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ARTICLEOPENEfcientsearchofcompositionalspaceforhybridorganic–inorganicperovskitesviaBayesianoptimizationHenryC.
Herbol1,WeiciHu2,PeterFrazier2,PauletteClancy3andMatthiasPoloczek4Acceleratedsearches,madepossiblebymachinelearningtechniques,areofgrowinginterestinmaterialsdiscovery.
Asuitablecaseinvolvesthesolutionprocessingofcomponentsthatultimatelyformthinlmsofsolarcellmaterialsknownashybridorganic–inorganicperovskites(HOIPs).
Thenumberofmolecularspeciesthatcombineinsolutiontoformtheselmsconstitutesanoverwhelminglylarge"compositional"space(attimes,exceeding500,000possiblecombinations).
SelectingaHOIPwithdesirablecharacteristicsinvolveschoosingdifferentcations,halides,andsolventblendsfromadiversepaletteofoptions.
Anunguidedsearchbyexperimentalinvestigationsormolecularsimulationsisprohibitivelyexpensive.
Inthiswork,weproposeaBayesianoptimizationmethodthatusesanapplication-specickerneltoovercomechallengeswheredataisscarce,andinwhichthesearchspaceisgivenbybinaryvariablesindicatingwhetheraconstituentispresentornot.
WedemonstratethattheproposedapproachidentiesHOIPswiththetargetedmaximumintermolecularbindingenergybetweenHOIPsaltandsolventatconsiderablylowercostthanpreviousstate-of-the-artBayesianoptimizationmethodologyandatafractionofthetime(lessthan10%)neededtocompleteanexhaustivesearch.
Wendanoptimalcompositionwithin15±10iterationsinaHOIPcompositionalspacecontaining72combinations,andwithin31±9iterationswhenconsideringmixedhalides(240combinations).
ExhaustivequantummechanicalsimulationsofallpossiblecombinationswereusedtovalidatetheoptimalpredictionfromaBayesianoptimizationapproach.
ThispaperdemonstratesthepotentialoftheBayesianoptimizationmethodologyreportedherefornewmaterialsdiscovery.
npjComputationalMaterials(2018)4:51;doi:10.
1038/s41524-018-0106-7INTRODUCTIONHybridorganic–inorganicperovskites(HOIPs)areanexcitingclassofemergentmaterialsthatexhibitextremelypromisingphoto-voltaic(PV)properties.
1Amongsolutionprocessablesolarcellmaterials,HOIPsare—byfar—themostefcientPVdevices,withover22%photovoltaicconversionefciencies(PCEs).
2Thiscombinationofhighefciencyandeaseoffabricationpromisesamorereadilyscalableapproachtosolarcellmanufacturing.
3,4HOIPsformasub-setofalargerclassofperovskitematerials,allofwhichexhibitanABX3conguration.
InHOIPs,theB-sitesareoccupiedbymetalcations(invariably,Pb,butsometimesSn,oramixtureofboth).
A-sitecationscanbeorganicorinorganicinnature,andaretypicallymethylammonium(MA),formamidinium(FA),orcesium(Cs).
Xdenotesachoiceofhalide(Cl,Br,and/orI).
Theseperovskitesareusuallyprocessedinasolventblend(S0:S1).
1,4,5Inthispaper,wewillfocusonthefollowingcommonlyusedHOIPsolvents:Tetrahydrothiophene1-oxide(THTO),dimethylsulfoxide(DMSO),dimethylformamide(DMF),N-methyl-2-pyrrolidone(NMP),γ-butyrolactone(GBL),acetone,methacro-lein,andnitromethane.
AnexceedinglylargecombinatorialspacethusexistsfromwhichHOIPsaltscanbefabricated.
Inprinciple,anycombinationofthesespeciesarepossible,suchasmixing5%CsIwiththebinaryHOIPblendof(FAPbI3)0.
83and(MAPbBr3)0.
17.
4,6–8Inaddition,thechoiceofsolventiscriticalforproducinghigh-qualitylms;hence,awidevarietyofsolvents,bothpureandblended,havebeenstudied(e.
g.
,DMSO,DMF,GBL,NMP).
9–12Despiteintenseexperimentalscrutinyofmanycombinationsofthesespeciesanddifferentsolventprocessingprotocols,thereisnowaytoknowwhetheracurrentlyuntested,buthigherperforming,materialmightexist,onecomprisedofanalternativecombinationofA-sitecation,B-sitecation,halide,andsolventblend.
Weillustratethiscombinatorialgrowthofthesearchspace:SupposewechooseourHOIPcandidatefromamongthreeA-sitecations(MA,FA,Cs),oneB-sitecation(Pb),threeX-sitehalides,allofwhichareconstrainedby10%increments(e.
g.
,MA0.
9Cs0.
1isallowed,butnotMA0.
99Cs0.
01),andallowingbinaryblendsofsolvents(fromareducedlistofonlyeightsolventchoices)of13asthesmallestratioincrements(allowingfor1:1:1asthelargestmixture).
Inthatcase,wewouldbefacedwithconsidering522,720possiblecombinationsofspeciesfromwhichtoformouridealHOIPmaterial.
Weareinterestedinunderstandingmolecular-scaleinteractionsinsolution.
Unfortunately,nosufcientlyaccurateparameterizedforceeldcurrentlyexiststomodelHOIPsystems,essentiallyremovingmoleculardynamics(MD)asaviableinformationsourceforthishigh-delityinvestigation.
13,14Asaresult,wewillrestrictourselvestousingaccurateabinitiodensityfunctionaltheory(DFT)tocapturethisinformation.
15,16Toprobethesolutionprocessingofspeciesthatultimatelyformthinlmsofperovskites,understandingthesolubilityofHOIPreagentsinaso-called"bath"solventisanimportantrststep.
Received:24April2018Revised:9August2018Accepted:15August20181DepartmentofMaterialsScienceandEngineering,CornellUniversity,Ithaca,NY,USA;2SchoolofOperationsResearchandInformationEngineering,CornellUniversity,Ithaca,NY,USA;3RobertF.
SmithSchoolofChemicalandBiomolecularEngineering,CornellUniversity,Ithaca,NY,USAand4DepartmentofSystemsandIndustrialEngineering,UniversityofArizona,Tucson,AZ,USACorrespondence:MatthiasPoloczek(poloczek@email.
arizona.
edu)www.
nature.
com/npjcompumatsPublishedinpartnershipwiththeShanghaiInstituteofCeramicsoftheChineseAcademyofSciencesIdeally,optimizingtheenthalpyofsolvationwouldbeusedasthemetricbywhichtoevaluatetheidealsetofspeciesthatleadstothegreatestsolubility.
However,anaccurateestimateoftheenthalpyofsolvationrequirestheconsiderationofafullsolvationshellaroundtheleadsalt,whichcantakeweeks(ormonths)tocomputeusingDFT.
AnappealingalternativeistousetheUnsaturatedMayerbondorder(UMBO),laidoutbyStevensonetal.
,17asameasureofeffectivesolubility.
Incontrasttoenthalpyofsolvationcalculations,theUMBOtypicallytakesonlyhourstocompute,withthecalculationtimedependingonthecompositionofthemixtureandtheleveloftheoryusedfortheDFT.
However,theUMBOisadescriptorthatwassolelydesignedforpuresolvents,notsolventblends.
Assuch,agoodstartingobjectiveforvalidationpurposeswouldbethelimitingcaseoftheenthalpyofsolvation:theintermolecularbindingenergybetweenaperovskitesalt(ABX3)andapuresolvent(S0).
RESULTSInthersttestcase,wesearchedtheperovskitecompositionalspaceforanABX3combinationwithoptimalintermolecularbindingenergytoasolventmolecule,S0.
Notethatonlycompositionswith"pure"halidesareconsidered,thatis,allthreehalidesarethesameelement.
Eachsystemcontainsasinglehalide,cation,centralion,andapuresolventmolecule.
Hence,thereexistsatotalnumberof72combinations.
Usingourapproach,thePhysicalAnalyticspipeLine(PAL),theintermolecularbindingenergyofapurehalideABX3S0wasoptimized,asshowninFig.
1.
The"optimal"mixturewasfoundtobeMAPbI3inTHTO.
Inthesecondtestcase,afull-scaleBayesianoptimizationwasrunwhilealsoconsideringmixedhalidesystems(e.
g.
,aBr/Cl/Imix).
Thisinvolvesatotalof240possiblecombinations.
AnexhaustivesearchofalloptionswasmadeusingDFTcalculationstoprovideatarget(optimal)bindingenergyforvalidationpurposes,asintestcase1.
Startingwitharandomlysampledcombination,weiterativelyupdateourhyperparametersuntil10totalsampleshavebeenmade.
Wethenoptimizehyperpara-meterseverysubsequent10iterationsuntilwefoundtheHOIPcompositionthatminimizedtheABX3S0intermolecularbindingenergy(maximizingitsmagnitude).
AsseeninFigs.
2and3,theBayesianoptimizationmethodidentiedaHOIP-solventcombina-tionthatproducedamaximaltargetvaluewithin50iterations,withameanof31andastandarddeviationof9(averagedover1000replications).
Wealsocomparetheresultsofournewmethodagainstastate-of-the-artoptimizationapproach,pySMAC,aswellastwoBayesianoptimizationmethodsthatuseGaussianprocess(GP)-basedmodelsfortheobjectiveanddifferintheirhandlingofcategoricalvariables(seebelowfordetails).
Moreover,wecomparetoarandomsearch:18pySMAC19,20isahighlyoptimizedBayesianoptimizationtechniqueoriginallydevelopedfortheautomatedcongura-tionofmathematicaloptimizationalgorithms.
Itsstatisticalmodel,basedonrandomforests,21,22isabletohandleFig.
1PredictionsofthebindingenergybetweenABX3andthreesolventmoleculesduringasinglerunofPALtondthelowestenergyforagivenABX3composition(A∈{MA,FA,Cs}andX∈{Cl,Br,I}).
ThebluelinedepictsthebindingenergyofthecompositionrecommendedbyPALafteriterationt=1,…,50tothesolvent.
Theredlinegivestheoptimalbindingenergyofall72combinations,foundbyDFTcalculations,thusidentifyingtheoptimalABX3S0mixture.
PALndstheoptimalcombinationafterabout12iterations(16%ofthe72possibleoptions).
Thisparticulardemonstrationwasperformedforvesolvents.
ThemoleculeshownisanexampleofonesuchcombinationFig.
2Testcase2:BayesianoptimizationoftheABX3HOIP-solventintermolecularbindingenergy(A∈{MA,FA,Cs}andX∈{Cl,Br,I}).
Inthisinstance,weconsideredmixedhalides.
All240possiblecombinationsofthebindingenergyweresimulatedusingDFTcalculationsinordertondthemostnegative(andmostfavorable)bindingenergy,whichisshownasahorizontalredline.
Here,theoptimalvalueforthebindingenergywasachievedinunder20iterations(thismayvarydependingoninitialsampledpoints)Fig.
3PerformanceevaluationofPAL,SimpleBO,HutterBO,pySMAC,andrandomsearchesfortheminimizationoftheintermolecularABX3-solventbindingenergy(maximizingitsstrength).
44PALobtainsaglobaloptimumbindingenergyatconsiderablylowercostthantheothermethods.
Ashadedregionof±2standarderrors(SE)isshownforeachmethodEfcientsearchofcompositionalspaceforhybrid.
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2npjComputationalMaterials(2018)51PublishedinpartnershipwiththeShanghaiInstituteofCeramicsoftheChineseAcademyofSciences1234567890():,;categoricalvariables,makingiteligibletooptimizeovercombinatorialspaces.
pySMACalsoemploystheexpectedimprovementacquisitioncriterion.
SimpleBOfollowsastandardapproachinBayesianoptimiza-tionandusesastandardGPmodelcombinedwiththeexpectedimprovementacquisitionfunction.
NotethatthelatterisalsoemployedbyPAL.
SimpleBOincorporatescategoricalvariablesvia"one-hot"encodingcommonlyusedinBayesianoptimization,forexample,inGoogle'svizier.
23HutterBOusesatailoredkernelforcategoricalvariablesthatwasproposedbyHutterinhisPh.
D.
thesis24andfollowsSimpleBOotherwise.
Randomsearch18picksanas-yetunevaluatedcombinationuniformlyatrandomineachiteration.
Theoptimizationprocesswasreplicatedformultiplerandomlychoseninitialdatasets.
PAL,SimpleBO,HutterBO,andpySMACwererunforathousandreplicationseach,andRandomforonemillionreplications.
TheresultsoftheseevaluationsaredepictedinFig.
3andTable1.
Thex-axisofFig.
3showsthenumberofobservedpointsofeachalgorithmandthey-axisthebindingenergyofthebestcompositionfoundaftertiteration.
Weplotthehighestfoundmeanbindingenergy±2standarderrors.
WeobservedthatpySMACinitiallyobtainsbettercombinationsthananyothermethod;however,PALconsistentlyobtainstheglobaloptimumrst.
Thatis,PALinvolvesthelowestnumberofDFTcomputationstondthebestperovskitecomposition.
Further,wecanseethebenetofourstatisticalmodelwhencomparingPALtoSimpleBO,inwhichPALachievesglobaloptimizationinaunderhalfthenumberofiterationsthanthatofSimpleBO(whencomparing99.
9thpercentile).
HutterBOperformsworsethanSimpleBO.
ThisissomewhatunexpectedasHutterBOusesatailoredcovariancefunctiondesignedforcombinationsofreal-valuedandcategoricalvariables,whereasSimpleBOusesanadhocencoding.
NotethatwhilebothGP-basedoptimizationmethodsperformworsethanpySMACinitially,theyrequirelessstepstondanoptimumonaverage(seeTable1fordetails).
AllmethodsbeatRandomsearcheventually.
DISCUSSIONWehaveproposedanewBayesianoptimizationapproachthatovercomeschallengesuniquetomaterialsdiscovery.
WhileBayesianoptimizationiscommonlyusedforproblemsinmachinelearningandengineeringwithcontinuousrectangularsearchspaces,thespecicperovskitedesignproblemthatweconsider,incommonwithmanychemicaldesignproblems,hassearchspaceswithbinaryvariables,indicatingthepresenceorabsenceofaconstituentspecies.
Anothercriticalchallengeinthissystemisthatthedataareparticularlyscarcesincetheevaluationofasingledesignisoftenexpensive,whetherdeterminedcomputationallyorexperimentally.
Forchallengingsituationslikethese,wehavedemonstratedthepromiseofaBayesianoptimizationapproachthatmaximizes(orminimizes)anobjectivefunction,inthiscase,maximizingtheintermolecularbindingenergy,toexplorethecombinatorialspaceofHOIPleadionsolvation.
Ourtwotestcaseshavedemonstratedanimprovementofcloseto90%comparedtothatofanexhaustivesearchof240possiblecombinationsofmixed-halideHOIPspace.
ThesameoptimalcompositionswerefoundwithboththeBayesianoptimizationpredictionandtheabinitiocalculations.
Inaddition,someofthetopperformerspredictedbytheBayesianoptimizationapproachhavebeenfoundexperimen-tallytobepromisingHOIPmaterials.
Other,moreunexpected,candidatespromisepotentiallyhigh-performingABX3S0mix-turesthatwesuggestshouldbetestedusingexperimentalstudiestovalidatetheseBayesianoptimization/DFTpredictions.
TheimprovementtoBayesianoptimizationusingourLinearBeliefmodel,laidoutinthesection"LinearBeliefmodel",suggeststhatatomiccomponentscontributeadditivelytowardfunctionsofcohesiveenergy.
Thiscanbeseenbytheconsiderableimprove-mentofPALoveralternativebenchmarkedmethods(pySMAC,SimpleBO,HutterBO,andRandom).
WhilewedemonstrateourworkinthecontextofHOIPs,theproposedmethodologyrepresentsanadvancethatisbroadlyapplicableforothermaterialsdesignproblemsandconstitutesapowerfulnewtoolfortheoptimizationofthedesignofchemicalsystems.
ApplyingBayesianmaximizationoftheintermolecularbindingenergyallowedustopredicttheoptimalHOIP–Solventmixture(inapurehalidesystem),whichwasfoundtobeMAPbI3inTHTO.
Thispredictioncorrespondswellwiththeprevalenceofexperi-mentalstudiesofMAPbI3intheliterature,inwhichithasbeenconsideredoneofthebest-performingcompositions.
3,25,26Inaddition,experimentsbyChoietal.
haveshowntheparticularefcacyoftheTHTOadditivetodissolvePb,corroboratingtheseBO-guided/DFT-drivenpredictions.
27Thecaliberofthepredictionforthemixedhalidescaseismoredifculttoevaluatesincetheexperimentalcommunityhasnotyetdecidedonanoptimalcomposition,probablybecausethenumberofoptionsisoverwhelming.
OurBayesianoptimizationstudiespredictedthemaximizedintermolecularbindingenergytooccurbetweenFAPbI2ClandTHTO.
ThisspecicmixtureofhalidesinconjunctionwithFAisunlikethosenormallystudiedexperimentally.
This,initself,isaninterestingresultsincetheBayesianoptimizationhasclearlyfounda"disruptive"prediction.
Oncemore,wendthatTHTOisasuperioradditivefordissolvingABX3.
26,27ItisimportanttonotethattheBayesiansearchproducesarankedlistinwhichanumberofcandidatesmayproduceclose-to-optimalsuggestionsthatmeetthetargetvaluewithintheuncertaintyofthemethodology.
Thatistrueinthiscase,wherefourcompositionsarethehighestperformerswithbindingenergieswithin0.
84kJ/mol(0.
2kcal/mol)ofeachother,makingthemessentiallyinseparable.
ThesecandidatesareFAPbI2Cl,MAPbI3(anexperimentallyknownhighperformer),andCsPbI3inTHTO,andnallyCsPbICl2inDMSO.
Allthesecandidatesarepredictedtobeexcellentchoices,andnoticethattwoofthefourfavorasingle(ratherthanamixed)halide.
Therankedlistservesasecond,practicalpurpose,itpredictscombinationsnotworthtestinginthelaboratoryifcomplexationisthemetricforsuccess.
ThechoiceoftheobjectivefunctioniscrucialwhenemployingBayesianoptimizationandisauser-suppliedaspectofourmethod.
If,forexample,wehadchosentooptimizethephotovoltaicconversionefciencies(PCE),thenthesolventwouldnotbeafactor,asthecrystalexhibitingthePCEissolvent-free;thisassumesthatasinglecrystalisformedduringthesolutionTable1.
ComparisonofPALtothestate-of-the-artMethodMeanSTD99.
9thpercentilePAL30.
88.
645SimpleBO47.
719.
696HutterBO51.
621.
9104pySMAC61.
634.
7160Random119.
369.
3239ComparisonoftheperformanceofPALisdoneagainststate-of-the-artoptimizationapproaches,pySMAC,SimpleBO,HutterBO,aswellasarandomsearchofABX3compositions.
Therstcolumnshowstheaveragenumberofiterationsrequiredtondacompositionwithoptimalmixed-halideABX3S0intermolecularbindingenergy.
Thestandarddeviationofthisnumberisstatedincolumn2,whilethethirdcolumngivesthe99.
9thpercentile.
PALrequiresconsiderablyfewerDFTcalculationstondanoptimalsolution,onaverage,andshowsarobustperformancewithitssignicantlylower99.
9thpercentileEfcientsearchofcompositionalspaceforhybrid.
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3PublishedinpartnershipwiththeShanghaiInstituteofCeramicsoftheChineseAcademyofSciencesnpjComputationalMaterials(2018)51processing,inwhichthepresenceofsolventisimportant.
IftheobjectiveweretondthemostreadilycrystallizedHOIP–Solventcombination,wewouldneedtotakemorethantheintermole-cularbindingenergyintoaccount.
Ourmodelcanreadilybeextendedtotakeintoaccountmixedionsandmixedhalides/cations(e.
g.
,92%MAand8%Cs)andbeyonda1:1:1ratioofhalidemixing.
Ourfutureworkwillinvestigatemixedsolvents,forwhichmixingruleswillneedtobedenedinordertoprovideappropriatesolventdescriptorsofdensityandpermittivity,oranalternativedescriptor.
METHODSInthissection,weoutlinethedesignofPAL,whichincorporatesanytestcaseswemayultimatelywanttostudy,theBayesianoptimizationapproach,andnallythewaythatthePALcodeimplementsthisBayesianapproach.
TestcasesDFTcalculationsaresufcientlycomputationallyintensivethatanexhaustivecomputationalsearchwillgenerallybeintractable,unlessthecompositionalspaceisconstrainedsufcientlytomakeavalidationstudypossible.
Weconductedtwosuchtractablyconstrained,proof-of-conceptstudiesthathighlighttheefcacyofusingaBayesianoptimizationapproachtoacceleratethesearchforanoptimalHOIPcomposition.
Purehalides.
Ourrstproof-of-conceptwassmallenoughinscopetoallowustovalidatetheBayesianoptimizationpredictionoftheoptimalHOIPcompositionbycomparisontoanexhaustivesearchusingDFT.
Thistestcasesoughttheoptimalpurehalideperovskitethatmaximizestheleadion'ssolubility.
Inourmodel,aperovskite"composition"consistsoffourmajorcomponents:ahalideanion,anA-sitecation,onepuresolvent,andaleadcation(B-site)thatservesasthecentralionofthecluster.
ThehalideanionsareselectedamongI,Br,orCl.
TheA-sitecationcanbeMA,FA,orCs.
Forthesolvent,weconsideredonlyoneofeightcommonpuresolventchoices:THTO,DMSO,DMF,NMP,GBL,acetone,methacrolein,andnitromethane.
ThesolubilitywasapproximatedastheintermolecularbindingenergybetweenABX3andNsolventmolecules(forbenchmarkingpurposes,3waschosen),whichwecanreasonablyassumetobepositivelycorrelatedwiththeenthalpyofsolvation(Hsolv).
Agivenperovskitecompositioncanbedescribedbyaneight-dimensionalvectorx2X,whereXf0;1g6RR.
x1:6arebinaryvaluesthattogetherindicatethepresenceofaparticularhalideinthesolution.
WerequirethatP3i1xi1,andthenon-zeroxiindicatesthatthecorrespondinghalideispresent.
Forexample,(0,1,0)indicatesaperovskitesaltofABBr3.
x4:6arealsobinaryvaluesthattogetherindicatethepresenceofaparticularcation.
Similarly,werequireP6i4xi1,inwhichthenon-zeroxiindicatesthatthecorrespondingcationispresent.
x7andx8arereal-valuedsolventdescriptorswithx7beingtherelativedielectric(εr)andx8thedensity(ρ).
Sincetheleadionispresentineverycomposition,itisomittedhereinthedescriptionofx.
Notethatdifferentcentralions(e.
g.
,SninsteadofPb)couldbeaddedtothedescriptioninasimilarfashion.
Weremarkthatweneglectx8whenwecomparePALtotheotherbaselinemethods,whichresultsinaslightlyworseperformanceofPAL.
Thatis,thePALdescribedinthissectionperformsevensomewhatbetterthanintheperformanceplot.
ThisisdonetoallowforafaircomparisonbetweenPALandpySMAC,aspySMACdoesnotsupportdiscretevariablethattakevalueinanitesetofpairsofreals.
Mixedhalides.
Forthemorechallengingmixedhalidecasestudy,weextendedtheperovskitecompositionvectorxtobex2Y,whereYf0;1g12RR.
x1:9arebinaryvaluesthattogetherindicatethepresenceofaparticularhalideinthesolution.
WerequirethatP3i1xi=P6i4xi=P9i7xi=1,andthenon-zeroxiindicatesthatthecorrespond-inghalideispresentinthesolution.
Forexample,(0,1,0,0,1,0,1,0,0)indicatesaperovskitesaltofABBr2I.
x10:12arealsobinaryvaluesthattogetherindicatethepresenceofaparticularcation.
Similarly,werequireP12i10xi1andthenon-zeroxiindicatesthatthecorrespondingcationispresentinthesolution.
x13andx14arereal-valuedsolventdescriptorswithx13beingtherelativedielectric(εr)andx14thedensity(ρ).
Sincetheleadionispresentineverycomposition,itisagainomittedfromx.
Onceagain,weneglectx14whenbenchmarkingPALwiththestate-of-the-artmethods.
BayesianoptimizationBayesianoptimizationhasemergedasapowerfultechniquetondanoptimizerofexpensive-to-evaluatefunctions.
28,29Itconsistsoftwocomponents:asurrogatemodelfortheobjectivefunctionandanacquisitionfunctionforselectingthenextsamplepoint.
TheBayesianoptimizationalgorithmproceedsasfollows:1.
Usinganinitaldataset,estimatethehyperparametersofthepriorforthecomponentsoftheLinearBeliefmodel.
Whenbenchmarking,startwithonlyasinglepoint.
2.
Computetheposteriorprobabilitydistribution,giventhepriorandtheinitialdata.
3.
Untilthebudgetforsamplesisexhausted,doiniterationt:(a)Selectthenextcombinationx(t)tosampleviatheacquisitionfunction.
(b)Evaluatetheobjectivevalueofx(t).
(c)Updatetheposteriorwiththenewobservation.
(d)Onaregularschedule,estimatethehyperparametersagain,usingallgatheredobservations.
4.
Returnthebestfoundcombination.
Thedecisiontorecommendthecombinationwiththebestobjectivevalueamongthesampledcandidatesisconservativefromadecision-theoreticperspectiveandnaturalfornoise-freeobservations.
Inwhatfollows,weproposeastatisticalmodeltailoredforperovskitecompositions.
Fortheacquisitionfunction,weselectedtheexpectedimprovementapproachthatisknowntoperformwellwhenobjectivevaluescanbeobservedwithoutnoise,ashere.
30,31LinearBeliefmodel.
Weassumethatthefourcomponents(Pb,A-sitecation,halide,andsolvent)contributetothebindingenergyofaperovskite-solventmixtureinanadditivemanner.
Theactualeffectofeachcomponentisunknownandwillbeinferredfromobservations(i.
e.
,fromDFTcalculationsofthebindingenergy).
ThisisakintothegeneralizedFree-Wilsonapproachthatwassuccessfullyusedtopredictbiologicalactivity;however,wefurtherexpandonthisapproachbyalsodrawingfromaGaussianProcess(GP).
32WeapplyBayesianlinearregressionandsupposethatthebindingenergyofthecompositioncorrespondingtoxisgivenbyV(x),asoutlinedinEq.
(1):VxXni1αixiβxζfxε;xρ:(1)HerenindicatesthenumberofbinarydescriptorsfortheABX3salt:6inpurehalidesand12inmixedhalides.
αiisthecontributiontothebindingenergyofximadebythepresenceofhalidesandcations,andisassumedtobethesameforallhalidesandcations.
β(x)capturesthedeviationofV(x)fromthelinearmodel.
ζrepresentsthecontributionofthecommonleadcluster.
f(xε,xρ)capturesthecontributionofthesolventS0tothebindingenergy(inpurehalidesxε=x7andxρ=x8,inmixedhalidesxε=x13andxρ=x14).
Thecontributionofthesolventasafunctionofthedielectricandthedensityisrepresentedbyxεandxρ,respectively.
Priordistribution.
InaccordancewiththeBayesianparadigm,weplaceapriordistributiononthecomponentsoftheregressionmodel.
Wesupposethateachαi$Nμα;σ2α,i.
e.
,thecoefcientsareindependentandidenticallydistributed,conditionedonμαandσ2α.
Moreover,wesupposethatthenon-linearcorrectionforeveryxhaspriorβx$N0;σ2βandthatζ$Nμζ;σ2ζ.
WeemployGPregressiontocapturethecontributionofthesolventblend.
Wesupposethatf(·)isdrawnfromaGPwithpriormeanfunctionμ0(·)andcovarianceΣ0(·,·).
Sinceζcapturestheinvariantcontributionfromthecentralion,itisreasonableforustoassumeμ0(·)=0.
Finally,weassumedifferentsolventswillbehavesimilarlyiftheirdescriptorsaresimilar.
Thisisformalizedintheuseofthe5/2MatérnKernel.
LetSi=(xε,i,xρ,i)andrPdi1liS1;iS2;i2qdenotethe"similarity"betweensolventsS1andS2,withdimensionsweightedbyli.
Notethatinourmodeld=2.
ThecovarianceoftwosolventblendsS1andS2isthengivenas:Σ0S1;S2σ2m15pr135r2e5pr:(2)Efcientsearchofcompositionalspaceforhybrid.
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Herboletal.
4npjComputationalMaterials(2018)51PublishedinpartnershipwiththeShanghaiInstituteofCeramicsoftheChineseAcademyofSciencesThus,theproposedgenerativemodelhasthehyperparameters:Θμα;σα;σβ;μζ;σζ;σm;l0;l1;:::;ld:(3)OurpriordistributiononV1,V2,…,Vnismultivariatenormalwithmeanμ0andcovariancefunctionΣ0:V1V2.
.
.
VN0BBBB@1CCCCA$Nμ0;Σ0:(4)Notethatμ0atxequals:μ0xEPn1μαxiμζμ0!
n3μαμζ0;andthecovarianceΣ0x;x0ofxandx′underthepriorisgivenby:Σ0x;x0CovVx;Vx0Pni1σ2α1xix0i1σ2β1xx0σ2ζΣ0xε;xρ;x0ε;x0ρσ2αx1:njix01:nσ2βImσ2ζJmΣ0Sx;Sx0;(5)whereImisthem*midentitymatrix,andJmisthem*mmatrixofones.
Hereweintroducethebra-ketnotationforacolumnvector,|x,orarowvector,x|.
Assuch,x|xwouldbeascalar,and|xx|anm*mmatrix,wheremisthelengthofthe|xvector.
Finally,subscriptsareinclusive;thatis,x1:3|=(x1,x2,x3).
Hyperparameterttingandposterior.
Thehyperparametersofourmodel,assummarizedinEq.
(3),areestimatedfromdata.
Fortheexperimentalevaluation,weusedamaximumlikelihoodestimate,usinganinitialdatasetsampledaccordingtoaLatinHypercubedesignandsubsequentlyoptimizedviaanL-BFGS-Bapproach.
Theoptimizedsampledhyperpara-metersthatbehavethebestarethenchosen.
Wesupposethattheobjectivevalueofanyxcanbeobservedwithoutnoise.
Thentheposteriordistributionoff(·),thecontributionofthesolventblend,ismultivariatenormalandsoistheposteriordistributionofthecoefcientsoftheregressionmodel.
33Validationofthemodel.
WevalidatetheLinearBeliefmodeldescribedabovebytheleave-one-outcross-validation(LOO-CV)method(seeFig.
4).
Moreover,leave-p-outcross-validation(LpO-CV)wasalsoperformedinwhichp=100datapointsareremovedfromthetrainingsetforvalidationpurposes(seeFig.
4).
ThesevalidationsdemonstratethattheproposedregressionmodelwellrepresentstheintermolecularbindingenergiesfromDFT.
ThePALThePAL,illustratedinFig.
5,isaPython-basedframeworkthatwehavedesignedforthetaskofmaximizingachosencomputationalobjectiveasafunctionofamaterial'scomposition.
Bayesianoptimizationlendsitselftoconsiderationofexpensiveobjectivefunctionsand,assuch,providesseveralpossibilitiesforDFTcalculations.
28,29Thus,PALcombinesthefunctionalityofBayesianoptimization,usingthegenerativemodelproposedinthesection"Bayesianoptimization",witharenedmethodtocalculatetheenthalpyofsolvationorothermetricofsuccess.
InordertoensuretherobustnessofPAL,thecodewaswrittentoaccommodatecalculationsofotheroptions,including(1)solvation,(2)intermolecularattractions,(3)sterichinderance,and(4)molecularproperties.
Assuch,thefollowingchoicesareconsideredfortheobjectivefunction:1.
TheUnsaturatedMayerBondOrder(UMBO)withinasolventmolecule(fromthemostpolaratomtoitsnearestintra-molecularneighboringatom).
2.
Theintermolecularbindingenergies.
3.
Thevolumeandeccentricityofasolventorsolute,denedbyaMinimumVolume-EnclosingEllipse(MVEE).
344.
Thedipolemomentofasolvent.
UnsaturatedMayerbondorder.
InordertocapturetheUMBObetweensolventandsolute,wherethesoluteissomeHOIPspeciesABX3(withAbeingacation,BbeingeitherPborSn,andXahalide),wemustrstisolateastablegeometriccongurationofthesemolecules.
ThisisaccomplishedbyequilibratingalargeboxofsolventsusingMD,ofwhichasmallersub-systemisfurtherequilibrated(withinMD),andnallythespecicmotif(withinthesub-system)isgeometry-optimizedinDFT.
TheinitialtwoMDstepsareperformedinordertoautomatethegenerationofanadequatestartingcongurationforthesubsequentDFToptimization.
Tomakethelargesolventbox,weusedthepackmolsoftwaretopacktogetherasingleleadsaltsoluteina25*25*253boxofsolvent.
35Fromthisstartingpoint,theMDcalculationsweredoneinLAMMPS,apopularMDsoftware,withtheOPLS-AAForceField.
36,37SinceweplantoperformanaccurateDFTgeometryoptimizationofaproposedconguration,weallowforthefollowingassumptionswithinourMDsimulations:1.
Wecanrunanequilibrationusingthemicrocanonical,NVE,ensemblewithalimitedstepsizeasawaytocircumventtheriskofanunexpectedlypoorstartingstate.
2.
WeneedonlyruntheNVEandisobaric-isothermal,NPT,Fig.
4Cross-validationusingaLOO-CVmethod(left)andaLpO-CVmethod(right).
TheblacklineshowsthevaluesobtainedfromDFTcalculations.
Thebluelineshowsthepredictivemeanoftheproposedregressionmodel,whosehyperparametersweretrainedoneither(1)alldatapointsexcludingtheonebeingpredicted(LOO-CV)or(2)140datapoints,andthentestedagainstthe100remaining.
They-axisshowstheintermolecularbindingenergyEfcientsearchofcompositionalspaceforhybrid.
.
.
H.
C.
Herboletal.
5PublishedinpartnershipwiththeShanghaiInstituteofCeramicsoftheChineseAcademyofSciencesnpjComputationalMaterials(2018)51equilibrationsfor10,000iterations(10ps).
3.
Thelargebox(inMD)isequilibratedusingtheNVEensemblewitha0.
1limitedstepsize,andisfollowedupbyequilibrationusingtheNPTensembleat300Kand1atm.
4.
Thesmallerbox(inMD)isequilibratedusingtheNVEensemblewitha0.
001limitedstepsize,andisfollowedupbyanotherNVEensembleequilibrationwitha0.
01limitedstepsize.
5.
WeapproximatePbasBainOPLS-AA,astheyhavethesameoxidationstate(+2).
ThisisbecausePbisnotparameterizedinthisforceeld.
6.
WeholdABX3constantbyremovingforcesthroughouttheMDsimulations.
7.
Atimestepof1.
0fsisreasonable,especiallywiththelimitedstepsizesoftheNVEcalculations.
ThesmallerMDsimulationboxisderivedfromthelargersimulationbox(post-equilibration)byrstdeterminingtheintermoleculardistancesfromthesoluteion(inourcase,Pb)tothemostpolaratomwithinthesolvents(typicallyoxygen).
Fromthis,eithertheclosestNsolventscanbechosen,orallsolventswithinagivenradius.
InthecasewhereweusetheUMBOasthebasisforourobjectivefunction,weconsidertheclosestsolventmolecule.
ThenalgeometryisthenoptimizedusingtheB97-D3GGAbasissetwithGrimme'sdispersioncorrection.
38,39Thedef2-TZVPbasissetisused,withaneffectivecorepotentialforthePbion.
40Wheneverasolventissimulated,bothanexplicitsolventandimplicitsolventmodel(usingtheCOSMOpackage)areused.
41Thisisdonebecausetheimplicitmodelisalow-costmethodtoimprove/incorporatetheelectronicpolarizationofthesolute.
42Thesystem,aswellasindividualcomponents(soluteandsolvent),areoptimizedintheOrcaDFTpackage,withthenalUMBObeingreturnedasanoutput.
43Bindingenergy.
InasimilarfashiontotheUMBOcalculation,webeginwithanequilibrationofalargesolventbox.
Here,however,insteadofconsideringonlyonesolvent,weisolatetheNnearestneighborsolventmoleculestoagivenABX3solute.
Finally,inordertoaccomplishabindingenergycalculation,wealsoneedtheenergyoftheindividualsystemsofinterest.
InthecaseofABX3orasinglesolvent,wesimplyoptimizethemoleculeindependentlyinDFT.
However,inthecaseofNmixedsolvents,wemustusethesameprocedureasthatusedwhenisolatingastablegeometriccongurationforthesolvent-solutesystem:alargeMDequilibrationofasolventbox,followedbyasmallerMDequilibrationofNsolvents,followedbyaDFTgeometryoptimizationofthenalsystem.
Assuch,PALiscapableofcalculatingthefollowingintermolecularbindingenergies:1.
S0vs.
S0,2.
S0vs.
S1,3.
S0vs.
ABX3,and4.
S0,S1,…,Snvs.
ABX3.
Sterichinderance.
Sterichinderance,shortenedhereafteras"sterics",capturestheeffectofvolumeexclusioneffectsduetothepresenceofthemolecules.
Thiscancauseissueswithmolecularpackingandinducestrainwithinlargemolecularsystems.
Inthecaseofnucleation,stericscomesintoplayasthephysicalrestrictionsinvolvedwhenreactivespeciesapproachoneanother.
Assuch,deningorderparameterstocapturetheeffectsofstericsinrelationto(a)solvationor(b)reactivityisrelevant.
OnesuchapproachistodeneaMVEEaroundmoleculestoabstracttheglobularityofthemoleculeandultimatelysimplifytheproblem.
34Withanellipsedescribingthemolecule,avarietyofmathematicalrepresentationsarenowavailable,includingvolume,eccentricity,andsurfacearea.
Alone,thesevaluesmaybeofuse,butitislikelythatacombinationwithothermolecularpropertieswouldproveevenmoreuseful.
Molecularproperties.
Beyondcalculationsofmolecularinteractions(e.
g.
,theUMBO,bindingenergy,orsterichinderance),moleculesthemselvespossesscharacteristicproperties(suchaspolarizationanddensity)thatcontributetotheirbulkbehavior.
SomeofthesepropertiesremainaccessibleattheDFTlevel,allowingustousethemwithinsuitablyconstructedobjectivefunctions(suchastheGutmanndonornumber,whichwasrecentlyshowntobeapotentialindicatorofPbsolubility).
9Althoughithasbeenshownthatnotallcommonlyusedmolecularpropertiesdirectlycorrelatewithbulkpropertiesofinterest,anobjectivefunctioncouplingthemtomolecularinteractionsmayproveuseful.
17DATAAVAILABILITYThecodeforthisarticleandtheresultsoftheDFTcalculationsarepubliclyavailableathttps://github.
com/clancyLab/NCM2018.
ACKNOWLEDGEMENTSThisworkwaspartiallysupportedbytheCornellCenterforMaterialsResearchwithfundingfromtheNSFMRSECprogram(DMR-1719875)throughaseedfundingaward.
H.
H.
andM.
P.
werepartiallysupportedbythisaward.
P.
C.
,P.
F.
,W.
H.
,andM.
P.
werepartiallysupportedbyNSF(CMMI-1536895,DMR-1719875,DMR-1120296,CMMI-1254298)andbyAFOSR(FA9550-15-1-0038).
TheCornellInstituteforComputationalScienceandEngineeringisthankedforaccesstotheircomputingresources.
AUTHORCONTRIBUTIONSH.
C.
H.
implementedPALinitsentirety(automatedMD-DFTobjectivefunction),andincorporatedtheBayesianoptimizationcodeintothepackage.
W.
H.
implementedFig.
5ThePhysicalAnalyticspipeLine.
Ontheleft,objectivefunctionsfromsimulationsarefedintotheBayesianoptimizationcode(middleoftheschematic)toproducerankedpredictionsofABX3–Solventmixturesofinterest(rightmostcartoon).
Experimentalsourcesofinformation(topschematic)canultimatelybeusedforfurthervalidation,aswellasfurtherne-tuningofthestatisticalmodelEfcientsearchofcompositionalspaceforhybrid.
.
.
H.
C.
Herboletal.
6npjComputationalMaterials(2018)51PublishedinpartnershipwiththeShanghaiInstituteofCeramicsoftheChineseAcademyofSciencestheinitialproof-of-conceptpurehalideBayesianoptimizationcode,andH.
C.
H.
furtherextendedittoincorporatemixedhalides.
P.
F.
helpedcorrectinaccuraciesduringtheimplementationoftheBayesianoptimization.
M.
P.
andP.
C.
wereresponsiblefortheoveralldesignofthestudyandpreparationofthemanuscript.
M.
P.
wasalsoresponsibleforsuggestingthetestsofthestatisticalsignicanceoftheresults.
H.
C.
H.
,M.
P.
,andP.
C.
wrotethemanuscript.
M.
P.
istheguarantorforthismanuscript.
ADDITIONALINFORMATIONSupplementaryinformationaccompaniesthepaperonthenpjComputationalMaterialswebsite(https://doi.
org/10.
1038/s41524-018-0106-7).
Competinginterests:Theauthorsdeclarenocompetinginterests.
Publisher'snote:SpringerNatureremainsneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalafliations.
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TheAuthor(s)2018Efcientsearchofcompositionalspaceforhybrid.
.
.
H.
C.
Herboletal.
7PublishedinpartnershipwiththeShanghaiInstituteofCeramicsoftheChineseAcademyofSciencesnpjComputationalMaterials(2018)51