated as~\\cite{Mazin_PRL1999}\n\\begin{equation}\nP_\\mathrm{F}=\\frac{n_\\mathrm{F}^\\uparrow-n_\\mathrm{F}^\\downarrow}{n_\\mathrm{F}^\\uparrow+n_\\mathrm{F}^\\downarrow}\\:,\n\\end{equation}\nand expresses the ability of a metal to sustain spin-polarized currents~\\cite{Coey_JPDAP_2004}. We find a broad distribution of\n$P_\\mathrm{F}$s with values ranging from 0.93 (Co$_2$VZn) to 0.06 (Pd$_2$MnCu). None of the HAs display\nhalf-metallicity, and in general their spin-polarization is similar to those of the elementary 3$d$ magnets\n(Fe, Co and Ni).\n\nWe then calculate the entropic temperature~\\cite{nmatHT,monsterPGM,curtarolo:art98}, $T_\\mathrm{S}$. For simplicity we give\nthe definition for a $XY$ binary alloy, although all our calculations are performed for its ternary equivalent,\n\\begin{equation}\nT_\\mathrm{S}=\\max_i\\left[\\frac{H_{\\mathrm{f}}(X_{x_i}Y_{1-x_i})}{k_\\mathrm{B}[x_i\\log x_i+(1-x_i)\\log(1-x_i)]}\n\\right]\\:,\n\\end{equation}\nwhere $k_\\mathrm{B}$ is the Boltzmann constant and $i$ counts all the stable compounds in the $XY$ binary system.\nEffectively $T_\\mathrm{S}$ is a concentration-maximized formation enthalpy weighted by the inverse of its ideal entropic\ncontribution (random alloy). It measures the ability of an ordered phase to resist deterioration into a temperature-driven,\nentropically-promoted, disordered mixture. The sign of $T_\\mathrm{S}$ is chosen such that a positive temperature\nis needed for competing against the compound stability (note that $T_\\mathrm{S}<0$ if $H_{\\mathrm{f}}>0$), and one expects\n$T_\\mathrm{S}\\rightarrow0$ for a compound spontaneously decomposing into a disordered mixture.\nIf we analyze the $T_\\mathrm{S}$ distribution for all the intermetallic HAs with $H_{\\mathrm{f}}<0$\n(8776 compounds) we find the behavior to closely follow that of a two-parameter Weibull distribution with a shape of\n1.13 and a scale of 2585.63 (see histogram in Figure~\\ref{fig:art109:histogram_full}). The same distribution for the 248 stable intermetallic HAs is\nrather uniform in the range 1,000-10,000~K and presents a maximum at around 3,500~K. A similar trend is observed\nfor the 20 stable magnetic HAs, suggesting that several of them may be highly disordered.\n\nFinally, Table~\\ref{tab:art109:BLtab} includes an estimate of the magnetic ordering temperatures, $T_\\mathrm{C}$. These\nhave been calculated based on available experimental data. Namely we have collected the experimental\n$T_\\mathrm{C}$'s of approximately 40 known magnetic Heusler compounds (see Section~\\ref{subsec:art109:tc_known_heuslers}) and performed a linear\nregression correlating the experimental $T_\\mathrm{C}$'s with a range of calculated electronic structure properties,\nnamely equilibrium volume, magnetic moment per formula unit, spin-decomposition and number of valence\nelectrons. The regression is possible only for those compounds for which the set of available experimental data\nis large enough, namely for Co$_2YZ$ and $X_2$Mn$Z$ HAs. We have trained the regression over the existing\ndata and found that for the two classes Co$_2YZ$ and $X_2$Mn$Z$ the typical error in the $T_\\mathrm{C}$ estimate\nis in the range of 50~K, which is taken as our uncertainty.\n\n\\subsection{Discussion}\n\nWe have found three different classes of stable magnetic HAs, namely Co$_2YZ$, $X_2$Mn$Z$ and Mn$_2YZ$. In\naddition we have predicted also Rh$_2$FeZn to be stable. This is rather unique since there are no other HAs with Fe\nin octahedral coordination and no magnetic ions at the tetrahedral positions.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.65\\linewidth]{fig121}\n\\mycaption[Slater-Pauling curve for magnetic HAs of the form Co$_2YZ$.]\n{The magnetic moment per formula unit, $m$,\nis plotted against the number of valence electron, $N_\\mathrm{V}$, in the left panel, while $T_\\mathrm{C}$ is displayed\non the right. Red symbols corresponds to predicted HAs, while the black ones to existing materials. For the sake of clarity\nseveral compounds have been named collectively on the picture. Co$_2AB$ 1: Co$_2$FeGa, Co$_2$FeAl, Co$_2$MnSi,\nCo$_2$MnGe, Co$_2$MnSn; Co$_2AB$ 2: Co$_2$TaAl, Co$_2$ZrAl, Co$_2$HfGa, Co$_2$HfAl, Co$_2$TaGa;\nCo$_2AB$ 3: Co$_2$ZrAl, Co$_2$HfAl, Co$_2$HfGa, Co$_2$TaGa.}\n\\label{fig:art109:Co2XY}\n\\end{figure}\n\nThe first class is Co$_2YZ$, a class which is already populated by about 25 known compounds all lying on the\nSlater-Pauling curve~\\cite{Graf_PSSC_2011}. Our analysis reveals four new stable alloys, three of them with the low valence\nelectron counts of 25 (Co$_2$VZn, Co$_2$NbZn, Co$_2$TaZn) and one, Co$_2$MnTi, presenting the large count\nof 29. The regression correctly places these four on the Slater-Pauling curve (see Figure~\\ref{fig:art109:Co2XY}) and predicts\nfor Co$_2$MnTi the remarkably high $T_\\mathrm{C}$ of 940~K. This is a rather interesting since only about\ntwo dozen magnets are known to have a $T_\\mathrm{C}$ in that range~\\cite{CoeyBook}. Therefore, the discovery\nof Co$_2$MnTi has to be considered as exceptional. The other three new compounds in this class are all predicted to\nhave a $T_\\mathrm{C}$ around 200~K, but two of them become non-magnetic upon tetragonal distortion leaving\nonly Co$_2$VZn magnetic ($T_\\mathrm{C}\\sim228$~K).\n\nThe second class is $X_2$Mn$Z$ in which we find 13 new stable magnets, most of them including a 4$d$ ion (Ru, Rh\nand Pd) in the tetrahedral $X$ position. In general, these compounds have a magnetic moment per formula unit ranging between\n4~$\\mu_\\mathrm{B}$ and 5~$\\mu_\\mathrm{B}$, consistent with the nominal 2+ valence of Mn in octahedral coordination.\nThe regression, run against 18 existing compounds of which 13 are with $X$ = Ru, Rh or Pd, establishes a correlation\nbetween the Mn-Mn nearest neighbors distance, $d_\\mathrm{Mn-Mn}$, and $T_\\mathrm{C}$ as shown in\nFigure~\\ref{fig:art109:X2MnY}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.65\\linewidth]{fig122}\n\\mycaption[Magnetic data for $X_2$Mn$Z$ magnets.]\n{$T_\\mathrm{C}$ (left) and magnetic moment per formula unit (right)\nas a function of the Mn-Mn distance, $d_\\mathrm{Mn-Mn}$.\nNote that the $T_\\mathrm{C}$ is limited to about 550~K and\npeaks at a volume of about 60~\\AA$^3$. In contrast the magnetic moment is approximately constant with values in between\n4~$\\mu_\\mathrm{B}$ and 5~$\\mu_\\mathrm{B}$. Close circles (with associated chemical compositions) correspond to the\npredicted compounds, while the other symbols correspond to experimental data. Different colors correspond to different\nnumber of valence electrons, $N_\\mathrm{V}$. Blue chemical formulas correspond to compound displaying tetragonal\ndistortion. The two red lines are Castelliz-Konamata curves, while the black one is to guide the eye.}\n\\label{fig:art109:X2MnY}\n\\end{figure}\n\nWe find that $T_\\mathrm{C}$ is a non-monotonic function of $d_\\mathrm{Mn-Mn}$ with a single maximum at\n$d_0$$\\sim$4.4~\\AA\\ corresponding to a temperature of 550~K (the maximum coincides approximately with\nCu$_2$MnSn). The only apparent exception to such trend is the prototypical Cu$_2$MnAl, which displays a large\n$T_\\mathrm{C}$ and relatively small $d_\\mathrm{Mn-Mn}$~\\cite{Oxley1963}. A strong sensitivity of the $T_\\mathrm{C}$\nof Mn-containing compounds to $d_\\mathrm{Mn-Mn}$ was observed long time ago and rationalized in an empirical\n$T_\\mathrm{C}$-$d_\\mathrm{Mn-Mn}$ curve by Castelliz~\\cite{Castelliz_ZM_1955}. This predicts that $T_\\mathrm{C}$ is not\nmonotonically dependent on $d_\\mathrm{Mn-Mn}$ and has a maximum at around $d_\\mathrm{Mn-Mn}=3.6$. The curve\nhas been validated for a number of HAs and it has been used to explain the positive pressure coefficient of\n$T_\\mathrm{C}$, $(1\/T_\\mathrm{C})(\\mathrm{d}T_\\mathrm{C}\/\\mathrm{d}P)$, found, for instance, in\nRh$_2$MnSn~\\cite{Adachi200437}. Refinements of the Castelliz curve predict that the rate of change of $T_\\mathrm{C}$\nwith $d_\\mathrm{Mn-Mn}$ in HAs is related to the valence count~\\cite{Kanomata_JMMM_1987}, although the position\nof the maximum is not. In general the results of Figure~\\ref{fig:art109:X2MnY}, including several experimental data, seems to contradict\nthe picture since a monotonically decreasing $T_\\mathrm{C}$ is expected for any $d_\\mathrm{Mn-Mn}>3.6$~\\AA, \\nobreak\\mbox{\\it i.e.},\npractically for any HAs of the form $X_2$Mn$Z$. There are a few possible reasons for such disagreement. Firstly, the\nCastelliz curve assumes that only Mn presents a magnetic moment, which is unlikely since many of the $X_2$Mn$Z$\ncompounds of Figure~\\ref{fig:art109:X2MnY} have Rh or Pd in the X position, two highly spin-polarizable ions. Secondly,\nmany HAs in Figure~\\ref{fig:art109:X2MnY} present various levels of disorder, meaning that Mn-Mn pairs separated\nby less than the nominal $d_\\mathrm{Mn-Mn}$ are likely to be present in actual samples. We then propose that the\ntrend of Figure~\\ref{fig:art109:X2MnY} (see dashed black lines) represents a new empirical curve, valid for $X_2$Mn$Z$ HAs,\nand taking into account such effects.\n\nThe last class of predicted magnetic HAs is populated by Mn$_2YZ$ compounds. These have recently\nreceived significant attention because of their high $T_\\mathrm{C}$ and the possibility of displaying tetragonal\ndistortion and hence large magneto-crystalline anisotropy~\\cite{Kreiner2014}. Experimentally when the 4$c$ position\nis occupied by an element from group III, IV or V one finds the regular Heusler structure if the atomic number of\nthe $Y$ ion is smaller than that of Mn, $Z$($Y$)$<$$Z$(Mn), and the inverse one for $Z$($Y$)$>$$Z$(Mn). To date only\nMn$_2$VAl and Mn$_2$VGa have been grown with a $Y$ element lighter than Mn, so that except those two all other\nMn$_2YZ$ HAs crystallize with the inverse structure (see Figure~\\ref{fig:art109:Mn2YZ}). In the case of the two regular\nHAs, Mn$_2$VAl and Mn$_2$VGa, the magnetic order is ferrimagnetic with the two Mn ions at the tetrahedral\nsites being anti-ferromagnetically coupled to V~\\cite{Nakamichi1983,Itoh1983,Kumar2008}. In contrast for\nthe inverse Mn$_2$-based HAs the antiferromagnetic alignment is between the two Mn ions and the magnetic\nground state then depends on whether there are other magnetic ions in the compound. In general, however, site disorder is\nnot uncommon (see Section~\\ref{subsec:art109:tet_disorder_Mn2PtPd}) and so is tetragonal distortion, so that the picture becomes more complicated. There are\nalso some complex cases, such as that of Mn$_3$Ga, presenting a ground state with a non-collinear arrangement of\nboth the spin and angular momentum~\\cite{Rode_PRB_2013}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.65\\linewidth]{fig123}\n\\mycaption[Enthalpy of formation difference between the regular and inverse Heusler structure, $\\Delta H_\\mathrm{RI}$,\nfor Mn$_2$-containing compounds as a function of the cell volume.]\n{The solid red squares (with chemical formulas) are\nthe predicted stable intermetallic materials, while the open red squares are existing compounds. For completeness we\nalso include data for Co$_2$-based HAs, again with open symbols for existing compounds and solid one for predicted.\nIn brackets beside the chemical formulas we report the value for the entropic temperature, $T_\\mathrm{S}$, in {\\small (K)}.}\n\\label{fig:art109:Mn2YZ}\n\\end{figure}\n\nIf we now turn our attention to the predicted compounds we find five stable compositions of which three match the\n$\\delta_{\\mathrm{sc}}^{30}$ robustness criterion. Most intriguingly the regular {\\it Fm$\\overline{3}$m} structure\nappears to be the ground state for all the compounds, regardless of their chemical composition. This sets Mn$_2$-based\nintermetallic compounds aside from those with elements from the main groups. In Figure~\\ref{fig:art109:Mn2YZ} we present the\nenthalpy of formation difference between the regular and the inverse structure, $\\Delta H_\\mathrm{RI}=H_\\mathrm{f,R}-H_\\mathrm{f,I}$,\nfor the computed and the experimentally known Mn$_2$-based HAs, together with their $T_\\mathrm{S}$ and reference\ndata for Co$_2$-based alloys. In general we find that $\\Delta H_\\mathrm{RI}$ for the Mn$_2YZ$ class is significantly smaller\nthan for the Co$_2YZ$ one. In fact there are cases, \\nobreak\\mbox{\\it e.g.}, Mn$_2$PtGa and Mn$_2$PtIn, in which the two phases are almost\ndegenerate and different magnetic configurations can favor one over the other. Overall, one then expects such compounds\nto be highly disordered. Finally, we take a look at the magnetic ground state. In all cases the compounds present some\ndegree of antiferromagnetic coupling, which results in either a zero-moment ground state when Mn is the only magnetic ion,\nand in a ferrimagnetic configuration when other magnetic ions are present.\n\nThe last step in our approach consists in validating the theoretical predictions by experiments. We have attempted the\nsynthesis of four HAs, namely Co$_2$MnTi, Mn$_2$PtPd, Mn$_2$PtCo and Mn$_2$PtV. Co$_2$MnTi\nis chosen because of its high Curie temperature, while among the Mn$_2$-based alloys we have selected two\npresenting ferrimagnetic ground state (Mn$_2$PtCo and Mn$_2$PtV) and one meeting the stringent\n$\\delta_{\\mathrm{sc}}^{30}$ robustness criterion (Mn$_2$PtPd). The alloys have been prepared by arc melting in high-purity Ar, with\nthe ingots being remelted four times to ensure homogeneity. An excess of 3 \\% wt. Mn is added in order to\ncompensate for Mn losses during arc melting (see Section~\\ref{subsec:art109:exp_data_Mn2_based} for details). Structural characterization has been carried\nout by powder X-ray diffraction (XRD), while magnetic measurements were made using a superconducting magnetometer\nin a field of up to 5~T. Furthermore, the microstructure has been analyzed by scanning electron microscopy\nof the polished bulk samples, while the compositions are determined by Energy Dispersive X-ray (EDX)\nspectroscopy.\n\nTwo of the four HAs have been successfully synthesized, Co$_2$MnTi and Mn$_2$PtPd, while the other two,\nMn$_2$PtCo and Mn$_2$PtV, decompose into binary compounds (see Section~\\ref{subsec:art109:exp_data_Mn2_based} for details).\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig124}\n\\mycaption[Experimental magnetic characterization of Co$_2$MnTi.]\n{(\\textbf{a}) magnetization curve at 4~K and 300~K (inset: zero-field cooled magnetization\ncurve as a function of temperature in magnetic field of 1~T);\n(\\textbf{b}) XRD spectrum (inset: EDX chemical composition\nanalysis).\nCo$_2$MnTi crystallizes in a single {\\it Fm$\\overline{3}$m} phase corresponding to a regular Heusler.\nThe $T_\\mathrm{C}$ extrapolated from the magnetization curve is around 900~K.}\n\\label{fig:art109:Co2MnTi}\n\\end{figure}\n\nIn Figure~\\ref{fig:art109:Co2MnTi} we present the structural and magnetic characterization of Co$_2$MnTi. It crystallizes in\nthe regular {\\it Fm$\\overline{3}$m} Heusler structure with no evidence of secondary phases and a lattice parameter of\n$a=5.89$~\\AA\\, in close agreement with theory, $a=5.84$~\\AA. The magnetization curve displays little temperature\ndependence and a saturation moment of 4.29~$\\mu_\\mathrm{B}$\/f.u. at 4~K, fully consistent with the calculated\nferromagnetic ground state (see Table~\\ref{tab:art109:BLtab}). Most notably, the $T_\\mathrm{C}$ extrapolated from the\nzero-field cooled magnetization curve in a field of 1~T is found to be 938~K, essentially identical that predicted\nby our regression, 940~K.\nThis is a remarkable result, since it is the first time that a new high-temperature ferromagnet\nhas been discovered by HT means.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig125}\n\\mycaption[Experimental magnetic characterization of Mn$_2$PtPd.]\n{(\\textbf{a}) field cooled and zero-field cooled magnetization curve as a function of temperature in\na magnetic field of 0.1~T (inset: magnetization curve at 4~K and 300~K);\n(\\textbf{b}) XRD spectrum (inset: EDX chemical\ncomposition analysis). Mn$_2$PtPd crystallizes in a single {\\it I}4{\\it\/mmm} (TiAl$_3$-type) phase corresponding\nto a regular tetragonal distorted Heusler.\nSEM images confirm that the bulk sample is mainly of Mn$_2$PtPd\ncomposition (gray color) with a small amount of a secondary Mn-O inclusions, which have spherical shape of\ndiameter 400-900~nm and do not appear in the XRD spectrum.}\n\\label{fig:art109:Mn2PtPd}\n\\end{figure}\n\nAlso in the case of Mn$_2$PtPd a single phase is found without evidence of decomposition. The XRD pattern\n[Figure~\\ref{fig:art109:Mn2PtPd}(b)] corresponds to a tetragonally-distorted regular Heusler with space group {\\it I}4{\\it\/mmm}\n(TiAl$_3$-type) and lattice parameters $a=4.03$~\\AA\\ and $c=7.24$~\\AA. Our magnetic data show a magnetic\ntransition at $\\sim$320~K, which shifts to a slightly higher temperature upon field cooling [Figure~\\ref{fig:art109:Mn2PtPd}(a)].\nMagnetization curves at room temperature and 4~K show no hysteresis or spontaneous magnetization indicating\nthat the compound is antiferromagnetic at low temperature.\nFrom Table~\\ref{tab:art109:BLtab} it will appear that the only difference between the calculated and experimental\ndata for Mn$_2$PtPd concerns the tetragonal distortion. However, the search for tetragonal distortion reported in the\ntable was performed only for the ferromagnetic state. Further analysis for the antiferromagnetic ground state (see Section~\\ref{subsec:art109:tet_disorder_Mn2PtPd})\nreveals that indeed Mn$_2$PtPd is antiferromagnetic and tetragonal distorted with a $c\/a$ ratio of around 1.3, in good\nagreement with experiments.\n\n\\subsection{Table of \\texorpdfstring{$T_\\mathrm{C}$}{Tc} of known Heusler alloys} \\label{subsec:art109:tc_known_heuslers}\n\nHere we present experimental data, collected from the literature, for known magnetic Heusler alloys.\nThese data have been used to perform the regression used to extract the $T_\\mathrm{C}$ of the\nnew predicted compounds.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Summary Table for the magnetic Heusler alloys of the type Co$_2XY$.]\n{Here are reported the compound,\nthe magnetic moment per formula unit, $m$, and the experimental $T_\\mathrm{C}$, together with the\nappropriate reference. The quantities labeled with a `*' have been used to run the\nregression.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r|r|r|r}\nmaterial & $m$\/f.u. ($\\mu_\\mathrm{B}$) & $m^*$\/f.u. ($\\mu_\\mathrm{B}$) & $T_\\mathrm{C}$~{\\small (K)}\\ & $T_\\mathrm{C}^*$~{\\small (K)}\\ & source & reference \\\\\n\\hline\nCo$_2$TiAl & 0.74 & 0.74 & 134 & 134 & Exp. & \\onlinecitesq{PhysRevB.76.024414} \\\\\nCo$_2$TiGa & 0.82 & 0.82 & 128 & 128 & Exp. & \\onlinecitesq{LandBorn1,Sasaki2001406} \\\\\nCo$_2$TiSi & 1.96 & 1.96 & 380 & 380 & Exp. & \\onlinecitesq{LandBorn1,Barth28092011} \\\\\nCo$_2$TiGe & 1.94 & 1.94 & 380 & 380 & Exp. & \\onlinecitesq{LandBorn1,Barth28092011} \\\\\nCo$_2$TiSn & 1.97 & 1.97 & 355 & 355 & Exp. & \\onlinecitesq{LandBorn1,Barth28092011} \\\\\nCo$_2$ZrSn & 1.56 & 1.56 & 448 & 448 & Exp. & \\onlinecitesq{Zhang2006255} \\\\\nCo$_2$VGa & 2.04 & 2.04 & 357 & 357 & Exp. & \\onlinecitesq{PhysRevB.76.024414,PhysRevB.82.144415} \\\\\nCo$_2$VSn & 1.21 & 1.21 & 95 & 95 & Exp. & \\onlinecitesq{PhysRevB.76.024414,0022-3727-40-6-S01} \\\\\nCo$_2$VAl & 1.86 & 1.86 & 342 & 342 & Exp. & \\onlinecitesq{LandBorn1,PhysRevB.82.144415} \\\\\nCo$_2$ZrAl & 0.74 & 0.74 & 185 & 185 & Exp. & \\onlinecitesq{LandBorn1,Kanomata200526} \\\\\nCo$_2$ZrSn & 1.51 & 1.51 & 444 & 444 & Exp. & \\onlinecitesq{LandBorn1} \\\\\nCo$_2$NbAl & 1.35 & 1.35 & 383 & 383 & Exp. & \\onlinecitesq{LandBorn1,0022-3727-40-6-S01} \\\\\nCo$_2$NbSn & 0.52 & 0.52 & 119 & 119 & Exp. & \\onlinecitesq{LandBorn1,PhysRevB.66.174428} \\\\\nCo$_2$HfAl & 0.81 & 0.81 & 193 & 193 & Exp. & \\onlinecitesq{LandBorn1} \\\\\nCo$_2$HfGa & 0.54 & 0.54 & 186 & 186 & Exp. & \\onlinecitesq{LandBorn1} \\\\\nCo$_2$HfSn & 1.55 & 1.55 & 394 & 394 & Exp. & \\onlinecitesq{LandBorn1} \\\\\nCo$_2$CrGa & 3.01 & 3.01 & 495 & 495 & Exp. & \\onlinecitesq{PhysRevB.76.024414} \\\\\nCo$_2$CrAl & 1.55 & 1.55 & 334 & 334 & Exp. & \\onlinecitesq{PhysRevB.76.024414,Hakimi20103443,JEPT2013Svyazhin} \\\\\nCo$_2$MnAl & 4.01-4.04 & 4.04 & 693-697 & 697 & Exp. & \\onlinecitesq{LandBorn1,PhysRevB.76.024414} \\\\\nCo$_2$MnGa & 4.05 & 4.05 & 694 & 694 & Exp. & \\onlinecitesq{LandBorn1} \\\\\nCo$_2$MnGe & 5.11 & 5.11 & 905 & 905 & Exp. & \\onlinecitesq{LandBorn1} \\\\\nCo$_2$MnSi & 4.90 & 4.90 & 985 & 985 & Exp. & \\onlinecitesq{PhysRevB.76.024414,LandBorn1} \\\\\nCo$_2$MnSn & 5.08 & 5.08 & 829 & 829 & Exp. & \\onlinecitesq{PhysRevB.76.024414,LandBorn1} \\\\\nCo$_2$FeSi & 6.00 & 6.00 & 1100 & 1100 & Exp. & \\onlinecitesq{PhysRevB.76.024414} \\\\\nCo$_2$FeAl & 4.96 & 4.96 & 1000 & 1000 & Exp. & \\onlinecitesq{Trudel:2013p323} \\\\\nCo$_2$FeGa & 5.15 & 5.15 & $>$1100 & 1100 & Exp. & \\onlinecitesq{Trudel:2013p323} \\\\\nCo$_2$TaAl & 0.75 & 0.75 & 260 & 260 & Exp. & \\onlinecitesq{Carbonari1996} \\\\\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Summary Table magnetic Heuslers of the type $X_2$Mn$Y$.]\n{Here are reported the compound,\nthe magnetic moment per formula unit, $m$, the experimental $T_\\mathrm{C}$, the volume of\nthe $F\\overline{4}3m$ cell, and the number of valence electrons per formula unit, $N_\\mathrm{V}$,\ntogether with the appropriate reference. The quantity labeled with a `*' are those, which have\nbeen used to run the regression.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r}\nmaterial & $m$\/f.u. ($\\mu_\\mathrm{B}$) & $m^*$\/f.u. ($\\mu_\\mathrm{B}$) & $T_\\mathrm{C}$~{\\small (K)}\\ & $T_\\mathrm{C}^*$~{\\small (K)}\\ & volume (\\AA$^3$) & $N_\\mathrm{V}$ & order & reference \\\\\n\\hline\nRh$_2$MnGe & 4.17-4.62 & 4.62 & 400-470 & 450 & 56.46 & 29 & FM & \\onlinecitesq{LandBorn1,Klaer2009,Suits1976,Adachi200437,Hames1971} \\\\\nRh$_2$MnSn & 3.10-3.93 & 3.10 & 412-431 & 412 & 62.22 & 29 & FM & \\onlinecitesq{LandBorn1,Suits1976,Adachi200437} \\\\\nRh$_2$MnPb & 4.12 & 4.12 & 338 & 338 & 65.58 & 29 & FM & \\onlinecitesq{LandBorn1,Suits1976} \\\\\nRh$_2$MnAl & 4.1 & 4.1 & 85-105 & 95 & 54.96 & 28 & FM & \\onlinecitesq{Wijn1,Suits1976} \\\\\nCu$_2$MnSn & 4.11 & 4.11 & 530 & 530 & 60.36 & 33 & FM & \\onlinecitesq{LandBorn1,Wijn1,Oxley1963} \\\\\nCu$_2$MnAl & 3.73-4.12 & 4.12 & 603 & 603 & 51.93 & 32 & FM & \\onlinecitesq{LandBorn1,Wijn1,Oxley1963} \\\\\nCu$_2$MnIn & 3.95 & 3.95 & 510 & 510 & 59.45 & 32& FM & \\onlinecitesq{Oxley1963,Coles1949} \\\\\nPd$_2$MnAl & 4.4 & 4.4 & 240 & 240 & 58.89 & 30 & AFM & \\onlinecitesq{Wijn1,Webster1968} \\\\\nPd$_2$MnSn & 4.23 & 4.23 & 189 & 189 & 66.00 & 31 & FM & \\onlinecitesq{LandBorn1,Wijn1,Campbell1977,Webster1967} \\\\\nPd$_2$MnSb & 4.40 & 4.40 & 247 & 247 & 67.58 & 32 & FM & \\onlinecitesq{LandBorn1,Wijn1,Webster1967} \\\\\nPd$_2$MnGe & 3.2 & 3.2 & 170 & 170 & 60.49 & 31 & FM & \\onlinecitesq{Wijn1} \\\\\nPd$_2$MnIn & 4.3 & 4.3 & 142 & 142 & 65.88 & 30 & AFM & \\onlinecitesq{Wijn1,Webster1967} \\\\\nAu$_2$MnAl & 4.2 & 4.2 & 233 & 233 & 65.37 & 32 & FM & \\onlinecitesq{Wijn1,Bacon1967} \\\\\nAu$_2$MnZn & 4.6 & 4.6 & 253 & 253 & 65.32 & 31 & FM & \\onlinecitesq{Wijn1,Bacon1973} \\\\\nRu$_2$MnGe & 3.2-3.8 & 3.8 & 316 & 316 & 54.33 & 27 & AFMII & \\onlinecitesq{Kanomata2006,Gotoh1995} \\\\\nRu$_2$MnSi & 2.8 & 2.8 & 313 & 313 & 51.82 & 27 & AFMII & \\onlinecitesq{Kanomata2006} \\\\\nRu$_2$MnSb & 3.9-4.4 & 4.4 & 195 & 195 & 58.98 & 28 & AFMII & \\onlinecitesq{Kanomata2006,Gotoh1995} \\\\\nRu$_2$MnSn & 2.8 & 2.8 & 296 & 296 & 58.92 & 27 & AFMII & \\onlinecitesq{Kanomata2006} \\\\\n\\end{tabular}}\n\\end{table}\n\n\\newcommand{\\heuslerstabfootone}{\nNote that the tetragonal phase is obtained when annealing at $400^\\circ$C.\nA higher annealing temperature of $800^\\circ$C results in a disorder pseudo-cubic phase.\nNo magnetic data are available for this second phase.}\n\\newcommand{\\heuslerstabfoottwo}{\nNote that Mn$_2$NiGa is a shape memory alloy, displaying a martensitic transformation at a critical temperature\n$T_{\\mathrm{m}}=270$~K.\nThe structure is cubic for $T>T_{\\mathrm{m}}$ and tetragonal for $T0$).]\n{The continuous red line is our best fit to a two-parameter Weibull distribution with a shape of 1.13\nand a scale of 2585.63.}\n\\label{fig:art109:histogram_full}\n\\end{figure}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig134}\n\\mycaption[Histogram of the entropic temperature, $T_\\mathrm{S}$, for all the 248 intermetallic Heuslers estimated stable\nafter the construction of the convex hull diagrams for the ternary phase.]\n{The red lines indicate three compounds present in the {\\small ICSD}\\ database.}\n\\end{figure}\n\n\\clearpage\n\n\\subsection{Tetragonal distortion for \\texorpdfstring{Mn$_2$}{Mn2}PtPd} \\label{subsec:art109:tet_disorder_Mn2PtPd}\n\nThe total energy of Mn$_2$PtPd is calculated for different $c\/a$ ratio (and constant volume) for both the\nferromagnetic and antiferromagnetic state. Note that, while in the ferromagnetic configuration the energy\nminimum is found for the cubic solution, in the antiferromagnetic case (lower in energy) this is found for\n$c\/a=1.3$, in agreement with the experimental data.\n\n\\vspace{1cm}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.5\\linewidth]{fig135}\n\\mycaption{Total energy as a function of the $c\/a$ ratio for Mn$_2$PtPd calculated with {\\small GGA}-{\\small DFT}.}\n\\end{figure}\n\n\\clearpage\n\n\\subsection{List of all stable intermetallic Heuslers}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers (1\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nZn$_2$AgAu &\t64.64\t&\t-0.15\t&\t1723\t\\\\\nPd$_2$AgCd &\t68.13\t&\t-0.27\t&\t2958\t\\\\\nAg$_2$CdSc\t&\t77.7808\t&\t-0.248922\t&\t2778.26\t\\\\\nAg$_2$CdY\t&\t85.8372\t&\t-0.301022\t&\t3359.76\t\\\\\nAg$_2$CdZr\t&\t77.9156\t&\t-0.098514\t&\t1099.53\t\\\\\nHg$_2$AgLa\t&\t99.102\t&\t-0.392808\t&\t4384.2\t\\\\\nPd$_2$AgHg\t&\t69.3436\t&\t-0.146835\t&\t1638.85\t\\\\\nHg$_2$AgSc\t&\t81.9652\t&\t-0.256216\t&\t2859.68\t\\\\\nSc$_2$AgHg\t&\t84.1676\t&\t-0.364257\t&\t4065.54\t\\\\\nHg$_2$AgY\t&\t89.648\t&\t-0.349733\t&\t3903.43\t\\\\\nSc$_2$AgOs\t&\t72.1864\t&\t-0.376927\t&\t4206.95\t\\\\\nSc$_2$AgRu\t&\t72.4904\t&\t-0.44129\t&\t4925.31\t\\\\\nY$_2$AgRu\t&\t87.0664\t&\t-0.346082\t&\t3862.68\t\\\\\nAu$_2$CdLa\t&\t94.5472\t&\t-0.66943\t&\t7471.63\t\\\\\nPd$_2$AuCd\t&\t68.5296\t&\t-0.301286\t&\t3362.7\t\\\\\nAu$_2$CdY\t&\t85.436\t&\t-0.674423\t&\t7527.36\t\\\\\nAu$_2$CdZr\t&\t78.2676\t&\t-0.457602\t&\t5107.38\t\\\\\nCu$_2$AuPd\t&\t57.166\t&\t-0.115899\t&\t1293.57\t\\\\\nAu$_2$CuZn\t&\t62.4144\t&\t-0.142872\t&\t1594.62\t\\\\\nAu$_2$HfZn\t&\t71.7456\t&\t-0.438785\t&\t4897.35\t\\\\\nAu$_2$HgLa\t&\t94.9036\t&\t-0.627046\t&\t6998.57\t\\\\\nPd$_2$AuHg\t&\t69.7384\t&\t-0.162896\t&\t1818.12\t\\\\\nZn$_2$AuRh\t&\t59.324\t&\t-0.312353\t&\t3486.23\t\\\\\nSc$_2$AuRu\t&\t71.926\t&\t-0.675774\t&\t7542.43\t\\\\\nAu$_2$TiZn\t&\t66.8216\t&\t-0.352571\t&\t3935.11\t\\\\\nAu$_2$ZnZr\t&\t73.2548\t&\t-0.467891\t&\t5222.21\t\\\\\nCu$_2$CdZr\t&\t65.8668\t&\t-0.155451\t&\t1735.01\t\\\\\nRh$_2$CdHf\t&\t67.1976\t&\t-0.68254\t&\t7617.94\t\\\\\nHg$_2$CdLa\t&\t103.328\t&\t-0.460008\t&\t5134.23\t\\\\\nHg$_2$CdSc\t&\t86.474\t&\t-0.265346\t&\t2961.58\t\\\\\nHg$_2$CdY\t&\t94.0524\t&\t-0.381128\t&\t4253.84\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_1}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (2\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nPd$_2$CdSc\t&\t70.7684\t&\t-0.725422\t&\t8096.56\t\\\\\nPd$_2$CdY\t&\t78.308\t&\t-0.731543\t&\t8164.88\t\\\\\nPd$_2$CdZr\t&\t72.2028\t&\t-0.58712\t&\t6552.94\t\\\\\nRh$_2$CdSc\t&\t67.0008\t&\t-0.622274\t&\t6945.31\t\\\\\nRh$_2$CdZr\t&\t68.4788\t&\t-0.627501\t&\t7003.65\t\\\\\nHf$_2$CoRe\t&\t66.8792\t&\t-0.412526\t&\t4604.27\t\\\\\nCo$_2$HfSc\t&\t61.3956\t&\t-0.38894\t&\t4341.02\t\\\\\nHf$_2$CoTc\t&\t66.1292\t&\t-0.493898\t&\t5512.48\t\\\\\nCo$_2$HfZn\t&\t53.9212\t&\t-0.326005\t&\t3638.6\t\\\\\nSc$_2$CoIr\t&\t64.4424\t&\t-0.71918\t&\t8026.89\t\\\\\nTi$_2$CoIr\t&\t56.8924\t&\t-0.622184\t&\t6944.3\t\\\\\nTi$_2$CoMn\t&\t52.0108\t&\t-0.382265\t&\t4266.53\t\\\\\nTi$_2$CoRe\t&\t56.7704\t&\t-0.444075\t&\t4956.4\t\\\\\nSc$_2$CoRu\t&\t63.6324\t&\t-0.467309\t&\t5215.72\t\\\\\nTi$_2$CoTc\t&\t56.0352\t&\t-0.510928\t&\t5702.56\t\\\\\nZr$_2$CoTc\t&\t68.3008\t&\t-0.359379\t&\t4011.09\t\\\\\nCo$_2$TiZn\t&\t48.8244\t&\t-0.350328\t&\t3910.07\t\\\\\nCo$_2$ZnZr\t&\t55.166\t&\t-0.268346\t&\t2995.06\t\\\\\nV$_2$CrFe\t&\t47.8092\t&\t-0.167619\t&\t1870.82\t\\\\\nTi$_2$CrIr\t&\t57.4292\t&\t-0.551684\t&\t6157.44\t\\\\\nV$_2$CrMn\t&\t48.2312\t&\t-0.193973\t&\t2164.97\t\\\\\nNb$_2$CrOs\t&\t62.176\t&\t-0.200243\t&\t2234.95\t\\\\\nTa$_2$CrOs\t&\t62.2812\t&\t-0.311877\t&\t3480.92\t\\\\\nV$_2$CrOs\t&\t52.3748\t&\t-0.302942\t&\t3381.19\t\\\\\nV$_2$CrRe\t&\t53.0104\t&\t-0.258046\t&\t2880.1\t\\\\\nTa$_2$CrRu\t&\t61.7168\t&\t-0.280556\t&\t3131.34\t\\\\\nV$_2$CrRu\t&\t51.9164\t&\t-0.25086\t&\t2799.9\t\\\\\nHf$_2$CuRe\t&\t69.632\t&\t-0.296279\t&\t3306.82\t\\\\\nHf$_2$CuTc\t&\t69.12\t&\t-0.339081\t&\t3784.54\t\\\\\nCu$_2$HfZn\t&\t58.7964\t&\t-0.19888\t&\t2219.74\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_2}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (3\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nSc$_2$CuIr\t&\t68.0976\t&\t-0.699208\t&\t7803.98\t\\\\\nSc$_2$CuOs\t&\t67.3716\t&\t-0.408716\t&\t4561.75\t\\\\\nZr$_2$CuOs\t&\t71.076\t&\t-0.345336\t&\t3854.35\t\\\\\nPd$_2$CuZn\t&\t56.2556\t&\t-0.403379\t&\t4502.19\t\\\\\nSc$_2$CuPt\t&\t70.4452\t&\t-0.801364\t&\t8944.16\t\\\\\nRh$_2$CuTa\t&\t58.3456\t&\t-0.455697\t&\t5086.11\t\\\\\nSc$_2$CuRu\t&\t67.33\t&\t-0.46985\t&\t5244.08\t\\\\\nY$_2$CuRu\t&\t81.6212\t&\t-0.318052\t&\t3549.83\t\\\\\nZr$_2$CuTc\t&\t71.3516\t&\t-0.26889\t&\t3001.13\t\\\\\nCu$_2$TiZn\t&\t53.8756\t&\t-0.169069\t&\t1887\t\\\\\nCu$_2$ZnZr\t&\t60.172\t&\t-0.223658\t&\t2496.28\t\\\\\nHf$_2$FeOs\t&\t65.9256\t&\t-0.524889\t&\t5858.38\t\\\\\nTi$_2$FeMn\t&\t51.8856\t&\t-0.336061\t&\t3750.83\t\\\\\nTi$_2$FeOs\t&\t55.8712\t&\t-0.568209\t&\t6341.88\t\\\\\nHf$_2$IrMn\t&\t66.5552\t&\t-0.641543\t&\t7160.37\t\\\\\nHf$_2$IrMo\t&\t70.62\t&\t-0.605585\t&\t6759.04\t\\\\\nHf$_2$IrRe\t&\t69.8952\t&\t-0.743454\t&\t8297.82\t\\\\\nHf$_2$IrTc\t&\t69.3832\t&\t-0.854328\t&\t9535.3\t\\\\\nIr$_2$HfZn\t&\t63.0396\t&\t-0.732469\t&\t8175.21\t\\\\\nHf$_2$MoRh\t&\t70.6316\t&\t-0.529099\t&\t5905.36\t\\\\\nTc$_2$HfMo\t&\t64.7628\t&\t-0.293247\t&\t3272.98\t\\\\\nTc$_2$HfNb\t&\t67.0276\t&\t-0.447695\t&\t4996.8\t\\\\\nNi$_2$HfZn\t&\t55.6964\t&\t-0.431443\t&\t4815.41\t\\\\\nHf$_2$OsRu\t&\t68.636\t&\t-0.769146\t&\t8584.58\t\\\\\nOs$_2$HfSc\t&\t67.5148\t&\t-0.560224\t&\t6252.76\t\\\\\nHf$_2$OsTc\t&\t69.0408\t&\t-0.626591\t&\t6993.49\t\\\\\nHf$_2$PdRe\t&\t71.6136\t&\t-0.559388\t&\t6243.42\t\\\\\nHf$_2$PdTc\t&\t71.1996\t&\t-0.620052\t&\t6920.51\t\\\\\nPd$_2$HfZn\t&\t65.598\t&\t-0.675223\t&\t7536.29\t\\\\\nHf$_2$ReRh\t&\t69.9132\t&\t-0.700075\t&\t7813.66\t\\\\\nHf$_2$ReZn\t&\t71.8108\t&\t-0.299023\t&\t3337.45\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_3}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (4\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nHf$_2$RhTc\t&\t69.3144\t&\t-0.78918\t&\t8808.18\t\\\\\nRh$_2$HfZn\t&\t61.8484\t&\t-0.857463\t&\t9570.3\t\\\\\nRu$_2$HfSc\t&\t66.8988\t&\t-0.728346\t&\t8129.19\t\\\\\nHf$_2$RuTc\t&\t68.6544\t&\t-0.669049\t&\t7467.38\t\\\\\nTc$_2$HfTa\t&\t66.898\t&\t-0.509943\t&\t5691.57\t\\\\\nTc$_2$HfW\t&\t64.88\t&\t-0.346078\t&\t3862.64\t\\\\\nTi$_2$IrMn\t&\t56.4032\t&\t-0.694021\t&\t7746.08\t\\\\\nTi$_2$IrMo\t&\t61.0552\t&\t-0.626827\t&\t6996.13\t\\\\\nSc$_2$IrNi\t&\t65.8112\t&\t-0.801053\t&\t8940.69\t\\\\\nSc$_2$IrPd\t&\t69.9072\t&\t-0.991899\t&\t11070.8\t\\\\\nY$_2$IrPd\t&\t83.818\t&\t-0.881856\t&\t9842.55\t\\\\\nTi$_2$IrRe\t&\t60.1416\t&\t-0.756525\t&\t8443.71\t\\\\\nSc$_2$IrRh\t&\t67.5008\t&\t-1.04502\t&\t11663.7\t\\\\\nY$_2$IrRh\t&\t81.2688\t&\t-0.841385\t&\t9390.84\t\\\\\nSc$_2$IrRu\t&\t66.5136\t&\t-0.830031\t&\t9264.12\t\\\\\nSc$_2$IrZn\t&\t70.6392\t&\t-0.724073\t&\t8081.51\t\\\\\nTi$_2$IrTc\t&\t59.6952\t&\t-0.840269\t&\t9378.39\t\\\\\nZr$_2$IrTc\t&\t71.522\t&\t-0.693835\t&\t7744.01\t\\\\\nIr$_2$TiZn\t&\t57.6528\t&\t-0.695974\t&\t7767.89\t\\\\\nIr$_2$ZnZr\t&\t64.3208\t&\t-0.634848\t&\t7085.65\t\\\\\nMn$_2$NbTi\t&\t54.8624\t&\t-0.227403\t&\t2538.09\t\\\\\nTi$_2$MnNi\t&\t53.3032\t&\t-0.342964\t&\t3827.88\t\\\\\nTi$_2$MnOs\t&\t56.2816\t&\t-0.502285\t&\t5606.09\t\\\\\nTi$_2$MnRh\t&\t56.0512\t&\t-0.577568\t&\t6446.34\t\\\\\nMn$_2$TaTi\t&\t54.9728\t&\t-0.27885\t&\t3112.3\t\\\\\nMn$_2$TiV\t&\t49.6572\t&\t-0.274813\t&\t3067.23\t\\\\\nMn$_2$TiW\t&\t52.8752\t&\t-0.237692\t&\t2652.92\t\\\\\nNb$_2$MoOs\t&\t65.8108\t&\t-0.281237\t&\t3138.94\t\\\\\nNb$_2$MoRe\t&\t66.5\t&\t-0.250455\t&\t2795.37\t\\\\\nNb$_2$MoRu\t&\t65.5172\t&\t-0.256633\t&\t2864.32\t\\\\\nMo$_2$NbTa\t&\t67.5352\t&\t-0.166161\t&\t1854.55\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_4}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (5\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nNb$_2$MoTc\t&\t66.0904\t&\t-0.252979\t&\t2823.54\t\\\\\nMo$_2$NbW\t&\t65.644\t&\t-0.113515\t&\t1266.96\t\\\\\nTi$_2$MoNi\t&\t58.4756\t&\t-0.291743\t&\t3256.19\t\\\\\nTa$_2$MoOs\t&\t65.8048\t&\t-0.393033\t&\t4386.71\t\\\\\nV$_2$MoOs\t&\t56.3172\t&\t-0.312104\t&\t3483.45\t\\\\\nTi$_2$MoPd\t&\t62.484\t&\t-0.393388\t&\t4390.67\t\\\\\nTi$_2$MoPt\t&\t62.1896\t&\t-0.647475\t&\t7226.58\t\\\\\nTa$_2$MoRe\t&\t66.5052\t&\t-0.341358\t&\t3809.96\t\\\\\nRe$_2$MoTi\t&\t61.2988\t&\t-0.294221\t&\t3283.85\t\\\\\nV$_2$MoRe\t&\t56.986\t&\t-0.280895\t&\t3135.12\t\\\\\nTi$_2$MoRh\t&\t60.9012\t&\t-0.515086\t&\t5748.97\t\\\\\nTa$_2$MoRu\t&\t65.4864\t&\t-0.37169\t&\t4148.5\t\\\\\nV$_2$MoRu\t&\t56.0544\t&\t-0.266878\t&\t2978.68\t\\\\\nTa$_2$MoTc\t&\t66.1236\t&\t-0.348888\t&\t3894\t\\\\\nMo$_2$TaW\t&\t65.598\t&\t-0.138309\t&\t1543.69\t\\\\\nTc$_2$MoTi\t&\t60.4636\t&\t-0.348792\t&\t3892.93\t\\\\\nMo$_2$TiW\t&\t63.4916\t&\t-0.13852\t&\t1546.05\t\\\\\nMo$_2$VW\t&\t61.3504\t&\t-0.111196\t&\t1241.07\t\\\\\nOs$_2$NbSc\t&\t65.272\t&\t-0.455798\t&\t5087.24\t\\\\\nTa$_2$NbOs\t&\t67.6176\t&\t-0.32288\t&\t3603.72\t\\\\\nNb$_2$OsW\t&\t66.2136\t&\t-0.199311\t&\t2224.54\t\\\\\nRe$_2$NbTa\t&\t65.8404\t&\t-0.370123\t&\t4131.01\t\\\\\nNb$_2$ReTc\t&\t65.302\t&\t-0.339447\t&\t3788.63\t\\\\\nRe$_2$NbTi\t&\t63.3304\t&\t-0.398599\t&\t4448.84\t\\\\\nRh$_2$NbZn\t&\t60.0404\t&\t-0.492704\t&\t5499.15\t\\\\\nRu$_2$NbSc\t&\t64.652\t&\t-0.549806\t&\t6136.48\t\\\\\nTa$_2$NbRu\t&\t67.4116\t&\t-0.270899\t&\t3023.55\t\\\\\nRu$_2$NbZn\t&\t59.4144\t&\t-0.275852\t&\t3078.83\t\\\\\nTc$_2$NbTa\t&\t64.8712\t&\t-0.435369\t&\t4859.23\t\\\\\nTc$_2$NbTi\t&\t62.4432\t&\t-0.468812\t&\t5232.5\t\\\\\nTc$_2$NbZr\t&\t67.9524\t&\t-0.372639\t&\t4159.09\t\\\\\nSc$_2$NiOs\t&\t65.1596\t&\t-0.499591\t&\t5576.03\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_5}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (6\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nSc$_2$NiPt\t&\t68.0772\t&\t-0.888355\t&\t9915.08\t\\\\\nTi$_2$NiRe\t&\t57.7528\t&\t-0.434031\t&\t4844.29\t\\\\\nZn$_2$NiRh\t&\t52.1512\t&\t-0.344857\t&\t3849.01\t\\\\\nSc$_2$NiRu\t&\t64.7916\t&\t-0.579752\t&\t6470.71\t\\\\\nTi$_2$NiTc\t&\t57.1568\t&\t-0.486273\t&\t5427.38\t\\\\\nNi$_2$TiZn\t&\t50.568\t&\t-0.405025\t&\t4520.55\t\\\\\nSc$_2$OsPd\t&\t68.9692\t&\t-0.693907\t&\t7744.82\t\\\\\nSc$_2$OsPt\t&\t68.3896\t&\t-0.8455\t&\t9436.77\t\\\\\nTa$_2$OsRe\t&\t65.1812\t&\t-0.351313\t&\t3921.07\t\\\\\nTi$_2$OsRu\t&\t59.0072\t&\t-0.744346\t&\t8307.77\t\\\\\nZr$_2$OsRu\t&\t70.7628\t&\t-0.593618\t&\t6625.47\t\\\\\nOs$_2$ScTa\t&\t65.0652\t&\t-0.533786\t&\t5957.68\t\\\\\nSc$_2$OsZn\t&\t69.8296\t&\t-0.439858\t&\t4909.33\t\\\\\nOs$_2$ScZr\t&\t68.6728\t&\t-0.476955\t&\t5323.38\t\\\\\nTa$_2$OsTc\t&\t64.6524\t&\t-0.405699\t&\t4528.07\t\\\\\nOs$_2$TaTi\t&\t62.2512\t&\t-0.496833\t&\t5545.25\t\\\\\nTa$_2$OsW\t&\t66.2384\t&\t-0.299962\t&\t3347.93\t\\\\\nTi$_2$OsTc\t&\t59.5392\t&\t-0.632543\t&\t7059.92\t\\\\\nV$_2$OsTc\t&\t55.1728\t&\t-0.345037\t&\t3851.02\t\\\\\nZr$_2$OsTc\t&\t71.0824\t&\t-0.476841\t&\t5322.1\t\\\\\nSc$_2$PdPt\t&\t72.2524\t&\t-1.08971\t&\t12162.5\t\\\\\nZn$_2$PdRh\t&\t56.2356\t&\t-0.51684\t&\t5768.54\t\\\\\nSc$_2$PdRu\t&\t68.874\t&\t-0.78368\t&\t8746.79\t\\\\\nPd$_2$ScZn\t&\t64.8108\t&\t-0.783946\t&\t8749.75\t\\\\\nTi$_2$PdTc\t&\t61.282\t&\t-0.542858\t&\t6058.93\t\\\\\nZr$_2$PdTc\t&\t73.412\t&\t-0.522887\t&\t5836.03\t\\\\\nPd$_2$TiZn\t&\t60.4704\t&\t-0.57928\t&\t6465.44\t\\\\\nPd$_2$ZnZr\t&\t67.0388\t&\t-0.641322\t&\t7157.91\t\\\\\nZn$_2$PtRh\t&\t56.5432\t&\t-0.518725\t&\t5789.58\t\\\\\nSc$_2$PtRu\t&\t68.1764\t&\t-0.962407\t&\t10741.6\t\\\\\nPt$_2$ScZn\t&\t65.1192\t&\t-0.926966\t&\t10346\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_6}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (7\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nSc$_2$PtZn\t&\t73.0916\t&\t-0.836692\t&\t9338.47\t\\\\\nZn$_2$PtSc\t&\t62.8012\t&\t-0.667899\t&\t7454.54\t\\\\\nTi$_2$PtTc\t&\t61.1772\t&\t-0.743086\t&\t8293.71\t\\\\\nTi$_2$ReRh\t&\t60.04\t&\t-0.667826\t&\t7453.72\t\\\\\nTa$_2$ReRu\t&\t64.9088\t&\t-0.376608\t&\t4203.39\t\\\\\nTa$_2$ReTc\t&\t65.2984\t&\t-0.465143\t&\t5191.55\t\\\\\nRe$_2$TaTi\t&\t63.47\t&\t-0.457933\t&\t5111.07\t\\\\\nTa$_2$ReW\t&\t66.8768\t&\t-0.278984\t&\t3113.79\t\\\\\nRe$_2$TiV\t&\t58.2544\t&\t-0.402985\t&\t4497.78\t\\\\\nRe$_2$TiW\t&\t61.6096\t&\t-0.350499\t&\t3911.97\t\\\\\nTi$_2$ReZn\t&\t61.3876\t&\t-0.316406\t&\t3531.47\t\\\\\nSc$_2$RhRu\t&\t66.6712\t&\t-0.818639\t&\t9136.97\t\\\\\nRh$_2$ScZn\t&\t60.8824\t&\t-0.779193\t&\t8696.71\t\\\\\nRh$_2$TaZn\t&\t59.9676\t&\t-0.548351\t&\t6120.24\t\\\\\nTi$_2$RhTc\t&\t59.5088\t&\t-0.741616\t&\t8277.31\t\\\\\nZr$_2$RhTc\t&\t71.5328\t&\t-0.64339\t&\t7180.99\t\\\\\nRh$_2$TiZn\t&\t56.792\t&\t-0.783097\t&\t8740.29\t\\\\\nRh$_2$VZn\t&\t55.0032\t&\t-0.416055\t&\t4643.66\t\\\\\nRh$_2$ZnZr\t&\t63.148\t&\t-0.778072\t&\t8684.2\t\\\\\nRu$_2$ScTa\t&\t64.4184\t&\t-0.625766\t&\t6984.29\t\\\\\nRu$_2$ScTi\t&\t62.122\t&\t-0.656051\t&\t7322.31\t\\\\\nRu$_2$ScV\t&\t59.5772\t&\t-0.460194\t&\t5136.31\t\\\\\nSc$_2$RuZn\t&\t70.06\t&\t-0.491623\t&\t5487.1\t\\\\\nRu$_2$ScZr\t&\t68.1104\t&\t-0.649445\t&\t7248.57\t\\\\\nTa$_2$RuTc\t&\t64.312\t&\t-0.412004\t&\t4598.45\t\\\\\nRu$_2$TaTi\t&\t61.542\t&\t-0.554291\t&\t6186.54\t\\\\\nTa$_2$RuW\t&\t66.0004\t&\t-0.285381\t&\t3185.19\t\\\\\nRu$_2$TaY&\t70.2656\t&\t-0.340037\t&\t3795.22\t\\\\\nRu$_2$TaZn\t&\t59.4956\t&\t-0.344438\t&\t3844.33\t\\\\\nTi$_2$RuTc\t&\t59.1864\t&\t-0.643868\t&\t7186.32\t\\\\\nV$_2$RuTc\t&\t54.8572\t&\t-0.320533\t&\t3577.52\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_7}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (8\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nZr$_2$RuTc\t&\t70.7428\t&\t-0.524685\t&\t5856.11\t\\\\\nRu$_2$VZn\t&\t54.4836\t&\t-0.218631\t&\t2440.18\t\\\\\nRu$_2$WZn\t&\t57.7132\t&\t-0.126584\t&\t1412.82\t\\\\\nTc$_2$TaTi\t&\t62.5436\t&\t-0.530531\t&\t5921.35\t\\\\\nTc$_2$TaZr\t&\t67.8028\t&\t-0.431941\t&\t4820.97\t\\\\\nTc$_2$TiV\t&\t57.4784\t&\t-0.450416\t&\t5027.17\t\\\\\nTc$_2$TiW\t&\t60.7084\t&\t-0.403323\t&\t4501.56\t\\\\\nTi$_2$TcZn\t&\t61.0412\t&\t-0.346197\t&\t3863.97\t\\\\\nTc$_2$WZr\t&\t65.7664\t&\t-0.258379\t&\t2883.81\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_8}\n\\end{table}\n\n\\subsection{Conclusion}\n\nIn conclusion we have demonstrated a new systematic pathway to the discovery of novel magnetic materials. We have\ncreated an extensive library of Heusler compounds including about 250,000 structures. For the sub-class of intermetallic\nalloys we have been able to establish the materials stability against decomposition of 20 novel magnetic HAs,\nbelonging to Co$_2YZ$, Mn$_2YZ$ and $X_2$Mn$Z$ classes. A simple machine learning method, correlating calculated\nmicroscopic electronic structure quantities with macroscopic measured properties, has been used to predict the magnetic\n$T_\\mathrm{C}$ of such compounds. The method has been put to the test with the experimental synthesis of four\ncompounds and validated by the growth of two. In particular we have discovered a new high-performance ferromagnet,\nCo$_2$MnTi and a tetragonally distorted antiferromagnet, Mn$_2$PtPd. Our method offers a new high-throughput tool\nfor the discovery of new magnets, which can now be applied to other structural families, opening new possibilities for\ndesigning materials for energy, data storage and spintronics applications.\n\\clearpage\n\\chapter{Conclusion}\nModeling approaches promise a direct and systematic path to materials discovery.\nTo justify their application, these methods need to bridge several gaps:\n\\textbf{i.} prediction of synthesizability (as property prediction\/optimization becomes irrelevant if the material cannot form),\n\\textbf{ii.} treatment of more ``real-world'' phenomena (\\nobreak\\mbox{\\it vs.}\\ the ideal systems modeled \\nobreak\\mbox{\\it ab initio}),\nand\n\\textbf{iii.} identification of structure-property relationships (harnessing the information for practical design rules).\nRecent progress has been driven by data-centric approaches~\\cite{curtarolo:art13}\nfacilitated by large, programmatically-accessible materials databases.\n\nFrameworks like {\\small AFLOW}~\\cite{curtarolo:art49,curtarolo:art53,curtarolo:art57,aflowBZ,curtarolo:art63,aflowPAPER,curtarolo:art85,curtarolo:art110,monsterPGM,aflowANRL,aflowPI}\\ have characterized millions of compounds\nwithout the need for laborious human intervention~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}.\nCombinatorial exploration of various structure prototypes offers a means for sampling\ncandidate stable structures~\\cite{curtarolo:art130,aflowANRL}.\nThe gamut of extractable features derives from electronic, magnetic, chemical, crystallographic, thermomechanical,\nand thermodynamic characterizations --- each warranting\nrobust algorithms\nthat scale with the panoply of structures in the database.\nFor example, convenient definitions for the primitive cell representation~\\cite{aflowPAPER} and\nhigh-symmetry Brillouin Zone path~\\cite{aflowBZ} have not only standardized electronic structure calculations,\nbut also optimized their computation.\nMoreover, careful treatment of spatial tolerance and proper validation schemes have finally\nenabled accurate and autonomous determination of the\ncomplete symmetry profile of crystals~\\cite{curtarolo:art134}.\nElasticity~\\cite{curtarolo:art115} and phonon~\\cite{aflowPAPER,curtarolo:art114,curtarolo:art119,curtarolo:art125}\ncalculations are incredibly sensitive to the quality of the symmetry analysis.\nThe scheme resolves experimentally-validated space groups and\naccommodates even the most skewed unit cells, meeting the demand for high-throughput thermomechanical characterizations.\n\nThe development of the {\\sf \\AFLOW.org}\\ repository has motivated both\nbroad-scale thermodynamic formability modeling and adoption of {\\small ML}\\ algorithms.\nEnsembles of ordered phases are successfully employed to\n\\textbf{i.} construct phase diagrams forecasting stability~\\cite{curtarolo:art146}\nand\n\\textbf{ii.} formulate descriptors and models to predict the formation\/properties of disordered materials~\\cite{curtarolo:art110}.\nThese methods go beyond standard modeling approaches, leveraging\nseveral \\nobreak\\mbox{\\it ab-initio}\\ calculations in each analysis\nand encouraging\nthe continued expansion of these large materials databases.\n\nAs the proliferation of high-throughput approaches\nincreases the wealth of data in the field, the gap between accumulated-information and derived-knowledge widens.\nThe divergence must be addressed autonomously, reciprocating the pace of data generation.\n{\\small ML}\\ models\nare constructed for rapid predictions and exposing subtle\/hidden trends\nthat would have otherwise evaded human detection\/understanding.\nUseful examples include models\npredicting electronic and thermomechanical properties\nfrom basic features of the structure and composition, \\nobreak\\mbox{\\it i.e.}, not requiring additional calculations\nor experiments, affording easy integration into virtually any materials design workflow~\\cite{curtarolo:art124}.\n\n{\\small ML}\\ models are also employed to identify meaningful correlations among materials\/properties,\nleading to enhanced understanding of fundamental physical mechanisms.\nFor many phenomena, the connection between the arrangement of elements into solid compounds\nand the observed macroscopic behavior is still largely unknown, as with\nhigh-temperature superconductors.\nThese materials are particularly difficult to address within automated \\nobreak\\mbox{\\it ab-initio}\\ frameworks because\nthe underlying {\\small DFT}\\ theory fails to capture the strong interactions and correlations\nresponsible for the effect~\\cite{DFT}.\nHowever, as demonstrated by the materials cartography approach~\\cite{curtarolo:art94},\nother similarities between materials, such as the electronic density of states and band structure,\ncan be exploited to reveal interesting candidates.\nAlternatively, {\\small DFT}\\ data can be avoided altogether.\nInstead, models have been constructed leveraging empirical\ninformation retrieved from the SuperCon database~\\cite{SuperCon} for more than 12,000 materials~\\cite{curtarolo:art137}.\nDistinct driving mechanisms are resolved by comparing important features\nof a general model, trained on all data,\nwith that of family-specific models, trained on\nlow-$T_{\\mathrm{c}}$, cuprate, and iron-based superconductors, respectively.\n\nStructure-property relationships have also been resolved\nin perovskites ($ABX_{3}$ where $X$ = F and O) for high-temperature thermoelectric applications~\\cite{curtarolo:art120}.\nThe thermal conductivity of fluorides is strongly influenced by substitutions of the $B$ site,\nwhile in oxides the same is true for the $A$ site --- presenting a useful engineering opportunity.\nFor example, to mitigate costs in device production, substitutions in the less influential site\ncan be expected not to affect the thermoelectric performance.\n\nFinally, thermodynamic descriptors and regression analyses among classes of ground-state compounds\ncontributed to the screening of 36,540 Heusler compounds for new magnetic systems~\\cite{curtarolo:art109}.\nAn attempt to synthesize four candidates yielded two novel materials.\nOf these, Co$_{2}$MnTi promises to be a high-performance ferromagnet with $T_{\\mathrm{C}}=938$~K,\nas predicted by the Slater-Pauling curve --- illustrating the predictive power of data-driven approaches.\nThese methods will accelerate the path to synthesis and, ultimately,\ntransform the practice of traditional materials discovery to one of rational and autonomous materials design.\n\\clearpage\n\n\\newcommand{Ozoli\\c{n}\\v{s}}{Ozoli\\c{n}\\v{s}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nOver the last decades, multi-agent systems have been considered in a variety of applications such as connectivity and formation control \\cite{magnus}\nor coverage \\cite{coverage}. The complexity of these applications has motivated the need of an expressive language, capable of describing complex task specifications for planning and control synthesis. \n\nRecently, extensive interest has been shown in planning under high-level task specifications expressed by Linear Temporal Logic (LTL) \\cite{tl1,tl2}. In these methods the temporal formula, the environment and the agent dynamics are abstracted into finite-transition systems. Then, graph-based methods are employed to find a discrete path satisfying the LTL specifications which is finally followed using continuous control laws. An important limitation of the aforementioned methods is the increasing computational complexity as the number of the agents in the team becomes larger. Towards minimizing the computational costs, large effort has been devoted to the decomposition of a global LTL formula into local LTL tasks whose satisfaction depends on subsets of agents. Existing methods, applied to heterogeneous agents \\cite{dimos2,ltl_services}, most often employ exhausting automata-based approaches \\cite{dimos1,dimos2,ltl_services} or more recently, cross-entropy optimization methods limited, though, to homogeneous agents \\cite{ltldec_entropy}. \n\nAll methods presented so far consider the satisfaction of LTL tasks without explicit time constraints. On the other hand, Signal Temporal Logic (STL) \\cite{stl} can express complex tasks under strict deadlines. An advantage of STL over LTL, is the robust semantics \\cite{stlrob, fainekos} it offers that allow the evaluation of the satisfaction of the task over a continuous-time signal, rendering the abstractions of the agents' dynamics obsolete. \n\nExisting methods for planning under STL specifications consider a global STL formula and find plans as solutions to computationally prohibitive MILPs \\cite{mpc_raman,mpc_sadra} or to scalable convex programs \\cite{lars2,kunal}. Other approaches compose \\cite{decentralized} or assume the existence \\cite{lars_linear} of local STL tasks whose satisfaction involves only a small subset of agents. This facilitates the design of decentralized frameworks that are inherently more robust to agents' failures and often cheaper in terms of communication. Towards decentralized control under global task specifications, a satisfiability modulo theories (SMT) approach has been proposed in \\cite{stldec2} for tasks described in caSTL, in which both the global formula and the team of agents is decomposed. Here, the decomposition is based on a set of services required for each task and a set of utility functions specifying the capabilities of the agents. Nevertheless, the decomposition of a global STL formula in continuous space and time remains an open problem.\n\nIn this paper we propose a novel framework for the decomposition of a global STL formula imposed on a multi-agent system into a set of local tasks when the team of agents is a-priori divided into disjoint sub-teams. The goal of the decomposition is to make the satisfaction of every local task dependent only to a subset of agents that belong to the same sub-team. Initially, the predicate functions corresponding to STL formulas forming the local tasks are parameterized as functions of the infinity norm of the agents' states while their parameters are found as part of the solution to a convex program that aims at maximizing the volume of their zero level-set. Although the choice of the parametric family of the predicate functions is not restrictive, our current choice allows us to draw conclusions on the volume of a continuous state-space set by incorporating a finite, but possibly large, number of constraints in the convex program. The number of these constraints differs per global STL task but depends solely on the number of the agents' states involved in its satisfaction. Two definitions of the local tasks that differ on the definition of the STL tasks originating from eventually formulas are introduced. Finally, for both definitions the satisfaction of the global STL formula is proven when the conjunction of the local tasks is satisfied.\n\nThe remainder of the paper is as follows: Section II includes the preliminaries and problem formulation. Section III introduces the proposed method for STL decomposition. Simulations are shown in Section IV and conclusions are summarized in Section V.\n\n\n\\section{Preliminaries and Problem Formulation}\nThe set of real and non-negative real numbers are denoted by $\\mathbb{R}$ and $\\mathbb{R}_{\\geq 0}$ respectively. True and false are denoted by $\\top, \\bot$ respectively. Scalars and vectors are denoted by non-bold and bold letters respectively. The infinity norm of a vector $\\mathbf{x} \\in \\mathbb{R}^n$ is defined as $\\Vert \\mathbf{x} \\Vert_{\\infty}=\\max_i\\vert \\mathbf{x}_i \\vert$, where $\\mathbf{x}=\\begin{bmatrix} \\mathbf{x}_1 & \\ldots & \\mathbf{x}_n \\end{bmatrix}^T$.\nGiven a finite set $ \\mathcal{V}$, $\\prod_{k\\in \\mathcal{V}}\\mathbb{X}_k$ denotes the Cartesian product of the sets $\\mathbb{X}_k, k\\in \\mathcal{V}$. Given a rectangular matrix $A\\in M_{n\\times m}(\\mathbb{R})$ we define the set $A\\mathbb{X}$ as $A\\mathbb{X}=\\{A\\mathbf{x}: \\mathbf{x} \\in \\mathbb{X}\\}$. A square matrix $P\\in M_n(\\{0,1\\})$ is called a \\textit{permutation matrix} \\cite[Ch. 0.9.5]{horn} if exactly\none entry in each row and column is equal to 1 and all other entries are 0. Consider the vectors $\\mathbf{x} \\in \\mathbb{R}^n, \\mathbf{y}\\in \\mathbb{R}^m$ with $n\\leq m$ satisfying $\\mathbf{x}=B\\mathbf{y}$. The matrix $B=[b_{ij}]$ is called a \\textit{selection matrix} if it has the following properties: 1) $b_{ij}\\in \\{0,1\\}$, 2) $\\sum_{j=1}^m b_{ij}=1, \\forall i=1,\\ldots,n$ and 3) $\\sum_{i=1}^n b_{ij}=1, \\forall j=1,\\ldots,m$.\n\n\\subsection{Signal Temporal Logic (STL)}\nSignal Temporal Logic (STL) determines whether a predicate $\\mu$ is true or false. The validity of each predicate $\\mu$ is evaluated based on a continuously differentiable function $h:\\mathbb{R}^n \\rightarrow \\mathbb{R}$ as follows:\n\\begin{equation*}\n \\mu=\\begin{cases} \\top, &h(\\mathbf{x}) \\geq 0 \\\\ \\bot, & h(\\mathbf{x})< 0 \\end{cases}\n\\end{equation*}\nfor $\\mathbf{x} \\in \\mathbb{R}^n$. The basic STL formulas are given by the grammar:\n$$ \\phi:= \\top \\; | \\; \\psi \\;| \\;\\neg \\phi \\; | \\; \\phi_1 \\land \\phi_2 \\; |\\; \\mathcal{G}_{[a,b]} \\phi \\;| \\; \\mathcal{F}_{[a,b]} \\phi \\;| \\; \\phi_1 \\; \\mathcal{U}_{[a,b]} \\; \\phi_2 $$\nwhere $\\phi_1, \\phi_2 $ are STL formulas and $\\mathcal{G}_{[a,b]},\\; \\mathcal{F}_{[a,b]}, \\; \\mathcal{U}_{[a,b]}$ is the always, eventually and until operator defined over the interval $[a,b]$ with $0 \\leq a \\leq b$. Let $ \\mathbf{x} \\models \\phi$ denote the satisfaction of the formula $\\phi$ by a signal $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{R}^n$. The formula $\\phi$ is satisfiable if $\\exists \\; \\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{R}^n$ such that $\\mathbf{x} \\models \\phi$. The STL semantics for a signal $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{R}^n$ are recursively given and can be found, e.g., in \\cite{lars2}. STL is equipped with robustness metrics determining how robustly an STL formula $\\phi$ is satisfied at time $t$ by a signal $\\mathbf{x}$. These semantics are defined as follows \\cite{stlrob, fainekos}: $\\rho^{\\mu}(\\mathbf{x},t)=h(\\mathbf{x}(t))$, $\\rho^{\\neg \\phi}(\\mathbf{x},t)=-\\rho^{\\phi}(\\mathbf{x},t)$, $ \\rho^{\\phi_1 \\wedge \\phi_2}(\\mathbf{x},t)=\\min(\\rho^{\\phi_1}(\\mathbf{x},t),\\rho^{\\phi_2}(\\mathbf{x},t))$, $\\rho^{\\phi_1 \\; \\mathcal{U}_{[a,b]} \\; \\phi_2}(\\mathbf{x},t)=\\max_{t_1 \\in [t+a,t+b]} \\min(\\rho^{\\phi_2}(\\mathbf{x},t_1), \\min_{t_2\\in [t,t_1]} \\rho^{\\phi_1}(\\mathbf{x},t_2)) $, $\\rho^{\\mathcal{F}_{[a,b]} \\phi}(\\mathbf{x},t)=\\max_{t_1\\in [t+a,t+b]} \\rho^{\\phi}(\\mathbf{x},t_1) $, $\\rho^{\\mathcal{G}_{[a,b]} \\phi}(\\mathbf{x},t)=\\min_{t_1\\in [t+a,t+b]} \\rho^{\\phi}(\\mathbf{x},t_1)$. Finally, it should be noted that $\\mathbf{x} \\models \\phi$ if $\\rho^{\\phi}(\\mathbf{x},0)>0$.\n\n\n\\subsection{Problem Formulation}\n\nIn this work we consider the following STL fragment:\n\\begin{subequations}\n\\begin{align}\n \\psi &:= \\; \\mu \\;| \\;\\neg \\mu \\label{eq:f1} \\\\\n \\varphi &:= \\mathcal{G}_{[a,b]} \\psi \\;| \\; \\mathcal{F}_{[a,b]} \\psi \\label{eq:f2}\\\\\n \\phi&:=\\bigwedge_{i=1}^{p} \\varphi_i \\label{eq:f3}\n\\end{align}\n\\end{subequations}\nwhere $0\\leq a \\leq b < \\infty$ and $p\\geq 1$.\n\\begin{remark}\nThe STL fragment defined by \\eqref{eq:f1}-\\eqref{eq:f3} is expressive enough to accommodate until STL formulas of the form $\\varphi=\\psi_1 \\mathcal{U}_{[a,b]} \\psi_2$ where $\\psi_i, i=1,2$ are defined by \\eqref{eq:f1}. By definition, for any $t^* \\in [a,b]$ the until formula $\\varphi=\\psi_1 \\mathcal{U}_{[a,b]} \\psi_2$ can be written as $\\varphi=\\mathcal{G}_{[a,t^*]} \\psi_1 \\wedge \\mathcal{F}_{[t^*,t^*]} \\psi_2$. Hence, if for a given time instant $t^* \\in [a,b]$ there exists a signal $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{R}^n$ such that $\\mathbf{x} \\models \\big(\\mathcal{G}_{[a,t^*]} \\psi_1 \\wedge \\mathcal{F}_{[t^*,t^*]} \\psi_2\\big)$ then $\\mathbf{x} \\models \\varphi$.\n\\end{remark}\n\nConsider a team of $R$ agents with each agent identified by its index $k \\in \\mathcal{V}=\\{1, \\ldots, R\\}$. For every agent $k$ let $ \\mathbf{x}_k \\in \\mathbb{X}_k$ denote its state vector, where $\\mathbb{X}_k\\subseteq \\mathbb{R}^{\\bar{n}_k} $ is a known, bounded, convex set for every $k\\in \\mathcal{V}$. Let $n=\\sum_{k\\in \\mathcal{V}} \\bar{n}_k$ and $\\mathbf{x}=\\begin{bmatrix} \\mathbf{x}_1^T & \\ldots & \\mathbf{x}_R^T \\end{bmatrix}^T \\in \\mathbb{X}$ where $\\mathbb{X}=\\prod_{k\\in \\mathcal{V}} \\mathbb{X}_k$ is convex as the Cartesian product of convex sets. Assume that the agents are decomposed in $v$ smaller teams $\\{\\mathcal{V}_1, \\ldots, \\mathcal{V}_v\\}$, $\\mathcal{V}_l \\subseteq \\mathcal{V}, l=1,\\ldots,v$ that are disjoint, i.e., for any $l_1, l_2\\in \\{1,\\ldots, v\\}$ with $l_1\\neq l_2$ it holds that $\\mathcal{V}_{l_1}\\cap \\mathcal{V}_{l_2}=\\emptyset$ and satisfy $\\bigcup_{l=1}^v \\mathcal{V}_l=\\mathcal{V}$.\n\nConsider a global STL formula $\\phi$ of the form \\eqref{eq:f3} with $\\mathcal{I}= \\{1,\\ldots, p\\}$ and sub-formulas $\\varphi_i, \\; i\\in \\mathcal{I}$ satisfying \\eqref{eq:f1}-\\eqref{eq:f2}. Let $[a_i,b_i]$ be the interval of satisfaction associated with the temporal operator of $\\varphi_i, i\\in \\mathcal{I}$ and define the sets of always and eventually formulas of $\\phi$ as $\\mathcal{I}_{\\mathcal{G}}=\\big\\{i\\in \\mathcal{I}: \\varphi_i=\\mathcal{G}_{[a_i,b_i]} \\psi_i\\big\\}$ and $\\mathcal{I}_{\\mathcal{F}}=\\big\\{i\\in \\mathcal{I}: \\varphi_i=\\mathcal{F}_{[a_i,b_i]} \\psi_i\\big\\}$ respectively. Observe that by definition of the STL fragment in \\eqref{eq:f1}-\\eqref{eq:f3} it holds that $\\mathcal{I}=\\mathcal{I}_{\\mathcal{G}} \\cup \\mathcal{I}_{\\mathcal{F}}$. Assume without loss of generality that the satisfaction of each $\\varphi_i, i\\in \\mathcal{I}$ depends on multiple agents of different teams $\\mathcal{V}_l$ and let $V_i\\subseteq \\{1,\\ldots,v\\}, i\\in \\mathcal{I}$ denote the set of indices of the agents' groups that have at least one member contributing to the satisfaction of $\\varphi_i$. Since $\\phi$ is a global task its satisfaction requires agents to be fully aware of the actions of their peers. However, in real-time scenarios communication between all agents may often be hard to establish, especially when the working environment of the agents is large. Addressing this problem, in this paper we propose decomposing the initial task $\\phi$ into local tasks the satisfaction of which depends only on the agents in the same team $\\mathcal{V}_l$. This problem is formally introduced as:\n\n\\begin{problem}\nGiven a global STL formula $\\phi$ defined by \\eqref{eq:f3} and the disjoint sets of agents $\\mathcal{V}_l, l=1,\\ldots,v$ satisfying $\\bigcup_{l=1}^v \\mathcal{V}_l=\\mathcal{V}$ find STL formulas $\\phi_1, \\ldots, \\phi_v$ such that: 1) each STL formula $\\phi_l$ depends on the agents in $\\mathcal{V}_l$ and 2) $\\mathbf{x} \\models \\big(\\phi_1\\wedge \\ldots\\wedge \\phi_v\\big) \\Rightarrow \\mathbf{x} \\models \\phi$ if such $\\mathbf{x}:\\mathbb{R}_{\\geq 0}\\rightarrow \\mathbb{X}$ exists.\n\\end{problem}\n\n\\section{Decomposition of STL Formulas}\n\nIn this Section we design a number of STL tasks the satisfaction of which depends on a known subset of agents. Consider the formula $\\phi$ defined by \\eqref{eq:f3}. Let the predicate function $h_i:\\mathbb{X} \\rightarrow \\mathbb{R}$ associated with the formula $\\varphi_i,i\\in \\mathcal{I}$. Then, the zero level-set of $h_i(\\mathbf{x})$ is defined as follows:\n\\begin{equation}\n \\mathcal{S}_i=\\{\\mathbf{x} \\in \\mathbb{X}: h_i(\\mathbf{x})\\geq 0\\} \\label{eq:levelset}\n\\end{equation}\nHere, we assume that $h_i(\\mathbf{x}),i\\in \\mathcal{I}$ is a function whose value may depend on the states of all agents in $\\mathcal{V}$. As a result guaranteeing the satisfaction of $\\varphi_i, i\\in \\mathcal{I}$ may require the knowledge of all agents' actions and thus global communication. In real-time scenarios communication among all agents can become costly or hard due to packet losses or communications delays. On the other hand, decentralized approaches allow agents to communicate with a subset of their peers and optimize their actions with respect to a limited number of agents thus improving the computational complexity of the problem. \n\nIn the context of STL control synthesis a decentralized approach involves the requirement of assigning to agents tasks whose satisfaction depends only to a subset of agents with established communication links , i.e., to $\\mathcal{V}_l, l=1,\\ldots,v$ while guaranteeing the satisfaction of the global task $\\phi$. To that end, in this paper we propose a set of STL tasks $\\phi_l=\\bigwedge_{q_i=1}^{p_l} \\bar{\\varphi}_{q_i}^l,\\; l=1,\\ldots,v$ whose satisfaction depends on the corresponding set of agents $\\mathcal{V}_l$. Here, $\\bar{\\varphi}_{q_i}^l$ denotes the $q_i^l$-th formula of $\\phi_l$ that is considered to be the result of the decomposition of the sub-formula $\\varphi_i$ of \\eqref{eq:f3}. If it is clear from context, we may omit the subscript of the index $q_i\\in \\{1,\\ldots,p_l\\}$. \n\nLet $\\mathbf{z}_l \\in \\mathcal{Z}_l \\subset \\mathbb{R}^{n_l}$ be the states of the agents in $\\mathcal{V}_l$ where $n_l=\\sum_{k\\in \\mathcal{V}_l} \\bar{n}_k$ and $\\mathcal{Z}_l=\\prod_{k\\in \\mathcal{V}_l} \\mathbb{X}_k$. The vector $\\mathbf{z}_l, l=1,\\ldots,v$ can be obtained from $\\mathbf{x}$ using the following equation:\n\\begin{equation}\n \\mathbf{z}_l=E_l \\mathbf{x} \\label{eq:z2x}\n\\end{equation}\nwhere $E_l \\in M_{n_l\\times n}(\\{0,1\\})$ is a selection matrix. Additionally, the vector $\\mathbf{x}$ can be written with respect to the vectors $\\mathbf{z}_l, l=1,\\ldots,v$ as:\n\\begin{equation}\n \\mathbf{x}=A \\mathbf{z} \\label{eq:permutation}\n\\end{equation}\nwhere $\\mathbf{z}=\\begin{bmatrix} \\mathbf{z}_1^T & \\ldots & \\mathbf{z}_v^T \\end{bmatrix}^T$ and $A\\in M_n(\\{0,1\\})$ is an appropriately chosen permutation matrix. Let $[a_{q}^l,b_{q}^l]$ and $h_{q}^l:\\mathcal{Z}_l\\rightarrow \\mathbb{R}, q=1,\\ldots,p_l, \\; l=1,\\ldots v$ denote the interval of satisfaction and predicate function corresponding to $\\bar{\\varphi}_{q}^l$ respectively. Here, for every $l=1,\\ldots,v$ we assume that $h_{q_i}^l(\\mathbf{z}_l)=h_{q_i}^l(\\mathbf{z}_l;\\bm{\\theta}_i^l), q_i=1,\\ldots,p_l$ belongs to a known family of functions and its value depends on a set of parameters $\\bm{\\theta}_i^l\\in \\Theta_i^l \\subseteq \\mathbb{R}^{m_i^l}$ to be tuned towards maximizing the volume of the zero level-set of $h_{q_i}^l(\\mathbf{z}_l)$ defined as:\n\\begin{equation}\n S_{q_i}^l=\\big\\{\\mathbf{z}_l \\in \\mathcal{Z}_l: h_{q_i}^l(\\mathbf{z}_l)\\geq 0 \\big\\} \\label{eq:set}\n\\end{equation}\n\nBased on the above we propose the following method for designing $\\phi_l,l=1,\\ldots,v$:\n\\begin{theorem}\nConsider the global STL formula $\\phi$ defined by \\eqref{eq:f1}-\\eqref{eq:f3} and the predicate function $h_i(\\mathbf{x})$ associated to $\\varphi_i, i\\in \\mathcal{I}$. Assume that $\\mathcal{S}_i\\neq \\emptyset$, where $\\mathcal{S}_i, i\\in \\mathcal{I}$ is defined in \\eqref{eq:levelset}. For every $i\\in \\mathcal{I}$ derive the functions $h_{q_i}^l(\\mathbf{z}_l)$ as solutions to the following optimization problem:\n\\begin{subequations}\\label{eq:dec}\n\\begin{align}\n \\max_{\\bm{\\theta}_i^l\\in \\Theta_i^l, l\\in V_i} \\sum_{l\\in V_i}\\textit{vol}(S_{q_i}^l) \\tag{\\ref{eq:dec}}\n \\end{align}\nsubject to:\n\\begin{align}\n \\mathbf{z}_l&\\in S_{q_i}^l, \\quad l\\in V_i\\\\\n \\mathbf{x}&\\in \\mathcal{S}_i \\label{eq:basiceq}\\\\\n \\mathbf{z}_l&=E_l \\mathbf{x}, \\quad l\\in V_i\n\\end{align}\n\\end{subequations}\nwhere $\\textit{vol}(S_{q_i}^l)$ denotes the volume of the set $S_{q_i}^l$ defined in \\eqref{eq:set}. For every $l=1,\\ldots,v$ define the formulas $\\bar{\\varphi}_{q_i}^l$ as follows:\n\\begin{equation}\n \\bar{\\varphi}_{q_i}^l=\\begin{cases} \\mathcal{F}_{[a_{q_i}^l,b_{q_i}^l]} \\bar{\\mu}_{q_i}^l, \\quad i \\in \\mathcal{I}_{\\mathcal{F}}\\\\\\mathcal{G}_{[a_{q_i}^l,b_{q_i}^l]} \\bar{\\mu}_{q_i}^l, \\quad i\\in \\mathcal{I}_{\\mathcal{G}}\n \\end{cases} \\label{eq:newformula}\n\\end{equation}\nwith \n\\begin{subequations}\n\\begin{align}\n [a_{q_i}^l,b_{q_i}^l]&=\\begin{cases}[t_i,t_i], \\quad i \\in \\mathcal{I}_{\\mathcal{F}}\\\\ [a_i,b_i],\\quad i\\in \\mathcal{I}_{\\mathcal{G}} \\end{cases} \\label{eq:interval}\\\\\n \\bar{\\mu}_{q_i}^l&=\\begin{cases} \\top, & h_{q_i}^l(\\mathbf{z}_l)\\geq 0 \\\\ \\bot, & h_{q_i}^l(\\mathbf{z}_l)< 0 \\end{cases} \\label{eq:predicate}\n\\end{align}\n\\end{subequations}\nwhere $\\mathcal{I}=\\mathcal{I}_{\\mathcal{G}} \\cup \\mathcal{I}_{\\mathcal{F}}$, $t_i\\in [a_i,b_i]$ and $[a_i,b_i]$ is the interval of satisfaction associated with each $\\varphi_i$ of the global formula $\\phi$. Let $\\phi_l=\\bigwedge_{q_i=1}^{p_l} \\bar{\\varphi}_{q_i}^l$, $l=1,\\ldots,v$. If there exists $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{X}$ such that $\\rho^{\\phi_1\\wedge \\ldots\\wedge \\phi_v}(\\mathbf{x},0)>0$, then $\\rho^\\phi(\\mathbf{x},0)>0$.\n\\end{theorem}\n\n\\begin{proof}\nFor every $i\\in \\mathcal{I}$, \\eqref{eq:dec} aims at maximizing the volume of the $S_{q_i}^l,l\\in V_i$ which underapproximates the projection set of $\\mathcal{S}_i$ onto $\\mathcal{Z}_l$. Since $\\mathcal{S}_i\\neq \\emptyset$ for every $i\\in \\mathcal{I}$, \\eqref{eq:dec} is always feasible.\nBy definition of the robust semantics and the definition of the min operator it holds that:\n\\begin{equation*}\n \\rho^{\\phi_1\\wedge \\ldots\\wedge \\phi_v}(\\mathbf{x},0)\\leq \\rho^{\\phi_l}(\\mathbf{x},0), \\; l=1,\\ldots,v\n\\end{equation*}\nAs a result if there exists $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{X}$ such that $\\rho^{\\phi_1\\wedge \\ldots\\wedge \\phi_v}(\\mathbf{x},0)>0$ then $\\rho^{\\phi_l}(\\mathbf{x},0)> 0$ for every $l=1,\\ldots,v$. By design, the satisfaction of $\\phi_l$ depends on a subset of agents, thus $\\rho^{\\phi_l}(\\mathbf{x},0)=\\rho^{\\phi_l}(\\mathbf{z}_l,0)> 0$ where $\\mathbf{z}_l$ satisfies \\eqref{eq:z2x}. Then, by the definition of the robust semantics for every $l=1,\\ldots,v$ and $ q_i=1,\\ldots,p_l$ it holds that:\n\\begin{equation*}\n 0< \\rho^{\\phi_l}(\\mathbf{z}_l,0)=\\min_{q_i=1,\\ldots,p_l} \\rho^{\\bar{\\varphi}_{q_i}^l}(\\mathbf{z}_l,0)\\leq \\rho^{\\bar{\\varphi}_{q_i}^l}(\\mathbf{z}_l,0)\n\\end{equation*}\nIf $i\\in \\mathcal{I}_{\\mathcal{G}}$, then $\\varphi_{q_i}^l$ is an always formula. Hence due to \\eqref{eq:newformula}, $ \\rho^{\\bar{\\varphi}_{q_i}^l}(\\mathbf{z}_l,0)>0$ implies $h_{q_i}^l(\\mathbf{z}_l(t))> 0$ for every $t\\in [a_i,b_i]$, $l\\in V_i$. Since $h_{q_i}^l(\\mathbf{z}_l), l\\in V_i$ is a feasible solution of \\eqref{eq:dec}, it holds that $h_i(\\mathbf{x}(t))>0, \\forall t\\in [a_i,b_i]$ where $\\mathbf{x}(t)=A\\mathbf{z}(t)$ and $\\mathbf{z}(t)=\\begin{bmatrix}\\mathbf{z}_1^T(t) &\\ldots & \\mathbf{z}_v^T(t) \\end{bmatrix}^T$. Hence, $\\rho^{\\varphi_i}(\\mathbf{x},0)>0$. If $i\\in \\mathcal{I}_{\\mathcal{F}}$, then due to \\eqref{eq:newformula} and \\eqref{eq:interval}-\\eqref{eq:predicate}, for every $l\\in V_i$ it holds that: $ \\rho^{\\bar{\\varphi}_{q_i}^l}(\\mathbf{z}_l,0)=h_{q_i}^l(\\mathbf{z}_l(t_i))>0$. Following a similar argument as before, we can conclude that $h_i(\\mathbf{x}(t_i))>0$ where $\\mathbf{x}(t_i)=A\\mathbf{z}(t_i)$. This implies that $\\max_{t\\in [a_i,b_i]}h_i(\\mathbf{x}(t))\\geq h_i(\\mathbf{x}(t_i))>0$ leading to $\\rho^{\\varphi_i}(\\mathbf{x},0)>0$. Then, the result follows by the fact that $\\rho^{\\phi}(\\mathbf{x},0)=\\min\\big( \\min_{i\\in \\mathcal{I}_{\\mathcal{G}}} \\rho^{\\varphi_i}(\\mathbf{x},0), \\min_{i\\in \\mathcal{I}_{\\mathcal{F}}} \\rho^{\\varphi_i}(\\mathbf{x},0)\\big)$.\n\\end{proof}\n\nIn problem \\eqref{eq:dec} the goal is to maximize the volume of set $S_{q_i}^l$ by exhaustively evaluating $h_{q_i}^l(\\mathbf{z}_l)$ over the continuous set $\\mathcal{Z}_l$, which is in practice intractable. Another limitation of the proposed problem is often the lack of a known formula for computing the volume of a set, unless $h_{q_i}^l(\\mathbf{z}_l)$ belongs to a specific class of functions such as the class of ellipsoids.\n\nAiming at reducing the computational complexity of the STL decomposition problem described above, we propose a convex formulation for designing the predicate functions corresponding to \\eqref{eq:newformula} for every $l=1,\\ldots,v$. The computational benefits of the proposed approach are related to the number of points in $\\mathcal{Z}_l, l\\in V_i$ that are considered for evaluation of the satisfaction of \\eqref{eq:basiceq}. More specifically, contrary to \\eqref{eq:dec}, in this approach only a finite number of points is evaluated that depends on the number of states $\\mathbf{x}_k$ of the agents $k\\in \\mathcal{V}_l, l\\in V_i$ involved in the satisfaction of $h_i(\\mathbf{x})$. Let $d_i^l\\geq 1$ be the number of states in $\\mathbf{z}_l, l\\in V_i$ contributing to $h_i(\\mathbf{x})$. Since the global formula is a-priori given, the elements of $\\mathbf{z}_l , l\\in V_i$ on which the predicate function $h_i(\\mathbf{x})$ depends are known. Hence, we may write $h_i(\\mathbf{x}),i \\in \\mathcal{I}$ as:\n\\begin{subequations}\n\\begin{align}\n h_i(\\mathbf{x})&=h_i(\\mathbf{y}_{\\alpha(1)},\\ldots,\\mathbf{y}_{\\alpha(\\vert V_i \\vert)}) \\label{eq:dependency2}\\\\\n\\mathbf{y}_{\\alpha(c)}&=B_i^{\\alpha(c)} \\mathbf{z}_{\\alpha(c)}, \\quad c=1,\\ldots, \\vert V_i \\vert \\label{eq:dependency}\n\\end{align}\n\\end{subequations}\nwhere $\\alpha: \\{1,\\ldots,\\vert V_i \\vert\\} \\rightarrow V_i$ is an injective function defined as $\\alpha(c)=l$ and where $B_i^{\\alpha(c)}\\in M_{d_i^l\\times n_l}(\\{0,1\\})$ is an appropriate selection matrix and $\\mathbf{z}_{\\alpha(c)} \\in \\mathcal{Z}_{\\alpha(c)}$.\n\nBased on the above, we can consider a special class of concave functions of the following form:\n\\begin{equation}\n h_{q_i}^l(\\mathbf{z}_l)=r_{q_i}^l-\\Vert B_i^l(\\mathbf{z}_l-\\mathbf{c}_{q_i}^l) \\Vert_{\\infty}, \\quad q_i=1,\\ldots,p_l \\label{eq:prinf}\n\\end{equation}\nwhere $r_{q_i}^l\\in \\mathbb{R}_{\\geq 0}$, $\\mathbf{c}_{q_i}^l\\in \\mathcal{Z}_l$ and $B_i^l\\in M_{d_i^l\\times n_l}(\\{0,1\\})$ is the same selection matrix considered in \\eqref{eq:dependency} with $\\alpha(c)=l$.\nLet $J_{q_i}^l\\subseteq \\{1,\\ldots,n_l\\}$ denote the set of indices of the columns of $B_i^l$ with non-zero entries. Given the predicate functions defined by \\eqref{eq:prinf}, it follows that:\n\\begin{equation}\n h_{q_i}^l(\\mathbf{z}_l)\\geq 0 \\Leftrightarrow \\quad \\mathbf{z}_l(\\eta) \\in [-r_{q_i}^l+\\mathbf{c}_{q_i}^l(\\eta),r_{q_i}^l+\\mathbf{c}_{q_i}^l(\\eta)]\n\\end{equation}\nfor every $\\eta \\in J_{q_i}^l$ where $\\mathbf{z}_l(\\eta),\\mathbf{c}_{q_i}^l(\\eta)$ denote the $\\eta$-th element of the vectors $\\mathbf{z}_l,\\mathbf{c}_{q_i}^l$ respectively. For every $i\\in\\mathcal{I}$ and $l\\in\\{1,\\ldots,v\\}$ consider the following set of vectors:\n\\begin{equation}\n\\begin{split}\n \\mathcal{P}_i^l=\\big\\{\\bm{\\xi}\\in \\mathcal{Z}_l: \\bm{\\xi}(\\eta)&=-r_{q_i}^l+\\mathbf{c}_{q_i}^l(\\eta) \\; \\text{or} \\\\ \\bm{\\xi}(\\eta)&=r_{q_i}^l+\\mathbf{c}_{q_i}^l(\\eta), \\eta \\in J_{q_i}^l \\big\\} \\label{eq:vertexset}\n \\end{split}\n\\end{equation}\nwhere $\\bm{\\xi}(\\eta)$ denotes the $\\eta$-th element of $\\bm{\\xi}$. If $r_{q_i}^l\\geq 0$, the set $B_i^l\\mathcal{P}_i^l$ consists of the vertices of a hypercube in $\\mathbb{R}^{d_i^l}$ of edge length $r_{q_i}^l$ and center $\\mathbf{c}_{q_i}^l$.\nHence, its cardinality will be equal to $2^{d_i^l}$. To guarantee the convexity of the proposed problem we pose the following assumption:\n\n\\begin{assumption}\nFor every $i\\in \\mathcal{I}$ the predicate function $h_i(\\mathbf{x})$ is concave in $\\mathbb{X}$.\n\\end{assumption}\n\n\\begin{theorem}\nConsider the global STL formula $\\phi$ defined by \\eqref{eq:f1}-\\eqref{eq:f3} and the predicate functions $h_i(\\mathbf{x}), i\\in \\mathcal{I}$ associated to $\\varphi_i$. Let Assumption 1 hold. For every $i\\in \\mathcal{I}$ assume that $\\mathcal{S}_i\\neq \\emptyset$, where $\\mathcal{S}_i$ is defined in \\eqref{eq:levelset}. Consider the functions $ h_{q_i}^l(\\mathbf{z}_l), \\; q_i=1,\\ldots,p_l, \\; l=1,\\ldots,v$ defined by \\eqref{eq:prinf} where $\\mathbf{c}_{q_i}^l,r_{q_i}^l$ are parameters found as the solution to the following optimization problem:\n\\begin{subequations}\\label{eq:convex}\n\\begin{align}\n \\max_{\\mathbf{c}_{q_i}^l,r_{q_i}^l} \\sum_{l\\in V_i} r_{q_i}^l \\tag{\\ref{eq:convex}}\n \\end{align}\nsubject to:\n\\begin{align}\n h_i(\\mathbf{y}_{\\alpha(1)},\\ldots,\\mathbf{y}_{\\alpha(\\vert V_i \\vert)}) &\\geq 0 \\\\\n \\mathbf{y}_{\\alpha(c)}&\\in B_i^{\\alpha(c)}\\mathcal{P}_i^{\\alpha(c)}, \\quad c=1,\\ldots,\\vert V_i \\vert\n\\end{align}\n\\end{subequations}\nwhere $\\alpha: \\{1,\\ldots,\\vert V_i \\vert\\} \\rightarrow V_i$ and $B_i^{\\alpha(c)}\\in M_{d_i^l\\times n_l}(\\{0,1\\})$ is the injective function and selection matrix respectively considered in \\eqref{eq:dependency2}-\\eqref{eq:dependency} and $\\mathcal{P}_i^{\\alpha(c)}$ is the set defined by \\eqref{eq:vertexset} for every $\\alpha(c)=l\\in V_i$.\nFor every $l=1,\\ldots,v$ define the formulas $\\bar{\\varphi}_{q_i}^l$ based on \\eqref{eq:newformula} and \\eqref{eq:interval}-\\eqref{eq:predicate}\nand consider the decomposed STL formulas $\\phi_l=\\bigwedge_{q_i=1}^{p_l} \\bar{\\varphi}_{q_i}^l$, $l=1,\\ldots,v$. If there exists $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{X}$ such that $\\rho^{\\phi_1\\wedge \\ldots\\wedge \\phi_v}(\\mathbf{x},0)>0$, then $\\rho^\\phi(\\mathbf{x},0)>0$.\n\\end{theorem}\n\\begin{proof}\nFor every $i\\in \\mathcal{I}$, \\eqref{eq:convex} finds the maximum volume sets $B_i^lS_{q_i}^l,l\\in V_i$ which are underapproximations of the projection sets of $\\mathcal{S}_i$ onto $B_i^l\\mathcal{Z}_l$. Since $\\mathcal{S}_i\\neq \\emptyset$ for every $i\\in \\mathcal{I}$, \\eqref{eq:convex} is always feasible. To simplify notation let the sets $W=\\big\\{\\mathbf{y}: \\mathbf{y}_{l}=B_i^{l}\\mathbf{z}_{l}, \\; \\mathbf{z}_l \\in S_{q_i}^l, \\; l\\in V_i\\big\\}$ and $W^\\prime=\\big\\{\\mathbf{y}: \\mathbf{y}_{l}=B_i^{l}\\bm{\\xi}_l, \\; \\bm{\\xi}_l\\in \\mathcal{P}_i^l, \\; l\\in V_i\\big\\}$ where $\\mathbf{y}=\\begin{bmatrix} \\mathbf{y}_{\\alpha(1)}^T & \\ldots & \\mathbf{y}_{\\alpha(\\vert V_i \\vert)}^T \\end{bmatrix}^T$. The sets $B_i^lS_{q_i}^l$ are convex since they are projection sets of the zero-level sets of the concave function $h_{q_i}^l(\\mathbf{z}_l)$ defined by \\eqref{eq:prinf}. Hence, $W$ is convex as the Castesian product of convex sets. By Caratheodory's theorem \\cite[Th. 17.1]{rockafellar} every point $\\mathbf{y}\\in W$ can be written as a convex combination of $d_i+1$ points where $d_i=dim(W)=\\sum_{l\\in V_i} d_i^l$. Observe that $W^\\prime \\subset W$ with $\\vert W^\\prime \\vert= 2^{d_i}> d_i$. Applying Caratheodory's theorem, we write any point $\\mathbf{y}\\in W$ as a convex combination of the form: $\\mathbf{y}=\\sum_{j=1}^{d_i+1} \\lambda_j \\mathbf{y}_j^\\prime$ where $\\mathbf{y}_j^\\prime \\in W^\\prime, \\; \\lambda_j\\geq 0 $ and $ \\sum_{j=1}^{d_i+1}\\lambda_j=1$. By feasibility of \\eqref{eq:convex} and due to Assumption 1 we can conclude that $h_i(\\mathbf{y}_{\\alpha(1)},\\ldots,\\mathbf{y}_{\\alpha(\\vert V_i \\vert)})\\geq 0$ for any $\\mathbf{y}\\in W$ with $\\mathbf{y}=\\begin{bmatrix} \\mathbf{y}_{\\alpha(1)}^T & \\ldots & \\mathbf{y}_{\\alpha(\\vert V_i \\vert)}^T \\end{bmatrix}^T$. The rest of the proof is similar to that of Theorem 1.\n\\end{proof}\n\nFor $i\\in \\mathcal{I}_{\\mathcal{F}}$ the new STL tasks, defined by \\eqref{eq:newformula}, are expected to be satisfied at a specific time instant $t_i\\in [a_i,b_i]$ which is considered a designer's choice. However, in many cases pre-determining the time instant of satisfaction of a formula may lead to conservatism and reduced performance. An alternative would be to allow satisfaction of the local formulas over time intervals $[a_q^l,b_q^l]\\subseteq [a_i,b_i]$. Then, in order to guarantee the satisfaction of the global formula we can define the local tasks corresponding to $\\varphi_i, i\\in \\mathcal{I}_{\\mathcal{F}}$ as STL tasks of the form $\\mathcal{G}_{[a_{q_i}^l,b_{q_i}^l]} \\mu$.\nThis is depicted in the following Proposition:\n\\begin{figure*}[!t]\n \\centering\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{ s11}\n \\caption{Trajectories of agents 1,4,5}\n \\label{fig:ev1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{s12}\n \\caption{Trajectories of agents 2,3}\n \\label{fig:ev2}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{s13}\n \\caption{Barrier Function Evolution}\n \\label{fig:ev3}\n \\end{subfigure}\n \\caption{Agents' Trajectories under the local STL tasks defined based on \\eqref{eq:newformula}, \\eqref{eq:interval}-\\eqref{eq:predicate} and Barrier Function Evolution}\n \\label{fig:evall}\n\\end{figure*}\n\\begin{proposition}\nConsider the global STL formula $\\phi$ defined by \\eqref{eq:f1}-\\eqref{eq:f3}. Let Assumption 1 hold. For every $i\\in \\mathcal{I}$ assume that $\\mathcal{S}_i\\neq \\emptyset$, where $\\mathcal{S}_i$ is defined by \\eqref{eq:levelset}. For every $i\\in \\mathcal{I}$ consider the functions $h_{q_i}^l(\\mathbf{z}_l), l\\in V_i$ defined by \\eqref{eq:prinf} with their parameters found as solutions to \\eqref{eq:convex}. Let the STL formula $\\bar{\\varphi}_{q_i}^l$ be defined as: \n\\begin{equation}\n \\bar{\\varphi}_{q_i}^l= \\mathcal{G}_{[a_{q_i}^l,b_{q_i}^l]} \\bar{\\mu}_{q_i}^l \\label{eq:always}\n\\end{equation}\nwhere\n\\begin{equation}\n [a_{q_i}^l,b_{q_i}^l]\\begin{cases}\\subseteq[a_i,b_i], \\quad i \\in \\mathcal{I}_{\\mathcal{F}}\\\\= [a_i,b_i],\\quad i\\in \\mathcal{I}_{\\mathcal{G}} \\end{cases} \\label{eq:alwaysint}\n\\end{equation}\nand $\\bar{\\mu}_{q_i}^l$ is a predicate defined by \\eqref{eq:predicate}. Let $\\phi_l=\\bigwedge_{q_i=1}^{p_l} \\bar{\\varphi}_{q_i}^l$, $l=1,\\ldots,v$. If there exists $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{X}$ such that $\\rho^{\\phi_1\\wedge \\ldots\\wedge \\phi_v}(\\mathbf{x},0)>0$, then $\\rho^\\phi(\\mathbf{x},0)>0$.\n\\end{proposition}\n\n\\begin{proof}\nFor $i\\in \\mathcal{I}_{\\mathcal{G}}$ the proof follows similar arguments to Theorem 1. For $i\\in \\mathcal{I}_{\\mathcal{F}}$, if $\\rho^{\\phi_1\\wedge \\ldots\\wedge \\phi_v}(\\mathbf{x},0)>0$ then, by \\eqref{eq:always}-\\eqref{eq:alwaysint} and the definition of the robust semantics, $\\rho^{\\bar{\\varphi}_{q_i}^l}(\\mathbf{z}_l,0)>0$ implies $ h_{q_i}^l(\\mathbf{z}_l(t))>0$ for every $ t\\in [a_{q_i}^l,b_{q_i}^l]$ and $l\\in V_i$. Since $ h_{q_i}^l(\\mathbf{z}_l), l \\in V_i$ are feasible solutions of \\eqref{eq:convex} we may conclude that $h_i(\\mathbf{x}(t))>0$, where $\\mathbf{x}(t)=A\\mathbf{z}(t)$ and $\\mathbf{z}(t)=\\begin{bmatrix}\\mathbf{z}_1^T(t) &\\ldots & \\mathbf{z}_v^T(t) \\end{bmatrix}^T$, for every $t\\in [a_{q_i}^l,b_{q_i}^l]\\subseteq [a_i,b_i]$. Hence, $\\rho^{\\varphi_i}(\\mathbf{x},0)=\\max_{t\\in [a_i,b_i]} h_i(\\mathbf{x}(t))\\geq \\max_{t\\in [a_{q_i}^l,b_{q_i}^l]} h_i(\\mathbf{x}(t))>0$. The rest of the proof is similar to that of Theorem 1.\n\\end{proof}\n\n\\begin{figure*}[!t]\n \\centering\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{ s21}\n \\caption{Trajectories of agents 1,4,5}\n \\label{fig:alw1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{s22}\n \\caption{Trajectories of agents 2,3}\n \\label{fig:alw2}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{s23}\n \\caption{Barrier Function Evolution}\n \\label{fig:alw3}\n \\end{subfigure}\n \\caption{Agents' Trajectories under the local STL tasks defined based on \\eqref{eq:predicate}, \\eqref{eq:always} and \\eqref{eq:alwaysint} and Barrier Function Evolution}\n \\label{fig:alwall}\n\\end{figure*}\n\\section{Simulations}\nConsider a team of $R=5$ agents. Without loss of generality the team is decomposed in 5 sub-teams: $\\mathcal{V}_k=\\{k\\}, k\\in \\mathcal{V}$. The agents' states $\\mathbf{x}_k, k\\in \\mathcal{V}$ evolve over time based on the following equation:\n\\begin{equation*}\n \\dot{\\mathbf{x}}_k=A_k\\mathbf{x}_k+\\mathbf{u}_k , \\; k=1,\\ldots,5\n\\end{equation*}\nwhere $A_k=\\begin{bmatrix}-0.5 &0\\\\1 & -1 \\end{bmatrix}$ for every $ k\\in \\{1,2,5\\}$ and $A_k=\\begin{bmatrix}-1 &-1\\\\0 & -3 \\end{bmatrix}$ for $k\\in \\{3,4\\}$. The states and inputs of the agents are subject to constraints, i.e., $\\mathbf{x}_k \\in \\mathbb{X}$, $\\mathbf{u}_k \\in \\mathbb{U}$ where $\\mathbb{X}=\\{\\mathbf{x}\\in \\mathbb{R}^2: \\Vert \\mathbf{x} \\Vert_2 \\leq d_x\\}$, $\\mathbb{U}=\\{\\mathbf{u}\\in \\mathbb{R}^2: \\Vert \\mathbf{u} \\Vert_2 \\leq d_u\\}$, $d_x=1$ and $d_u=5$. Consider the global STL formula $\\phi=\\bigwedge_{i=1}^4 \\varphi_i$ where $\\varphi_i,i\\in \\mathcal{I}$ are defined as: $ \\varphi_1=\\mathcal{G}_{[0,2.1]}( \\Vert \\mathbf{x}_1-\\mathbf{x}_2-p_x \\Vert_2^2 \\leq 0.1)$, $\\varphi_2= \\mathcal{G}_{[2,4]}( \\Vert \\mathbf{x}_3-\\mathbf{x}_4 \\Vert_{2}^2 \\leq 0.2)$, $ \\varphi_3= \\mathcal{F}_{[3,7]}(\\Vert \\mathbf{x}_5-\\mathbf{x}_4\\Vert_{P_1}^2 \\leq 0.2)$ and $\\varphi_4= \\mathcal{F}_{[8,10]}(\\Vert \\mathbf{x}_5-\\mathbf{x}_2\\Vert_{P_2}^2 \\leq 0.25)$, where $p_x=\\begin{bmatrix} 0.3 & 0.5\\end{bmatrix}^T$ and $P_1=\\text{diag}(4,1), \\; P_2=\\text{diag}(0.1,0.4)$ are positive definite weight matrices. Since the predicate functions corresponding to $\\varphi_i, i\\in \\mathcal{I}$ are quadratic, the proposed problem \\eqref{eq:convex} becomes a Quadratically Constrained Quadratic Program (QCQP) and is efficiently solved using \nthe \\textit{Opti Toolbox} \\cite{opti}. The average computational time of the QCQPs is 0.052sec on an Intel Core i7-8665U with 16GB RAM using MATLAB. \n\nTo verify the validity of Theorem 2 and Proposition 1 we design agents' trajectories using the MPC scheme proposed in \\cite{ecc} with a sampling frequency of 10 Hz and optimization horizon length $N=1$. Each agent $k$ solves a local MPC problem without communicating with its peers since the satisfaction of the assigned tasks depends only on its own behavior. Here, a single, time-varying barrier $b_k(\\mathbf{x}_k,t), k\\in \\mathcal{V}$ is considered and designed offline encoding the local STL task specifications $\\phi_k$ corresponding to $\\mathcal{V}_k$. For every subtask of $\\phi_k$ a temporal behavior is designed for agent $k$ such that the satisfaction of $\\phi_k$ with a robustness value $r=0.005$ is guaranteed when $b_k(\\mathbf{x}_k,t)\\geq 0$ is true for every $t\\in [0,10]$. For details on the design of the barrier function $b_k(\\mathbf{x}_k,t)$ see \\cite{lars_linear,ecc}. The local STL task $\\phi_k$ assigned to each agent $k$ is defined by \\eqref{eq:newformula}, \\eqref{eq:interval}-\\eqref{eq:predicate} as follows:\n\\begin{align*}\n \\phi_1&=\\mathcal{G}_{[0,2.1]}\\;\\bar{\\mu}_1^1=\\bar{\\varphi}_1^1\\\\\n \\phi_2&= (\\mathcal{G}_{[0,2.1]}\\;\\bar{\\mu}_1^2)\\wedge(\\mathcal{F}_{[9,9]}\\; \\bar{\\mu}_4^2)=\\bar{\\varphi}_1^2 \\wedge \\bar{\\varphi}_4^2\\\\\n \\phi_3&= \\mathcal{G}_{[2,4]}\\;\\bar{\\mu}_2^3=\\bar{\\varphi}_2^3 \\\\\n \\phi_4&= (\\mathcal{G}_{[2,4]}\\;\\bar{\\mu}_2^4)\\wedge(\\mathcal{F}_{[7,7]}\\; \\bar{\\mu}_3^4)=\\bar{\\varphi}_2^4 \\wedge \\bar{\\varphi}_3^4\\\\\n \\phi_5&= (\\mathcal{F}_{[7,7]}\\;\\bar{\\mu}_3^5)\\wedge(\\mathcal{F}_{[9,9]}\\; \\bar{\\mu}_4^5)=\\bar{\\varphi}_3^5 \\wedge \\bar{\\varphi}_4^5\n\\end{align*}\nIn Figure \\eqref{fig:ev1}, \\eqref{fig:ev2} the agents' trajectories and the zero level sets $S_q^k$ of the predicate functions $h_q^k(\\mathbf{x}_k)$ are shown when the parameters $c_q^k,r_q^k$ are found as solutions to \\eqref{eq:convex}. Since the agents move on $\\mathbb{R}^2$ and $r_q^k\\neq 0$ for every $q$ and $k$ the zero level sets define square areas with edge length $r_q^k$. In Figure \\eqref{fig:ev3} the evolution of the local barrier functions $b_k(\\mathbf{x}_k,t)$ is shown. Since $\\min_k \\inf_{t\\in [0,10]} b_k(\\mathbf{x}_k,t)\\geq 5.38\\cdot 10^{-4}$ is true, we can conclude that $\\rho^{\\phi_k}(\\mathbf{x}_k,0)\\geq 0.005$ for every $k\\in \\mathcal{V}$. To validate Theorem 1 and given the trajectories of the agents found by the local MPC controllers we aim at designing a barrier function $b_c(\\mathbf{x},t)$ encoding the global specifications described by $\\phi$ and evaluating its value over the interval $[0,10]$. If $b_c(\\mathbf{x},t)\\geq 0$ is true for every $t\\in [0,10]$, then the global formula $\\phi$ is satisfied. From Figure \\eqref{fig:ev3} we have that $\\inf_{t\\in [0,10]} b_c(\\mathbf{x},t)\\geq 0.0234$. Hence, $\\mathbf{x} \\models \\phi$.\n\nNext, we consider the alternative definition of the local tasks as described in Proposition 1. Observe that the local tasks $\\phi_1, \\phi_3$ remain the same. The new local tasks $\\phi_2, \\phi_4, \\phi_5$ are defined as: $\\phi_2=\\bar{\\varphi}_1^2 \\wedge\\bar{\\varphi}_4^2$, $\\phi_4=\\bar{\\varphi}_2^4 \\wedge \\bar{\\varphi}_3^4$ and $\\phi_5=\\bar{\\varphi}_3^5 \\wedge \\bar{\\varphi}_4^5 $, where $\\bar{\\varphi}_1^2=\\mathcal{G}_{[0,2.1]}\\;\\bar{\\mu}_1^2$, $\\bar{\\varphi}_4^2=\\mathcal{G}_{[9,10]}\\; \\bar{\\mu}_4^2$, $\\bar{\\varphi}_2^4=\\mathcal{G}_{[2,4]}\\;\\bar{\\mu}_2^4$, $\\bar{\\varphi}_3^4=\\mathcal{G}_{[5,7]}\\; \\bar{\\mu}_3^4$, $\\bar{\\varphi}_3^5=\\mathcal{G}_{[5,7]}\\;\\bar{\\mu}_3^5$ and $\\bar{\\varphi}_4^5=\\mathcal{G}_{[9,10]}\\; \\bar{\\mu}_4^5$. In Figure \\eqref{fig:alw1} and \\eqref{fig:alw2} the agents' trajectories are shown. Following a similar procedure as before, we design a set of local barrier functions $b_k(\\mathbf{x}_k,t)$ and a function $b_c(\\mathbf{x},t)$ with robustness $r=0.005$. Based on Figure \\eqref{fig:alw3}, $\\min_k \\inf_{t\\in [0,10]} b_k(\\mathbf{x}_k,t)\\geq 5.17\\cdot 10^{-4}$ implying $\\mathbf{x}_k \\models \\phi_k, k \\in \\mathcal{V}$. Additionally, it holds that $\\inf_{t\\in [0,10]} b_c(\\mathbf{x},t)\\geq 0.0234$. Hence, $\\rho^{\\phi}(\\mathbf{x},0)\\geq 0.005$.\n\n\\section{Conclusions}\nIn this work a global STL formula is decomposed to a set of local STL tasks whose satisfaction depends on an a-priori chosen subset of agents. The predicate functions of the new formulas are chosen as functions of the infinity norm of the agents' states. A convex optimization problem is, then, designed for optimizing their parameters towards increasing the volume of their zero level-sets. Two alternatives are proposed for defining the local STL tasks in both of which the interval of satisfaction corresponding to the eventually formulas is considered a designer's choice. Future work will consider a more sophisticated framework for choosing the interval of satisfaction of the formulas aiming at increasing the total robustness of the task.\n\n\n\n \n \n \n \n \n\n\n\n\n\n\n\n\n\n\n\\printbibliography\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:intro} \nThe replicator equation is the most widely applied dynamics among evolutionary game models. It also gives the first dynamical model in evolutionary game theory \\cite{TJ}, thereby establishing an important connection to theoretical explanations of animal behaviors \\cite{MP}. For this application, the derivation of the equation considers a large well-mixed population. The individuals implement strategies from a finite set $S$ with $\\#S\\geq 2$, such that the payoff to $\\sigma\\in S$ from playing against $\\sigma'\\in S$ is $\\Pi(\\sigma,\\sigma')$. In the continuum, the density $X_\\sigma$ of strategy $\\sigma$ evolves with a per capita rate given by the difference between its payoff \n\\begin{align}\\label{def:Fsigma}\nF_{\\sigma}(X)=\\sum_{\\sigma'\\in S}\\Pi(\\sigma,\\sigma')X_{\\sigma'}\n\\end{align}\nand the average payoff of the population, where $X=(X_{\\sigma'};\\sigma'\\in S)$.\n Hence, the vector density process of strategy obeys the following replicator equation:\n\\begin{align}\\label{eq:replicator}\n\\dot{X}_\\sigma=X_\\sigma\\left(F_\\sigma(X)-\\sum_{\\sigma''\\in S}F_{\\sigma''}(X)X_{\\sigma''}\\right),\\quad \\sigma\\in S.\n\\end{align}\nThis equation is a point of departure for studying connections between the payoff matrix $(\\Pi(\\sigma,\\sigma'))_{\\sigma,\\sigma'\\in S}$, and the equilibrium states of the model by methods from dynamical systems. See \\cite{Cressman,HS} for an introduction and more properties. The replicator equation also arises from the Lotka--Volterra equation of ecology and Fisher's fundamental theorem of natural selection \\cite{HS,SS}. \n\nIn this paper, we consider the stochastic evolutionary game dynamics in large finite structured populations. Our goal is to prove that the vector density processes of strategy converge to the replicator equation. In this direction, one of the major results in the biological literature is the convergence to the replicator equation on large random regular graphs~\\cite{ON}. The authors further conjecture that their approximations extend to more general graphs \\cite[Section~5]{ON}. To obtain the proofs, we view the model as a perturbation of the voter model, since this viewpoint has made possible several mathematical results for it (e.g. \\cite{CDP:2,C:BC, CD,CMN,ALCFYN,C:EGT}). Our starting point here is the method in \\cite{C:EGT}, extended from \\cite{CCC,CC}, for proving the diffusion approximations of the time-changed density processes of strategy under \\emph{weak selection}. In that context, the corresponding perturbations away from the voter model use strengths typically given by $w=\\mathcal O(1\/N)$, where $N$ is the population size. \n\nThe questions from \\cite{ON} nevertheless concern very different properties. The crucial step of the method in \\cite{C:EGT} develops along the equivalence of probability laws in the limit between the evolutionary game model and the voter model. Now, this property breaks down for nontrivial parameters according to the limiting equation from \\cite{ON}; distributional limits of the density processes under the evolutionary game and the voter model degenerate to delta distributions of distinct deterministic functions as solutions of differential equations. As will be explained in this introduction, the convergence to the replicator equation also requires the different range of perturbation strengths $w$ satisfying $1\/N\\ll w\\ll 1$. This stronger perturbation implies weaker relations between the two models, and thus, calls for perturbation estimates of the evolutionary game model by the voter model generalizing those in \\cite{C:EGT}. With this change of perturbation strengths, the choice of time changes for the density processes and the characterization of coefficients of the limiting equation are the further tasks to be settled.\n\nBefore further explanations of the main results of \\cite{ON,C:EGT}, let us specify the evolutionary game model considered throughout this paper. First, to define spatial structure, we impose directed weights $q(x,y)$ on all pairs of sites $x$ and $y$ in a given population of size $N$. We assume that $q$ is an irreducible probability transition kernel with a zero trace $\\sum_{x}q(x,x)=0$. The perturbation strength $w>0$ defines the selection intensity of the model in the following form: for an individual at site $x$ using strategy $\\xi(x)$, its interactions with the neighbors determine the fitness as the sum $1-w+w\\sum_{y}q(x,y)\\Pi\\big(\\xi(x),\\xi(y)\\big)$, where $\\xi(y)$ denotes the strategy held by the neighbor at $y$. Under the condition of positive fitness by tuning the selection intensity appropriately, the death-birth updating requires that in a transition of state, an individual is chosen to die with rate $1$. Then the neighbors compete to fill in the site by reproduction with probability proportional to the fitness. Although the main results of this paper extend to include mutations of strategies, we relegate this additional mechanism until Section~\\ref{sec:mainresults}. \n\nBesides the case of mean-field populations, the density processes of strategy play a significant role in the biological literature for studying equilibrium states of spatial evolutionary games. Here and throughout this paper, we refer the density of $\\sigma$ under population configuration $\\xi$ to the weighted sum $\\sum_x \\pi(x)\\mathds 1_\\sigma\\big( \\xi(x)\\big)$, where $\\pi$ is the stationary distribution associated with the transition probability $q$. For such macroscopic descriptions of the model, the critical issue arises from the non-closure of the stochastic equations. The density processes are projections of the whole system, and in general, the density functions are not Markov functions in the sense of \\cite{RP}. More specifically, for the evolutionary game with death-birth updating introduced above, the microscopic dynamics from pairwise interactions determine the densities' dynamics. It is neither clear how to reduce the densities' dynamics analytically in the associated Kolmogorov equations. See \\cite[Section~1]{C:EGT} for more details on these issues and the physics method discussed below. \n\nOne of the main results in \\cite{ON} shows that for selection intensities $w\\ll 1$, the \\emph{expected} density processes on large random $k$-regular graphs for any integer $k\\geq 3$ approximately obey the following extended form of the replicator equation :\n\\begin{align}\\label{eq:replicatorext}\n\\dot{X}_\\sigma=wX_\\sigma\\left(F_\\sigma(X)+\\widetilde{F}_\\sigma(X)-\\sum_{\\sigma''\\in S}F_{\\sigma''}(X)X_{\\sigma''}\\right),\\quad \\sigma\\in S.\n\\end{align}\nHere, $F_\\sigma(X)$ and $\\widetilde{F}_\\sigma(X)$ are linear functions in $X=(X_{\\sigma'};\\sigma'\\in S)$ such that the constant coefficients are explicit in the payoff matrix and the graph degree $k$. See \\cite[Equations (22) and (36)]{ON}. Note that \\eqref{eq:replicatorext} underlines the nontrivial effect of spatial structure, since the coefficients are very different from those for the replicator equation \\eqref{def:Fsigma} in mean-field populations. For the derivation of \\eqref{eq:replicatorext}, \\cite{ON} applies the physics method of pair approximation. It enables the asymptotic closure of the equations of the density processes by certain moment closure approximations and circumvents the fundamental issues discussed above. Moreover, based on computer simulations from \\cite{OHLN}, the authors of \\cite{ON} conjecture that the approximate replicator equation for the density processes applies to many non-regular graphs, provided that the constant graph degree $k$ in the coefficients of the replicator equation is replaced by the corresponding average degree. In approaching this conjecture, it is still not clear on how the average degrees of graphs enter. The method in this paper does not extend to this generality either. On the other hand, even within the scope of large random regular graphs, the constant graph degrees and the locally regular tree-like property seem essential in \\cite{ON}. We notice that locally tree-like spatial structures are known to be useful to pair approximations in general \\cite{SF}. \n\nIn the case of two strategies, the supplementary information (SI) of \\cite{OHLN} shows that the density processes of a fixed one approximate the Wright--Fisher diffusion with drift. The derivation also applies pair approximations on large random regular graphs, although it is noticeably different from the derivation in \\cite{ON} for the replicator equation on graphs. (A slow-fast dynamical system for the density and a certain rapidly convergent local density is considered in \\cite[SI]{OHLN}.) It is neither clear how to justify the derivation in \\cite[SI]{OHLN} mathematically. On the other hand, the diffusion approximations of the density processes can be proven on large finite spatial structures subject to appropriate, but general, conditions \\cite[Theorem~4.6]{C:EGT} that include random regular graphs as a special case. See \\cite{CFM06,CFM08} for mathematical investigations of moment closure in other spatial biological models and some general discussions, among other mathematical works in this direction.\n\nThe method in \\cite{C:EGT} begins with the aforementioned asymptotic equivalence of probability laws via perturbations for $w=\\mathcal O(1\/N)$ (not just equivalence of laws of the density processes). The $1\/N$-threshold is sharp such that the critical case yields nontrivial drifts of the limiting diffusions. This relation reduces the convergence of the game density processes to a convergence problem of the voter model. For the latter, fast mixing of spatial structure ensures approximations of the coalescence times in the ancestral lineages by analogous exponential random variables from large mean-field populations. This method goes back to \\cite{Cox}. Moreover, for the voter density processes, the relevant coalescence times can be reduced to the meeting times for two independent copies of the stationary Markov chains over the populations. The almost exponentiality of hitting times \\cite{Aldous:AE, AB,AF:MC} applies to these times and leads to the classical diffusivity in the voter density processes in general spatial populations \\cite{CCC,CC}. The first moments of these meeting times are also used to time change the densities for the convergence. \n\nBesides the methods, the convergence results in \\cite[Theorem~4.6]{C:EGT} for the game density processes under the specific setting of large random regular graphs and the payoff matrices for prisoner's dilemma games are closely related to the replicator equation on graphs from \\cite{ON}. See \\eqref{prisoner} for these payoff matrices and \\cite{C:MT} for the exact asymptotics $N(k-1)\/[2(k-2)]$, $N\\to\\infty$, of the expected meeting times on the large random $k$-regular graphs. In this case, the diffusion approximations in \\cite[SI]{OHLN} hold to the degree of matching constants if the time changes are formally undone \\cite{C:MT}. (See also \\cite[Remark~3.1]{C:MT} for a correction of inaccuracy in \\cite{C:EGT} on passing limits along random regular graphs.) This standpoint extends to a recovery of the replicator equation on graphs from \\cite{ON} by a similar formal argument. It shows that these results in \\cite{ON,OHLN}, both due to pair approximations, are algebraically consistent with each other. See the end of Section~\\ref{sec:mainresults} for details and the second main result discussed below for further comparison. In addition to its own interest, the replicator equation on graphs concerns a unified characterization of the evolutionary game within an enlarged range of selection intensities as mentioned above. \n\nThe main results of this paper obtain the convergence to the replicator equation under the above specific setting, in addition to extensions to general spatial populations and payoff matrices. Multiple strategies and mutations are allowed. See Theorem~\\ref{thm:main} and Corollary~\\ref{cor:symmetric}. For the extended context, the first main result [Theorem~\\ref{thm:main} (1${^\\circ}$)] proves the convergence of the vector density processes of strategy under the following assumptions. We require that the stationary distributions associated with the spatial structures are asymptotically comparable to the uniform distributions (see \\eqref{cond:pi}), and these spatial structures allow for suitable time changes of the density processes and suitable selection intensities (Definition~\\ref{def:admissible}). Here, for the typical eligible populations, the time changes can range in $1\\ll\\theta \\ll N$. The selection intensities are \\emph{of the inverse order} so that $1\/N\\ll w\\ll1 $. Then the precise limiting equation is given by \\eqref{eq:replicatorext}, with the selection intensity $w$ replaced by a constant $w_\\infty$ as a limit of the parameters. The proof also determines $F_\\sigma(X)$ and $\\widetilde{F}_\\sigma(X)$ for \\eqref{eq:replicatorext}:\n\\begin{align}\nF_\\sigma(X)&=\\overline{\\kappa}_{0|2|3}\\sum_{\\sigma'\\in S}\\Pi(\\sigma,\\sigma')X_{\\sigma'},\\label{F1}\\\\\n\\begin{split}\n\\widetilde{F}_\\sigma(X)&=(\\overline{\\kappa}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3})\\Pi(\\sigma,\\sigma)\\\\\n&\\quad +\\sum_{\\sigma'\\in S}({\\overline{\\kappa}}_{(0,3)|2}-{\\overline{\\kappa}}_{0|2|3})[\\Pi(\\sigma,\\sigma')-\\Pi(\\sigma',\\sigma)]X_{\\sigma'}\\\\\n&\\quad -(\\overline{\\kappa}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3})\\sum_{\\sigma'\\in S}\\Pi(\\sigma',\\sigma')X_{\\sigma'}.\\label{F2}\n\\end{split}\n\\end{align}\nHere, $\\overline{\\kappa}_{(2,3)|0}$, ${\\overline{\\kappa}}_{(0,3)|2}$, and $\\overline{\\kappa}_{0|2|3}$ are nonnegative constants defined by the asymptotics of some coalescent characteristics of the spatial structures. See Section~\\ref{sec:slow} for the definitions of these constants. \n\nThe first main result [Theorem~\\ref{thm:main} (1${^\\circ}$)] has the meaning of spatial universality as the diffusion approximations of the voter model and the evolutionary game in \\cite{CCC,CC,C:EGT}, although it does not recover the explicit equations obtained in \\cite{ON} on large random regular graphs under general payoff matrices. The conditions do not require convergence of local geometry as in the large discrete tori and large random regular graphs. The spatial structures can remain sparse in the limit, which is in stark contrast to the usual assumptions for proving scaling limits of particle systems. The locally tree-like property usually assumed in pair approximations is not required either. Based on these properties, the first main result [Theorem~\\ref{thm:main} (1${^\\circ}$)] gives an answer in the positive for the conjecture in \\cite{ON} to the degree of using constants that may depend implicitly on the space: The approximations of the expected density processes by the replicator equation extend to many non-regular graphs, whenever the initial conditions converge deterministically. \n \nTo further the formal comparison mentioned above with the approximate Wright--Fisher diffusion from \\cite[SI]{OHLN}, the second main result [Theorem~\\ref{thm:main} (2${^\\circ}$)] considers one additional aspect for the convergence of the density processes. In this part, the normalized fluctuations are proven to converge to a vector centered Gaussian martingale [Theorem~\\ref{thm:main} (2${^\\circ}$)]. The quadratic covariation is the Wright--Fisher diffusion matrix in the limiting densities $X$: $\\int_0^{ t} X_\\sigma(s)[\\delta_{\\sigma,\\sigma'}-X_{\\sigma'}(s)]\\d s$, $\\sigma,\\sigma'\\in S$, where $\\delta_{\\sigma,\\sigma'}$ are the Kronecker deltas. For the case of only two strategies on large random regular graphs, this covariation formally recovers the approximate Wright--Fisher diffusion term from \\cite[SI]{OHLN}. Note that this result and the convergence to the replicator equation do not imply the diffusion approximations of the density processes. \n\nIn the rest of this introduction, we explain the proof of the first main result. Its investigation raises all the central technical issues pointed out above. First, the lack of an asymptotic equivalence of probability laws is resolved via the populations' microscopic dynamics driving the density processes. Duhamel's principle replaces the pathwise, global change of measure method in \\cite{C:EGT} and shows the irrelevance of selection intensities in the microscopic dynamics (Proposition~\\ref{prop:duhamel}). This approach then links to the decorrelation proven in \\cite[Section~4]{CCC} for some ``local'' meeting time distributions, from the ancestral lineages, driving the dynamics of the voter density processes. Here, local meeting times refer to those where the initial conditions of the Markov chains are within fixed numbers of edges. \n\nThe decorrelation property from \\cite{CCC} shows that the probability distributions of those particular local meeting times for general populations converge to nontrivial convex combinations of the delta distributions at zero and infinity. The time scales are slower than those for the diffusion approximations. In particular, the exponential distribution is not just absent. No distributions with a nonzero mass between zero and infinity arise in the limit. Informally speaking, the decorrelation occurs at time scales between the period when details of the spatial structures dominate and the period when the almost exponentiality \\cite{Aldous:AE, AB,AF:MC} plays a role. To us, this presence of multiple time scales in the evolutionary dynamics is reminiscent of the slow-fast dynamical system in \\cite[SI]{OHLN}.\n\nFor the convergence to the replicator equation, the choice of the time changes for the densities and the characterization of the limiting equation use the decorrelation from \\cite{CCC} and its extensions. First, the time changes can only grow slower than those for the diffusion approximations since the limiting trajectories are less rougher. This requirement relates the convergence to the decorrelation. We are now interested in proving the best possible range of growing time changes for the decorrelation, not just using the particular ones from \\cite{CCC}. After all, in \\cite{ON}, the replicator equation is expected to be present within the broad range $w\\ll 1$, and our argument requires the selection intensities to be of the inverse order of the time changes. Moreover, the application of Duhamel's principle mentioned above leads to the entrance of various local meeting times more than those for the voter densities in \\cite{CCC}. Simultaneous decorrelation in these local meeting times is essential for getting a deterministic limiting differential equation: This property involves asymptotic path regularity of the density processes. The constant coefficients in \\eqref{F1} and \\eqref{F2} also arise as the weights at infinity in the limiting local meeting time distributions for the typical eligible populations. See Sections~\\ref{sec:slow} and \\ref{sec:eqn} for the related proofs. \\medskip \n\n\\noindent {\\bf Organization.} Section~\\ref{sec:mainresults} introduces the evolutionary game model and the voter model analytically and discusses the main results (Theorem~\\ref{thm:main} and Corollary~\\ref{cor:symmetric}). In Section~\\ref{sec:dynamics}, we define the voter model and the evolutionary game model as semimartingales and briefly explain the role of the coalescing duality. In Section~\\ref{sec:slow}, we quantify the time changes in proving the main results and characterize the coefficients of the limiting equation. Section~\\ref{sec:eqn} is devoted to the main arguments of the proofs of Theorem~\\ref{thm:main} and Corollary~\\ref{cor:symmetric}. Finally, Section~\\ref{sec:coal} presents some auxiliary results for coalescing Markov chains.\\medskip\n\n\\noindent {\\bf Acknowledgments}\nThe author would like to thank Lea Popovic for comments on earlier drafts and Sabin Lessard for pointing out several references from the literature. Support from the Simons Foundation before the author's present position and from the Natural Science and Engineering Research Council of Canada is gratefully acknowledged. \n\n\n\n\n\\section{Main results}\\label{sec:mainresults}\nIn this section, we introduce the stochastic spatial evolutionary game with death-birth updating in more detail. A discussion of the main results of this paper then follows. To be consistent with the viewpoint of voter model perturbations and the neutral role of the voter model, strategies will be called {\\bf types} in the rest of this paper. The settings here and in the next section are adapted from those in \\cite{CCC,CC,C:EGT} to the context of evolutionary games with multiple types. \n\nRecall that a discrete spatial structure considered in this paper is given by an irreducible, reversible probability kernel $q$ on a finite nonempty set $E$ such that ${\\sf tr}(q)=\\sum_{x\\in E}q(x,x)=0$. Write $N=\\#E$ and $\\pi$ for the unique stationary distribution of $q$. The interactions of individuals are defined by a payoff matrix $\\Pi=(\\Pi(\\sigma,\\sigma'))_{\\sigma,\\sigma'\\in S}$ of real entries. Fix $\\overline{w}\\in (0,\\infty)$ such that\n\\begin{align}\\label{def:wbar}\nw+w\\sum_{z\\in E}q(y,z)|\\Pi\\big(\\xi(y),\\xi(z)\\big)|<1,\\quad \\forall\\;w\\in [0,\\overline{w}],\\;y\\in E.\n\\end{align}\nThen the following perturbed transition probability is used to update types of individuals due to interactions:\n\\begin{align}\nq^w(x,y,\\xi)&\\stackrel{\\rm def}{=} \\frac{q(x,y)\\left[(1-w)+w\\sum_{z\\in E}q(y,z)\\Pi\\big(\\xi(y),\\xi(z)\\big)\\right]}{\\sum_{y'\\in E}q(x,y')\\left[(1-w)+w\\sum_{z\\in E}q(y',z)\\Pi\\big(\\xi(y'),\\xi(z)\\big)\\right]}.\\label{def:qw}\n\\end{align}\nWith these updates and the updates based on a mutation measure $\\mu$ on $S$, two types of configurations $\\xi^{x,y},\\xi^{x|\\sigma}\\in S^E$ result. They are obtained from $\\xi\\in S^E$ by changing only the type at $x$ such that $\\xi^{x,y}(x)=\\xi(y)$ and $\\xi^{x|\\sigma}(x)=\\sigma$. Hence, the evolutionary game $(\\xi_t)$ is a Markov jump process with a generator given by\n\\begin{align}\\label{def:Lw}\n\\begin{split}\n\\mathsf L^{w} H(\\xi)=&\\sum_{x,y\\in E}q^w(x,y,\\xi)[H(\\xi^{x,y})-H(\\xi)]\\\\\n&+\\sum_{x\\in E}\\int_S [H(\\xi^{x|\\sigma})-H(\\xi)]\\d \\mu(\\sigma),\\quad H:S\\to {\\Bbb R}.\n\\end{split}\n\\end{align}\nThe first sum on the right-hand side of \\eqref{def:Lw} governs changes of types due to selection, and the second sum is responsible for mutations. Given $\\xi\\in S^E$ and a probability distribution $\\nu$ on $S^E$ as initial conditions, we write $\\P^w_\\xi$ and ${\\mathbb E}^w_\\xi$, or $\\P^w_\\nu$ and ${\\mathbb E}^w_\\nu$, under the laws associated with $\\mathsf L^w$. For $w=0$, the generator $\\mathsf L^{w}$ is reduced to the generator $\\mathsf L$ of the multi-type voter model with mutation, and the notation $\\P$ and ${\\mathbb E}$ is used.\n\n\n\nThe object in this paper is the vector density processes $p(\\xi_t)=(p_\\sigma(\\xi_t);\\sigma\\in S)$ for the evolutionary game with death-birth updating. Here, the density function of $\\sigma\\in S$ is given by\n\\begin{align}\\label{def:density}\np_\\sigma(\\xi)=\\sum_{x\\in E}\\mathds 1_\\sigma\\circ\\xi(x)\\pi(x),\n\\end{align}\nwhere $f\\circ \\xi(x)=f(\\xi(x))$. Under $\\P^w$, $p_\\sigma(\\xi_t)$ admits a semimartingale decomosition:\n\\begin{align}\\label{density:dynamics}\np_\\sigma(\\xi_t)=p_\\sigma(\\xi_0)+A_\\sigma(t)+M_\\sigma(t),\n\\end{align}\nwhere $A_\\sigma(t)=\\int_0^t \\mathsf L^w p_\\sigma(\\xi_s)\\d s $. In the sequel, we study the convergence of the vector density processes and the martingales $M_\\sigma$ separately, along an appropriate sequence of discrete spatial structures $(E_n,q^{(n)})$ with $N_n=\\#E_n\\to\\infty$. \\medskip\n\n\n\\noindent {\\bf Convention for superscripts and subscripts.} Objects associated with $(E_n,q^{(n)})$ will carry either superscripts ``$(n)$'' or subscripts ``$n$'', although additional properties may be assumed so that these objects are not just based on $(E_n,q^{(n)})$. Otherwise, we refer to a fixed spatial structure $(E,q)$. \\hfill $\\blacksquare$\\medskip\n\n\n\nFor the main theorem, we choose parameters as time changes for the density processes, mutation measures, and selection intensities. The choice is according to the underlying discrete spatial structures. We use $\\nu_n(\\mathds 1)=\\sum_{x\\in E_n}\\pi^{(n)}(x)^2$ and the first moment $\\gamma_n$ of the first meeting time of two independent stationary rate-$1$ $q^{(n)}$-Markov chains. The other characteristic of the spatial structure considers the mixing time $\\mathbf t^{(n)}_{\\rm mix}$ of the $q^{(n)}$-Markov chains and the spectral gap $\\mathbf g_n$ as follows. Recall that the semigroup of the continuous-time rate-$1$ $q^{(n)}$-Markov chain is given by $({\\rm e}^{t(q^{(n)}-1)};t\\geq 0)$. With\n\\begin{align}\\label{def:dE}\nd_{E_n}(t)=\\max_{x\\in E_n}\\big\\|{\\rm e}^{t(q^{(n)}-1)}(x,\\cdot)-\\pi^{(n)}\\big\\|_{\\rm TV}\n\\end{align}\nfor $\\|\\cdot\\|_{\\rm TV}$ denoting the total variation distance, we choose\n\\begin{align}\n\\label{def:tmix}\n\\mathbf t^{(n)}_{\\rm mix}=\\inf\\{t\\geq 0;d_{E_n}(t)\\leq (2{\\rm e})^{-1}\\}.\n\\end{align}\nThe spectral gap $\\mathbf g_n$ is the distance between the largest and second largest eigenvalues of $q^{(n)}$. \n\n\n\\begin{defi}\\label{def:admissible}\nFor all $n\\geq 1$, let $\\theta_n\\in (0,\\infty)$ be a time change, $\\mu_n$ a mutation measure on $S$, and $w_n\\in [0,\\overline{w}]$. The sequence $(\\theta_n,\\mu_n,w_n)$ is said to be {\\bf admissible} if all of the following conditions hold. First, $(\\theta_n)$ satisfies\n\\begin{align}\\label{cond1:thetan}\n\\lim_{n\\to\\infty}\\theta_n=\\infty,\\quad \\lim_{n\\to\\infty}\\frac{\\theta_n}{\\gamma_n}<\\infty,\\quad \n\\lim_{n\\to\\infty}\\gamma_n\\nu_n(\\mathds 1){\\rm e}^{-t\\theta_n}= 0,\\quad \\forall\\;t\\in (0,\\infty),\n\\end{align}\nand at least one of the two mixing conditions holds: \n\\begin{align}\\label{cond2:thetan}\n\\lim_{n\\to\\infty}\\gamma_n\\nu_n(\\mathds 1){\\rm e}^{-\\mathbf g_n \\theta_n}=0\\quad\\mbox{or}\\quad \\lim_{n\\to\\infty}\\frac{\\mathbf t^{(n)}_{\\rm mix}}{\\theta_n}[1+\\log^+(\\gamma_n\/\\mathbf t^{(n)}_{\\rm mix})]=0,\n\\end{align}\nwhere $\\log^+\\alpha=\\log (\\max\\{\\alpha,1\\})$. Second, we require the following limits for $(\\mu_n)$ and $(w_n)$: \n\\begin{align}\n& \\lim_{n\\to\\infty} \\mu_n(\\sigma)\\theta_n=\\mu_\\infty(\\sigma)<\\infty,\\quad \\forall\\;\\sigma\\in S;\\label{def:mun}\\\\\n& \\lim_{n\\to\\infty}\\ w_n=0,\\quad \n\\lim_{n\\to\\infty}\\frac{w_n\\theta_n }{2\\gamma_n\\nu_n(\\mathds 1)}=w_\\infty<\\infty,\\quad \\limsup_{n\\to\\infty}w_n\\theta_n<\\infty.\n\\label{def:wn}\n\\end{align}\n\\end{defi}\n\n\n\nAnother condition of the main theorem requires that $\\sup_nN_n\\max_{x\\in E_n}\\pi^{(n)}(x)<\\infty$, which implies $\\gamma_n\\geq \\mathcal O(N_n)$ (see \\eqref{ergodic} or \\cite[(3.21)]{CCC} for details). In this context, the admissible $\\theta_n$ has the following effects. If $\\lim_{n}\\theta_n\/\\gamma_n\\in (0,\\infty)$, the time-changed density processes $p_1(\\xi_{\\theta_nt})$ of the voter model converge to the Wright--Fisher diffusion \\cite{CCC,CC}. Moreover, the density processes of the evolutionary game converge to the same diffusion but with a drift \\cite[Theorem~4.6]{C:EGT}. These diffusion approximations hold under the mixing conditions slightly different from those in \\eqref{cond2:thetan}. Therefore, assuming $\\lim_{n}\\theta_n\/\\gamma_n=0$ in \\eqref{cond1:sn} has the heuristic that the time-changed density processes have paths less rougher in the limit, and so, do not converge to diffusion processes. Note that this variation of time scales can be contrasted with, e.g., the context considered in \\cite{EN:80} where, among other results, the discrete processes converge to the equilibrium states of the limiting process due to faster time changes.\n\nThe other conditions for the admissible sequences mainly consider the typical case of ``transient'' spatial structures. The kernels are characterized by the condition $\\sup_n \\gamma_n\\nu_n(\\mathds 1)<\\infty$ \\cite[Remark~2.4]{CCC}. In this case, \\eqref{cond1:thetan} can be satisfied by any sequence $(\\theta_n)$ such that $1\\ll \\theta_n\\ll N_n$, and \\eqref{def:mun} and \\eqref{def:wn} allow for nonzero $\\mu_\\infty$ and $w_\\infty$. The somewhat tedious condition in \\eqref{def:wn} simplifies drastically, and we get $N_n^{-1}\\ll w_n\\ll 1$ when $\\lim_n w_n\\theta_n$ is nonzero. As for the mixing conditions in \\eqref{cond2:thetan}, they can pose severe limitations if the spatial structures are ``recurrent'' ($\\sup_n \\gamma_n\\nu_n(\\mathds 1)=\\infty$). In this case, we may not be able to find admissible sequences such that $w_\\infty>0$, so that the limiting equation to be presented below only allows constant solutions in the absence of mutation. \nFor example, the two-dimensional discrete tori satisfy $\\gamma_n\\nu_n(\\mathds 1)\\sim C\\log N_n$, $\\mathbf t^{(n)}_{\\rm mix}\\leq \\mathcal O(N_n)$ and $\\mathbf g_n=\\mathcal O(1\/N_n)$. See \\cite{Cox} and \\cite[Theorem~10.13 on p.133, Theorem~5.5 on p.66 and Section~12.3.1 on p.157]{LPW}. We can choose $\\theta_n=N_n(\\log\\log N_n)^2$ to satisfy \\eqref{cond1:thetan} with $\\lim_n\\theta_n\/\\gamma_n=0$, and the first mixing condition in \\eqref{cond2:thetan}. But now the admissible $(w_n)$ only gives $w_\\infty=0$. We notice that a similar restriction is pointed out in \\cite{Cox:Feller} on the low density scaling limits of the biased voter model, where the limit is Feller's branching diffusion with drift. \n\nFrom now on, we write $\\pi_{\\min}=\\min_{x\\in E}\\pi(x)$ and $\\pi_{\\max}=\\max_{x\\in E}\\pi(x)$ for the stationary distribution $\\pi$ of $(E,q)$. The main theorem stated below shows a law of large numbers type convergence for the density processes and a central limit theorem type convergence for the fluctuations. These two results do not combine to give the diffusion approximation of the density processes proven in \\cite{C:EGT}. \n \n\n\n\\begin{thm}\\label{thm:main}\nLet $(E_n,q^{(n)})$ be a sequence of irreducible, reversible probability kernels defined on finite sets with $N_n=\\#E_n\\to\\infty$. Assume the following conditions:\n\\begin{enumerate}\n\\item [\\rm (a)] Let $\\nu_n$ be a probability measure on $S^{E_n}$ such that $\\nu_n(\\xi;p(\\xi)\\in \\cdot)$ converges in distribution to a probability measure $\\overline{\\nu}_\\infty$ on $[0,1]^S$.\n\\item [\\rm (b)] It holds that \n\\begin{align}\\label{cond:pi}\n0<\\liminf_{n\\to\\infty}N_n\\pi^{(n)}_{\\min}\\leq \\limsup_{n\\to\\infty}N_n\\pi^{(n)}_{\\max}<\\infty.\n\\end{align}\n\\item [\\rm (c)] The limits in \\eqref{def:|kell1|ell2} and \\eqref{def:|||} defining the nonnegative constants $\\overline{\\kappa}_{(2,3)|0}$, $\\overline{\\kappa}_{(0,3)|2}$ and ${\\overline{\\kappa}}_{0|2|3}$ exist. These constants depend only on space. \n\\item [\\rm (d)] We can choose an admissible sequence $(\\theta_n,\\mu_n,w_n)$ as in Definition~\\ref{def:admissible} such that $\\lim_n\\theta_n\/\\gamma_n=0$.\n\\end{enumerate}\nThen the following convergence in distribution of processes holds:\n\\begin{enumerate}\n\\item [\\rm (1${^\\circ}$)] The sequence of the vector density processes $\\big(p(\\xi_{\\theta_nt}),\\P^{w_n}_{\\nu_n}\\big)$ converges to the solution $X$ of the following differential equation with the random initial condition $\\P(X_0\\in \\cdot)=\\overline{\\nu}_\\infty$:\n\\begin{align}\n\\begin{split}\\label{p1:lim}\n\\dot{X}_\\sigma&=w_\\infty X_\\sigma\\left(F_\\sigma(X)+\\widetilde{F}_\\sigma(X)-\\sum_{\\sigma''\\in S}F_{\\sigma''}(X)X_{\\sigma''}\\right)\\\\\n&\\quad\\; +\\mu_\\infty(\\sigma)(1-X_\\sigma)-\\mu_\\infty(S\\setminus\\{\\sigma\\}) X_\\sigma ,\\quad \\sigma\\in S,\n\\end{split}\n\\end{align}\nwhere $F_\\sigma(X)$ and $\\widetilde{F}_\\sigma(X)$ are linear functions in $X$ defined by \\eqref{F1} and \\eqref{F2}. \nMoreover, the sum of the ${\\overline{\\kappa}}$-constants in $F_\\sigma(X)$ and $\\widetilde{F}_\\sigma(X)$ is nontrivial to the following degree: \n\\begin{align}\\label{kappa:>}\n(\\overline{\\kappa}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3})+ \\overline{\\kappa}_{0|2|3}+({\\overline{\\kappa}}_{(0,3)|2}-{\\overline{\\kappa}}_{0|2|3})\\in (0,\\infty).\n\\end{align}\n\n\\item [\\rm (2${^\\circ}$)] Recall the vector martingale defined by \\eqref{density:dynamics}, and set $M^{(n)}_\\sigma(t)=(M_\\sigma(\\theta_nt);\\sigma\\in S)$ under $\\P^{w_n}_{\\nu_n}$. If, moreover, $\\lim_n \\gamma_n\\nu_n(\\mathds 1)\/\\theta_n=0$ holds, then $(\\gamma_n\/\\theta_n)^{1\/2}M^{(n)}$ converges to a vector centered Gaussian martingale with quadratic covariation $(\\int_0^tX_\\sigma(s)[\\delta_{\\sigma,\\sigma'}-X_{\\sigma'}(s)]\\d s;\\sigma,\\sigma'\\in S) $. \n\\end{enumerate}\n\\end{thm}\n\n\n\n\n\n\nWe present the proof of Theorem~\\ref{thm:main} in Section~\\ref{sec:eqn}. The existence of the limits in condition (c) is proven in Proposition~\\ref{prop:kell}. See Lemma~\\ref{lem:tight} for the additional condition in Theorem~\\ref{thm:main} (2${^\\circ}$).\n\n\n\nTo illustrate Theorem~\\ref{thm:main}, we consider the generalized prisoner's dilemma matrix in the rest of this section. The matrix is for games among individuals of two types : \n\\begin{align}\\label{prisoner}\n\\Pi=\n\\bordermatrix{~ & 1& 0 \\cr\n 1 & b-c & -c \\cr\n 0 & b & 0 \\cr}\n\\end{align}\nfor real entries $b,c$. (The usual prisoner's dilemma matrix requires $b>c>0$.) The proof of the following corollary also appears in Section~\\ref{sec:eqn}.\n\n\\begin{cor}\\label{cor:symmetric}\nLet conditions {\\rm (a)--(d)} of Theorem~\\ref{thm:main} be in force and $\\Pi$ be given by \\eqref{prisoner}. If, moreover, $q^{(n)}$ are symmetric ($q^{(n)}(x,y)\\equiv q^{(n)}(y,x)$) and \n\\begin{align}\\label{cond:q2}\n\\lim_{n\\to\\infty}\\gamma_n\\nu_n(\\mathds 1)\\pi^{(n)}\\{x\\in E_n;q^{(n),2}(x,x)\\neq q^{(\\infty),2}\\}=0\n\\end{align}\nfor some constant $q^{(\\infty),2}$, then the differential equation for $X_1=1-X_0$ takes a simpler form:\n\\begin{align}\\label{eq:X1}\n\\dot{X}_1=w_\\infty(bq^{(\\infty),2}-c)X_1(1-X_1)+\\mu_\\infty(1)(1-X_1)-\\mu_\\infty(0) X_1.\n\\end{align}\n\\end{cor}\n\\medskip\n\n\n\nCorollary~\\ref{cor:symmetric} applies to large random $k$-regular graph for a fixed integer $k\\geq 3$, with $q^{(\\infty),2}=1\/k$ and $\\gamma_n\/N_n\\to (k-1)\/[2(k-2)]$ (see \\eqref{MUU:RG} and the discussion there). Additionally, $(\\theta_n)$ can be chosen to be any sequence such that $1\\ll \\theta_n\\ll N_n$, and $(w_n)$ can be any such that $(w_n\\theta_n)$ converges in $[0,\\infty)$. See \\cite{C:MT} and Section~\\ref{sec:rrg}. (More precisely, the application needs to pass limits along subsequences, since these graphs are randomly chosen.) Assume the absence of mutation. Then in this case, one can \\emph{formally} recover the replicator equation \\eqref{eq:X1} from the drift term of the approximate Wright--Fisher diffusion in \\cite[SI]{OHLN} as follows. For the density process $p_1(\\xi_t)$ under $\\P^{w_n}$, that drift term reads\n\\begin{align}\\label{drift}\nw_n\\cdot \\frac{(k-2) (b-ck)}{k(k-1)}p_1(\\xi_t)[1-p_1(\\xi_t)].\n\\end{align}\nNote that $\\gamma_n\\approx N_n(k-1)\/[2(k-2)]$ as mentioned above and the choice in \\eqref{def:wn} of $w_n$ gives $w_n \\approx w_\\infty 2\\gamma_nN_n^{-1}\/\\theta_n$. By using these approximations and multiplying the foregoing drift term by $\\theta_n$ as a time change, we get the approximate drift $w_\\infty (b\/k-c)p_1(\\xi_{\\theta_nt})[1-p_1(\\xi_{\\theta_nt})]$ of $p_1(\\xi_{\\theta_nt})$. This approximation recovers \\eqref{eq:X1}. The same formal argument can be used to recover the noise coefficient in Theorem~\\ref{thm:main} (2${^\\circ}$). See also \\cite[Remark~4.10]{C:EGT} for the case of diffusion approximations.\n\n\n\n\n\\section{Semimartingale dynamics}\\label{sec:dynamics}\nIn this section, we define the voter model and the evolutionary game model as solutions to stochastic integral equations driven by point processes. Then we view these equations in terms of semimartingales and identify some leading order terms for the forthcoming perturbation argument. We recall the coalescing duality for the voter model briefly at the end of this section. \n\n\nFirst, given a triplet $(E,q,\\mu)$, an equivalent characterization of the corresponding voter model is given as follows. Introduce independent $(\\mathscr F_t)$-Poisson processes $\\{\\Lambda(x,y);x,y\\in E\\}$ and $\\{\\Lambda^\\sigma_t(x);\\sigma\\in S,x\\in E\\}$ such that\n\\begin{align}\n\\begin{split}\\label{rates}\n\\Lambda_t(x,y)& \\quad\\mbox{with rate}\\quad {\\mathbb E}[\\Lambda_1(x,y)]=q(x,y)\\quad\\mbox{and}\\\\\n\\Lambda^\\sigma_t(x)&\\quad \\mbox{with rate}\\quad {\\mathbb E}[\\Lambda^\\sigma_1(x)]=\\mu(\\sigma),\\quad x,y\\in E,\\;\\sigma\\in S.\n \\end{split}\n\\end{align}\nThese jump processes are defined on a complete filtered probability space $\\big(\\Omega,\\mathscr F,(\\mathscr F_t),\\P\\big)$. Then given an initial condition $\\xi_0\\in S^E$, the $(E,q,\\mu)$-voter model can be defined as the pathwise unique $S^E$-valued solution of the following stochastic integral equations \\cite{CDP,MT}: for $x\\in E$ and $\\sigma\\in S$, \n\\begin{align}\n\\begin{split}\n\\mathds 1_\\sigma\\circ \\xi_t(x)&=\\mathds 1_\\sigma\\circ \\xi_0(x)+\\sum_{y\\in E}\\int_0^t [\\mathds 1_\\sigma\\circ \\xi_{s-}(y)-\\mathds 1_\\sigma\\circ \\xi_{s-}(x)]\\d \\Lambda_s(x,y)\\\\\n&{\\quad \\,}+\\int_0^t \\mathds 1_{\\sigma_{S\\setminus\\{\\sigma\\}}}\\circ \\xi_{s-}(x)\\d \\Lambda^\\sigma_s(x)-\\sum_{\\sigma'\\in S\\setminus\\{\\sigma\\}}\\int_0^t\\mathds 1_\\sigma\\circ \\xi_{s-}(x)\\d \\Lambda^{\\sigma'}_s(x).\n\\label{eq:voter}\n\\end{split}\n\\end{align}\nHence, the type at $x$ is replaced and changed to the type at $y$\nwhen $\\Lambda(x,y)$ jumps, \nand the type seen at $x$ is $\\sigma$ right after $\\Lambda^\\sigma(x)$ jumps.\n\n\n\n\nRecall that the rates of the evolutionary game are defined by \\eqref{def:qw}. With the choice of $\\overline{w}$ from \\eqref{def:wbar}, $q^w(x,y,\\xi)>0$ if and only if $q(x,y)>0$. Hence, Girsanov's theorem for point processes \\cite[Section~III.3]{JS} can be applied to change the intensities of the Poisson processes $\\Lambda(x,y)$ to $q^w(x,y,\\xi)$ such that under a probability measure $\\P^w$ equivalent to $\\P$ on $\\mathscr F_t$ for all $t\\geq 0$, \n\\begin{align}\\label{mg:hat}\n\\widehat{\\Lambda}_t(x,y)\\stackrel{\\rm def}{=}\\Lambda_t(x,y)-\\int_0^t q^w(x,y,\\xi_s)\\d s\\quad\\&\\quad \\widehat{\\Lambda}_t^\\sigma(x)\\stackrel{\\rm def}{=}\\Lambda^\\sigma_t(x)-\\mu(\\sigma)t\n\\end{align}\nare $(\\mathscr F_t,\\P^w)$-martingales. See \\cite[Section~2]{C:EGT} for the explicit form of $D^w$ when $S=\\{0,1\\}$. Since all of $\\widehat{\\Lambda}(x,y)$ and $\\widehat{\\Lambda}^\\sigma(x)$ do not jump simultaneously under $\\P^w$ by the absolute continuity with respect to $\\P$, the product of any distinct two of them has a zero predictable quadratic variation \\cite[Theorem~4.2, Proposition~4.50, and Theorem~4.52 in Chapter~I]{JS}.\n\n\nThe point processes defined above now allows for straightforward representations of the dynamics of the density processes. By (\\ref{eq:voter}), \n\\begin{align}\n\\begin{split}\np_\\sigma(\\xi_t)\n&=p_\\sigma(\\xi_0)+\\sum_{x,y\\in E}\\pi(x)\\int_0^t \\big[\\mathds 1_{\\sigma}\\circ\\xi_{s-}(y)-\\mathds 1_\\sigma\\circ\\xi_{s-}(x)\\big]\\d\\Lambda_s(x,y)\\\\\n&\\quad +\n\\sum_{x\\in E}\\pi(x)\\int_0^t \\mathds 1_{S\\setminus\\{\\sigma\\}}\\circ\\xi_{s-}(x)\\d \\Lambda^\\sigma_s(x)\\\\\n&\\quad -\\sum_{\\sigma'\\in S\\setminus\\{\\sigma\\}}\\sum_{x\\in E}\\pi(x)\\int_0^t \\mathds 1_{\\sigma}\\circ\\xi_{s-}(x)\\d \\Lambda^{\\sigma'}_s(x).\\label{dynamics:p1}\n\\end{split}\n\\end{align}\nTo obtain the limiting semimartingale for the density processes, we use the foregoing equation to derive the explicit semimartingale decompositions of the density processes. \n\nTo obtain these explicit decompositions, first, note that the dynamics of $p_\\sigma(\\xi_t)$ under $\\P^w$ relies on various kinds of frequencies and densities as follows. For all $x\\in E,\\xi\\in S^E$ and $\\sigma,\\sigma_1,\\sigma_2\\in S$, we set\n\\begin{align}\\label{def:Well}\n\\begin{split}\nf_\\sigma(x,\\xi)&=\\sum_{y\\in E}q(x,y)\\mathds 1_{\\sigma}\\circ \\xi(y),\\\\\n f_{\\sigma_1\\sigma_2}(x,\\xi)&= \\sum_{y\\in E}q(x,y)\\mathds 1_{\\sigma_1}(y)\\sum_{z\\in E}q(y,z)\\mathds 1_{\\sigma_2}\\circ\\xi(z),\\\\\nf_{\\bullet\\sigma}(x,\\xi)&= \\sum_{y\\in E}q(x,y)\\sum_{z\\in E}q(y,z)\\mathds 1_{\\sigma}\\circ \\xi(z),\\quad \n\\overline{f}(\\xi)=\\sum_{x\\in E}\\pi(x)f(x,\\xi).\n\\end{split}\n\\end{align} \nTo minimize the use of the summation notation, we also express these functions in terms of stationary \\emph{discrete-time} $q$-Markov chains $\\{U_\\ell;\\ell\\in \\Bbb Z_+\\}$ and $\\{U'_\\ell;\\ell\\in \\Bbb Z_+\\}$ with $U_0=U_0'$ such that conditioned on $U_0$, the two chains are independent. Additionally, let $(U,U')\\sim \\pi\\otimes \\pi$ and $(V,V')$ be distributed as\n\\begin{align}\\label{def:VV'}\n \\P(V=x,V'=y)=\\frac{\\nu(x,y)}{\\nu(\\mathds 1)},\\quad x,y\\in E,\n\\end{align}\nfor $\\nu(x,y)=\\pi(x)^2q(x,y)$ and $\\nu(\\mathds 1)=\\sum_{x,y}\\nu(x,y)=\\sum_x \\pi(x)^2$. (When $q$ is symmetric, $\\nu(\\mathds 1)$ reduces to $N^{-1}$.) For example, $\\overline{f_{\\sigma_1}f_{\\sigma_2\\sigma_3}}={\\mathbb E}[\\mathds 1_{\\sigma_1}\\circ \\xi(U_1')\\mathds 1_{\\sigma_2}\\circ \\xi(U_1)\\mathds 1_{\\sigma_3}\\circ \\xi(U_2)]$. We also set\n\\begin{align}\\label{def:p10}\np_{\\sigma\\sigma'}(\\xi)={\\mathbb E}[\\mathds 1_\\sigma\\circ \\xi(V)\\mathds 1_{\\sigma'}\\circ \\xi(V')].\n\\end{align}\n\n\n\nSecond, we turn to algebraic identities that determine the leading order terms for the forthcoming perturbation arguments. For $w\\in [0,\\overline{w}]$, the kernel $q^w$ defined by \\eqref{def:qw} can be expanded to the second order in $w$ as follows:\n\\begin{align}\nq^w(x,y,\\xi)&=q(x,y)\\frac{1-wB(y,\\xi)}{1-wA(x,\\xi)}\\notag\\\\\n&=q(x,y)+\\sum_{i=1}^\\infty w^iq(x,y)[A(x,\\xi)-B(y,\\xi)]A(x,\\xi)^{i-1}\\notag\\\\\n&=\nq(x,y)+wq(x,y)[A(x,\\xi)-B(y,\\xi)]+w^2q(x,y)R^w(x,y,\\xi),\\label{qwq:exp}\n\\end{align}\nwhere \n\\begin{align*}\nA(x,\\xi)&=1-\\sum_{z\\in E}q(x,z)\\sum_{z'\\in E}q(z,z')\\Pi\\big(\\xi(z),\\xi(z')\\big),\\\\\nB(y,\\xi)&=1-\\sum_{z\\in E}q(y,z)\\Pi\\big(\\xi(y),\\xi(z)\\big),\n\\end{align*}\nand $R^w$ is uniform bounded in $w\\in [0,\\overline{w}],x,y,\\xi,(E,q)$.\n\n\n\\begin{lem}\\label{lem:D}\nFor all $\\xi\\in S^E$ and $\\sigma\\in S$,\n\\begin{align}\n\\overline{D}_\\sigma(\\xi)&\\!\\stackrel{\\rm def}{=} \\sum_{x,y\\in E}\\pi(x)\\big[\\mathds 1_\\sigma\\circ\\xi(y)-\\mathds 1_\\sigma\\circ\\xi(x)\\big]q(x,y)[A(x,\\xi)-B(y,\\xi)]\\label{eq:Dsigma0}\\\\\n&=\\sum_{\\stackrel{\\scriptstyle \\sigma_0,\\sigma_3\\in S}{ \\sigma_0\\neq\\sigma}}\\Pi(\\sigma,\\sigma_3)\\overline{f_{\\sigma_0}f_{\\sigma\\sigma_3}}(\\xi)-\\sum_{\\stackrel{\\scriptstyle \\sigma_2,\\sigma_3\\in S}{\\sigma_2\\neq\\sigma}}\\Pi(\\sigma_2,\\sigma_3)\\overline{f_{\\sigma}f_{\\sigma_2\\sigma_3}}(\\xi).\\label{eq:Dsigma}\n\\end{align}\nIn particular, if $\\Pi$ is given by \\eqref{prisoner},\nthen \n\\begin{align}\\label{eq:Dsigma1}\n\\overline{D}_1(\\xi)=b\\overline{f_{1}f_{ \\bullet 0}}(\\xi)-b\\overline{f_{10}}(\\xi)-c\\overline{f_1f_{0}}(\\xi).\n\\end{align}\n\\end{lem}\n\\begin{proof}\nBy using the reversibility of $q$ and taking $y$ in \\eqref{eq:Dsigma0} as the state of $U_0$ in the sequence $\\{U_\\ell\\}$ defined above, we can compute $\\overline{D}_\\sigma$ as\n\\begin{align}\n\\begin{split}\\label{Dbar:cal}\n\\overline{D}_\\sigma(\\xi)\n&=-\\sum_{x,y\\in E}\\pi(x)\\mathds 1_\\sigma\\circ\\xi(y)q(x,y)\\sum_{z\\in E}q(x,z)\\sum_{z'\\in E}q(z,z')\\Pi\\big(\\xi(z),\\xi(z')\\big)\\\\\n&{\\quad \\,}+\\sum_{x,y\\in E}\\pi(x)\\mathds 1_\\sigma\\circ\\xi(y)q(x,y)\\sum_{z\\in E}q(y,z)\\Pi\\big(\\xi(y),\\xi(z)\\big)\\\\\n&{\\quad \\,}+\\sum_{x,y\\in E}\\pi(x)\\mathds 1_\\sigma\\circ\\xi(x)q(x,y)\\sum_{z\\in E}q(y,z)\\Pi\\big(\\xi(y),\\xi(z)\\big)\\\\\n&{\\quad \\,}-\\sum_{x,y\\in E}\\pi(x)\\mathds 1_\\sigma\\circ\\xi(x)q(x,y)\\sum_{z\\in E}q(x,z)\\sum_{z'\\in E}q(z,z')\\Pi\\big(\\xi(z),\\xi(z')\\big)\n\\end{split}\\\\\n&=-{\\mathbb E}\\left[\\mathds 1_\\sigma\\circ\\xi(U_{0})\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right]+{\\mathbb E}\\left[\\mathds 1_\\sigma\\circ\\xi(U_{2})\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right]\\label{Dbar:cal1}\\\\\n&=-{\\mathbb E}\\left[\\mathds 1_\\sigma\\circ\\xi(U_{0})\\mathds 1_\\sigma\\circ\\xi(U_2)\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right]\\notag\\\\\n&\\quad -{\\mathbb E}\\left[\\mathds 1_\\sigma\\circ\\xi(U_{0})\\mathds 1_{S\\setminus \\{\\sigma\\}}\\circ\\xi(U_2)\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right]\\notag\\\\\n&\\quad +{\\mathbb E}\\left[\\mathds 1_\\sigma\\circ\\xi(U_{2})\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right]\\notag\\\\\n&={\\mathbb E}\\left[\\mathds 1_{S\\setminus\\{\\sigma\\}}\\circ\\xi(U_{0})\\mathds 1_\\sigma\\circ\\xi(U_{2})\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right]\\notag\\\\\n&\\quad -{\\mathbb E}\\left[\\mathds 1_\\sigma\\circ\\xi(U_{0})\\mathds 1_{S\\setminus \\{\\sigma\\}}\\circ\\xi(U_2)\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right].\\notag\n\\end{align}\nHere, we use the reversibility of $q$ with respect to $\\pi$ to cancel\nthe last two terms in \\eqref{Dbar:cal} and write the first term in \\eqref{Dbar:cal} as the first term in \\eqref{Dbar:cal1}. \nSee \\cite[Lemma~1 on p.8]{CMN} for the case of two types. \n\n\nThe proof of \\eqref{eq:Dsigma1} appears in \\cite[Lemma~7.1]{C:EGT}. Now \\eqref{eq:Dsigma} allows for a quick proof:\n$\\overline{D}_1(\\xi)=(b-c)\\overline{f_0f_{11}}-c\\overline{f_0f_{10}}-b\\overline{f_1f_{01}}$. Then we use the identities $\\overline{f_0f_{11}}+\\overline{f_0f_{10}}=\\overline{f_0f_1}$, $\\overline{f_0f_{11}}+\\overline{f_0f_{01}}=\\overline{f_0f_{\\bullet 1}}$, and $\\overline{f_0f_{01}}+\\overline{f_0f_{01}}=\\overline{f_{01}}$. This calculation will be used in the proof of Corollary~\\ref{cor:symmetric}.\n\\end{proof}\n\nWe are ready to state the explicit semimartingale decompositions of the density processes and identify the leading order terms. \nFrom \\eqref{dynamics:p1}, \\eqref{qwq:exp} and the martingales in \\eqref{mg:hat}, we obtain the following decompositions extended from \\eqref{density:dynamics}:\n\\begin{align}\\label{psigma:dec}\np_\\sigma(\\xi_t)=p_\\sigma(\\xi_0)+A_\\sigma(t)+M_\\sigma(t)=p_\\sigma(\\xi_0)+I_\\sigma(t)+R_\\sigma(t)+M_\\sigma(t),\n\\end{align}\nwhere\n\\begin{align}\nI_\\sigma(t)&=w\\int_0^t \\overline{D}_\\sigma(\\xi_s)\\d s+\\int_0^t \\Bigg(\\mu(\\sigma)\\sum_{\\sigma'\n\\in S\\setminus\\{\\sigma\\}}p_{\\sigma'}(\\xi_s)-\\mu(S\\setminus\\{\\sigma\\}) p_\\sigma(\\xi_s)\\Bigg)\\d s,\n\\label{def:I}\\\\\nR_\\sigma(t)&=w^2\\sum_{x,y\\in E}\\pi(x)\\int_0^t \\big[\\mathds 1_\\sigma\\circ\\xi_{s}(y)-\\mathds 1_\\sigma\\circ\\xi_{s}(x)\\big]q(x,y)R^w(x,y,\\xi_s)\\d s,\\label{def:R}\\\\\n\\begin{split}\nM_\\sigma(t)&=\\sum_{x,y\\in E}\\pi(x)\\int_0^t \\big[\\mathds 1_\\sigma\\circ\\xi_{s-}(y)-\\mathds 1_\\sigma\\circ\\xi_{s-}(x)\\big]\\d\\widehat{\\Lambda}_s(x,y) \\\\\n&\\quad +\\sum_{x\\in E}\\pi(x)\\int_0^t \\mathds 1_{S\\setminus\\{\\sigma\\}}\\circ\\xi_{s-}(x)\\d \\widehat{\\Lambda}^\\sigma_s(x)\\\\\n&\\quad -\\sum_{\\sigma'\\in S\\setminus\\{\\sigma\\}}\\sum_{x\\in E}\\pi(x)\\int_0^t \\mathds 1_{\\sigma}\\circ\\xi_{s-}(x)\\d \\widehat{\\Lambda}^{\\sigma'}_s(x).\\label{def:M}\n\\end{split}\n\\end{align}\nBy \\eqref{mg:hat}, the predictable quadratic variations and covariations of $M_\\sigma$ and $M_{\\sigma'}$, for $\\sigma\\neq \\sigma'$, are \n\\begin{align}\n\\begin{split}\\label{def:}\n \\langle M_\\sigma,M_{\\sigma}\\rangle_t\n&=\\sum_{x,y\\in E}\\pi(x)^2\\int_0^t \\big\\{\\mathds 1_\\sigma\\circ\\xi_{s}(y)[1-\\mathds 1_{\\sigma}\\circ \\xi_s(x)]\\\\\n&\\hspace{-.5cm} +[1-\\mathds 1_{\\sigma}\\circ \\xi_s(y)]\\mathds 1_\\sigma\\circ\\xi_{s}(x)\\big\\} q^w(x,y,\\xi_{s})\\d s\\\\\n&\\hspace{-.5cm} +\\sum_{x\\in E}\\pi(x)^2\\int_0^t\\big[ \\mathds 1_{S\\setminus\\{\\sigma\\}}\\circ\\xi_{s-}(x)\\mu(\\sigma)+\\mathds 1_\\sigma\\circ\\xi_{s}(x)\\mu\\big(S\\setminus\\{\\sigma\\}\\big)\\big] \\d s,\n\\end{split}\\\\\n\\begin{split}\\label{def:}\n\\langle M_\\sigma,M_{\\sigma'}\\rangle_t\n&=-\\sum_{x,y\\in E}\\pi(x)^2\\int_0^t \\big[\\mathds 1_\\sigma\\circ\\xi_{s}(y)\\mathds 1_{\\sigma'}\\circ\\xi_s(x)\\\\\n&\\quad +\\mathds 1_\\sigma\\circ\\xi_s(y)\\mathds 1_{\\sigma'}\\circ\\xi_{s}(x)\\big] q^w(x,y,\\xi_{s})\\d s\\\\\n& \\quad -\\sum_{x\\in E}\\pi(x)^2\\int_0^t\\big[ \\mathds 1_{S\\setminus\\{\\sigma\\}}\\circ\\xi_{s-}(x) \\mathds 1_{\\sigma}\\circ\\xi_{s-}(x)\\mu(\\sigma)\\\\\n&\\quad +\\mathds 1_{S\\setminus\\{\\sigma'\\}}\\circ\\xi_{s-}(x) \\mathds 1_{\\sigma'}\\circ\\xi_{s-}(x)\\mu(\\sigma')\\big] \\d s.\n\\end{split}\n\\end{align}\nIn Section~\\ref{sec:eqn}, the above equations play the central role in characterizing the limiting density processes.\n\n\nFor this study, we apply the coalescing duality between $(E,q,\\mu)$-voter model and the coalescing rate-$1$ $q$-Markov chains $\\{B^x;x\\in E\\}$, where $B^x_0=x$. These chains move independently before meeting, and for any $x,y\\in E$, $B^x=B^y$ after their first meeting time\n$ M_{x,y}=\\inf\\{t\\geq 0;B^x_t=B^y_t\\}$. In the absence of mutation, the duality is given by\n\\begin{align}\\label{dual:1}\n{\\mathbb E}\\left[\\prod_{i=1}^n \\mathds 1_{\\sigma_i}\\circ \\xi_0(B^{x_i}_t)\\right]={\\mathbb E}_{\\xi_0}\\left[\\prod_{i=1}^n \\mathds 1_{\\sigma_i}\\circ\\xi_t(x_i)\\right] \n\\end{align}\nfor all $\\xi_0\\in S^E$, $\\sigma_1,\\cdots,\\sigma_n\\in S$, distinct $x_1,\\cdots,x_n \\in E$ and $n\\in \\Bbb N$. See the proof of Proposition~\\ref{prop:mutation} for the foregoing identity and the extension to the case with mutations. \n\nWithout mutation, the density process is a martingale under the voter model by \\eqref{density:dynamics}, and it follows from \\eqref{def:M} and \\eqref{def:} that, for any $\\sigma\\neq \\sigma'$, \n\\begin{align}\\label{eq:p1p0-voter}\n{\\mathbb E}_\\xi^0[p_\\sigma(\\xi_{t})p_{\\sigma'}(\\xi_{t})]=p_\\sigma(\\xi)p_{\\sigma'}(\\xi)-\\nu(\\mathds 1)\\int_0^t {\\mathbb E}_\\xi^0[p_{\\sigma\\sigma'}(\\xi_{s})+p_{\\sigma'\\sigma}(\\xi_s)]\\d s.\n\\end{align}\nFor the present problem, the central application of this dual relation is the foregoing identity \\cite{CCC}. Let the random variables defined below \\eqref{def:Well} to represent frequencies and densities be independent of the coalescing Markov chains. Then the foregoing equality implies that\n\\begin{align}\\label{ergodic}\n\\P(M_{U,U'}>t)=1-\\nu(\\mathds 1)-2\\nu(\\mathds 1)\\int_0^t \\P(M_{V,V'}>s)\\d s,\\quad \\forall\\;t\\geq 0.\n\\end{align}\nSee \\cite[Corollary~4.2]{CCC} and \\cite[Section~3.5.3]{AF:MC}.\nThis identity for meeting times has several important applications to the diffusion approximation of the voter model density processes. See \\cite[Sections~3 and 4]{CCC} and \\cite{CC}. \n\n\n\n\n\n\\section{Decorrelation in the ancestral lineage distributions}\\label{sec:slow}\nThis section is devoted to a study of degenerate limits of meeting time distributions. Here, we consider meeting times defined on a sequence of spatial structures $(E_n,q^{(n)})$ as before. According to the coalescing duality, these distributions are part of the ancestral line distributions of the voter model, and by approximation, the ancestral line distributions of the evolutionary game. On the other hand, these meeting times encode the typical local geometry of the space, but in a rough manner. With the study of these distributions, the main results of this section (Propositions~~\\ref{prop:sn-selection} and \\ref{prop:kell}) extend to the choice of appropriate time scaling constants and the characterization of the limiting density processes. These properties are crucial to the forthcoming limit theorems. \n\n\nOur direction in this section can be outlined in more detail as follows.\nRecall the auxiliary random variables defined below \\eqref{def:Well}, which are introduced to represent frequencies and densities. Under mild mixing conditions similar to those in \\eqref{cond2:thetan} with $\\gamma_n$ replaced by $\\theta_n$ \n and the condition $\\nu_n(\\mathds 1)\\to 0$, the sequence $\\P^{(n)}(M_{V,V'}\/\\gamma_n\\in \\cdot)$ is known to converge. The limiting distribution is a convex combination of the delta distribution at zero and an exponential distribution. Moreover, one can choose \\emph{some} $s_n\\to\\infty$ such that $s_n\/\\gamma_n\\to0 $ and the following $t$-independent limit exists:\n\\begin{align}\\label{cond:kappa0}\n\\overline{\\kappa}_0\\,\\stackrel{\\rm def}{=}\\,\n\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{V,V'}>s_n t),\\quad\\forall\\;t\\in (0,\\infty)\n\\end{align}\nwith $\\overline{\\kappa}_0=1$. See \\cite[Corollary~4.2 and Proposition~4.3]{CCC} for these results. As an extension of this existence result, our first goal in this section is to introduce \\emph{sufficient} conditions for these sequences $(s_n)$. Specifically, we require that the limit \\eqref{cond:kappa0} exists with $\\overline{\\kappa}_0\\in (0,\\infty)$. See Section~\\ref{sec:dec1}. The following is enough for the existence and the applications in the next section. \n\n\n\\begin{defi}\\label{def:slow}\nWe say that $(s_n)$ is a {\\bf slow sequence} if \n\\begin{align}\\label{cond1:sn}\n\\lim_{n\\to\\infty}s_n=\\infty,\\quad \\lim_{n\\to\\infty}\\frac{s_n}{\\gamma_n}=0,\\quad \n\\lim_{n\\to\\infty}\\gamma_n\\nu_n(\\mathds 1){\\rm e}^{-ts_n}= 0,\\quad \\forall\\;t\\in (0,\\infty),\n\\end{align}\nand at least one of the two mixing conditions holds: \n\\begin{align}\\label{cond2:sn}\n\\lim_{n\\to\\infty}\\gamma_n\\nu_n(\\mathds 1){\\rm e}^{-\\mathbf g_n s_n}=0\\quad\\mbox{or}\\quad \\lim_{n\\to\\infty}\\frac{\\mathbf t^{(n)}_{\\rm mix}}{s_n}[1+\\log^+(\\gamma_n\/\\mathbf t^{(n)}_{\\rm mix})]=0.\n\\end{align}\n\\end{defi}\n\n\n\n\nOur second goal is to extend the existence of the limit \\eqref{cond:kappa0} to the existence of analogous time-independent limits for other meeting time distributions: for integers $\\ell\\geq 1$, $\\ell_0,\\ell_1,\\ell_2\\geq 0$ with $\\ell_0,\\ell_1,\\ell_2$ all distinct, and all $t\\in (0,\\infty)$, \n\\begin{align}\\label{def:kell}\n\\overline{\\kappa}_\\ell &\\,\\stackrel{\\rm def}{=}\\,\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}>s_n t);\\\\\n\\label{def:|kell1|ell2}\n\\overline{\\kappa}_{(\\ell_0,\\ell_1)|\\ell_2}&\\,\\stackrel{\\rm def}{=}\\,\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}\\big(M_{U_{\\ell_0},U_{\\ell_1}}>s_n t,M_{U_{\\ell_1},U_{\\ell_2}}>s_n t\\big);\\\\\n\\label{def:|||}\n\\overline{\\kappa}_{\\ell_0|\\ell_1|\\ell_2}&\\,\\stackrel{\\rm def}{=}\\,\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}\\big(M_{U_{\\ell_0},U_{\\ell_1}}>s_n t,M_{U_{\\ell_1},U_{\\ell_2}}>s_n t,M_{U_{\\ell_0},U_{\\ell_2}}>s_n t\\big).\n\\end{align}\nThe extension to $\\overline{\\kappa}_1$ is straightforward if we allow passing limits along subsequences. Indeed,\nit follows from the definition of $\\{U_\\ell\\}$ and $(V,V')$ that\n\\begin{align}\\label{ineq:MUVcompare}\n\\frac{\\pi_{\\min}}{\\pi_{\\max}}\\P(M_{V,V'}\\in \\Gamma)\\leq \\P(M_{U_0,U_1}\\in \\Gamma)\\leq \n\\frac{\\pi_{\\max}}{\\pi_{\\min}}\\P(M_{V,V'}\\in \\Gamma),\\quad\\forall\\; \\Gamma\\in \\mathscr B({\\Bbb R}_+).\n\\end{align}\nHence, by taking a subsequence of $(E_n,q^{(n)})$ if necessary,\n\\eqref{cond:kappa0} and condition (a) of Theorem~\\ref{thm:main} imply the existence of the limit $\\overline{\\kappa}_1$. \n\n\nIn Section~\\ref{sec:higher-order}, we prove the existence of the other limits $\\overline{\\kappa}_\\ell$, $\\ell\\geq 2$. \nMore precisely, we prove tightness results as in the case of $\\overline{\\kappa}_1$ so that the limits may be passed along subsequences. We also prove that the limits $\\overline{\\kappa}_\\ell$, $\\ell\\geq 2$, are in $(0,\\infty)$. Note that in proving these results, we do not impose convergence of local geometry as in the case of discrete tori or random regular graphs. \n\n\n\n\\subsection{Mixing conditions for local meeting times}\\label{sec:dec1}\nTo apply mixing conditions to meeting times, first, we recall some basic properties of the spectral gap and the mixing time for the product of the continuous-time $q$-Markov chains. Note that by coupling the product chain with initial condition $(x,y)$ after the two coordinates meet, we get the coalescing chain $(B^x,B^y)$ defined before \\eqref{eq:p1p0-voter}. \n\n\nNow, the discrete-time chain for the product chain has a transition matrix such that each of the coordinates is allowed to change with equal probability. Hence, the spectral gap is given by $\\widetilde{\\mathbf g}=\\mathbf g\/2$ \\cite[Corollary~12.12 on p.161]{LPW}.\nIf $(\\widetilde{q}_t)$ denotes the semigroup of the product chain, then\n\\begin{align}\\label{product:bdd}\n\\sup_{(x,y)\\in E\\times E}\\big\\|\\widetilde{q}_t\\big((x,y),\\cdot\\big)-\\pi\\otimes \\pi\\big\\|_{\\rm TV}\\leq 2d_{E}(t),\n\\end{align}\nwhere $d_E$ is the total variation distance defined by \\eqref{def:dE}. Additionally, it follows from the definition \nthe mixing time in \\eqref{def:tmix} that \n\\begin{align}\\label{ineq:tmix}\nd_E(k\\mathbf t_{\\rm mix})\\leq {\\rm e}^{-k},\\quad \\forall\\;k\\in \\Bbb N\n\\end{align}\n\\cite[Section~4.5 on p.55]{LPW}. By the last two displays, the analogous mixing time $\\widetilde{\\mathbf t}_{\\rm mix}$ of the product chain satisfies \n\\begin{align}\\label{compare:mix} \n \\widetilde{\\mathbf t}_{\\rm mix}\\leq 3\\mathbf t_{\\rm mix}.\n\\end{align}\nWe are ready to prove the first main result of Section~\\ref{sec:slow}. Note that under the condition $\\sup_nN_n\\pi^{(n)}_{\\max}<\\infty$ (see the discussion below \\eqref{def:wn}), the first condition in \\eqref{cond2:sn} implies the first one in \\eqref{sn:lim}. \n\n\n\n\\begin{prop}\\label{prop:sn-selection}\nSuppose that $(s_n)$ satisfies \\eqref{cond1:sn} and at least one of the following mixing conditions:\n\\begin{align}\\label{sn:lim}\n\\lim_{n\\to\\infty}\\mathbf g_n s_n=\\infty\\quad\\mbox{or}\\quad \\lim_{n\\to\\infty}\\frac{\\mathbf t^{(n)}_{\\rm mix}}{s_n}[1+\\log^+(\\gamma_n\/\\mathbf t^{(n)}_{\\rm mix})]=0.\n\\end{align}\nThen \\eqref{cond:kappa0} holds with $\\overline{\\kappa}_0=1$. \n\\end{prop}\n\\begin{proof}\nWrite $f_n(t)=\\P^{(n)}(M_{U,U'}>t)$ and $g_n(t)=\\P^{(n)}(M_{V,V'}>t)$. The required result is proved in two steps. \\medskip \n\n\n\\noindent {\\bf Step 1.} We start with a preliminary result: for all $t_0\\in[0,\\infty)$ and $\\mu\\in(0,\\infty)$,\n\\begin{align}\\label{eq:Lap_nun}\n&\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\n\\int_0^\\infty {\\rm e}^{-\\mu t}g_n\\big(s_n(t+t_0)\\big)\\d t=\\frac{1}{\\mu}.\n\\end{align}\n\n\nTo obtain \\eqref{eq:Lap_nun}, first, we derive a representation of the integrals in \\eqref{eq:Lap_nun} by $f_n(t)$. \nNote that \\eqref{ergodic} under the $q^{(n)}$-chain takes the following form:\n\\[\nf_n(s_nt)=\n1-\\nu_n(\\mathds 1)-2\\nu_n(\\mathds 1)s_n\\int_0^t g_n(s_ns)\\d s,\\quad t\\geq 0.\n\\]\nHence, for any fixed $0\\leq t_0<\\infty$,\n\\begin{align}\\label{delta:f}\nf_n(s_n(t+t_0))-f_n(s_nt_0)=-2\\nu_n(\\mathds 1)s_n\\int_{0}^{t} g_n\\big(s_n(s+t_0)\\big)\\d s,\\quad t\\geq 0.\n\\end{align}\nTaking Laplace transforms of both sides of the last equality, we get, for $\\mu>0$,\n\\begin{align}\n\\int_{0}^\\infty {\\rm e}^{-\\mu t}\\big[f_n\\big(s_n(t+t_0)\\big)-f_n(s_nt_0)\\big]\\d t&=-2\\nu_n(\\mathds 1)s_n\\int_0^\\infty {\\rm e}^{-\\mu t}\\int_0^{t}g_n\\big(s_n(s+t_0)\\big)\\d s\\d t\\notag\\\\\n&=-\\frac{2\\nu_n(\\mathds 1)s_n}{\\mu}\\int_0^\\infty {\\rm e}^{-\\mu t}g_n\\big(s_n (t+t_0)\\big)\\d t,\\notag\n\\end{align}\nwhere the last integral coincides with the integral in \\eqref{eq:Lap_nun}. \n\n\nNext, rewrite the last equality as\n\\begin{align}\n&\\quad 2\\gamma_n\\nu_n(\\mathds 1)\n\\int_0^\\infty {\\rm e}^{-\\mu t}g_n\\big(s_n (t+t_0)\\big)\\d t\\notag\\\\\n&= -\\frac{\\gamma_n\\mu}{s_n} \\int_0^\\infty {\\rm e}^{-\\mu t}\\big[f_n \\big(s_n(t+t_0)\\big)-f_n(s_nt_0)\\big]\\d t\\notag\\\\\n\\begin{split}\n&=-\\frac{\\gamma_n\\mu}{s_n} \\int_0^\\infty {\\rm e}^{-\\mu t}\\Big\\{\\big[f_n \\big(s_n(t+t_0)\\big)-f_n(s_nt_0)\\big]-\n[{\\rm e}^{-s_n(t+t_0)\/\\gamma_n}-{\\rm e}^{-s_nt_0\/\\gamma_n}]\n\\Big\\}\\d t \\\\\n&\\quad +\\frac{ {\\rm e}^{-s_nt_0\/\\gamma_n}}{\\mu+s_n\/\\gamma_n}.\\label{eq:sn-selection0}\n\\end{split}\n\\end{align}\nThe last term tends to $1\/\\mu$ since $s_n\/\\gamma_n\\to 0$. To take the limit of the integral term in \\eqref{eq:sn-selection0}, we use the first mixing condition in \\eqref{cond2:sn}. In this case, a bound for exponential approximations of the distributions of $M_{U,U'}$ \\cite[Proposition~3.23]{AF:MC} gives\n\\begin{align}\\label{eq:sn-selection1}\n\\begin{split}\n&\\left|\\frac{\\gamma_n\\mu}{s_n} \\int_0^\\infty {\\rm e}^{-\\mu t}\\Big\\{\\big[f_n \\big(s_n(t+t_0)\\big)-f_n(s_nt_0)\\big]-\n[{\\rm e}^{-s_n(t+t_0)\/\\gamma_n}-{\\rm e}^{-s_nt_0\/\\gamma_n}]\n\\Big\\}\\d t\\right|\\\\\n&\\leq \\frac{2}{\\widetilde{\\mathbf g}_n s_n}\\xrightarrow[n\\to\\infty]{} 0\n\\end{split}\n\\end{align}\nnow that $\\widetilde{\\mathbf g}_n=\\mathbf g_n\/2$.\nAlternatively, by a different bound from \\cite[Theorem~1.4]{Aldous:AE}, the foregoing inequality holds with the bound replaced by \n\\begin{align}\\label{eq:sn-selection2}\n\\frac{C_{\\ref{eq:sn-selection2}}\\widetilde{\\mathbf t}^{(n)}_{\\rm mix}}{s_n}\\big[1+\\log^+(\\gamma_n\/\\widetilde{\\mathbf t}^{(n)}_{\\rm mix})\\big]\n\\leq \\frac{C_{\\ref{eq:sn-selection2}}\\cdot 3\\mathbf t^{(n)}_{\\rm mix}}{s_n}\\big[1+\\log^+(\\gamma_n\/(3\\mathbf t^{(n)}_{\\rm mix}))\\big]\n\\end{align}\nby \\eqref{compare:mix}, the monotonicity of $x\\mapsto x(1+\\log (x^{-1}\\vee 1))$ on $(0,\\infty)$, where $C_{\\ref{eq:sn-selection2}}$ is independent of the $q^{(n)}$-chains.The last term in \\eqref{eq:sn-selection2} tends to zero by the second mixing condition in \\eqref{cond2:sn}. \n\nFinally, we apply \\eqref{eq:sn-selection1} and \\eqref{eq:sn-selection2} to \\eqref{eq:sn-selection0}. Since the last term in \\eqref{eq:sn-selection0} tends to $1\/\\mu$, we have proved \\eqref{eq:Lap_nun}. \\medskip\n\n\n\\noindent {\\bf Step 2.} We are ready to prove the existence of the limit in \\eqref{cond:kappa0} and its independence of $t$.\nFirst, note that since $g_n$ is decreasing, we have\n\\[\n2\\gamma_n\\nu_n(\\mathds 1){\\rm e}^{-\\mu t}g_n\\big(s_n(t+t_0)\\big)\\leq \\frac{1}{t}2\\gamma_n\\nu_n(\\mathds 1)\\int_0^t {\\rm e}^{-\\mu s}g_n\\big(s_n(s+t_0)\\big)\\d s,\\quad\\forall\\;t,t_0\\in (0,\\infty),\n\\]\nwhereas the last integral is bounded by the same integral with the upper limit $t$ of integration replaced by $\\infty$. By the \\eqref{eq:sn-selection0} and the convergence proven for it in the preceding step, the last inequality implies that $t\\mapsto 2\\gamma_n\\nu_n(\\mathds 1){\\rm e}^{-\\mu t}g_n(s_nt)$, $n\\geq 1$, are uniformly bounded on $[a,\\infty)$, for any $a\\in (0,\\infty)$. Hence, by Helly's selection theorem, every subsequence of $\\{t\\mapsto 2\\gamma_n\\nu_n(\\mathds 1)g_n(s_nt)\\}$ has a further subsequence, say indexed by $n_j$, such that for some left-continuous function $g_\\infty$ on $(0,\\infty)$,\n\\begin{align}\\label{def:ginfty}\n\\lim_{j\\to\\infty}2\\gamma_{n_j}\\nu_{n_j}(\\mathds 1)g_{n_j}(s_{n_j}t)= g_\\infty(t),\\quad \\forall\\; t\\in (0,\\infty). \n\\end{align}\nMoreover, this convergence holds boundedly on compact subsets of $ (0,\\infty)$ in $t$.\n\nTo find $g_\\infty$, note that, as in \\eqref{eq:sn-selection1} and \\eqref{eq:sn-selection2}, either of the mixing conditions \\eqref{cond2:sn} implies that for fixed $0t)&={\\rm e}^{-2t}\\P(U_0\\neq U_\\ell)+\\int_0^t 2{\\rm e}^{-2(t-s)}\\P(M_{U_0,U_{\\ell+1}}>s)\\d s\\\\\n&\\quad -\\int_0^t 2{\\rm e}^{-2(t-s)}\\sum_{x,y\\in E}\\pi(x)q^{\\ell}(x,x)q(x,y)\\P(M_{x,y}>s)\\d s.\n\\end{split}\n\\end{align}\n\\end{lem}\n\\begin{proof}\nSince $M_{x,x}\\equiv 0$ and $(U_0,U_\\ell)$ is independent of the meeting times, conditioning on $(U_0,U_\\ell)$ gives $\\P(M_{U_0,U_\\ell}>t)=\\P(M_{U_0,U_\\ell}>t,U_0\\neq U_\\ell)$. Conditioning on the first update time of $(B^{U_0},B^{U_\\ell})$, which is an exponential variable with mean $1\/2$, yields\n\\begin{align}\\label{eq:st1}\n\\P(M_{U_0,U_\\ell}>t)&={\\rm e}^{-2t}\\P(U_0\\neq U_\\ell)+\\int_0^t 2{\\rm e}^{-2(t-s)}\\P(U_0\\neq U_{\\ell},M_{U_0,U_{\\ell+1}}>s)\\d s.\n\\end{align}\nHere, the initial condition $(U_0,U_{\\ell+1})$ in the last term follows from transferring the first transition of state of $(B^{U_0},B^{U_\\ell})$ to the initial condition. We also use the stationarity of $\\{U_\\ell;\\ell\\geq 0\\}$ when that first transition is made by $B^{U_0}$. To rewrite the integral term in \\eqref{eq:st1}, note that \n\\begin{align*}\n\\P(U_0\\neq U_\\ell,U_0=x,U_{\\ell+1}=y)\n&=\\P(U_0=x,U_{\\ell+1}=y)-\\P(U_0=U_\\ell,U_0=x,U_{\\ell+1}=y)\\\\\n&=\\pi(x)q^{\\ell+1}(x,y)-\\pi(x)q^{\\ell}(x,x)q(x,y)\n\\end{align*}\nso that\n\\begin{align}\\label{eq:st3}\n\\begin{split}\n\\P(U_0\\neq U_{\\ell},M_{U_0,U_{\\ell+1}}>s)=&\\P(M_{U_0,U_{\\ell+1}}>s)\\\\\n&-\\sum_{x,y\\in E}\\pi(x)q^{\\ell}(x,x)q(x,y)\\P(M_{x,y}>s).\n\\end{split}\n\\end{align}\nApplying \\eqref{eq:st3} to (\\ref{eq:st1}) yields\n (\\ref{eq:shifttime0}). \n\\end{proof}\n\n\n\nWe are ready to prove the existence of the limits in \\eqref{def:kell} and \\eqref{def:|kell1|ell2}.\n\n\\begin{prop}\\label{prop:kell}\nFor any sequence $(s_n)$ satisfying \\eqref{cond1:sn}, we have the following properties:\n\\begin{enumerate}\n\\item [\\hypertarget{prop:kell1}{\\rm (1${^\\circ}$)}] For any integer $\\ell\\geq 2$, every subsequence of $(E_n,q^{(n)})$ contains a further subsequence such that the limit in \\eqref{def:kell}\nexists in $[\\overline{\\kappa}_1,\\ell\\overline{\\kappa}_1]$ and is independent of $t\\in (0,\\infty)$.\n\n\\item [\\hypertarget{prop:kell2}{\\rm (2${^\\circ}$)}] Without taking any subsequence, \\eqref{def:kell} holds for $\\ell=2$ with $\\overline{\\kappa}_2=\\overline{\\kappa}_1$. \n\\item [\\hypertarget{prop:kell3}{\\rm (3${^\\circ}$)}] Suppose that \\eqref{cond:q2} holds for some constant $q^{(\\infty),2}$.\nThen without taking any subsequence, \\eqref{def:kell} holds $\\overline{\\kappa}_3=(1+q^{(\\infty),2})\\overline{\\kappa}_1$.\n\n\\item [\\hypertarget{prop:kell4}{\\rm (4${^\\circ}$)}] \nFor all distinct nonnegative integers $\\ell_0,\\ell_1,\\ell_2$, it holds that \n\\[\n{\\overline{\\kappa}}_{(\\ell_1,\\ell_2)|\\ell_0}+{\\overline{\\kappa}}_{(\\ell_0,\\ell_1)|\\ell_2}-{\\overline{\\kappa}}_{\\ell_0|\\ell_1|\\ell_2}={\\overline{\\kappa}}_{|\\ell_2-\\ell_0|},\n\\] \nprovided that all of the limits defining these constants exist.\n\n\\end{enumerate}\n\\end{prop}\n\\begin{proof} \n(1${^\\circ}$) To lighten notation in the rest of this proof but only in this proof, write $A_\\ell=\\P(U_0\\neq U_\\ell)$, $J_\\ell$ for $M_{U_0,U_\\ell}$, \n\\[\nB_\\ell=\\sum_{x,y\\in E}\\pi(x)q^\\ell(x,x)q(x,y),\n\\]\nand $K_\\ell$ for the first meeting time for the pair of coalescing Markov chains where the initial condition is distributed independently as $B_\\ell^{-1}\\pi(x)q^\\ell(x,x)q(x,y)$ provided that $B_\\ell\\neq 0$. We set $K_\\ell$ to be an arbitrary random variable.\n\n\n\nFix an integer $\\ell\\geq 1$. If $\\mathbf e$ is an independent exponential variable with mean $1$, then \\eqref{eq:shifttime0} can be written as \n\\[\n\\P(J_\\ell>t)=A_\\ell\\P(\\tfrac{1}{2} \\mathbf e>t)+\\P(J_{\\ell+1}+\\tfrac{1}{2} \\mathbf e>t,\\tfrac{1}{2} \\mathbf e\\leq t)-B_\\ell\\P(K_{\\ell}+\\tfrac{1}{2} \\mathbf e>t,\\tfrac{1}{2} \\mathbf e\\leq t).\n\\]\nAfter rearrangement, the foregoing equality yields\n\\begin{align*}\n\\P(J_{\\ell+1}+\\tfrac{1}{2} \\mathbf e>t)\n&=\\P(J_\\ell>t)+B_\\ell\\P(K_{\\ell}+\\tfrac{1}{2} \\mathbf e>t)+(1-A_\\ell-B_\\ell)\\P(\\tfrac{1}{2} \\mathbf e>t).\n\\end{align*}\nHence, for all left-open intervals $\\Gamma\\subset (0,\\infty)$, \n\\begin{align}\\label{id:JK}\n\\begin{split}\n&\\quad\\, \\P(J_{\\ell+1}+\\tfrac{1}{2} \\mathbf e\\in \\Gamma)+(A_\\ell+B_\\ell)\\P(\\tfrac{1}{2} \\mathbf e\\in \\Gamma)\\\\\n&=\\P(J_\\ell\\in \\Gamma)+B_\\ell\\P(K_{\\ell}+\\tfrac{1}{2} \\mathbf e\\in \\Gamma)+\\P(\\tfrac{1}{2} \\mathbf e\\in \\Gamma).\n\\end{split}\n\\end{align}\nSince $q^{\\ell}(x,x)\\leq 1$,\nwe have $B_\\ell\\P(K_\\ell\\in \\cdot)\\leq \\P(J_1\\in \\cdot)$, and so, the foregoing identity gives\n\\begin{align}\\label{id:JK1}\n\\begin{split}\n&\\quad \\P(J_{\\ell+1}+\\tfrac{1}{2} \\mathbf e\\in \\Gamma)+(A_\\ell+B_\\ell)\\P(\\tfrac{1}{2} \\mathbf e\\in \\Gamma)\\\\\n&\\leq \\P(J_\\ell\\in \\Gamma)+\\P(J_{1}+\\tfrac{1}{2} \\mathbf e\\in \\Gamma)+\\P(\\tfrac{1}{2} \\mathbf e\\in \\Gamma).\n\\end{split}\n\\end{align}\n\nWe are ready to prove the required result. \nFor any $0s_n a)\\leq (\\ell+1) \\overline{\\kappa}_1,\\quad \\forall\\;t\\in (0,\\infty).\n\\end{align}\nOn the other hand,\nsince $\\P(J_{\\ell+1}+\\tfrac{1}{2} \\mathbf e\\in \\cdot)+|A_\\ell+B_\\ell-1|\\P(\\tfrac{1}{2}\\mathbf e\\in \\cdot)\\geq \\P(J_\\ell\\in \\cdot)$ by \\eqref{id:JK}, it follows from \\eqref{def:kell} with $\\ell=1$ and an argument similar to the one leading to \\eqref{eq:kell1} that \n\\begin{align}\n\\liminf_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(J_{\\ell+1}>s_n t)\\geq \\overline{\\kappa}_1>0,\\quad\\forall\\;t\\in (0,\\infty).\\label{eq:kell4}\n\\end{align}\n\n\n\nCombining \\eqref{eq:kell3} and \\eqref{eq:kell4}, we deduce that for fixed $t_0\\in (0,\\infty)$, any subsequence of the numbers\n$2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(J_{\\ell+1}>s_nt_0)$ has a further subsequence that converges in $[\\overline{\\kappa}_1,(\\ell+1)\\overline{\\kappa}_1]$. By \\eqref{eq:kell1}, this limit extends to the existence of the limit of the corresponding subsequence of \n$2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(J_{\\ell+1}>s_nt)$ \nfor any $t\\in (0,\\infty)$, and all of these limits for different $t$ are equal.\nWe have proved \\eqref{def:kell}. \\medskip \n \n\n\\noindent (2${^\\circ}$) Note that $B_1^{(n)}=0$ since ${\\sf tr}(q^{(n)})=0$ by assumption. \nThen an inspection of \\eqref{eq:kell0} shows the second limit superior on the right-hand side there can be dropped. The rest of the argument in (2${^\\circ}$), especially \\eqref{eq:kell3} and \\eqref{eq:kell4}, can be adapted accordingly to get\nthe required identity. \\medskip\n \n\\noindent (3${^\\circ}$) The proof is done again by improving the argument for \\eqref{eq:kell3} and \\eqref{eq:kell4}, but now using \\eqref{id:JK} with $\\ell=2$. In doing so, we also use the following implication of \\eqref{cond:q2}:\n\\[\n\\lim_{n\\to\\infty}\\sup_{s\\geq 0}\\gamma_n\\nu_n(\\mathds 1)\\big|B^{(n)}_2\\P^{(n)}(K_2>s)-q^{(\\infty),2}\\P^{(n)}(J_1>s)\\big|=0,\n\\]\nwhich follows since the distributions of $K_2$ and $J_1$ differ by the initial conditions. \\medskip \n\n\\noindent (4${^\\circ}$) By the definitions in \\eqref{def:kell}--\\eqref{def:|||}, we have\n\\begin{align*}\n&{\\quad \\,} ({\\overline{\\kappa}}_{(\\ell_1,\\ell_2)|\\ell_0}-{\\overline{\\kappa}}_{\\ell_0|\\ell_1|\\ell_2})+({\\overline{\\kappa}}_{(\\ell_0,\\ell_1)|\\ell_2}-{\\overline{\\kappa}}_{\\ell_0|\\ell_1|\\ell_2})+{\\overline{\\kappa}}_{\\ell_0|\\ell_1|\\ell_2}\\\\\n&=\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_{\\ell_0},U_{\\ell_1}}>s_n t,M_{U_{\\ell_1},U_{\\ell_2}}\\leq s_n t,M_{U_{\\ell_0},U_{\\ell_2}}>s_n t)\\\\\n&\\quad\\, +\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1) \\P^{(n)}(M_{U_{\\ell_0},U_{\\ell_1}}\\leq s_n t,M_{U_{\\ell_1},U_{\\ell_2}}> s_n t,M_{U_{\\ell_0},U_{\\ell_2}}>s_n t)\\\\\n&\\quad \\,+\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_{\\ell_0},U_{\\ell_1}}>s_n t,M_{U_{\\ell_1},U_{\\ell_2}}>s_nt,M_{U_{\\ell_0},U_{\\ell_2}}>s_nt)\\\\\n&=\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_{\\ell_0},U_{\\ell_2}}>s_n t)={\\overline{\\kappa}}_{|\\ell_2-\\ell_0|}.\n\\end{align*}\nHere, the next to the last equality follows since on $\\{M_{U_{\\ell_0},U_{\\ell_2}}>s_n t\\}$, we cannot have both $M_{U_{\\ell_0},U_{\\ell_1}}\\leq s_n t$ and $M_{U_{\\ell_1},U_{\\ell_2}}\\leq s_n t$ by the coalescence of the Markov chains, and the last equality follows from the stationarity of the chain $\\{U_\\ell\\}$. The proof is complete.\n\\end{proof}\n\n\n\n\nWe close this subsection with another application of Lemma~\\ref{lem:MT}. It will be used in Section~\\ref{sec:eqn}.\n\n\\begin{prop}\\label{prop:Mcompare}\nLet $s_0\\in (2,\\infty)$.\nFor all integers $\\ell\\geq 1$ and all $t\\in (0,\\infty)$, it holds that \n\\begin{align}\\label{ineq:Cell}\n\\int_0^{t} \\P(M_{U_0,U_\\ell}>s_0s)\\d s\\leq \\sum_{j=1}^\\ell \\prod_{k=1}^{j-1} \\big(1-{\\rm e}^{-2^{k+1}t}\\big)^{-1}\\int_0^{2j t} \\P(M_{U_0,U_1}>s_0s)\\d s,\n\\end{align}\nwhere $\\prod_{k=i}^ja_k\\equiv 1$ for $js_0s)\\d s\\\\\n&\\leq \n\\P(M_{U_0,U_\\ell}>s_0 r)+\\int_0^{ r}2s_0 {\\rm e}^{-2s_0 ( r-s)}\\P(M_{U_0,U_1}>s_0s)\\d s.\n\\end{split}\n\\end{align}\nThe assumption $s_0\\in (2,\\infty)$ gives $t\\leq 2t(1-s_0^{-1})$, and so, for $h$ nonnegative and Borel measurable, \n\\begin{align}\n\\begin{split}\\label{Mcompare:2}\n &{\\quad \\,} (1-{\\rm e}^{-4t})\\int_0^{t}h(s_0 s)\\d s\\leq \\int_0^{2t(1-s_0^{-1})}h(s_0 s)(1-{\\rm e}^{-4t})\\d s\\\\\n &\\leq \\int_0^{2t} h(s_0 s)\\big(1-{\\rm e}^{-2s_0(2t-s)}\\big)\\d s=\\int_0^{2t}\\int_0^r 2s_0 {\\rm e}^{-2s_0(r- s)}h(s_0 s)\\d s\\d r \\\\&\\leq \\int_0^{2t} h(s_0s)\\d s.\n \\end{split}\n\\end{align}\nIntegrating both sides of \\eqref{Mcompare:1} over $[0,2t]$ and applying the first and last inequalities in \\eqref{Mcompare:2} give\n\\begin{align}\n&{\\quad \\,}(1-{\\rm e}^{-4t})\n\\int_0^{t} \\P(M_{U_0,U_{\\ell+1}}>s_0 s)\\d s\\notag\\\\\n&\\leq \n\\int_0^{2t} \\P(M_{U_0,U_\\ell}>s_0 s)\\d s+\\int_0^{2t} \\P(M_{U_0,U_1}>s_0 s)\\d s \\notag\\\\\n&\\leq \\sum_{j=1}^\\ell \\prod_{k=1}^{j-1} \\big(1-{\\rm e}^{-2^{k+2}t}\\big)^{-1}\\int_0^{2^{j+1} t} \\P(M_{U_0,U_1}>s_0s)\\d s +\\int_0^{2t} \\P(M_{U_0,U_1}>s_0 s)\\d s\\notag\\\\\n&\\leq \\sum_{j=2}^{\\ell+1} \\prod_{k=2}^{j-1} \\big(1-{\\rm e}^{-2^{k+1}t}\\big)^{-1}\\int_0^{2^{j} t} \\P(M_{U_0,U_1}>s_0s)\\d s +\\int_0^{2t} \\P(M_{U_0,U_1}>s_0 s)\\d s,\\label{Mcompare:final}\n\\end{align}\nwhere the second inequality follows from induction. Dividing both sides of \\eqref{Mcompare:final} by $(1-{\\rm e}^{-4t})$ proves \\eqref{ineq:Cell} for $\\ell$ replaced by $\\ell+1$. Hence, \\eqref{ineq:Cell} holds for all $\\ell\\geq 1$ by induction. \n\\end{proof}\n\n\n\\section{Convergence of the vector density processes}\\label{sec:eqn}\nWe present the proofs of Theorem~\\ref{thm:main} and Corollary~\\ref{cor:symmetric} in this section. The key result is Proposition~\\ref{prop:duhamel} where we reduce the evolutionary game model to the voter model. {\\bf Throughout this section, \n conditions (a)--(d) of Theorem~\\ref{thm:main} are in force.} \n \n \n \n The other settings for this section are as follows. First, we write $I^{(n)}_\\sigma=I_\\sigma(\\theta_n t)$ for the process $I_\\sigma(t)$ defined by \\eqref{def:I}, when the underlying particle system is based on $(E_n,q^{(n)})$. This notation extends to the other processes in the decompositions \\eqref{psigma:dec} by using the same time change. Next, recall that $S$ denotes the type space. We will mostly consider $(\\sigma_0,\\sigma_2,\\sigma_3)\\in S\\times S\\times S$ such that $\\sigma_0\\neq \\sigma_2$. These triplets fit into the context of \\eqref{eq:Dsigma}, from which we will prove the limiting replicator equation in Theorem~\\ref{thm:main}. Additionally, given an admissible sequence $(\\theta_n,\\mu_n,w_n)$ such that $\\lim_n \\theta_n\/\\gamma_n=0$, we can choose a slow sequence $(s_n)$ (recall Definition~\\ref{def:slow}) such that \n\\begin{align}\n\\lim_{n\\to\\infty}\\frac{s_n}{\\theta_n}=0.\\label{sn:adm}\n\\end{align} \n\n\n\n\\subsection{Asymptotic closure of equations and path regularity}\\label{sec:closure}\nWe begin by showing that the leading order drift term $I_\\sigma^{(n)}$ in \\eqref{psigma:dec} can be asymptotically closed by the vector density process $(p_{\\sigma}(\\xi_{\\theta_nt});\\sigma\\in S)$. By \\eqref{def:I}, this term takes the following explicit form: \n\\begin{align}\\label{def:In}\n\\begin{split}\nI^{(n)}_\\sigma(t)&=w_n\\theta_n\\int_0^t \\overline{D}_\\sigma(\\xi_{\\theta_n s})\n\\d s\\\\\n&+\\int_0^t \\Bigg(\\theta_n\\mu_n(\\sigma)[1-p_{\\sigma}(\\xi_{\\theta_ns})]-\\theta_n\\mu_n(S\\setminus\\{\\sigma\\}) p_\\sigma(\\xi_{\\theta_ns})\\Bigg)\\d s.\n\\end{split}\n\\end{align}\nSpecifically, in terms of the explicit form of $\\overline{D}_\\sigma$ in \\eqref{eq:Dsigma}, our goal is to prove that \n\\begin{align}\\label{closure}\n\\lim_{n\\to\\infty}\\sup_{\\xi\\in S^{E_n}}{\\mathbb E}^{w_n}_{\\xi}\\left[\\left|\\int_0^tw_n\\theta_n \\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{\\theta_ns})-w_\\infty Q_{\\sigma_0,\\sigma_2\\sigma_3}\\big(p(\\xi_{\\theta_ns })\\big)\\d s \\right|\\right]=0,\n\\end{align}\nwhere $\\sigma_0\\neq \\sigma_2$, $w_\\infty$ is defined by \\eqref{def:wn}, and $Q_{\\sigma_0,\\sigma_2\\sigma_3}(X)$ is a polynomial in $X=(X_\\sigma)_{\\sigma\\in S}$ defined by\n\\begin{align}\n\\begin{split}\n\\textcolor{black}{Q_{\\sigma_0,\\sigma_2\\sigma_3}(X)}&\\stackrel{\\rm def}{=}\\mathds 1_{\\{\\sigma_2=\\sigma_3\\}}({\\overline{\\kappa}}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3})X_{\\sigma_0}X_{\\sigma_2}\\\\\n&\\quad +\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}(\\overline{\\kappa}_{(0,3)|2}-\\overline{\\kappa}_{0|2|3})X_{\\sigma_0}X_{\\sigma_2}\\\\\n&\\quad +\\overline{\\kappa}_{0|2|3}X_{\\sigma_0}X_{\\sigma_2}X_{\\sigma_3}.\n\\end{split}\n\\label{def:Qsigma}\n\\end{align}\nThe choice of $Q_{\\sigma_0,\\sigma_2\\sigma_3}$ is due to the proof of Lemma~\\ref{lem:RW}.\n\nThe proof of \\eqref{closure} begins with an inequality central to the proof of \\cite[Theorem~2.2]{CCC}, which goes back to \\cite{CMP} and is also central to the proof of \\cite[Lemma~4.2]{CC}. This inequality is presented in a general form for future references. In what follows, we write $a\\wedge b$ for $\\min\\{a,b\\}$. \n\n\n\\begin{prop}\\label{prop:L2}\nGiven a Polish space $E_0$ and $T\\in (0,\\infty)$, let $(X_t)_{0\\leq t\\leq T}$ be an $E_0$-valued Markov process with c\\'adl\\'ag paths. Let $f$ and $g$ be bounded Borel measurable functions defined on $E_0$. Suppose that $x\\mapsto {\\mathbb E}_x[f(X_t)]$ is Borel measurable, and for some bounded decreasing function $a(t)$, \n\\begin{align}\\label{def:a(t)}\n\\sup_{x\\in E_0}{\\mathbb E}_x[|f(X_t)|]\\leq a(t),\\quad \\forall\\;t\\in [0,T]. \n\\end{align}\nThen for all $0<2\\deltas_0 s)\\d s\\\\\n\\leq & \nC_{\\ref{UVbdd:1}}\\Bigg(\\sum_{j=1}^\\ell \\prod_{k=1}^{j-1} \\big(1-{\\rm e}^{-2^{k+1}t}\\big)^{-1}\\Bigg)\\left(\\frac{\\pi^{(n)}_{\\max}}{\\pi^{(n)}_{\\min}}\\right) \\left[\\ell t+\\min\\Bigg\\{ \\frac{1}{\\mathbf g_ns_0}, \\frac{\\mathbf t^{(n)}_{\\rm mix}}{s_0}[1+\\log^+(\\gamma_n\/\\mathbf t^{(n)}_{\\rm mix})]\\Bigg\\}\\right],\\label{UVbdd:1}\n\\end{split}\n\\end{align}\nwhere $C_{\\ref{UVbdd:1}}$ is a universal constant.\n\\end{lem}\n\\begin{proof}\nBy \\eqref{ineq:MUVcompare} and Proposition~\\ref{prop:Mcompare}, we obtain the following inequality:\n\\begin{align}\n&{\\quad \\,} 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^{t}\\P^{(n)}(M_{U_0,U_\\ell}>s_0 s)\\d s\\notag\\\\\n&\\leq\\frac{\\gamma_n}{s_0}\\cdot \\Bigg(\\sum_{j=1}^\\ell \\prod_{k=1}^{j-1} \\big(1-{\\rm e}^{-2^{k+1}t}\\big)^{-1}\\Bigg)\\left(\\frac{\\pi^{(n)}_{\\max}}{\\pi^{(n)}_{\\min}}\\right)\n2s_0\\nu_n(\\mathds 1)\\int_0^{2\\ell t}\\P^{(n)}(M_{V,V'}>s_0 s)\\d s\\notag\\\\\n\\begin{split}\n&=\\frac{\\gamma_n}{s_0} \\cdot \\Bigg(\\sum_{j=1}^\\ell \\prod_{k=1}^{j-1} \\big(1-{\\rm e}^{-2^{k+1}t}\\big)^{-1}\\Bigg)\\left(\\frac{\\pi^{(n)}_{\\max}}{\\pi^{(n)}_{\\min}}\\right)\\\\\n&\\quad \\times \\left[\\P^{(n)}(M_{U,U'}>0)-\\P^{(n)}(M_{U,U'}>2\\ell s_0 t)\\right]\\label{UVbdd:00}\n\\end{split}\\\\\n\\begin{split}\n&\\leq C_{\\ref{UVbdd:0}}\\frac{\\gamma_n}{s_0}\\cdot \\Bigg(\\sum_{j=1}^\\ell \\prod_{k=1}^{j-1} \\big(1-{\\rm e}^{-2^{k+1}t}\\big)^{-1}\\Bigg)\\left(\\frac{\\pi^{(n)}_{\\max}}{\\pi^{(n)}_{\\min}}\\right)\\\\\n&{\\quad \\,}\\times \\left[\\big(1-{\\rm e}^{-2\\ell s_0t\/\\gamma_n}\\big)+\\min\\Bigg\\{ \\frac{2}{\\widetilde{\\mathbf g}_n\\gamma_n}, \\frac{\\widetilde{\\mathbf t}^{(n)}_{\\rm mix}}{\\gamma_n}\\big[1+\\log^+(\\gamma_n\/\\widetilde{\\mathbf t}^{(n)}_{\\rm mix})\\big]\\Bigg\\}\\right] \\label{UVbdd:0}\n\\end{split}\n\\end{align}\nfor a universal constant $C_{\\ref{UVbdd:0}}$. Here, \\eqref{UVbdd:00} follows from \\eqref{ergodic}, and \\eqref{UVbdd:0} follows from the exponential approximation of $M_{U,U'}$ as in the proof of Proposition~\\ref{prop:sn-selection}. Recall the reduction of mixing of products chains to mixing of the coordinates as used in that proposition, and the inequality $1-{\\rm e}^{-x}\\leq x$ holds for all $x\\geq 0$. Hence, we obtain \\eqref{UVbdd:1} from \\eqref{UVbdd:0}. The proof is complete. \n\\end{proof}\n\n\n\n\n\\begin{prop}\\label{prop:duhamel}\nFix $(\\sigma_0,\\sigma_1,\\sigma_2,\\sigma_3)\\in S\\times S\\times S$ such that $\\sigma_0\\neq \\sigma_2$ and $\\sigma_0\\neq \\sigma_1$. \\medskip \n\n\\noindent {\\rm (1${^\\circ}$)}\nFor any $w\\in [0,\\overline{w}]$ and $t\\in(0,\\infty)$, the following estimates of the evolutionary game by the voter model holds: for some constant $C_{\\ref{LwL:0}}$ depending only on $\\Pi$,\n\\begin{align}\n&\\quad \n\\sup_{\\xi\\in S^{E}}\\Big|{\\mathbb E}^{w}_\\xi\\big[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{t})\\big]-{\\mathbb E}^0_\\xi\\big[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{t})\\big]\\Big|\\notag\\\\\n\\begin{split}\n&\\leq C_{\\ref{LwL:0}} w\\int_0^t \\P(M_{U_0,U_2}>s)\\d s\n+C_{\\ref{LwL:0}}w\\mu(\\mathds 1)\\int_0^t\\int_0^s\\P(M_{U_0,U_2}>r)\\d r\\d s;\\label{LwL:0}\n\\end{split}\\\\\n&\\quad \n\\sup_{\\xi\\in S^{E}}\\Big|{\\mathbb E}^{w}_\\xi\\big[\\,\\overline{ f_{\\sigma_0\\sigma_1}}(\\xi_{t})\\big]-{\\mathbb E}^0_\\xi\\big[\\,\\overline{ f_{\\sigma_0\\sigma_1}}(\\xi_{t})\\big]\\Big|\\notag\\\\\n\\begin{split}\n&\\leq C_{\\ref{LwL:0}} w\\int_0^t \\P(M_{U_0,U_1}>s)\\d s\n+C_{\\ref{LwL:0}}w\\mu(\\mathds 1)\\int_0^t\\int_0^s\\P(M_{U_0,U_1}>r)\\d r\\d s.\\label{LwL:001}\n\\end{split}\n\\end{align}\n \\medskip \n\n\\noindent {\\rm (2${^\\circ}$)} For any admissible sequence $(\\theta_n,\\mu_n,w_n)$ and $T\\in(0,\\infty)$, it holds that \n\\begin{align}\\label{LwL:1}\n\\begin{split}\n&\\lim_{n\\to\\infty}\\int_0^T\\sup_{\\xi\\in S^{E_n}}\\left|{\\mathbb E}^{w_n}_{\\xi}\\left[w_n\\theta_n \\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{2s_n t})\\right]-{\\mathbb E}^{0}_{\\xi}\\left[w_n\\theta_n\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{2s_n t})\\right]\\right|\\d t=0.\n\\end{split}\n\\end{align}\n\\end{prop}\n\\begin{proof}\n(1${^\\circ}$) Recall that the generator of $\\mathsf L^w$ of the evolutionary game is given by \\eqref{def:Lw}, and $\\mathsf L=\\mathsf L^0$ denotes the generator of the voter model. By Duhamel's principle \\cite[(2.15) in Chapter~1]{EK:MP},\n\\begin{align}\\label{expansion}\n{\\rm e}^{t\\mathsf L^w}H={\\rm e}^{t\\mathsf L }H+\\int_0^t{\\rm e}^{(t-s)\\mathsf L^w}(\\mathsf L^w-\\mathsf L){\\rm e}^{s\\mathsf L}H\\d s .\n\\end{align}\nHere, it follows from \\eqref{def:Lw} that\n\\begin{align}\n(\\mathsf L^w-\\mathsf L)H_1(\\xi)&=\\sum_{x,y\\in E}[q^w(x,y,\\xi)-q(x,y)][H_1(\\xi^{x,y})-H_1(\\xi)].\n\\label{LwL1}\n\\end{align}\nTo apply \\eqref{expansion} and \\eqref{LwL1}, we choose $H=\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}$ and $H_1={\\rm e}^{s\\mathsf L}\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}$. The following bound will be proved in Proposition~\\ref{prop:mutation} (2${^\\circ}$):\n\\begin{align}\\label{claim:w0}\n\\begin{split}\n&\\sup_{\\xi\\in S^E}|{\\rm e}^{s\\mathsf L}\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi^x)-{\\rm e}^{s\\mathsf L}\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi)|\\\\\n&\\leq \n\\sum_{\\ell\\in \\{0,2,3\\}}4\\P(M_{U_0,U_2}>s,B^{U_\\ell}_{s}= x)\\\\\n&\\quad +\\sum_{\\ell\\in \\{0,2,3\\}}4\\mu(\\mathds 1)\\int_0^s \\P(M_{U_0,U_2}>r,B^{U_\\ell}_s=x)\\d r.\n\\end{split}\n\\end{align}\n\n\n\nTo bound $(\\mathsf L^w-\\mathsf L){\\rm e}^{s\\mathsf L}H=(\\mathsf L^w-\\mathsf L)H_1$ in the expansion \\eqref{expansion}, notice that \n\\begin{align}\\label{qwq:bdd}\n|q^w(x,y,\\xi)-q(x,y)|\\leq C_{\\ref{qwq:bdd}}wq(x,y)\n\\end{align}\nby \\eqref{qwq:exp} for some $C_{\\ref{qwq:bdd}}$ depending only on $\\Pi$. Putting \\eqref{LwL1}, \\eqref{claim:w0} and \\eqref{qwq:bdd} together, we get\n\\begin{align*}\n&\\quad \\sup_{\\xi\\in S^E}| (\\mathsf L^w-\\mathsf L){\\rm e}^{s \\mathsf L}\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi)|\\\\\n&\\leq C_{\\ref{qwq:bdd}} 4w\\sum_{\\ell\\in \\{0,2,3\\}}\\sum_{x,y\\in E}q(x,y)\\P(M_{U_0,U_2}>s,B^{U_\\ell}_s=x)\\\\\n&{\\quad \\,} +C_{\\ref{qwq:bdd}} 4w\\mu(\\mathds 1)\\sum_{\\ell\\in \\{0,2,3\\}}\\sum_{x,y\\in E}q(x,y)\\int_0^s \\P(M_{U_0,U_2}>r,B^{U_\\ell}_s=x)\\d r\\\\\n&\\leq C_{\\ref{qwq:bdd}}12 w\\P(M_{U_0,U_2}>s)+C_{\\ref{qwq:bdd}}12 w\\mu(\\mathds 1)\\int_0^s \\P(M_{U_0,U_2}>r)\\d r.\n\\end{align*}\nSince ${\\rm e}^{(t-s)\\mathsf L^w}$ is a probability, the required inequality in \\eqref{LwL:1} follows upon applying the foregoing inequality to \\eqref{expansion}. We have proved \\eqref{LwL:0}. The proof of \\eqref{LwL:001} is almost the same if we use Proposition~\\ref{prop:mutation} (3${^\\circ}$) instead of Proposition~\\ref{prop:mutation} (2${^\\circ}$). The details are omitted. \\medskip \n\n\\noindent (2${^\\circ}$) By the first limit in \\eqref{def:wn} and \\eqref{LwL:0}, it is enough to show that all of the following limits hold:\n\\begin{align}\n\\label{problem:wn1}\n&\\lim_{n\\to\\infty}\nw_n(2s_n)\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^{T} \\P^{(n)}(M_{U_0,U_2}>2s_n s)\\d s=0;\\\\\n&\\lim_{n\\to\\infty}\n[w_n(2s_n)+1]\\cdot \\mu_n(\\mathds 1)(2s_n)\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^{T}\\P^{(n)}(M_{U_0,U_2}>2s_n s)\\d s=0\\label{problem:wn2}.\n\\end{align}\n(The limit \\eqref{problem:wn2} is stronger than needed but is convenient for the other proofs below.)\n\nTo get \\eqref{problem:wn1}, first, note that by \\eqref{cond2:sn}, \\eqref{sn:adm} and the limit superior in \\eqref{def:wn},\n\\begin{align}\\label{problem:wn1-1}\n\\lim_{n\\to\\infty}w_n(2s_n)\\cdot \\left[T+\\min\\Bigg\\{ \\frac{1}{\\mathbf g_n(2s_n)}, \\frac{\\mathbf t^{(n)}_{\\rm mix}}{2s_n}[1+\\log^+(\\gamma_n\/\\mathbf t^{(n)}_{\\rm mix})]\\Bigg\\}\\right]=0.\n\\end{align}\nWe get \\eqref{problem:wn1} from applying \\eqref{cond:pi} and \\eqref{problem:wn1-1} to \\eqref{UVbdd:1} with $s_0=2s_n$.\nFor \\eqref{problem:wn2}, \n $\\lim_n\\mu_n(\\mathds 1)(2s_n)=0$ by \\eqref{def:mun} and \\eqref{sn:adm}. The limit superior in \\eqref{def:wn}\n and \\eqref{sn:adm} give $\\limsup_n w_n(2s_n)<\\infty$. These two properties are enough for \\eqref{problem:wn2}.\nThe proof is complete. \n\\end{proof}\n\nTo satisfy \\eqref{def:a(t)} under the setting of \\eqref{setup}, we consider the sum of the right-hand side of \\eqref{LwL:0}, with $t$ replaced by $\\theta_n t$, and $\\sup_{\\xi\\in S^{E_n}}{\\mathbb E}^0_\\xi\\big[w_n\\theta_n\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{\\theta_n t})\\big]$. Moreover, this supremum can be bounded by using \\eqref{ineq:mutation} and $\\P(M_{U_0,U_2}>\\theta_n t)$, thanks to duality and the choice $\\sigma_0\\neq \\sigma_2$. Therefore, given $T\\in(0,\\infty)$, we set $a(t)=a_n(t)=\\sum_{\\ell=1}^3 a_{n,\\ell}(t)$ for $t\\in [0,T]$, where\n\\begin{align}\\label{def:an(t)}\n\\begin{split}\na_{n,\\ell}(t)&\\stackrel{\\rm def}{=} C_{\\ref{def:an(t)}} \\cdot \n \\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot w_n\\theta_n\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^T \\P^{(n)}(M_{U_0,U_2}>\\theta_n s)\\d s\\\\\n&{\\quad \\,} +C_{\\ref{def:an(t)}} \\cdot \\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}>\\theta_n t) \\\\\n&{\\quad \\,} +C_{\\ref{def:an(t)}} \\cdot \\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot (w_n\\theta_n+1)\\cdot \n\\mu_n(\\mathds 1)\\theta_n\\\\\n&\\quad \\quad \\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^T\\P^{(n)}(M_{U_0,U_\\ell}>\\theta_ns)\\d s\n\\end{split}\n\\end{align}\nand $C_{\\ref{def:an(t)}}$ depends only on $(\\Pi,T)$. \nFor any $n\\geq 1$, $t\\mapsto a_{n}(t)$ is bounded and decreasing on $[0,T]$, and\n\\begin{align*}\n\\sup_{\\xi\\in S^{E_n}}{\\mathbb E}^{w_n}_\\xi\\left[w_n\\theta_n\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{\\theta_nt})\\right]&\n\\leq a_n(t),\\quad \\forall\\;t\\in [0,T].\n\\end{align*}\nHence, the conditions of $a_n(t)$ required in Proposition~\\ref{prop:L2} hold. \n\n\nFor the proof of \\eqref{closure}, the next step is to show that under the setting of \\eqref{setup} and the above choice of $a(t)=a_n(t)$, the right-hand side of \\eqref{ineq:L2} vanishes as $n\\to\\infty$. For the first term on the right-hand side of \\eqref{ineq:L2}, proving $\\int_0^{\\delta_n}a_n(t)\\d t$ amounts to proving $\\int_0^{\\delta_n}a_{n,\\ell}(t)\\d t\\to 0$ for all $1\\leq \\ell\\leq 3$. For the latter limits, note that $\\delta_n\\to 0$ by \\eqref{sn:adm}. Also, a slight modification of the proofs of \\eqref{problem:wn1}--\\eqref{problem:wn2} shows that for the right-hand side of \\eqref{def:an(t)}, the first and last terms in are bounded in $n$, and the second term satisfies \n\\begin{align*}\n&\\quad \\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\int_0^{\\delta_n}\\P^{(n)}(M_{U_0,U_\\ell}>\\theta_n s)\\d s\\\\\n&=\\lim_{n\\to\\infty}\\frac{2s_n}{\\theta_n}\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^{1}\\P^{(n)}(M_{U_0,U_\\ell}>2s_n s)\\d s=0.\n\\end{align*}\nFor the second term in \\eqref{ineq:L2}, it is enough to show that $a_n(\\delta_n)$'s are bounded. From the above argument for the first term in \\eqref{ineq:L2}, this property follows if we use the second limit in \\eqref{def:wn} and note that\n\\[\n2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}>\\theta_n t)|_{t=\\delta_n}=2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}>2s_n )\\xrightarrow[n\\to\\infty]{} \\overline{\\kappa}_\\ell,\n\\]\nwhere the limit follows from Proposition~\\ref{prop:sn-selection} and Proposition~\\ref{prop:kell}. To use these propositions precisely, passing the foregoing limit actually requires that given any subsequence of $(E_n,q^{(n)})$, a suitable further subsequence is used. To lighten the exposition, we continue to suppress similar uses of subsequential limits. \n\n\nFor the third term in \\eqref{ineq:L2}, note that $\\delta_n\\to 0$ by \\eqref{sn:adm}, and the $g_n$'s in \\eqref{setup} are uniformly bounded in $n$. The last term in \\eqref{ineq:L2} is the major term. By \\eqref{setup} and Proposition~\\ref{prop:duhamel} (2${^\\circ}$), it remains to prove\n\\begin{align}\\label{voter:density}\n\\lim_{n\\to\\infty}\\sup_{\\xi\\in S^{E_n}}\\big|{\\mathbb E}^{0}_\\xi[w_n\\theta_n\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{2s_n})]-w_\\infty Q_{\\sigma_0,\\sigma_2\\sigma_3}\\big(p(\\xi)\\big)\\big|=0.\n\\end{align}\nFor the next lemma, recall that the total variation distance $d_E$ and the spectral gap $\\mathbf g$ are defined at the beginning of Section~\\ref{sec:mainresults}. Also, here and in what follows, we use the shorthand notation ${\\mathbb E}[Z;A]={\\mathbb E}[Z\\mathds 1_A]$. \n\n\n\n\\begin{lem}\\label{lem:RW}\nFix $(\\sigma_0,\\sigma_2,\\sigma_3)\\in S\\times S\\times S$ such that $\\sigma_0\\neq \\sigma_2$. \\medskip \n\n\\noindent {\\rm (1${^\\circ}$)}\nGiven any $0s,M_{U_0,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\\\\n&\\quad\\quad+\\mathds 1_{\\{\\sigma_2=\\sigma_3\\}}\\P(M_{U_0,U_2}>s,M_{U_2,U_3}>s,M_{U_0,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\\\\n&\\quad\\quad -\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}\\P(M_{U_0,U_2}>s,M_{U_2,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\\\\n&\\quad\\quad +\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}\\P(M_{U_0,U_2}>s,M_{U_2,U_3}>s,M_{U_0,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\\\\n&\\quad\\quad -\\P(M_{U_0,U_2}>s,M_{U_2,U_3}>s,M_{U_0,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)p_{\\sigma_3}(\\xi)\n\\Big|\\\\\n&\\leq C_{\\ref{ineq:RW}}\\sum_{\\ell=1}^3\\Gamma_\\ell(s,t),\\label{ineq:RW}\n\\end{split}\n\\end{align} \nwhere $C_{\\ref{ineq:RW}}$ is a universal constant and\n\\begin{align}\n\\begin{split}\\label{ineq:Well_TV}\n\\Gamma_\\ell(s,t)&\\stackrel{\\rm def}{=} \\P(M_{U_0,U_\\ell}\\in (s,t])\\\\\n&{\\quad \\,}+\\min\\left\\{\\sqrt{\\frac{\\pi_{\\max}}{\\nu(\\mathds 1)}}{\\rm e}^{-\\mathbf g(t-s)},\\P(M_{U_0,U_\\ell}>s)d_E(t-s)\\right\\}\\\\\n&{\\quad \\,} + \\big(1-{\\rm e}^{-2\\mu(\\mathds 1)t}\\big)\\P(M_{U_0,U_\\ell}>t)+\n\\mu(\\mathds 1)\\int_0^t\\P(M_{U_0,U_\\ell}>r)\\d r.\n\\end{split}\n\\end{align}\n{\\rm (2${^\\circ}$)} The limit in \\eqref{voter:density} holds. \n\\end{lem}\n\n\n\n\\begin{proof}\n(1${^\\circ}$) First, we consider the case that there is no mutation. Roughly speaking, the method of this proof is to express ${\\mathbb E}_\\xi^0\\big[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\big]$ in terms of coalescing Markov chains before any two coalesce. This way we can express the coalescing Markov chains as independent Markov chains and compute the asymptotics of ${\\mathbb E}_\\xi^0\\big[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\big]$ by the ${\\overline{\\kappa}}$-constants defined in \\eqref{def:kell}--\\eqref{def:|||}. This idea goes back to \\cite[Proposition~6.1]{CCC}.\n\n\nNow, by duality and the assumption $\\sigma_0\\neq \\sigma_2$, it holds that \n\\begin{align}\n&\\quad\\,{\\mathbb E}_\\xi^0\\big[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\big]\\notag\\\\\n&={\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\n\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t)\\mathds 1_{\\sigma_3}\\circ \\xi(B^{U_3}_t)]\\notag\\\\\n&=\\mathds 1_{\\{\\sigma_2=\\sigma_3\\}}{\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t);M_{U_0,U_2}>t,M_{U_2,U_3}\\leq t,M_{U_0,U_3}>t]\\notag\\\\\n&\\quad +\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}{\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t);M_{U_0,U_2}>t,M_{U_2,U_3}> t,M_{U_0,U_3}\\leq t]\\notag\\\\\n&\\quad +{\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t)\\mathds 1_{\\sigma_3}\\circ \\xi(B^{U_3}_t);M_{U_0,U_2}>t,M_{U_2,U_3}> t,M_{U_0,U_3}>t]\\notag\\\\\n&=\\mathds 1_{\\{\\sigma_2=\\sigma_3\\}}{\\rm I}+\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}{\\rm II}+{\\rm III}.\\label{def:III}\n\\end{align}\nWe can further write ${\\rm I}$ and ${\\rm II}$ as\n\\begin{align}\n\\begin{split}\\label{def:Iterm}\n{\\rm I}&={\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t);M_{U_0,U_2}>t,M_{U_0,U_3}>t]\\\\\n&\\quad \\,-{\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t);M_{U_0,U_2}>t,M_{U_2,U_3}> t,M_{U_0,U_3}>t]\\\\\n&={\\rm I'}-{\\rm I''},\n\\end{split}\\\\\n\\begin{split}\\label{def:IIterm}\n{\\rm II}&={\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t);M_{U_0,U_2}>t,M_{U_2,U_3}> t]\\\\\n&\\quad\\, -{\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t);M_{U_0,U_2}>t,M_{U_2,U_3}> t,M_{U_0,U_3}> t]\\\\\n&={\\rm II'}-{\\rm I''}.\n\\end{split}\n\\end{align}\nWe estimate ${\\rm I'},{\\rm I''}, {\\rm II'}$ and ${\\rm III}$ below, using the property that the coalescing Markov chains move independently before meeting.\n\n\n\nFirst, for $0s,M_{U_0,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\big|\\\\\n&\\leq{\\mathbb E}\\Big[\\left|{\\rm e}^{(t-s)(q-1)}\\mathds 1_{\\sigma_0}\\circ\\xi(B_s^{U_0}){\\rm e}^{(t-s)(q-1)}\\mathds 1_{\\sigma_2}\\circ\\xi(B_s^{U_2})-p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\right|\\\\\n&\\quad \\quad ;M_{U_0,U_2}>s,M_{U_0,U_3}>s\\Big]\\\\\n&{\\quad \\,}+ \\P(M_{U_0,U_2}\\in (s,t])+\\P(M_{U_0,U_3}\\in (s,t]).\n\\end{split}\n\\end{align}\nOn the event $\\{M_{U_0,U_2}>s,M_{U_0,U_3}>s\\}$, $(B^{U_0}_r)_{0\\leq r\\leq s}$ and $(B^{U_2}_r)_{0\\leq r\\leq s}$ are independent $q$-Markov chains and each chain is stationary by the assumption on $\\{U_\\ell\\}$. Since $p_\\sigma(\\xi)=\\sum_x\\mathds 1_\\sigma\\circ \\xi(x)\\pi(x)$, the expectation in \\eqref{def:I1} can be estimate as in the proof of \\cite[Proposition~6.1]{CCC}. We get\n\\begin{align}\n&{\\quad \\,}\\big|{\\rm I}'-\\P(M_{U_0,U_2}>s,M_{U_0,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\big|\\notag\\\\\n&\\leq \n\\min\\left\\{2\\sqrt{\\frac{\\pi_{\\max}}{\\nu(\\mathds 1)}}{\\rm e}^{-\\mathbf g(t-s)},4\\P(M_{U_0,U_2}>s,M_{U_0,U_3}>s)d_E(t-s)\\right\\}\\notag\\\\\n&{\\quad \\,}+ \\P(M_{U_0,U_2}\\in (s,t])+\\P(M_{U_0,U_3}\\in (s,t]).\\notag\n\\end{align}\nSimilar estimates apply to the other terms ${\\rm I''}$, ${\\rm II}'$ and ${\\rm III}$ in \\eqref{def:III}, \\eqref{def:Iterm} and \\eqref{def:IIterm}. \n\n\nApplying all of these estimates to \\eqref{def:III} proves \\eqref{ineq:RW} when there is no mutation. The additional terms in \\eqref{ineq:RW} arise when we include mutation and use \\eqref{ineq:mutation} again.\\medskip\n\n \n\\noindent (2${^\\circ}$) Recall the second limit in \\eqref{def:wn} and \\eqref{problem:wn2}.\nThen by \\eqref{ineq:RW} and \\eqref{ineq:Well_TV} with $t=2s_n$ and $s=s_n$, it suffices to show all of the following limits:\n\\begin{align}\n&\\lim_{n\\to\\infty} 2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}\\in (s_n,2s_n])=0,\\quad 1\\leq \\ell\\leq 3;\\label{problem:wn4}\\\\\n&\\lim_{n\\to\\infty}\\big(1-{\\rm e}^{-2\\mu_n(\\mathds 1)\\cdot (2s_n)}\\big)\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}>2s_n),\\quad 1\\leq \\ell\\leq 3;\\label{problem:wn5}\\\\\n&\\lim_{n\\to\\infty} \\Gamma_{n,\\ell}=0,\\quad 1\\leq \\ell\\leq 3,\\label{problem:wn6}\n\\end{align}\nwhere $\\Gamma_{n,\\ell}$ is given by the minimum of the following two terms:\n\\begin{align}\\label{def:Gamma}\n\\gamma_n\\nu_n(\\mathds 1)\\cdot \\sqrt{\\frac{\\pi^{(n)}_{\\max}}{\\nu_n(\\mathds 1)}}{\\rm e}^{-\\mathbf g_ns_n},\\quad\n 2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}>s_n)d_{E_n}(s_n) .\n\\end{align}\n\nTo see \\eqref{problem:wn4}, we simply use Propositions~\\ref{prop:sn-selection} and \\ref{prop:kell}. The limit in \\eqref{problem:wn5} follows from the same propositions, in addition to \\eqref{def:mun} and \\eqref{sn:adm}. For \\eqref{problem:wn6}, we consider the following two cases. When $\\Gamma_n$ is given by the first term in \\eqref{def:Gamma}, the required limit holds by \\eqref{cond:pi} and the first limit in \\eqref{cond2:sn}. When $\\Gamma_n$ is given by the other term in \\eqref{def:Gamma}, we first use Propositions~\\ref{prop:sn-selection} and \\ref{prop:kell}. Then note that the second limit in \\eqref{cond2:sn} implies $\\lim_{n}\\mathbf t^{(n)}_{{\\rm mix}}\/s_n=0$, and so, $\\lim_nd_{E_n}(s_n)=0$ by \\eqref{ineq:tmix}. We have proved \\eqref{problem:wn6}. The proof is complete.\n\\end{proof}\n\nUp to this point, we have proved the asymptotic closure of equation in the sense of \\eqref{closure}. Note that under \\eqref{setup}, the convergence of the last term in \\eqref{ineq:L2} also contributes to asymptotic path regularity of the density processes. \n\nThe next lemma proves the asymptotic path regularity more explicitly as tightness in the convergence results of Theorem~\\ref{thm:main}. The limit of the normalized martingale terms in Theorem~\\ref{thm:main} (2${^\\circ}$) is also proven. Here, recall that the density processes satisfy the decompositions in \\eqref{density:dynamics}. From now on, $\\xrightarrow[n\\to\\infty]{\\rm (d)}$ refers to convergence in distribution as $n\\to\\infty$. \n\n\\begin{lem}\\label{lem:tight}\nFix $\\sigma\\in S$. \n\n\\begin{enumerate}\n\\item [\\rm (1${^\\circ}$)] \nThe sequence of laws of $I_\\sigma^{(n)}$ as continuous processes under $\\P^{w_n}_{\\nu_n}$ is tight.\n\n\n\\item [\\rm (2${^\\circ}$)] \nThe sequence of laws of $\\mathds 1_{\\{w_n>0\\}}w_n^{-1}R_\\sigma^{(n)}$ as continuous processes under $\\P^{w_n}_{\\nu_n}$ is tight.\n\n\\item [\\rm (3${^\\circ}$)] The sequence of laws of $M_\\sigma^{(n)}$ as continuous processes under $\\P^{w_n}_{\\nu_n}$ converges to zero in distribution. \n\\end{enumerate}\nIf, in addition, $\\lim_n \\gamma_n\\nu_n(\\mathds 1)\/\\theta_n=0$, then the following holds.\n\\begin{enumerate}\n\\item [\\rm (4${^\\circ}$)] The sequence of laws of \n\\begin{align}\\label{def:product}\n\\left(\\left(\n\\frac{\\gamma_n}{\\theta_n}\\right)^{1\/2}M_\\sigma^{(n)}(t),\\;\\;\n\\frac{\\gamma_n}{\\theta_n}\\langle M_\\sigma^{(n)},M_\\sigma^{(n)}\\rangle_{t}-\\int_0^t p_\\sigma(\\xi_{\\theta_ns})[1-p_{\\sigma}(\\xi_{\\theta_ns})]\\d s;t\\geq 0\\right)\n\\end{align}\nas processes with c\\`adl\\`ag paths under $\\P^{w_n}_{\\nu_n}$ is $C$-tight, and the second coordinates \nconverge to zero in distribution as processes. Moreover, for all $T\\in(0,\\infty)$,\n\\begin{align}\\label{ineq:L2-bdd}\n\\sup_{n\\geq 1}\\sup_{t\\in [0,T]}\\sup_{\\xi\\in S^{E_n}}\\frac{\\gamma_n}{\\theta_n}{\\mathbb E}^{w_n}_{\\xi}\\big[M^{(n)}_\\sigma(t)^2\\big]<\\infty.\n\\end{align}\n\\item [\\rm (5${^\\circ}$)] For any $\\sigma'\\in S$ with $\\sigma'\\neq \\sigma$, the sequence \n\\begin{align}\\label{def:product1}\n\\left(\n\\frac{\\gamma_n}{\\theta_n}\\langle M_\\sigma^{(n)},M_{\\sigma'}^{(n)}\\rangle_{t}+\\int_0^t p_\\sigma(\\xi_{\\theta_ns})p_{\\sigma'}(\\xi_{\\theta_ns})\\d s;t\\geq 0\\right)\n\\end{align}\nunder $\\P^{w_n}_{\\nu_n}$ converges to zero in distribution as processes. \n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n(1${^\\circ}$) First, we show a bound for $\\sup_{\\xi\\in S^{E_n}}{\\mathbb E}_{\\xi}^{w_n}[|I^{(n)}_\\sigma(\\theta)|]$ explicitly in $\\theta$. By \\eqref{eq:Dsigma0},\n\\begin{align}\\label{Dsigma:22}\n|\\overline{D}_\\sigma(\\xi)|\\leq C_{\\ref{Dsigma:22}}\\sum_{\\stackrel{\\scriptstyle \\sigma'\\in S}{\\sigma'\\neq \\sigma}}\\overline{f_{\\sigma\\sigma'}}(\\xi)\n\\end{align}\nfor some constant $C_{\\ref{ineq:Itheta}}$ depending only on $\\Pi$ and $\\#S$. Indeed, for $q(x,y)>0$, $\\mathds 1_\\sigma\\circ\\xi(y)-\\mathds 1_\\sigma\\circ\\xi (x)\\neq 0$ implies that either $\\xi(x)$ or $\\xi(y)$ is $\\sigma$ but not both. By \\eqref{def:In} and \\eqref{Dsigma:22}, \n\\begin{align}\\label{In:continuity}\n\\begin{split}\n\\sup_{\\xi\\in S^{E_n}}{\\mathbb E}_{\\xi}^{w_n}[|I^{(n)}_\\sigma(\\theta)|]&\\leq C_{\\ref{Dsigma:22}}\\sum_{\\stackrel{\\scriptstyle \\sigma'\\in S}{\\sigma'\\neq \\sigma}}\\int_0^\\theta \\sup_{\\xi\\in S^{E_n}}{\\mathbb E}^{w_n}_{\\xi}\\left[w_n\\theta_n\\overline{f_{\\sigma\\sigma'}}(\\xi_{\\theta_n s})\\right]\\d s\\\\\n&\\quad +2\\mu_n(\\mathds 1)\\theta_n\\cdot \\theta.\n\\end{split}\n\\end{align}\nFurthermore, we can bound the expectations on the right-hand side of \\eqref{In:continuity} by using an analogue of the $a_n(t)$ in \\eqref{def:an(t)}, but now involving only the meeting time $M_{V,V'}$. Specifically, given $T\\in(0,\\infty)$, the following inequality holds for all $t\\in [0,T]$:\n\\begin{align}\\label{def:an'(t)}\n\\begin{split}\n&{\\quad \\,}{\\mathbb E}^{w_n}_{\\xi}\\left[w_n\\theta_n\\overline{f_{\\sigma\\sigma'}}(\\xi_{\\theta_n t})\\right]\\\\\n&\\leq C_{\\ref{def:an'(t)}} \\cdot \n \\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot w_n\\theta_n\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^T \\P^{(n)}(M_{V,V'}>\\theta_n s)\\d s\\\\\n&{\\quad \\,} +C_{\\ref{def:an'(t)}} \\cdot \\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{V,V'}>\\theta_n t) \\\\\n&{\\quad \\,} +C_{\\ref{def:an'(t)}} \\cdot \\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot (w_n\\theta_n+1)\\cdot\n\\mu_n(\\mathds 1)\\theta_n\\\\\n&{\\quad \\,}\\quad \\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^T\\P^{(n)}(M_{V,V'}>\\theta_ns)\\d s\n\\end{split}\n\\end{align}\nand $C_{\\ref{def:an'(t)}}$ depends only on $(\\Pi,T,\\sup_n\\pi^{(n)}_{\\max}\/\\pi^{(n)}_{\\min})$. To see \\eqref{def:an'(t)}, we combine \\eqref{LwL:001} and \\cite[Proposition~3.2]{CC} and then use \\eqref{cond:pi} and \\eqref{ineq:MUVcompare} to reduce probabilities of $M_{U_0,U_1}$ to probabilities of $M_{V,V'}$.\n\n\nNext, we show that \n\\begin{align}\\label{ineq:Itheta}\n\\lim_{\\theta\\searrow 0}\\limsup_{n\\to\\infty}\\sup_{\\xi\\in S^{E_n}}{\\mathbb E}_\\xi^{w_n}[|I^{(n)}_\\sigma(\\theta)|]=0.\n\\end{align}\nFirst, \\eqref{def:mun} readily gives the required limit of the last term of \\eqref{In:continuity}. We focus on the sum of integrals on the right-hand side of \\eqref{def:mun}. For each of these integrals, note that the first and last terms in \\eqref{def:an'(t)} are uniformly bounded in $n$ as in the case of \\eqref{def:an(t)}. The integral over $t\\in [0,\\theta]$ of the second term in \\eqref{def:an'(t)} satisfies \n\\begin{align}\\label{MVV':conv-tight}\n\\lim_{\\theta\\searrow 0}\\limsup_{n\\to\\infty}\\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot \\frac{2\\gamma_n}{\\theta_n}\\nu_n(\\mathds 1)\\int_0^{\\theta\\theta_n} \\P^{(n)}(M_{V,V'}> s)\\d s=0\n\\end{align}\nby the second limit of \\eqref{def:wn}, \\eqref{ergodic}, and \\eqref{fnsn:est} with $s_n$ replaced by $\\theta_n$ and with $t_2=\\theta$ and $t_1=0$ since $(\\theta_n)$ is also a slow sequence. (The use of $M_{V,V'}$ allows us to circumvent Lemma~\\ref{lem:UVbdd} due to the explosion of the bound in \\eqref{UVbdd:1} as $t\\to 0$.) We have proved \\eqref{ineq:Itheta}.\n\n\n\nFinally, the required tightness follows from \\eqref{ineq:Itheta}, the strong Markov property of the particle system and Aldous's criterion for tightness \\cite[Proposition~VI.4.5 on p.356]{JS}. The detail is similar to the proof of \\cite[Theorem~5.1 (1)]{CCC}. \\medskip\n\n\\noindent {\\rm (2${^\\circ}$)} Recall the equation \\eqref{def:R} of $R_\\sigma$, and the explicit form of $R^w$ can be read from \\eqref{qwq:exp}. Then by the same reason for \\eqref{Dsigma:22}, the coefficient of $R_\\sigma$ satisfies \n\\begin{align}\\label{bdd:R}\n\\sum_{x,y\\in E}\\pi(x)|\\mathds 1_\\sigma\\circ\\xi(y)-\\mathds 1_\\sigma\\circ\\xi (x)|q(x,y)|R^w(x,y,\\xi)|\\leq C_{\\ref{bdd:R}}\\sum_{\\sigma'\\in S}\\overline{f_{\\sigma\\sigma'}}(\\xi),\n\\end{align}\nwhere $C_{\\ref{bdd:R}}$ depends only on $\\Pi$ and $\\#S$. From \\eqref{bdd:R},\n the argument in (1${^\\circ}$) applies again. \n\\medskip\n\n\n\\noindent {\\rm (3${^\\circ}$)} The proof follows from a slight modification of the proof of (4${^\\circ}$) below even without the additional assumption $\\lim_n \\gamma_n\\nu_n(\\mathds 1)\/\\theta_n=0$. \\medskip \n\n\\noindent {\\rm (4${^\\circ}$)}\nWe start with the convergence of the second coordinate in \\eqref{def:product}. Define a density function $\\widetilde{p}_{\\sigma}(\\xi)$ on $S^{E_n}$ such that the stationary weights $\\pi^{(n)}(x)$ in $p_\\sigma(\\xi)$ are replaced by $\\pi^{(n)}(x)^2\/\\nu_n(\\mathds 1)$. From \\eqref{def:}, the following equality holds under $\\P^{w_n}_{\\xi}$ for all $\\xi\\in S^{E_n}$:\n\\begin{align}\\label{eq:Mn}\n\\begin{split}\n&\\quad \\frac{\\gamma_n}{\\theta_n}\\langle M_\\sigma^{(n)},M_{\\sigma}^{(n)}\\rangle_t\\\\\n&=\\gamma_n\\nu_n(\\mathds 1)\\int_0^t \\sum_{\\sigma'\\in S\\setminus\\{\\sigma\\}}[p_{\\sigma'\\sigma}(\\xi_{\\theta_ns})+p_{\\sigma\\sigma'}(\\xi_{\\theta_ns})]\\d s\\\\\n&\\quad +\\gamma_n\\nu_n(\\mathds 1)\\int_0^t \\Big([1-\\widetilde{p}_{\\sigma}(\\xi_{\\theta_ns})]\\mu_n(\\sigma)+\\widetilde{p}_\\sigma(\\xi_{\\theta_ns})\\mu_n(S\\setminus\\{\\sigma\\})\\Big)\\d s\\\\\n&\\quad + \\frac{\\gamma_n\\nu_n(\\mathds 1)}{\\theta_n}\\cdot w_n\\theta_n\\int_0^{t} \\widetilde{R}^{(n)}_{w_n}(\\xi_{\\theta_n s})\\d s.\n\\end{split}\n\\end{align}\nHere, $\\widetilde{R}^{(n)}_{w_n}$ can be bounded in the same way as \\eqref{Dsigma:22}. Note that due to the use of $\\widetilde{R}^{(n)}_{w_n}$, we only involve the first term $q(x,y)$ in the expansion \\eqref{qwq:exp} of $q^{w_n}(x,y,\\xi)$. \n\nLet us explain how the required convergence of the second coordinate in \\eqref{def:product} follows \\eqref{eq:Mn}. First, since $\\lim_n \\gamma_n\\nu_n(\\mathds 1)\/\\theta_n=0$ by assumption, the bound for $\\widetilde{R}^{(n)}_{w_n}$ mentioned above and the proof of (1${^\\circ}$) show that the continuous process defined by the last integral of \\eqref{eq:Mn} converges to zero in distribution. Next, for all $0\\leq T_0}. The details are omitted. \n\\end{proof}\n\n\n\n\n\\subsection{The replicator equation and the Wright--Fisher fluctuations}\nIn this subsection, we complete the proof of Theorem~\\ref{thm:main} and give the proof of Corollary~\\ref{cor:symmetric}.\\\\\n\n\n\\begin{proof}[Completion of the proof of Theorem~\\ref{thm:main}]\nBy \\eqref{eq:Dsigma}, \\eqref{def:In}, \\eqref{closure} and Lemma~\\ref{lem:tight} (1${^\\circ}$)--(3${^\\circ}$), we have proved that the following vector process converges to zero in distribution:\n\\begin{align*}\n&p_\\sigma(\\xi_{\\theta_n t})-\\int_0^tw_\\infty\\Bigg(\\sum_{\\stackrel{\\scriptstyle \\sigma_0,\\sigma_3\\in S}{ \\sigma_0\\neq\\sigma}}\\Pi(\\sigma,\\sigma_3) Q_{\\sigma_0,\\sigma\\sigma_3}\\big(p(\\xi_{\\theta_n s})\\big)-\\sum_{\\stackrel{\\scriptstyle \\sigma_2,\\sigma_3\\in S}{\\sigma_2\\neq\\sigma}}\\Pi(\\sigma_2,\\sigma_3) Q_{\\sigma,\\sigma_2\\sigma_3}\\big(p(\\xi_{\\theta_n s})\\big)\\Bigg)\\d s\\\\\n&{\\quad \\,}-\\int_0^t \\Bigg(\\mu_\\infty(\\sigma)[1-p_{\\sigma}(\\xi_{\\theta_ns})]-\\mu_\\infty(S\\setminus\\{\\sigma\\}) p_\\sigma(\\xi_{\\theta_ns})\\Bigg)\\d s,\\quad \\sigma\\in S,\n\\end{align*}\nwhere the polynomials $Q_{\\sigma_0,\\sigma_2\\sigma_3}$ are defined in \\eqref{def:Qsigma}. Hence, the sequence of laws of $p(\\xi_{\\theta_nt})$ is $C$-tight, and $p(\\xi_{\\theta_nt})$ converges in distribution to $X(t)$ as processes, where $X$ is the unique solution to the following system:\n\\begin{align}\\label{p1:lim0}\n\\dot{X}_\\sigma=w_\\infty Q_\\sigma(X)+\\mu_\\infty(\\sigma)(1-X_\\sigma)-\\mu_\\infty(S\\setminus\\{\\sigma\\}) X_\\sigma,\\quad \\sigma\\in S,\n\\end{align}\nand the polynomial $Q_\\sigma(X)$ in \\eqref{p1:lim0} is given by\n\\begin{align}\\label{def:grandQ}\nQ_\\sigma(X)&=\\sum_{\\stackrel{\\scriptstyle \\sigma_0,\\sigma_3\\in S}{\\sigma_0\\neq \\sigma}}\\Pi(\\sigma,\\sigma_3)Q_{\\sigma_0,\\sigma\\sigma_3}(X)-\\sum_{ \\stackrel{\\scriptstyle \\sigma_2,\\sigma_3\\in S}{\\sigma_2\\neq \\sigma}}\\Pi(\\sigma_2,\\sigma_3)Q_{\\sigma,\\sigma_2\\sigma_3}(X).\n\\end{align}\n\n\nTo simplify \\eqref{def:grandQ} to the required form in \\eqref{p1:lim}, note that the constraints $\\sigma_0\\neq \\sigma$ and $\\sigma_2\\neq \\sigma$ in \\eqref{def:grandQ} can be removed from the definition of $Q_\\sigma(X)$ by cancelling repeating terms. In doing so, we extend the definition $Q_{\\sigma_0,\\sigma_2\\sigma_3}(X)$ to $\\sigma_0=\\sigma_2$ by the same formula in \\eqref{def:Qsigma}, but only in this proof. We also lighten notation by the following: $A=\\overline{\\kappa}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3}$ and $B=\\overline{\\kappa}_{(0,3)|2}-\\overline{\\kappa}_{0|2|3}$ and $C=\\overline{\\kappa}_{0|2|3}$. Then by \\eqref{def:grandQ},\n\\begin{align*}\n&Q_\\sigma(X)=\\sum_{\\sigma_0,\\sigma_3\\in S}\\Pi(\\sigma,\\sigma_3)\\left.\\left(\\mathds 1_{\\{\\sigma_2=\\sigma_3\\}}AX_{\\sigma_0}X_{\\sigma_2}+\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}BX_{\\sigma_0}X_{\\sigma_2}+CX_{\\sigma_0}X_{\\sigma_2}X_{\\sigma_3}\\right)\\right|_{\\sigma_2=\\sigma}\\\\\n&\\quad \\;-\\sum_{\\sigma_2,\\sigma_3\\in S}\\Pi(\\sigma_2,\\sigma_3)\\left.\\left(\\mathds 1_{\\{\\sigma_2=\\sigma_3\\}}AX_{\\sigma_0}X_{\\sigma_2}+\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}BX_{\\sigma_0}X_{\\sigma_2}+CX_{\\sigma_0}X_{\\sigma_2}X_{\\sigma_3}\\right)\\right|_{\\sigma_0=\\sigma}\\\\\n&=X_\\sigma \\sum_{\\sigma_0\\in S}A\\Pi(\\sigma,\\sigma)X_{\\sigma_0}+X_\\sigma\\sum_{\\sigma_0\\in S}B\n\\Pi(\\sigma,\\sigma_0)X_{\\sigma_0}+X_\\sigma \\sum_{\\sigma_3\\in S}C\\Pi(\\sigma,\\sigma_3)X_{\\sigma_3}\\\\\n&\\quad \\;-X_\\sigma\\sum_{\\sigma_2\\in S}A\\Pi(\\sigma_2,\\sigma_2)X_{\\sigma_2}-X_\\sigma \\sum_{\\sigma_2\\in S}B\\Pi(\\sigma_2,\\sigma)X_{\\sigma_2}\\\\\n&\\quad -\nX_\\sigma\\sum_{\\sigma_2\\in S}\\Bigg(\\sum_{\\sigma_3\\in S}C\\Pi(\\sigma_2,\\sigma_3)X_{\\sigma_3}\\Bigg)X_{\\sigma_2}\\\\\n&=X_\\sigma\\Bigg(A\\Pi(\\sigma,\\sigma)+\\sum_{\\sigma'\\in S}B\n[\\Pi(\\sigma,\\sigma')-\\Pi(\\sigma',\\sigma)]X_{\\sigma'}+\\sum_{\\sigma'\\in S}C\\Pi(\\sigma,\\sigma')X_{\\sigma'}\\Bigg)\\\\\n&\\quad -X_\\sigma \\sum_{\\sigma'\\in S}\\Bigg(A\\Pi(\\sigma',\\sigma')+\\sum_{\\sigma''\\in S}C\\Pi(\\sigma',\\sigma'')X_{\\sigma''}\\Bigg)X_{\\sigma'}.\n\\end{align*}\nNote that we have used the property $\\sum_{\\sigma}X_{\\sigma}=1$ in the last two equalities. The last equality is enough for the required form in \\eqref{p1:lim} upon recalling \\eqref{p1:lim0} and involving the polynomials $F_\\sigma(X)$ and $\\widetilde{F}_\\sigma(X)$ in \\eqref{F1} and \\eqref{F2}. Moreover, \\eqref{kappa:>} holds by Proposition~\\ref{prop:sn-selection}, \\eqref{ineq:MUVcompare}, and Proposition~\\ref{prop:kell} (1${^\\circ}$) and (4${^\\circ}$). \n\nFor the proof of (2${^\\circ}$), notice that by (1${^\\circ}$) and Lemma~\\ref{lem:tight} (4${^\\circ}$)--(5${^\\circ}$), the following convergence of matrix processes holds:\n\\[\n\\left(\\frac{\\gamma_n}{\\theta_n}\\langle M_\\sigma^{(n)},M_{\\sigma'}^{(n)}\\rangle_t\\right)_{\\sigma,\\sigma'\\in S}\\xrightarrow[n\\to\\infty]{\\rm (d)}\\left(\\int_0^t X_\\sigma(s)[\\delta_{\\sigma,\\sigma'}-X_{\\sigma'}(s)]\\d s\\right)_{\\sigma,\\sigma'\\in S}.\n\\]\nBy this convergence and \\eqref{ineq:L2-bdd}, the standard martingale problem argument shows that every weakly convergent subsequence of $((\\gamma_n\/\\theta_n)^{1\/2} M_\\sigma^{(n)};\\sigma\\in S)$ converges to a continuous vector $L_2$-martingale $(M_\\sigma^{(\\infty)};\\sigma\\in S)$ with a quadratic variation matrix given by $(\\int_0^t X_\\sigma(s)[\\delta_{\\sigma,\\sigma'}-X_{\\sigma'}(s)]\\d s;\\sigma,\\sigma'\\in S)$. See \\cite[Proposition~1.12 in Chapter~IX on p.525]{JS} and the proof of \\cite[Theorem~1.10 in Chapter~XIII on pp.519--520]{RY}. Hence, the limiting vector martingale $(M_\\sigma^{(\\infty)};\\sigma\\in S)$ is a Gaussian process with covariance matrix $(\\int_0^t X_\\sigma(s)[\\delta_{\\sigma,\\sigma'}-X_{\\sigma'}(s)]\\d s;\\sigma,\\sigma'\\in S)$ \\cite[Exercise (1.14) in Chapter~V on p.186]{RY}. Moreover, by uniqueness in law of this Gaussian process, the convergence holds along the whole sequence of the vector martingale $(\\gamma_n\/\\theta_n)^{1\/2}M^{(n)}$. The proof is complete.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Corollary~\\ref{cor:symmetric}]\nWrite $u$ for $X_1$. In this case, $X_0=1-u$ and the polynomial $Q_1(X)$ defined by \\eqref{def:grandQ} simplifies to\n\\[\nQ_1=(b-c)Q_{0,11}-cQ_{0,10}-bQ_{1,01}=(Q_{0,11}-Q_{1,01})b-(Q_{0,11}+Q_{0,10})c.\n\\]\nBy \\eqref{def:Qsigma}, the coefficient of $c$ is given by \n\\begin{align}\\label{coeff:c}\n\\begin{split}\n-Q_{0,11}(X)-Q_{0,10}(X)&=-(\\overline{\\kappa}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3})(1-u)u-\\overline{\\kappa}_{0|2|3}(1-u)u^2\\\\\n&\\quad -(\\overline{\\kappa}_{(0,3)|2}-\\overline{\\kappa}_{0|2|3})(1-u)u-\\overline{\\kappa}_{0|2|3}(1-u)^2u,\n\\end{split}\n\\end{align}\nand the coefficient of $b$ is\n\\begin{align}\\label{coeff:b}\n\\begin{split}\nQ_{0,11}(X)-Q_{1,01}(X)&=({\\overline{\\kappa}}_{(2,3)|0}-{\\overline{\\kappa}}_{0|2|3})(1-u)u+\\overline{\\kappa}_{0|2|3}u^2(1-u)u^2\\\\\n&\\quad -(\\overline{\\kappa}_{(0,3)|2}-\\overline{\\kappa}_{0|2|3})(1-u)u-\\overline{\\kappa}_{0|2|3}(1-u)u^2.\n\\end{split}\n\\end{align}\n\n\n\nThese two coefficients can be simplified by using the definition of $Q_{\\sigma_0,\\sigma_2\\sigma_3}$ and Proposition~\\ref{prop:kell} (4${^\\circ}$), if we follow the algebra in the proof of Lemma~\\ref{lem:D} that simplifies \\eqref{eq:Dsigma} to \\eqref{eq:Dsigma1}. For example, a similar argument as in the proof of Lemma~\\ref{lem:RW} shows that \\eqref{voter:density} holds with \n\\[\nQ_{0,01}(X)=(\\overline{\\kappa}_{(0,2)|3}-\\overline{\\kappa}_{0|2|3})(1-u)u+\\overline{\\kappa}_{0|2|3}(1-u)^2u,\n\\]\nand so\n\\begin{align*}\nQ_{0,11}(X)+Q_{0,01}(X)&=(\\overline{\\kappa}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3})(1-u)u+\\overline{\\kappa}_{0|2|3}(1-u)u^2\\\\\n&{\\quad \\,} +(\\overline{\\kappa}_{(0,2)|3}-\\overline{\\kappa}_{0|2|3})(1-u)u+\\overline{\\kappa}_{0|2|3}(1-u)^2u \\\\\n&=(1-u)u{\\overline{\\kappa}}_{3},\n\\end{align*}\nwhere the last equality follows from Proposition~\\ref{prop:kell} (4${^\\circ}$). In this way, we can obtain from \\eqref{coeff:c} and \\eqref{coeff:b} that $Q_1(X)=[({\\overline{\\kappa}}_3-{\\overline{\\kappa}}_1)b-{\\overline{\\kappa}}_2 c](1-u)u$. Moreover, by Proposition~\\ref{prop:kell}, we can pass limit along the whole sequence to get this limiting polynomial $Q_1(X)$.\n \\end{proof}\n \n \n \n \n\\section{Further properties of coalescing lineage distributions}\\label{sec:coal}\n\n\\subsection{A comparison with mutations}\nIn this section, we prove some auxiliary results for the proof of Theorem~\\ref{thm:main}. The next proposition estimates the voter model $(\\xi_t)$ under $\\P^0$ by its selection mechanism, that is, by the updates from $\\{\\Lambda(x,y);x,y\\in E\\}$. The proof extends \\cite[Proposition~3.2]{CC}. Recall the notation in Section~\\ref{sec:dynamics} for the coalescing Markov chains. \n\n\\begin{prop}\\label{prop:mutation}\n{\\rm (1${^\\circ}$)} Let $f:S\\times S\\times S\\to [-1,1]$ be a function such that $f(\\sigma,\\sigma,\\cdot)=0$ for all $\\sigma\\in S$. \nThen for all $t\\in (0,\\infty)$ and $x,y,z\\in E$,\n\\begin{align}\\label{ineq:mutation}\n\\begin{split}\n& \\sup_{\\xi\\in S^E}\\Big|{\\mathbb E}^0_\\xi\\big[f\\big(\\xi_t(x),\\xi_t(y),\\xi_t(z)\\big)\\big]-{\\mathbb E}\\big[f\\big(\\xi(B^x_t),\\xi(B^y_t),\\xi(B^z_t)\\big)\\big]\\Big|\\\\\n&\\quad \\leq \\big(1-{\\rm e}^{-2\\mu(\\mathds 1)t}\\big)\\P(M_{x,y}>t)+\\mathds 1_{x\\neq y}\\big(1-{\\rm e}^{-\\mu(\\mathds 1)t}\\big)\\P(M_{x,z}\\wedge M_{y,z}>t)\\\\\n&\\quad \\quad +2\\mu(\\mathds 1)\\int_0^t \\P(M_{x,y}>s)\\d s+\\mathds 1_{x\\neq y}\\mu(\\mathds 1)\\int_0^t \\P(M_{x,z}\\wedge M_{y,z}>s)\\d s.\n\\end{split}\n\\end{align}\n{\\rm (2${^\\circ}$)} For all $(\\sigma_0,\\sigma_2,\\sigma_3)\\in S\\times S\\times S$ with $\\sigma_0\\neq \\sigma_2$, $t\\in(0,\\infty)$ and $x\\in E$,\n\\begin{align}\\label{claim:w0-mutation}\n\\begin{split}\n&{\\quad \\,}\\sup_{\\xi\\in S^E}\\big|{\\mathbb E}^0_{\\xi^x}\\left[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\right]-{\\mathbb E}^0_\\xi\\left[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\right]\\big|\\\\\n&\\leq \\sum_{\\ell\\in \\{0,2,3\\}}4 \\P(M_{U_0,U_2}>t,B^{U_\\ell}_t=x)\\\\\n&\\quad +\\sum_{\\ell\\in \\{0,2,3\\}} 4\\mu(\\mathds 1)\\int_0^t \\P(M_{U_0,U_2}>s,B^{U_\\ell}_t=x)\\d s.\n\\end{split}\n\\end{align}\n\\noindent {\\rm (3${^\\circ}$)} For all $(\\sigma_0,\\sigma_1)\\in S\\times S$ with $\\sigma_0\\neq \\sigma_1$, $t\\in(0,\\infty)$ and $x\\in E$,\n\\begin{align}\\label{claim:w0-mutation1}\n\\begin{split}\n&{\\quad \\,}\\sup_{\\xi\\in S^E}\\big|{\\mathbb E}^0_{\\xi^x}\\left[\\,\\overline{f_{\\sigma_0\\sigma_1}}(\\xi_t)\\right]-{\\mathbb E}^0_\\xi\\left[\\,\\overline{ f_{\\sigma_0\\sigma_1}}(\\xi_t)\\right]\\big|\\\\\n&\\leq \\sum_{\\ell\\in \\{0,1\\}}4 \\P(M_{U_0,U_1}>t,B^{U_\\ell}_t=x)\\\\\n&\\quad +\\sum_{\\ell\\in \\{0,1\\}} 4\\mu(\\mathds 1)\\int_0^t \\P(M_{U_0,U_1}>s,B^{U_\\ell}_t=x)\\d s.\n\\end{split}\n\\end{align}\n\\end{prop}\n\\mbox{}\n\nThe proof of this proposition extends the proof of \\cite[Proposition~3.2]{CC} and is based on the pathwise duality between the voter model and the coalescing Markov chains. The relation follows from time reversal of the stochastic integral equations in Section~\\ref{sec:mainresults} of the voter model. More specifically, for fixed $t\\in (0,\\infty)$, we define a system of coalescing $q$-Markov chains $\\{B^{a,t};a\\in E\\}$ such that in the absence of mutation, $B^{a,t}$ traces out the time-reversed ancestral line that determines the type at $(a,t)$ under the voter model. For example, if $s$ is the last jump time of $\\{\\Lambda_r(a,b);b\\in E,r\\in (0,t]\\}$ and $\\Lambda(a,c)$ causes this jump, the state of $B^{a,t}$ stays at $a$ before transitioning to $B^{a,t}_{t-s}=c$. Similarly, with the Poisson processes $\\Lambda^\\sigma$ driving the mutations, we can define $e(a,t)$ and $M(a,t)$ for the time and the type from the first mutation event on the trajectory of $B^{a,t}$, with $e(a,t)=\\infty$ if there is no mutation. Since $e(a,t)>t$ if and only if $e(a,t)=\\infty$, we have\n\\begin{align}\\label{prob:dual}\n\\xi_t(a)=M(a,t)\\mathds 1_{\\{e(a,t)\\leq t\\}}+\\xi\\big(B^{a,t}_t\\big)\\mathds 1_{\\{e(a,t)>t\\}},\\quad\\forall\\;a\\in E,\\quad\\mbox{$\\P^0_\\xi$-a.s.}\n\\end{align}\nMore details can be seen by modifying the description in \\cite[Section~6.1]{CC}. In the absence of mutation, this relation between the duality and the stochastic integral equations is known in \\cite{MT}. \n\n\nWe also observe two identities for the probability distributions of the mutation times $e(a,t)$'s when we condition on $\\mathscr G\\stackrel{\\rm def}{=}\\sigma(\\Lambda(a,b);a,b\\in E)$. Let $x,y\\in E$. Write $0=J_0t\\}.\n\\end{split}\n\\end{align}\nThen consider the corresponding differences for the left-hand side of \\eqref{ineq:mutation}:\n\\begin{align}\\label{def:Deltaj}\n\\Delta_j={\\mathbb E}^0_\\xi\\big[f\\big(\\xi_t(x),\\xi_t(y),\\xi_t(z)\\big);A_j\\big]-{\\mathbb E}\\big[f\\big(\\xi(B^x_t),\\xi(B^y_t),\\xi(B^z_t)\\big);A_j\\big],\\quad 1\\leq j\\leq 4.\n\\end{align}\nLet $\\mathbf e_1$ and $\\mathbf e_2$ be i.i.d. exponential random variables with mean $1\/\\mu(\\mathds 1)$. It follows from \\eqref{eq:e2} and the independence between selection and mutation that\n\\begin{align}\\label{ineq:Delta1}\n|\\Delta_1|&\\leq \\P(\\mathbf e_1\\wedge \\mathbf e_2\\leq t)\\P(M_{x,y}>t)= \\big(1-{\\rm e}^{-2\\mu(\\mathds 1)t}\\big)\\P(M_{x,y}>t),\\\\\n\\label{ineq:Delta2}\n|\\Delta_2|&\\leq\\int_0^t \\P(t\\geq M_{x,y}>s)\\P(\\mathbf e_1\\wedge \\mathbf e_2\\in \\d s)\\leq 2\\mu(\\mathds 1)\\int_0^t \\P(M_{x,y}>s)\\d s.\n\\end{align}\nOn $A_3$, $B^{x,t}_t=B^{y,t}_t$ by coalescence, and hence, $\\xi_t(x)=\\xi_t(y)$ by \\eqref{prob:dual}. It follows from the assumption on $f$ that both of the expectations defining $\\Delta_3$ are zero. \n\nTo bound $\\Delta_4$, fix $z\\in E$ and partition $A_4$ into the following four sets:\n\\begin{align*}\nA_{41}&=\\{e(x,t)\\wedge e(y,t)>t, e(z,t)\\leq tt, e(z,t)\\leq M_{x,z}\\wedge M_{y,z}\\leq t\\},\\\\\nA_{43}&=\\{e(x,t)\\wedge e(y,t)>t,M_{x,z}\\wedge M_{y,z}< e(z,t)\\leq t\\},\\\\\nA_{44}&=\\{e(x,t)\\wedge e(y,t)>t,e(z,t)>t\\}.\n\\end{align*}\nThen define $\\Delta_{4k}$ for $1\\leq k\\leq 4$ as in \\eqref{def:Deltaj} by replacing $A_j$ with $A_{4k}$. By \\eqref{eq:e1} and similar arguments for \\eqref{ineq:Delta1} and \\eqref{ineq:Delta2}, we get\n\\begin{align}\\label{ineq:Delta412}\n\\begin{split}\n|\\Delta_{41}|&\\leq \\mathds 1_{x\\neq y}\\big(1-{\\rm e}^{-\\mu(\\mathds 1)t}\\big)\\P(M_{x,z}\\wedge M_{y,z}>t),\\\\\n|\\Delta_{42}|&\\leq \\mathds 1_{x\\neq y}\\mu(\\mathds 1)\\int_0^t \\P(M_{x,z}\\wedge M_{y,z}>s)\\d s,\n\\end{split}\n\\end{align}\nwhere the use of the indicator function $\\mathds 1_{x\\neq y}$ follows from the assumption of $f$. For $\\Delta_{43}$, it is zero because $A_{43}=\\varnothing$. Indeed, on $\\{M_{x,z}\\wedge M_{y,z}< e(z,t)\\leq t\\}$, either $e(x,t)\\leq t$ or $e(y,t)\\leq t$ since either $e(x,t)=e(z,t)$ (if $M_{x,z}\\wedge M_{y,z}=M_{x,z}$) or $e(y,t)=e(z,t)$ (if $M_{x,z}\\wedge M_{y,z}=M_{y,z}$). Hence, $\\{M_{x,z}\\wedge M_{y,z}< e(z,t)\\leq t\\}$ does not intersect $\\{e(x,t)\\wedge e(y,t)>t\\}$. Finally, $\\Delta_{44}=0$ by \\eqref{prob:dual} now that the random variables being taken expectation are actually equal. \n\nIn summary, we have proved that $\\Delta_3=\\Delta_{43}=\\Delta_{44}=0$. In addition, $\\Delta_{1}$, $\\Delta_2$, $\\Delta_{41}$ and $\\Delta_{42}$ satisfy \\eqref{ineq:Delta1}, \\eqref{ineq:Delta2} and \\eqref{ineq:Delta412}. We have proved \\eqref{ineq:mutation}.\\medskip \n\n\n\\noindent (2${^\\circ}$) For the left-hand side of \\eqref{claim:w0-mutation}, we use \\eqref{prob:dual} to write\n\\begin{align}\n&{\\quad \\,}{\\mathbb E}^0_{\\xi^x}\\left[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\right]-{\\mathbb E}^0_\\xi\\left[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\right]\\notag\\\\\n\\begin{split}\\label{A1-A4:0000}\n&={\\mathbb E}\\Bigg[\\prod_{j\\in \\{0,2,3\\}}\\mathds 1_{\\sigma_j}\\Big(M(U_j,t)\\mathds 1_{\\{e(U_j,t)\\leq t\\}}+\\xi^x(B^{U_j,t}_{t})\\mathds 1_{\\{e(U_j,t)> t\\}}\\Big)\\\\\n&{\\quad \\,}-\\prod_{j\\in \\{0,2,3\\}}\\mathds 1_{\\sigma_j}\\Big(M(U_j,t)\\mathds 1_{\\{e(U_j,t)\\leq t\\}}+\\xi(B^{U_j,t}_{t})\\mathds 1_{\\{e(U_j,t)> t\\}}\\Big)\\Bigg].\n\\end{split}\n\\end{align}\nMutation neglects the role of the initial condition. \nHence, to get a nonzero value for the difference inside the foregoing expectation, we cannot have $e(U_j,t)\\leq t$ for all $j\\in \\{0,2,3\\}$.\n In this case, at least one of the sums\n$\\mathds 1_{\\sigma_j}\\circ \\xi(B^{U_j,t}_t)+\\mathds 1_{\\sigma_j}\\circ \\xi^x(B^{U_j,t}_t)$, $j\\in \\{0,2,3\\}$, has to be nonzero. We must have $B^{U_j,t}_t=x$ for some $j\\in \\{0,2,3\\}$. By bounding the indicator functions associated with $\\sigma_3$ by $1$, we obtain from \\eqref{A1-A4:0000} that \n\\begin{align}\n&{\\quad \\,}\\big|{\\mathbb E}^0_{\\xi^x}\\left[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\right]-{\\mathbb E}^0_\\xi\\left[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\right]\\big|\\notag\\\\\n&\\leq \\sum_{\\ell\\in \\{0,2,3\\}}\\left({\\mathbb E}_{\\xi^x}+{\\mathbb E}_{\\xi}\\right)\\Bigg[\\prod_{j\\in \\{0,2\\}}\\mathds 1_{\\sigma_j}\\circ \\xi_t(U_j);B^{U_\\ell,t}_t=x\\Bigg],\\quad\\forall\\;x\\in E.\\label{A1-A4:main1}\n\\end{align}\n\n\nThe method in (1${^\\circ}$) now enters to remove mutations in each of the two expectations in the $\\ell$-th summand of \\eqref{A1-A4:main1}. For $\\eta\\in S^E$, we consider\n\\begin{align}\\label{A1-A4-diff}\n{\\mathbb E}_\\eta\\Bigg[\\prod_{j\\in \\{0,2\\}}\\mathds 1_{\\sigma_j}\\circ \\xi_t(U_j);B^{U_\\ell,t}_t=x\\Bigg]-{\\mathbb E}\\Bigg[\\prod_{j\\in \\{0,2\\}}\\mathds 1_{\\sigma_j}\\circ \\eta(B^{U_j,t}_t);B^{U_\\ell,t}_t=x\\Bigg]\n\\end{align}\nand use only the partition in \\eqref{A1-A4} with $x=U_0$ and $y=U_2$. In this case, on $A_4$, the two products of the indicator functions in \\eqref{A1-A4-diff} are equal. Since $\\sigma_0\\neq \\sigma_2$ ensures that the second expectation in \\eqref{A1-A4-diff} can be bounded by $\\P(M_{U_0,U_2}>t,B^{U_\\ell}_t=x)$, \\eqref{A1-A4-diff} and a slight extension of \\eqref{ineq:Delta1} and \\eqref{ineq:Delta2} give\n\\begin{align}\n&{\\quad \\,}{\\mathbb E}_\\eta\\Bigg[\\prod_{j\\in \\{0,2\\}}\\mathds 1_{\\sigma_j}\\circ \\xi_t(U_j);B^{U_\\ell,t}_t=x\\Bigg]\\notag\\\\\n&\\leq \\P(M_{U_0,U_2}>t,B^{U_\\ell}_t=x)+\\big(1-{\\rm e}^{-2\\mu(\\mathds 1)t}\\big)\\P(M_{U_0,U_2}>t,B^{U_\\ell}_t=x)\\notag\\\\\n&{\\quad \\,}+2\\mu(\\mathds 1)\\int_0^t \\P(M_{U_0,U_2}>s,B^{U_\\ell}_t=x)\\d s,\\quad \\forall\\;\\eta\\in S^E,\\;x\\in E.\\label{A1-A4:main2}\n\\end{align}\nThe required inequality \\eqref{claim:w0-mutation} now follows from \\eqref{A1-A4:main1} and \\eqref{A1-A4:main2}.\\medskip \n\n\\noindent {\\rm (3${^\\circ}$)} The proof of \\eqref{claim:w0-mutation1} is almost the same as the proof of \\eqref{claim:w0-mutation} and is omitted. \n\\end{proof}\n\n\\subsection{Full decorrelation on large random regular graphs}\\label{sec:rrg}\nIn this subsection, we give a different proof of the explicit form of \\eqref{cond:kappa0} by using the graphs' local convergence. Throughout the rest of this subsection, we use the graph-theoretic terminologies from \\cite{Bollobas,Chung}.\n\nWe start with the definition of the random regular graphs. Fix an integer $k\\geq 3$. Choose a sequence $\\{N_n\\}$ of positive integers such that $N_n\\to\\infty$ and $k$-regular graphs (without loops and multiple edges) on $N_n$ vertices exist. The existence of $\\{N_n\\}$ follows from the Erd\\H{o}s--Gallai necessary and sufficient condition. Then the random $k$-regular graph on $N_n$ vertices is the graph $G_n$ chosen uniformly from the set of $k$-regular graphs with $N_n$ vertices. We assume that the randomness defining the graphs is collectively subject to the probability $\\mathbf P$ and the expectation $\\mathbf E$. \n\nFor applications to the evolutionary dynamics, we need two properties of random walks on the random graphs. See \\cite[Section~3]{C:MT} and the references there for more details. First, the random walks are asymptotically irreducible in the following sense:\n \\begin{align}\\label{prob:comp}\n\\mathbf P(G_n\\mbox{ has only one connected component})\\to 1\\quad\\mbox{ as $n\\to\\infty$.}\n\\end{align} \nThis property follows since the $\\mathbf P$-probability that $G_n$ has a nonzero spectral gap tends to one \\cite{Friedman, Bordenave}. See \\cite[Lemma~1.7 (d) on pp.6--7]{Chung} for connections between graph spectral gaps and numbers of connected components. Second, $G_n$ for large $n$ is locally like the infinite $k$-regular tree $G_\\infty$ in the following sense. Write $q^{(n),\\ell}(x,y)$ for the $\\ell$-step transition probability of random walk on $G_n$. For any $n, r\\in \\Bbb N$, write $\\mathcal T_n(r)$ for the set of vertices $x$ in $G_n$ such that the subgraph induced by vertices $y$ with $d(x,y)s_n''t)=1,\\quad\\forall\\;t\\in (0,\\infty); \\quad \\mathbf P\\mbox{-a.s.}\n\\end{align}\nand so, by \\eqref{MUU:RG} and \\cite[Proposition~4.3 (2)]{CCC}, \\eqref{gamma} with $s_n''$ replaced by $s_n$ holds. We obtain \\eqref{ass:sigman} from this limit and \\eqref{gamma0}. The proof is complete.\n\\end{proof}\n\n\\begin{rmk}\nMcKay \\cite[Theorem~1.1]{McKay} derives the limiting spectral measures of large random regular graphs. There the randomness of graphs only plays the role of inducing asymptotically deterministic properties. For the present case, we could have worked with given sequences of $k$-regular graphs and obtained the same limit if the graphs have spectral gaps bounded away from zero and are locally tree-like. (Dropping the locally tree-like assumption calls for a different evaluation of the limit.) We choose to work with the above context to explain how the randomness of graphs should be handled for the convergence of the evolutionary game model. \\hfill $\\blacksquare$\n\\end{rmk}\n\n\n \n \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\n\n\n\n\\section{Methods}\n\n\\begin{figure}[t]\n\t\\includegraphics[width=0.95\\textwidth]{workflow_v2}\n\t\\centering\n\t\\caption{Overall workflow of GraphMapper. It is composed of three stages: shape initialization, shape refine and relationship modeling.}\n\t\\label{fig:workflow}\n\\end{figure}\n\n\\subsection{Primitive Graph}\nThe primitive graph is a homogeneous undirected graph $G=\\{V, E\\}$. Here, $V$ is a $L \\times C$ matrix representing $L$ geometric primitives and their $C$ dimensional properties. Point and line segment are the two basic types of primitives used in this study. Typical property $C$ includes coordinates, direction, and image features. $E$ is an $L \\times L$ adjacency matrix representing a set of relationships between primitives in $L$. Topology reconstruction can be represented by estimating a primitive graph with point or line segment primitives and a pairwise connectivity matrix as the relationship matrix. Shape regularization can be represented by the consistency between primitive's directions and their relationships. Further discussion on primitive configuration for building and road mapping can be found in Section \\ref{section:apply}.\n\n\\subsection{Primitive Initialization}\nThe goal of the primitive initialization module is to propose a reasonable initial shape for following shape modeling stages. We first employ feature extractor $f_{imEnc}$ to extract high resolution image features $F$ of input image $I^{H\\times W}$. Similar to \\cite{Bai_2017_CVPR,Girard_2021_CVPR}, an offset head $f_{off}$ is added in parallel to semantic segmentation head $f_{seg}$ for improved awareness of shape prior for feature extractor. A sampler $f_{sample}$ is applied to the predicted segmentation mask to extract a set of initial primitives. Initial primitives should be sparse to reduce the percentage of simple relationships.\n\n\\paragraph{Sampler.}\nFor polygons, $f_{sample}$ traces polygons from the predicted mask. Douglas-Pecker algorithm \\cite{douglas_1973} is applied with a tight error threshold (1 pixel) on the traced contours to remove redundant points. The points or line segments in the simplified contour can be used as primitives. For poly-lines, Sat2Graph \\cite{He_2020_ECCV} predicts evenly spaced points on line segments of poly-lines by segmentation. Local maximum points are sampled and sparsified using NMS (Non-Maximum Suppression) \\cite{Papandreou_2017_CVPR}. We found this method tends to miss points that are distant from junctions\/overlays. We resolve this by sampling additional points from the road segmentation mask outputted from $f_{seg}$. To learn more semantics in road topology, junctions, and overlays are segmented in additional to evenly spaced points on line segments using $f_{kp}$. Our full sampling scheme takes a stratified approach: sample key points from every non-background channel of $f_{kp}$ output and $f_{seg}$ independently; combine all sampled key points with priority order: junctions > overlays > other segmented key points > points sampled from $f_{seg}$ so that topologically significant points are kept over non-significant points.\n\n\\subsection{Primitive Refinement}\nPrimitive refinement improves the accuracy of $V$, which can reduce the ambiguity of relationship learning. We found it essential for accurate relationship prediction, especially for small structures. Primitive refinement network takes image feature $F$ and initial primitives $V$ as inputs and outputs the deformation of primitives' coordinates and directions (point tangent direction or line segment direction). Image features at primitives' locations are pooled (point interpolation for point primitives; LOI \\cite{Zhou_2019_ICCV} for line segment primitives). The sin-cos position encoding from $f_{pos}$ and the pooled image features are concatenated and fed into a multi-layer multi-head-attention (MHA) network $f_{GL\\_SH}$ \\cite{carion_2020_ECCV} to generate global shape contextualized primitive features. Primitive deform head $f_{voff}$ and primitive direction head $f_{vdir}$ take this feature as input and each outputs a $L\\times 2$ matrix representing the offset and direction of each primitive. \n\n\\subsection{Relationship modeling}\nThe goal of relationship modeling is to compute the pairwise relationship matrix of the refined primitives $V'$. This stage uses the same network structure as the primitive refinement network, except for the relationship classifier $f_{vRel}$, which takes concatenated primitive pair features to classify the pair's relationships. \n\nIn road mapping, the optimal connectivity between two primitives depends on the existence of other primitives\nIt requires sorting primitives in an embedding space, where connected primitives are expected to be closer than disconnected primitives. Inspired by popular contrastive learning methods \\cite{NEURIPS2020_f3ada80d,Bardes2021,Chen_2021_CVPR}, L2 normalization is used to improve embedding quality, which we found is essential for accurate topology reconstruction. Pairwise L2-normalized features are concatenated and fed into classifier $f_{vRel}$ for relationship classification. \n\nStandard cross-entropy loss (Eq. \\ref{eq:loss_vrel}) is used to train relationship modeling network. In Eq. \\ref{eq:loss_vrel}, $C$ is the set of possible relationships, $\\hat{E}^{(c)}_{i,j}$ and $E^{(c)}_{i,j}$ are the predicted probability and ground truth probability of primitive pair $i,j$ at relationship $c$. Compared to using feature correlation for relationship classification, cross-entropy works slightly better through our experiments. Cross entropy also allows a consistent network structure for both shape regularization and topology reconstruction. Additional contrastive loss on feature distance is explored, but not shown to improve performance in our study.\n\n\\begin{equation}\n\t\\mathcal{L}_{vRel} = -\\frac{1}{(L-1)L}\\sum_{i=1}^{L}\\sum_{j=1,j\\neq i}^{L}\\sum_{c\\in C} E_{i,j}^{(c)}log(\\hat{E}^{(c)}_{i,j})\n\t\\label{eq:loss_vrel}\n\\end{equation}\n\n\\paragraph{Shape Regularization.} Shape regularization is modeled by adding a direction-relation consistency loss:\n\\begin{equation}\n\t\\begin{aligned}\n\tL_{consistency} &= \\frac{1}{|V|^2-|V|}\\sum_{c\\in C\\setminus 0}\\hat{E}^{(c)} * [\\cos(2A_c) - \\cos(2mod(V_{dir}' - V_{dir}'^T+2\\pi, \\pi))]^2\n\t\\end{aligned}\n\t\\label{eq:loss_reg}\n\\end{equation}\n, where $mod$ term represents the direction difference between primitives pairs normalized to range [0, $\\pi$); $A_c$ is the perfect angle for relation $c$ ($A_c=0$ for relationship between a primitive and itself). Eq. \\ref{eq:loss_reg} computes the mean squared error of surrogate angles weighted by relation probability. Surrogate angle will be explained in the following sub-section. It enables the network to directly generate regularized shapes without any post-processing. \n\n\n\\subsection{Training Tasks and Objectives} \\label{section:apply}\nGraphMapper is trained with multiple losses. Common losses used by shape regularization and topology reconstruction include:\n\\begin{itemize}\n\t\\item Cross entropy loss for image feature extraction and segmentation sub-net (Eq. \\ref{eq:loss_seg}):\n\t\\begin{equation}\n\t\t\\mathcal{L}_{seg} = -\\frac{1}{|I|}\\sum_{i\\in I}\\sum_{c\\in \\{0,1\\}}y_i^{(c)}log(\\hat{y}_i^{(c)}),\n\t\t\\label{eq:loss_seg}\n\t\\end{equation}\n\twhere $y$ and $\\hat{y}$ are $f_{seg}$ predicted segmentation prob and ground truth segmentation mask.\n\t\\item L2 loss for offset branches $f_{off}$ (Eq. \\ref{eq:loss_off}):\n\t\\begin{equation}\n\t\t\\mathcal{L}_{off} = \\sum_{i\\in I_{sub}}{\\|\\hat{o_i} - o_i \\|_2}, \n\t\t\\label{eq:loss_off}\n\t\\end{equation}\n\twhere $I_{sub}$ is the subset of pixels in a buffer region of boundaries or center lines of segmentation mask $y$. $\\hat{o}_i$ is the predicted offset by $f_{off}$.\n\t\\item Surrogate L2 loss for direction branch $f_{vdir}$ (Eq. \\ref{eq:loss_dir}). The discontinuity of rotation angles can lead to unstable learning \\cite{Zhou_2019_CVPR}, we regress a surrogate angle which is 2 times the actual angle.\n\t\\begin{equation}\n\t\t\\mathcal{L}_{vDir} = \\frac{1}{|V|}\\sum_{v\\in V}(\\cos(2V_{dir}') - \\cos(2V_{dir}^T))^2\n\t\t\\label{eq:loss_dir}\n\t\\end{equation}\n\t\\item Bi-projection loss $L_{bp}$ \\cite{CHEN2020114} is used for deformation $f_{vOff}$ (Eq. \\ref{eq:loss_voff}). It first matches the vertices in ground truth to its nearest predictions; the rest of the predicted vertices are matched to its nearest projection in ground truth shape. \n\t\\begin{equation}\n\t\t\\mathcal{L}_{vOff} = L_{bp}(V', V)\n\t\t\\label{eq:loss_voff}\n\t\\end{equation}\n\t\n\\end{itemize}\n\n\\paragraph{Building Mapping.} \nLine segments of simplified building contours are used as primitives. Relationship classification predicts whether two consecutive lines are inline. Finally, the total training loss is a weighted sum of ($\\mathcal{L}_{seg}, \\mathcal{L}_{off}, \\mathcal{L}_{vOff}, \\mathcal{L}_{vDir}, \\mathcal{L}_{vRel}, \\mathcal{L}_{consistency}$).\n\nTo reconstruct the building polygon, the resulting line segments in $V'$ are rotated around their center to the direction estimated in the shape refinement stage. For each pair of neighboring line segments: merge if parallel (angle < parallel angle threshold) and close-by (nearest point pair distance > shortest edge length), connecting near-ends if parallel and not close-by; extend to the intersection if not parallel. The shortest edge length can be set according to the requirements in mapping accuracy. The parallel angle threshold is set to 30\\textdegree.\n\n\\paragraph{Road Network Mapping.}\nRoad network reconstruction uses the point as primitives and connectivity as the pairwise relationship. Pixels within a buffer distance (3 meters) to the center-line are considered as the road surface. The final training loss for road network reconstruction is a weighted sum of ($\\mathcal{L}_{seg}, \\mathcal{L}_{off},\\mathcal{L}_{kp}, \\mathcal{L}_{vOff}, \\mathcal{L}_{vDir}, \\mathcal{L}_{vRel}$). Only point pairs within a certain distance are used for relationship classification loss computation, to balance the ratio of positive and negative point pairs.\n\nThe road network graph is reconstructed by connecting each point to its nearest $N$ neighbors in embedding space, where $N$ is 3 for junction and 2 for others. No post-processing is required.\n\n\\section{Introduction}\nMaps are vectorized and simplified representations of the real world. It is important for urban planning, navigation, disaster recovery, and large-scale surveys and census. In less developed regions, efficient map production is vital for combating poverty. Traditionally, map production fully relies on manual labeling, which is time-consuming and expensive. With the increasingly available high-resolution satellite images and increasing needs in global monitoring, environment protection, and urbanization, it is desired to produce large-scale vector maps with efficient methods.\n\nA variety of techniques have been developed to automatically extract vector maps from satellite images. These methods often follow a \"segmentation and modeling\" paradigm: using semantic segmentation to extract the target mask and shape modeling algorithms to extract vector representations from the mask. Semantic segmentation of satellite images has improved dramatically in recent years \\cite{MARMANIS2018158,Zhou_2018_CVPR_Workshops,Chen_2018_CVPR,Batra_2019_CVPR} with the advances in deep learning. State-of-the-art methods \\cite{He_2020_ECCV,Girard_2021_CVPR} still rely on heuristic rules optionally with optimization for shape modeling. These methods often require careful parameter tuning for practical use, which has limited their ability to handle various environments in large-scale mapping. Additionally, state-of-the-art methods are often designed for a specific type of mapping target, and multiple methods are needed for comprehensive vector mapping of multiple target types, which further increases parameter tuning effort. Finally, the separation of semantic segmentation and shape modeling causes error accumulation, which leads to performance dropping.\n\nEnd-to-end shape modeling methods use data-driven regularization and shape priors instead of heuristics, which is potentially more applicable in large-scale scenarios compared to heuristics-based methods. State-of-the-art end-to-end shape modeling methods formulate shape prediction as a point sequence prediction problem. Recurrent neural networks (RNNs) are used to predict points one by one in sequence \\cite{Castrejon_2017_CVPR, Acuna_2018_CVPR, CHEN2020114, Zorzi_2020_ICPR}. Unfortunately, these methods are difficult to train and practically outperformed by state-of-the-art heuristics-based methods.\nTo address these problems, we propose a unified framework for end-to-end vector mapping from satellite images with a novel generic shape representation named \"primitive graph\" that describes both geometric primitives and pairwise relationships between them. Compared to existing unified shape representations \\cite{Li_2019_ICCV} that only work for coordinates learning, such a representation enables explicit joint learning of geometry, shape regularization, and topology in one model. \n\nAccordingly, we propose a generic network named GraphMapper to progressively reconstruct primitive graphs. The overall workflow is shown in Fig. \\ref{fig:workflow}. Multi-head attention networks are used to encode global-shape-context for primitives, which encodes object-level shape context or image-level shape context to support primitive-level learning. One of the main challenges in relationship matrix prediction is the ambiguity in dynamic matching between predicted primitives and ground truth shapes for loss computation. By progressively refining the coordinates before relationship learning, the dynamic matching ambiguity is reduced along the training process, which we demonstrate is a crucial strategy of the proposed approach.\n\nGraphMapper enables shape regularization through learning. Specifically, reconstruction of shape geometry and regularization requires accurate coordinate regression and faithful parallel\/orthogonal relationships prediction. Primitive graph jointly explores the geometry regression and relationship classification and benefits both tasks by imposing inherent consistency.\nUnlike existing methods where shape regularization is enforced in a post-optimization stage, we define shape regularization as consistency between geometry and relationships in a primitive graph during training, thereby avoids tedious parameter tuning.\n\nAs for topological reconstruction, primitive graph provides compact and straightforward representation by directly representing topological connectivity as pairwise relationship.\nWe observe the challenge of accurate classification of connectivity relationships, whose performance is sensitive to probability threshold of predicted scores. Therefore, we consider topology reconstruction as a hidden space sorting problem, i.e., primitives are embedded into a space where connected primitives are closer in feature distance; topology is reconstructed by connecting every primitive to a certain number of nearest primitives in hidden feature space. We found this strategy significantly improves performance.\n\nWe mainly solve two major problems in vector mapping: building footprint mapping and road network mapping. Building footprint mapping requires regularized building footprint polygons, while road network mapping requires topologically correct and smooth poly-line graphs. We demonstrate that GraphMapper can adapt to both tasks well by simple reconfiguration. With simple post-processing or no post-processing, our method outperforms state-of-the-art methods by 10\\% in building footprint mapping and 8\\% in road network mapping.\n\\section{Experiments}\n\n\\subsection{Datasets And Metrics}\n\\paragraph{Building} (1) CrowdAI Mapping Challenge Dataset \\cite{mohanty2020deep} (CrowdAI dataset): it contains 341438 annotated aerial images of size 300 $\\times$ 300 pixels. The official train-valid-test splits are used\n\nThe commonly used mIOU (Mean Intersection Over Union) and AP (Average Precision) in semantic segmentation tasks cannot describe the cleanness of predictions at boundaries. Mean Max Tangent Angle Error (MMTE) \\cite{Girard_2021_CVPR} is adopted to evaluate the correctness of extracted vector shapes of buildings. MMTE computes the average max angle error of all line segments of each building over the entire dataset.\n\n\\paragraph{Road Network} (1) SpaceNet road dataset \\cite{vanetten2019spacenet}: it contains 2549 satellite images of size 1300 $\\times$ 1300 pixels with resolution around 0.3m. This dataset is challenging due to the diverse scenarios from 5 cities around the globe. The train-val-test splits used by Sat2Graph \\cite{He_2020_ECCV} are used, which uses 80\\% for training, 15\\% for testing, and 5\\% for validation. (2) City-Scale Dataset \\cite{He_2020_ECCV}: it contains 180 tiles of size 2000 $\\times$ 2000 with 1 meter spatial resolution. This dataset covers 20 U.S. cities with less diversity compared to the SpaceNet road dataset. The ground truth vector annotations were collected from OpenStreetMap \\cite{Mordechai_2008}. Follow \\cite{He_2020_ECCV}, images in both datasets are resized to 1 meter spatial resolution.\n\nRoad network topology is evaluated using TOPO \\cite{Biagioni_2012_TRR} and Average Path Length Similarity (APLS) \\cite{vanetten2019spacenet}. TOPO measures the similarity of sub-graphs randomly sampled from the inferred graph and ground truth graph within a certain distance of a seed location. The similarity of sub-graphs is quantified as positive or negative. Average precision, recall, and F1 scores are reported on randomly sampled seed points. APLS measures the difference of the shortest path between sampled point pairs on the inferred graph and ground truth graph. It sums up the path difference for all paths in the graph. \n\n\\subsection{Implementation Details}\n\\paragraph{Network structure.} We use the panoptic segmentation FPN \\cite{Kirillov_2019_CVPR} with ResNet101 backbone \\cite{He_2016_CVPR} pre-trained on Imagenet \\cite{ILSVRC15} as the multi-scale image feature extractor. Predictors $f_{off}, f_{seg}, f_{kp}$ share the same structure of 2$\\times$(Conv-BN-ReLU)-2$\\times$ConvTrans. The last two Transpose Convolution layers use stride $2$ and kernel size $3$ to generate output the same size of input image.\n\nFollowing the same setting in \\cite {carion_2020_ECCV}, position encoding, image features, segmentation logits, and key point logits are concatenated and fed into $f_{GL\\_SH}$. The pooled features of primitives are compressed using a 1x1 convolution projection layer to 256 channels. $f_{GL\\_SH}$ uses three MHA layers proposed in \\cite{Vaswani_2017_NIPS} for global shape feature encoding which uses 256-dimensional internal representations and output 256-dimensional features. $f_{voff}, f_{vdir}$ use the same structure as $f_{off}$. The kernel size of convolution layers in predictors is set to 1$\\times$1 for road topology reconstruction because of the unordered nature of point primitives. For building, line segment primitives are ordered in sequence, the kernel sizes are set to 3$\\times$3 to better use context information.\n\n\\paragraph{Training and testing.} Shape initialization stage is first trained without the shape modeling stages to provide reasonable initial shapes for shape modeling stages. Adam optimizer \\cite{adam} is used with batch size 32 and initial learning rate 1e-3. The learning rate will decrease 10 times twice when training loss plateaus. After the shape initialization stage is trained, all stages are trained together using Adam\\cite{adam} with batch size 1 and initial learning rate 1e-4. Weight decay is disabled. Training stops when the validation score stops increasing. Input crop size is 300x300 for building datasets and 448x448 for road datasets. Standard data augmentation techniques are used, including random rotation between [-30\\textdegree, 30\\textdegree], flipping, color jittering, and rescale between [0.7, 1.5]. The model with the best validation score is selected for testing. No test-time augmentation is used.\n\n\nOur implementation is based on Detectron2 \\cite{wu2019detectron2} and Pytorch \\cite{NEURIPS2019_bdbca288}. All experiments are conducted on a work station equipped with 1 Intel(R) Xeon(R) Gold 6278C CPU @ 2.60GHz and 4 NVIDIA Tesla T4 GPUs.\n\n\\subsection{Benchmark results}\n\n\\paragraph{Building footprint extraction.} The performance of our model and state-of-the-art methods are reported in Tab. \\ref{tab:building_main}. GraphMapper outperforms Frame Field Learning in both datasets by a large margin. As shown in Fig. \\ref{fig:building_main}, GraphMapper generates visually natural yet well-regularized shapes without parallel or perpendicular rectification. Small errors in semantic segmentation can be corrected by shape modeling. In Fig. \\ref{fig:building_main}(4), some gross errors in semantic segmentation are also corrected by shape modeling.\n\n\\begin{table}[ht]\n\t\\caption{Building evaluation results.}\n\t\\centering\n\t\\input{resource\/table_building_main.tex}\n\t\\label{tab:building_main}\n\\end{table}\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\textwidth]{building_main_vis.png}\n\t\\centering\n\t\\caption{Example building footprint extraction results. Top row is the simplified segmentation contour, bottom row is the GraphMapper output. (1) Round corners are corrected; (2) small errors in semantic segmentation are corrected by shape modeling; (3) irregular shape can be modeled correctly; (4) large segmentation errors are corrected by shape modeling.}\n\t\\label{fig:building_main}\n\\end{figure}\n\n\\paragraph{Road network extraction.} Evaluation results are reported in Tab. \\ref{tab:road_main}. With no post-processing, GraphMapper achieves superior performance on both TOPO and APLS compared to state-of-the-art methods. TOPO F1 is improved by 6-8 in absolute value compared to Sat2Graph\\cite{He_2020_ECCV}. GraphMapper is showing to fix the topologically disconnected scenarios in segmentation mask (Fig. \\ref{fig:road_main}(a,b)). Incorrectly predicted road surface key points (blue) and road junction points (red) are not showing to affect the shape modeling output (Fig. \\ref{fig:road_main}(a,b)). It suggests that the network is learning more complicated rules than relying on the predicted key point category for topology reconstruction.\n\\begin{table}[t]\n\t\\caption{Comparison of road TOPO and APLS evaluation metric.}\n\t\\centering\n\t\\input{resource\/table_road_main.tex}\n\t\\label{tab:road_main}\n\\end{table}\n\n\\begin{figure}[t]\n\t\\includegraphics[width=0.9\\textwidth]{road_main_vis.png}\n\t\\centering\n\t\\caption{Qualitative result of road network reconstruction. (1) Extracted road network visualized in yellow lines; (2) segmentation mask of (1); (3) a zoomed-in view of the red box in (1), blue points are predicted road surface points, red points are predicted road junctions; (4) segmentation mask of (3).}\n\t\\label{fig:road_main}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{minipage}[b]{.45\\textwidth}\n\t\\includegraphics[width=\\textwidth]{road_overlay_vis.png}\n\t\\centering\n\t\\caption{Qualitative results at road junctions and overlays.}\n\t\\label{fig:junction_and_overlays}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[b]{.45\\textwidth}\n\\includegraphics[width=0.9\\textwidth]{sort_vs_cls_sensitivity.png}\n\t\\centering\n\t\\caption{Threshold sensitivity for embedding space sorting and connectivity classification.}\n\t\\label{fig:sort_vs_cls}\n\\end{minipage}\n\\end{figure}\n\n\\subsection{Ablation Study and Discussion}\n\n\\paragraph{Shape embedding normalization and sorting.} The usefulness of shape embedding normalization and hidden-space sorting-based topology reconstruction are tested on the City-Scale dataset. The results are reported In Tab. \\ref{tab:road_ablation}. Shape embedding normalization improves road-mapping performance by 4.6 in TOPO F1 and 5.6 in APLS. Sorting in embedding space is shown to outperform standard relationship classification (-sort) and is more robust to connectivity threshold as shown in Fig. \\ref{fig:sort_vs_cls}.\n\\begin{table}\n\\begin{minipage}[t]{.49\\textwidth}\n\\centering\n\\caption{Ablation study on City-Scale dataset.}\n\\input{resource\/table_road_ablation.tex}\n\\label{tab:road_ablation}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{.49\\textwidth}\n\\centering\n\\caption{Direction accuracy of different stages on CrowdAI dataset.}\n\\input{resource\/table_building_ablation.tex}\n\\label{tab:building_ablation}\n\\end{minipage}\n\\end{table}\n\n\\paragraph{Shape regularization.} As the ground truth shape of buildings are regularized shapes manually created, MMTE can be used as a good indicator of shape regularity. Increasing direction-topology consistency loss weight from 0.001 to 0.1 is showing to to increase MMTE by 0.7\\textdegree (Tab. \\ref{tab:building_main}). The increase of shape regularity hurts mIOU in our experiments. Similar behavior has been reported in previous studies \\cite{CHEN2020114,Girard_2021_CVPR}. This may be related to the misalignment noise in the ground truth. \n\n\\paragraph{Accuracy of direction.} The MMTE of line direction of different stages are reported in Tab. \\ref{tab:building_ablation}. The final polygon uses the predicted direction in refine stage. It has MMTE 26.7\\textdegree, which is significantly better than the line direction derived from refined polygons which have MMTE 35.9\\textdegree. The big difference in MMTE is surprising as deformation and direction regression are coupled tasks sharing the same MHA shape feature encoder. It could be caused by the relatively accurate line direction and less accurate line location in training annotations. \n\n\\paragraph{Parameter tuning and sensitivity.} GraphMapper requires little parameter tuning for shape post-processing. In our building shape post-processing step, a shorted line length term is used to control the output simplicity. This term is not data-dependent but task-dependent. The value should be set according to the mapping task requirement in map accuracy. For road extraction, no post-processing is needed. GraphMapper is less sensitive to point sampling density compared to Sat2Graph \\cite{He_2020_ECCV} during our test. A possible reason is that GraphMapper is trained to work with the preset point density.\n\n\n\\paragraph{Limitations and future study.} \\label{limitations}\nOur primitive sampling method may remove important points when multiple roads are densely overlaid, which can cause missing connections at road overlays. Line segment prediction methods such as \\cite{Zhou_2019_ICCV} can be used to directly sample line segments as initial primitives. We leave this for future study. \n\nThe quality of dynamic ground truth generation is directly related to the network's performance. Incorrect matching for relationship classification is not rare in our study. Methods that do not rely on complicated dynamic ground truth generation will be our focus in the following studies.\n\n\n\\section{Conclusion}\nWe propose GraphMapper, an end-to-end model for unified vector mapping from satellite images based on primitive graphs. By converting vector mapping tasks into primitive graph estimation tasks, it can handle various topology reconstruction and shape regularization tasks. With simple post-processing or no post-processing, GraphMapper achieved state-of-the-art performance in both building footprint and road network mapping. This makes it easy to deploy for map production systems. We hope this can increase the mapping capability of less developed regions. \n\n\n\\section{Related Works}\n\\label{sec:relatedwork}\n\\paragraph{Building footprint mapping.} Modern building footprint extraction methods typically use semantic segmentation neural networks to extract building masks from satellite images. Shape rectification rules such as main direction alignment \\cite{MSBuilding}, parallel and perpendicular \\cite{sirko2021continentalscale} rectification are applied to the contours of segmented building masks to extract vector maps. Rule-based shape modeling methods are widely used in practice due to their simplicity and reasonable performance on simple buildings. Instead of using heuristic rules, Active Contour Model (ACM) or similar contour optimization techniques are developed to encourage building contours to be consistent with predicted energy fields, such as boundary probability, distance, and direction \\cite{Marcos_2018_CVPR, Cheng_2019_CVPR,Girard_2021_CVPR}. With energy field predictors highly coupled with semantic segmentation, the predicted directions or boundary probabilities are often highly consistent with segmentation masks, which limited their impact on output shapes. Several terms are used to balance shape smoothness and consistency to image semantics. There is no systematic approach to tuning these terms. ASIP \\cite{Li_2020_CVPR} learns to approximate shapes with low complexity polygons through a split and merge strategy. It can generate simple but not regularized shapes for vector mapping.\n\nAnother stream of work takes end-to-end approaches. PolygonRNN \\cite{Castrejon_2017_CVPR} first proposed to use LSTM (Long-Short-Term-Memory) to recurrently generate the points of a polygon in sequence. It defines a coarse grid on an image and classifies within which grid the next point should be located. To reduce the number of possible point locations, the grid is set to have low resolution, which limited the location accuracy of predicted polygons. PolygonRNN++ \\cite{Acuna_2018_CVPR} further improved PolygonRNN by predicting location offset in the grid. Polymapper \\cite{Li_2019_ICCV} improved on PolygonRNN by using ConvLSTM \\cite{NIPS2015_07563a3f} with additional global boundary mask and vertex mask for point prediction. Compared to state-of-the-art heuristics-based methods, RNN-based methods still lacks performance and are more difficult to train. \n\nInstead of recurrently generating shapes, PolygonCNN \\cite{CHEN2020114} and Polygon Transformer \\cite{Liang_2020_CVPR} predict point deformation of polygon contours. Polygon Transformer \\cite{Liang_2020_CVPR} uses Attention Network to compute shape features for deformation prediction of instance contours. These models are shown to improve instance segmentation boundaries. Similar to Polygon Transformer, we also use a transformer to encode global shape features, but with a few critical differences: our method generalizes to all different kinds of primitives, while Polygon Transformer only uses point; we aim at generic shape regularization and topology reconstruction, while Polygon Transformer is designed for semantic segmentation. \n\n\\paragraph{Road network mapping.}\nThe key challenge of road network mapping is topology reconstruction under scenarios such as shadows, tree blocking, or complex junctions and overlays. One stream of work focus on improving road segmentation, such as using CNN \\cite{Mnih_2010}, adding more context information modeling structure \\cite{Zhou_2018_CVPR_Workshops}, training with auxiliary perception loss \\cite{Mosinska_2018_CVPR}, self-supervised pre-training \\cite{singhBMVC18overhead}, and multi-stage road segment connection refinement \\cite{Batra_2019_CVPR}. Another stream of work focus on improving topology from imperfect segmentation results. \\citet{Mattyus_2017_ICCV} uses a binary decision classifier to predict the correctness of connections of nearby road endpoints from their image features. Local road structure and shape prior are not exploited. Graph-based methods \\cite{Bastani_2018_CVPR, Tan_2020_CVPR, Li_2019_ICCV} are able to use both shape prior and image information by iterative searching of the next point in the road graph, using CNNs or CNN-RNN structure. Compared to graph-based methods, our method generates a road graph in one forward run. This allows easier integration of global shape information and shapes regularization into the shape modeling process.\n\nRecently, Sat2Graph \\cite{He_2020_ECCV} achieved the state-of-the-art performance by connecting predicted key points along road direction using heuristic rules. Uniformly distributed points on road segments are predicted in a multi-task learning CNN together with point direction and road surface mask. The road network is reconstructed by connecting each key point to neighboring key points near its predicted directions. A list of shape post-processing steps with additional tunable parameters is performed to reduce loops and false connections. A fan radius threshold and fan spread angle threshold need to be tuned for connectivity estimation. We found Sat2Graph sensitive to these parameters and the accuracy of direction prediction, while our method does not need complicated parameter tuning or post-processing.","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction to Belle at KEKB}\nKEKB~\\cite{bib:KEKB} is an asymmetric-energy $e^+e^-$ collider operating at and near $\\Upsilon(4S)$ mass peak. As the only detector installed in KEKB, Belle detector has a good performance on momentum and vertex resolution, $K\/\\pi$ separation etc. A detailed description of the Belle detector can be found elsewhere~\\cite{bib:BelleDetector}. It has been ten years since the final full data set ($\\sim$1 ${\\rm ab^{-1}}$) was accumulated, however, fruitful results on physics are lasting to be produced. Here we select some recent charm results from Belle to present in this proceedings. \n\n\\section{Charm-mixing parameter $y_{CP}$ in $D^0\\toK_S^0 \\omega$}\nThe mixing parameter $y_{CP}$ is measured in $D^0$ decays to the $CP$-odd final state $K_S^0\\omega$ for the first time~\\cite{bib:ycp_D0ToKsOmega}.\nConsidering mixing parameters $|x|$ and $|y|\\ll1$, the decay-time dependence of $D^0$ to a $CP$ eigenstate is approximately exponential, $d\\Gamma\/dt \\propto e^{- \\Gamma(1+\\eta_{f} y_{cp})t}$ where $\\eta_f=+1$ ($-1$) for $CP$-even (-odd) decays.\nAlong with the decay rate in flavored eigenstate decays $d\\Gamma\/dt \\propto e^{-\\Gamma t}$, the $y_{CP}$ is determined by the decay proper-time value with the formula $y_{CP}=1-\\frac{\\tau(D^{0}\\toK^{-}\\pi^{+})}{\\tau(D^{0}\\toK_S^0\\omega)}$, where $D^{0}\\toK^{-}\\pi^{+}$ is the chosen normalization mode with flavor eigenstate final state. \n\n Based on the full Belle data sample of 976 $\\rm fb^{-1}$, we obtain 91 thousands of $D^{0}\\toK_S^0\\omega$ and 1.4 millions of reference mode $D^{0}\\toK^{-}\\pi^{+}$ in $M-\\Delta M$ signal region, where $M$ is the invariant mass of reconstructed $D^{0}$ and $\\Delta M$ is the mass difference of reconstructed $D^{*+}$ and $D^{0}$. \nUsing unbinned maximum-likelihood fits for lifetime on these two samples with high purities, the proper decay-time of $D^{0}$ is determined as $\\tau_{K_S^0\\omega}=(410.47\\pm 3.73)$ fs and $\\tau_{K\\pi}=(406.53\\pm 0.57)$ {\\rm fs}, as shown in Fig.~\\ref{fig:ycp}. \nThus, we calculate $y_{CP}=(0.96\\pm0.91\\pm 0.62^{+0.17}_{-0.00})\\%$, where the first uncertainty is statistical, the second is systematic due to event selection and background, and the last is due to possible presence of CP-even decays in the data sample. \nThis $y_{CP}$ result is consistent with the world average value. \nIn the future, comparing more precise measurements of $y_{CP}$ with that of $y$ may test the SM precisely or reveal new physics effects in the charm system.\n\n\\begin{figure}[!htpb]\n \\begin{centering}\n \\begin{overpic}[width=0.49\\textwidth,height=0.48\\textwidth]{tau_ycp1.png}\n \\put(20, 64){\\large(a)}\n \\end{overpic}~~\n \\begin{overpic}[width=0.49\\textwidth,height=0.48\\textwidth]{tau_ycp2.png}\n \\put(20, 64){\\large(b)}\n \\end{overpic}\n \\vskip-5pt\n \\caption{\\label{fig:ycp} The fit of $D^0$ proper lifetime: (a) $D^0\\toK_S^0\\omega$ and (b) $D^0\\toK^{-}\\pi^{+}$. The dashed red curves are the signal contribution, and the shaded surfaces beneath are the background estimated from $M-\\Delta M$ sidebands.}\n \\end{centering}\n\\end{figure}\n\n\n\n\n\\section{Dalitz-plot analysis of $D^{0}\\toK^{-}\\pi^{+}\\eta$ decays}\nThe understanding of hadronic charmed-meson decay is theoretically challenging due to the significant non-perturbative contributions, and input from experimental measurements thus plays an important role. A Dalitz-plot analysis of $D^0\\toK^{-}\\pi^{+}\\eta$ is performed for the first time at Belle based on 953 ${\\rm fb^{-1}}$ of data~\\cite{bib:PRD102012002}. \nUsing a $M$-$Q$ two-dimensional fit where $M$ is the invariant-mass of reconstructed $D^0$ meson, $M=M(K^{+}\\pi^{-}\\eta)$, and $Q$ is the released energy of $D^{*+}$ decay, $Q=M(K^{-}\\pi^{+}\\eta\\pi_s)-M-m_{\\pi_s}$, a signal yield of $105\\,197\\pm990$ is obtained in the signal region of $1.85~{\\rm GeV}\/c^2 < M < 1.88~{\\rm GeV}\/c^2$ \nand $5.35~{\\rm MeV}\/c^2 < Q < 6.35~{\\rm MeV}\/c^2$ with a high purity $(94.6\\pm0.9)\\%$. \nThe Dalitz plot is well described by a combination of the six resonant decay channels $\\bar{K}^{*}(892)^0\\eta$, $K^{-} a_0(980)^+$, $K^{-} a_2(1320)^+$, $\\bar{K}^{*}(1410)^0\\eta$, $K^{*}(1680)^-\\pi^{+}$ and $K_2^{*}(1980)^-\\pi^{+}$, together with $K\\pi$ and $K\\eta$ S-wave components, as shown in Fig.~\\ref{fig:nominal}.\nThe dominant contributions to the decay amplitude arise from $\\bar{K}^{*}(892)^{0}$, $a_0(980)^{+}$ \nand the $K\\pi$ S-wave component. The $K\\eta$ S-wave component, including $K_0^{*}(1430)^{-}$, \nis observed with a statistical significance of more than $30\\sigma$, and the decays \n$K^{*}(1680)^{-}\\toK^{-}\\eta$ and $K^{*}_2(1980)^{-}\\toK^{-}\\eta$ are observed for the first time \nand have statistical significances of $16\\sigma$ and $17\\sigma$, respectively. \n\n\\begin{figure}[!htpb]\n \\begin{centering}\n \\begin{overpic}[width=0.45\\textwidth]{exp_dlz_withcolor.eps}\n \\put(21, 65){\\large(a)}\n \\end{overpic}%\n \\begin{overpic}[width=0.42\\textwidth]{dlz_m2ksp0_lass.eps}\n \\put(22, 68){\\large(b)}\n \\end{overpic}\\\\\n \\begin{overpic}[width=0.42\\textwidth]{dlz_m2p0et_lass.eps}\n \\put(22, 68){\\large(c)}\n \\end{overpic}%\n \\begin{overpic}[width=0.42\\textwidth]{dlz_m2kset_lass.eps}\n \\put(22, 68){\\large(d)}\n \\end{overpic}%\n \\vskip-5pt\n \\caption{\\label{fig:nominal} The Dalitz plot of $D^{0}\\toK^{-}\\pi^{+}\\eta$ in (a) $M$-$Q$ signal region $1.85~{\\rm GeV}\/c^2 < M < 1.88~{\\rm GeV}\/c^2$ \nand $5.35~{\\rm MeV}\/c^2 < Q < 6.35~{\\rm MeV}\/c^2$, and projections on (b) $m_{K\\pi}^2$, (c) $m_{\\pi\\eta}^2$ and (d) $m_{K\\eta}^2$. In projections the fitted contributions of individual components are shown, along with contribution of combinatorial background (grey-filled) from sideband region.}\n \\end{centering}\n\\end{figure}\n\nWe extract the signal yield from the $D^0$ invariant mass distribution in $1.78~{\\rm GeV}\/c^2