Low-Cost Computing of the Thermophysical Properties of Organic − Inorganic Halide Perovskites by Density Functional Theory Combined with

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pubs.acs.org/JPCCArticleLow-CostComputingoftheThermophysicalPropertiesofOrganic−InorganicHalidePerovskitesbyDensityFunctionalTheoryCombinedwiththeThree-DimensionalReferenceInteractionSiteMethodTomoyasuYokoyama,*SatoruOhuchi,TaisukeMatsui,YukihiroKaneko,andTakaoSasagawaCiteThis:J.Phys.Chem.C2021,125,6601−6610ReadOnlineACCESSMetrics&MoreArticleRecommendations*sıSupportingInformationABSTRACT:Organic−inorganichybridmaterials(OIHMs)representedbymethylammoniumleadtriiodide(MAPbI3)andformamidiniumleadtriiodide(FAPbI3),whicharetheabsorptionlayerofperovskitesolarcells,haveattractedmuchattention.However,itisdifficulttocalculatethepropertiesofOIHMsbydensityfunctionaltheory(DFT)intermsofcomputationalcostbecausetheorganicmoleculesrotatewithintheinorganicframeworkatfinitetemperature.Here,usingcubicMAPbI3asanexample,wedemonstratethereductionofthecomputationalcostforOIHMsbyDFTcombinedwiththethree-dimensionalreferenceinteractionsitemethod(3D-RISM).TheRISMwasdevelopedtosimulatethestructuresandreactionsattheinterfaceofsolidsandliquids.BytreatingtheMAcationasapseudoliquid,wesuccessfullyreproducedthedistributionoftheMAcationexperimentallyobservedatafinitetemperature.WhentheRISMtemperaturewaschanged,thethermalexpansioncoefficientandtemperaturedependenceofthebandgapofMAPbI3werepredicted,andtheywereingoodagreementwithexperimentalreports.Inaddition,thelatticeconstant,thebandgap,andfreeenergyofmixingofMA1−xFAxPbI3canbeevaluatedbyDFT/3D-RISMwithoutusingaDownloadedviaUNIVOFNEWMEXICOonMay16,2021at07:59:01(UTC).supercell.Therefore,thismethodmakestheevaluationofthermophysicalpropertieswithouthigh-costcomputingpossible.Seehttps://pubs.acs.org/sharingguidelinesforoptionsonhowtolegitimatelysharepublishedarticles.■PbI3inorganiccageatfinitetemperature.Athightemperatures,INTRODUCTIONTheprocessingoforganic−inorganichybridmaterialsMAPbI3isstableinthecubicaristotypePm3̅mstructurewith1−6(OIHMs)hasattractedagreatdealofinterest.Thesynergythehighestsymmetry(Figure2a)becausetheMAcation15−17oftheorganicandinorganiccomponentshasbeencanfreelyrotate.Asthetemperaturedecreases,theMAdemonstratedinanumberofnaturallyformedhybridcationscannotfreelyrotateandbegintoformhydrogenbondsmaterials.Aninterestingexampleisorganic−inorganichalidewiththehalogenatoms.Thisorderingoftheorganiccationsperovskites.Thetypicalcompositionsofthesematerialsarecausesaphasetransition.18ThephasetransformationsfromCH3NH3PbI3(MAPbI3)andHC(MH2)2PbI3(FAPbI3).cubictotetragonalandtetragonaltoorthorhombicoccuratThesematerialshaveathree-dimensionalperovskitestructure330and161K,respectively.1−6,19,201−7representedbyABX3,whereAisanorganiccation,suchasCHNH+orHC(NH)+(Figure1),BisPb2+,andXisI−.3322Usingthesematerialsastheabsorptionlayerofsolarcells,theReceived:February8,2021powerconversionefficiencyofsuchsolarcellshasincreasedRevised:February19,2021from3.8%to25.2%inthepastdecade.8−14Published:March16,2021Interestingly,therotationalorganiccationdynamicsaffectsthecrystalstructureandelectronicandopticalpropertiesofOIHMs.ForMAPbI3,theorganiccationsrotatewithinthe©2021AmericanChemicalSocietyhttps://doi.org/10.1021/acs.jpcc.1c011716601J.Phys.Chem.C2021,125,6601−6610

1TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleorganicmoleculesaremixed,supercellsmustbemodeledconsideringnotonlythefreedomofrotationbutalsothe34,35freedomofthearrangementofthesites.Anothersemiempiricalmethodisthereferenceinteractionsitemethod(RISM),whichdescribesasolventpartasa39continuummodel.Thethree-dimensionalRISM(3D-RISM),whichimposes3DperiodicboundaryconditionsontheRISM,cancalculatethesolvationstructuresinconfined40,41nanospacesinframeworksystems.Recently,Nishiharaand42Otanireformulatedthe3D-RISMbyimposingaplanewaveandpseudopotential(PW−PP)methodbasedonKohn−ShamFigure1.Structuresofthe(a)CHNH+(MA)and(b)HC(NH)+3322densityfunctionaltheory(DFT).ThisDFT/3D-RISM(FA)cations.approachwasdevelopedtosimulatethestructuresandreactionsattheinterfaceofthesolidsandliquids.Whencalculatingthesolid−liquidinterfacebyDFT/3D-RISM,waterTherearemanytheoreticalreportsonorganic−inorganicmoleculesintheliquidcanbetreatedas3D-RISM,andsolidhalideperovskitesusingfirst-principlescalculations,which21−36inorganiccrystalscanbesimulatedattheDFTlevel.Asaprovidenewinsights.However,itisdifficulttocalculateresult,itwaspossibletocalculatethecrystalmaterialsatlowerthepropertiesofOIHMsonthebasisofthefirstprinciplescostthanwhenhandlingallwatermoleculesattheDFTlevel.consideringtherotationoftheorganicmolecules.BecausetheIthasbeenappliedtotheanalysisoftheelectrode/electrolytefirst-principlescalculationstreatamodelatatemperatureof043,44interfaceoflithiumionbatteries,electricdoublelayerK,itisnecessarytofixthepositionoftheorganicmolecules4445capacitors,andphotocatalysts.Thismethodallowsasolutewithintheinorganicframework.Ithasbeentheoretically42reportedthatchangingtheorientationoftheMAcationinmoleculeinabulkporousmaterial,suchaszeolite,tobeMAPbIchangestheelectronicandopticalproperties.21−24treatedbytheRISM.3Therefore,inordertomodelanorganicmoleculethattakesaInthiswork,weutilizedDFT/3D-RISMtosolvetherandomorientationatafinitetemperature,itisnecessarytoproblemofthecomputationalcostofOIHMs.Likewateraveragethecalculationresultsofthemodeloftheorganicmoleculesinliquids,wetreatedorganicmoleculesinOIHMsmoleculerotatedinvariousdirections.Ingeneral,alargebythe3D-RISM.Thiscreatesthefollowingfouradvantages.supercellunderperiodicboundaryconditionsneedstobe(i)Itcancalculateorganicmoleculesconsideringthedegreeofconstructed.25−30,34,35Althoughfirst-principlesmolecularfreedomofitsrotationatlowcost.Becauseorganicmoleculesdynamics(MD)simulationscantakeintoaccountthethermalcanbeexpressedasaprobabilitydistribution,itissufficienttomotionoforganicmoleculesatfinitetemperatures,25−30theycalculateonlyoneunitcellinsteadofseveralsupercellswithrequireconsiderablecomputationalcost.Therefore,thevariousdifferentmolecularorientations.(ii)TheelectronicevaluationofthepropertiesofOIHMsatfinitetemperaturestatecanbecalculatedwhilemaintainingthesymmetryisdifficultintermsofcomputationalcostforbothstaticandobservedatthefinitetemperature.Itisbecausethedynamicfirst-principlescalculations.distributionoforganicmoleculesbecomesisotropicbytreatingSemiempiricalcomputationisanimportanttechniquetoorganicmoleculesasacontinuummodel.(iii)Thetemper-37,38aturedependenceofthepropertiescanbecalculatedrealizelow-costcomputing.Recently,Uratanietal.reportedthatthedensity-functionaltight-binding(DFTB)consideringthethermalmotionoftheorganicmolecules.approach,whichisoneofthesemiempiricalmethods,realizedRISMcanexpresstheaveragethermalmotionofthemoleculesalow-costMDsimulation.Theyanalyzedtheformationbychangingthedistributionofthecontinuummodel.processofpolaronsinMAPbI3withalargesystemsizebytheTherefore,thethermalexpansioncoefficientandcrystalDFTB-MDcalculation.However,whenseveraltypesofstructurestabilityoftheOIHMsathightemperaturescanbeFigure2.(a)CubicMAPbI3modelusedintheDFTapproach.Alloftheatomsinthestructurearetreatedexplicitly.(b)ModelofcubicMAPbI3usedinDFT/3D-RISM.TheMAcationsaretreatedimplicitlyasacontinuumdistribution,andthePbandIatomsaretreatedexplicitly.6602https://doi.org/10.1021/acs.jpcc.1c01171J.Phys.Chem.C2021,125,6601−6610

2TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticlepredicted.(iv)Theaverageelectronicstatesofcompoundwherethefirsttermontheright-handsideistheLJforcefieldmixingwithdifferenttypesoforganicmoleculescanbeandthesecondtermistheelectrostaticpotential.TheLJcalculatedwithoutusingasupercell.ThiscanbedonebyparametersεAandσAaredefinedforeachsoluteatomA,andsettingthecontinuummodelratioatanarbitrarynonintegerthesummationisperformedforallsoluteatoms.Inthevalue.Thereby,thepropertiesofthemolecularmixedsystemelectrostaticpotential,vDFT(r)isthesumoftheHartreeoftheOIHMscanbecalculatedwithoutperformingMDpotentialandlocalpotentialcalculatedbyDFT.calculations,whichsignificantlyreducesthecomputationalNishiharaandOtani42developedavariantoftheDFT/3D-costs.RISMusingPW−PPasasolutesystem.Inthismethod,theHere,usingcubicMAPbI3asanexample,wedemonstratecorrelationfunctionsofthe3D-RISMaredefinedinaunitcellthereductionofthecomputationalcoststopredicttheofthePW−PPcalculation.Inaddition,theperiodicboundarythermophysicalpropertiesfororganic−inorganichalideperov-conditionisimposedonboththe3D-RISMandPW−PP.skitesbyDFT/3D-RISM.MAcationsweretreatedasaContrarytoanisolatedmoleculeasasoluteintheRISM,thiscontinuumdistribution(liquid)of3D-RISM,whereasPbandImethodallowsasolutemoleculeinabulkporousmaterial,atomsweretreatedasasolidframework.Wesuccessfullysuchaszeolite,tobetreatedbytheRISM.42ThenewlyreproducedthedistributionoftheMAcationexperimentallydevelopedhybridsolvationmodelsareimplementedintheobservedbyNeutrondiffractionatafinitetemperature.Whenfirst-principlessoftwareQuantumESPRESSO.48theRISMtemperaturewaschanged,thethermalexpansionUsingDFT/3D-RISM,wecalculatedcubicMAPbI,whose3coefficientandtemperaturedependenceofthebandgapofunitcellisshowninFigure2a.ThePb−IinorganicframeworkMAPbI3werepredicted,andtheresultswereingoodwascalculatedbyPW−PPwiththePerdew−Burke−Ernzerhofagreementwiththeexperimentalreports.Inaddition,we49exchange−correlationfunctionalrevisedforsolids(PBEsol).calculatedtheaverageelectronicstateofacompoundUltrasoftpseudopotentialswereusedfortheleadandiodinecontainingseveraltypesoforganicmoleculesbyDFT/3D-atoms.ThetotalchargeofthePb−Iframeworkwassetto−1RISM.Thelatticeconstant,thebandgap,andfreeenergyofpercell.SamplingintheBrillouinzonewasperformedusingthemixedmolecularsystem,MA1−xFAxPbI3,wereevaluatedthe8×8×8k-pointmeshoftheGammacenter.withoutusingasupercell.Therefore,thismethodmakesitTheMAcationinsideoftheframeworkwastreatedasapossibletoevaluatethermophysicalpropertieswithouthigh-continuumdistributionofthe3D-RISMusingtheclosureofcostcomputing.46themodelofKovalenkoandHirata.The3D-RISM■implementedinQuantumESPRESSOcodecancontrolnotMETHODonlythedensityofthesolutionmoleculesbutalsothenumber40,46KovalenkoandHiratadevelopedtheoriginalconceptofofthemperconfinedspace.Weusedthelatterfunctionasonethe3D-RISMwithapartiallylinearizedclosureequationtoorganiccationmoleculeperunitcellofPbI3.Theelectronsimulatethedistributionsofa“solvent”,suchaswaterandchargedensityandatomicshapeoftheMAcationwereions,arounda“solute”,suchasabiomolecule.Theinteraction50optimizedbytheGaussian09codeusingtheB3LYPdensitybetweenthesoluteandsolventisdescribedasaclassicalforce51functionaland6-311++G**basisset.ThetotalchargeofthefieldmodeledbytheLennard-Jones(LJ)potentialandpoint47MAcationwassetto+1permolecule.Usingthecalculatedchargestocalculateuαγ(r)bychargedensityandoptimizedmolecularstructure,theMAÄÅÅ126ÉÑÑcationwastreatedbythesimplepointchargemodelinthe3D-ÅÅÅÅloo()σσαγ++|ooloo()σσαγ|ooÑÑÑÑqqÅÅoo2oooo2ooÑÑαγRISM.TheLJparametersfortheorganicmoleculeswereuαγ()r=−4εεαγÅÅmoo}oomoo}ooÑÑ+calculatedbytheACPYPEcode52usingthegeneralAMBERÅÅÅÅoorrooooooÑÑÑÑr47ÅÅÇn~n~ÑÑÖforcefield(GAFF).TheLJparametersarediscussedintheSupportingInformation.Theoptimizedstructure,chargewherethefirsttermontheright-handsideistheLJpotentialdensity,andRISMcorrelationfunctionoftheFAcationandthesecondtermistheelectrostaticpotential.Thewereobtainedinthesameway.TheLJparametersofthesubscriptsαandγrefertoatomicsites;risthedistanceframeworkwerethoseoftheuniversalforcefield.53Thecutoffbetweenthesites.εαandσαaretheparametersoftheLJenergieswere100,211,and276Ryfortheelectronicwavepotential,andqαistheamountofcharge.Theseparametersarefunction,chargedensity,andsolventcorrelationfunction,constantanddefinedforeachatomicsite.respectively.TheaboveparametersweredeterminedbyToimprovemodelingofthesolutemolecule,anabinitiocalculatingtheconvergenceoftheparametersshowninFigureelectronicstructurecalculationwithaGaussianbasissetwas41S2.Theroot-mean-squarethresholdsoftheone-dimensionalappliedtoasolutemoleculebySatoandco-workers.This−8−6RISMand3D-RISMweresetto1×10and5×10,methodiscalledtheDFT/3D-RISMor3D-RISMself-respectively.consistentfield(3D-RISM-SCF).Inthismethod,theTheharmonicphonondispersionforMAPbI3wascalculatedCoulombicinteractionfromthepointchargesisreplacedby54usingthePHONOPYpackage.Thephonondispersion(fortheelectrostaticpotentialfromtheabinitiocalculation.TheqpointsawayfromtheBrillouinzonecenter,theΓpoint)inpotentialfunctionusedinthe3D-RISMcalculationuα(r)isthethecubicphasewasprobedina2×2×2supercell.Beforeinteractionbetweenthesolventatomicsiteαandsolute:thiscalculation,weperformedcompleteoptimizationoftheÄÅÅ126ÉÑÑlatticeconstant,shape,andatomicpositions.ThisprocedureÅÅÅÅloo()σσαα++AA|ooloo()σσ|ooÑÑÑÑu()r=∑4εεÅÅoom2oo}−oom2oo}ÑÑwasdonemanuallyasfollows,sincetheschemeforthe3D-ααAÅÅÅÅoo|−|rRoooo|−|rRooÑÑÑÑRISMandPW−PPcalculationshasnotbeenimplementedinAÅÅoonAoo~oonAoo~ÑÑÅÇÑÖQuantumESPRESSO.Whilemaintainingthecubiccell,thelatticeconstantwaschangedfrom6.3to6.5Åinstepsof0.01−qvDFT()rαÅ,andatomicrelaxationwasperformedtoobtainthemost6603https://doi.org/10.1021/acs.jpcc.1c01171J.Phys.Chem.C2021,125,6601−6610

3TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleFigure3.(a)Chargedensitydistributiononthe(001)planeobtainedbyDFT.(b)RISMchargedensitydistributionoftheMAcationonthe(001)planeobtainedbyDFT/3D-RISM.ElectronicdensityofstatesofcubicMAPbI3obtainedby(c)DFTand(d)DFT/3D-RISM.BandstructureofcubicMAPbI3obtainedby(e)DFTand(f)DFT/3D-RISM.Theoriginoftheenergyissettothevalencebandmaximum.Thedifferencebetweentheconductionbandminimumandvalencebandmaximumcorrespondstothebandgap(Eg).HarmonicphonondispersionofcubicMAPbI3obtainedby(g)DFTand(h)DFT/3D-RISM.TheRISMtemperaturewassetto330K.stablestructure.EachcalculatedstructurewasdrawnwithOntheotherhand,becausetheMAcationistreatedasa55VESTA.continuummodelinDFT/3D-RISM,thechargedensitydistributionisisotropicalongthex,y,andzaxes,which■RESULTSANDDISCUSSIONmeansthecubicsymmetryremainsthesame(Figure3b).TheUsingDFT/3D-RISM,itispossibletocalculatetheactualpositivelychargedregionsoftheMAcationarelocatedinthedistributionoftheMAcationatafinitetemperature.The[011]directiontothenegativelychargedIion.ThechargedensitydistributionsofMAPbI3onthe(100)planedistributionsoftheCandNatomsoftheMAcationobtainedobtainedbyconventionalDFTandDFT/3D-RISMareshownbyDFT/3D-RISMareshowninFigureS3.ThedistributionofinFigure3a,b.IntheDFTapproach,theMAcationsareNatomsisclosertotheIionsthanthatoftheCatoms.Thismodeledexplicitlyandorientedinthe[001]direction.Asaresult,mostofthechargesarelocalizednearNandCatoms,meansthatthe−NH3groupoftheMAcation,whichisforminganelectricdipolealongthe[001]direction(Figurepositivelycharged,isattractedtothenegativelychargedIion3a).Thesymmetryisloweredfromcube(Pm3̅m)totriclinicbytheCoulombinteraction.Anexperimentalneutron(P1)intheDFTmodelbyexplicitlytreatingtheMAcations.diffractionstructuralstudyreportedthattheMAcation6604https://doi.org/10.1021/acs.jpcc.1c01171J.Phys.Chem.C2021,125,6601−6610

4TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleFigure4.(a)RISMchargedensitydistributionsoftheMAcationsonthe(001)planeincubicMAPbI3obtainedbyDFT/3D-RISMatRISMtemperaturesof200,300,400,and500K.(b)TotalenergydependenceofthelatticeconstantofcubicMAPbI3whentheRISMtemperature(T)waschanged.Theoriginoftheenergyissettotheminimumvalueoftheenergywhenthelatticeconstantischanged.TheresultofT=0KwasobtainedbyDFT,andtheresultsforT>0KwereobtainedbyDFT/3D-RISM.Temperaturedependenceofthelatticeconstantsof(c)MAPbI359,61and(d)FAPbI3.RedcircleandblackcrossmarksindicatetheresultsobtainedbyDFT/3D-RISMandtheexperiment,respectively.15primarilyresidesalongthe[011]direction,whichagreeswithmolecule,CsPbI3maintainsthecubicPm3̅msymmetryevenourresults.afterthefulllatticerelaxationintheconventionalDFT.TheTheelectronicdensityofstates(DOS)andbandstructureresultsforCsPbI3areclosetothoseforMAPbI3obtainedbyofcubicMAPbI3obtainedbyDFTandDFT/3D-RISMareDFT/3D-RISM.ItshouldbenotedthatthechargesinCsPbI3showninFigure3c−f.Forcomparison,theywerecalculatedinareisotropicandlocalizedinthebodycenter,whereasthoseinaunitcellwiththesamecubiclatticeconstant(6.31Å),whichMAPbI3aredistributedinacentrosymmetricbutmoreistheexperimentallatticeconstantofMAPbI3at330K.Thecomplexstructure.Inanycase,DFT/3D-RISMenablesustoRISMtemperaturewassetto330K.AlthoughtheMAcationscalculatetheelectronicstatesofthehigh-temperaturephaseofaretreatedimplicitlyinDFT/3D-RISM,theelectronicstatesMAPbI3,whichhasbeenexperimentallyconfirmedtohavethenearthevalencebandmaximum(VBM)andconductionbandcubicsymmetryduetotherotationoftheMAmolecule.minimum(CBM)obtainedbyDFT/3D-RISMaresimilartoThephononbandstructuresofcubicMAPbI3calculatedbythoseobtainedbyDFT.Thebandgapof1.29eVobtainedbyDFTandDFT/3D-RISMareshowninFigure3g,h.TheRISMDFT/3D-RISMisclosetothatof1.26eVobtainedbyDFT.temperaturewassetto330K.ForDFT,theimaginaryThisisbecausethevalenceandconductionbandsofMAPbI3(negativeenergy)modeofthephononsisobservedattheRareformedbytheorbitalsofthePb−Iframework.Thepoint.ThisindicatesthepresenceofasaddlepointintheelectronicstateofPbI−afterremovaloftheMAcationfrompotentialenergysurfaceoftheatomicpositionsand,thus,that3theAsiteisshowninFigureS4.ThebandgapofPbI−is1.20thecubicstructureisnotthermodynamicallystable,whichisin331−33,36eV,whichissmallerthanthatofMAPbI3calculatedbyDFT/goodagreementwithprevioustheoreticalreports.This3D-RISMandDFT.Thisisbecauseofthespatialspreadingofisbecause,intheDFTapproach,theMAcationsareorientedtheelectronsintheinorganicframeworkwithouttheinfluenceinacertaindirection,resultinginthepolarizationofthecrystalofCoulombrepulsionfromtheAsiteinPbI−.Therefore,instructure.36However,ithasbeenexperimentallyreportedthat31−6,19,20DFT/3D-RISM,theRISMchargesreflecttotheelectronstatesthenonpolarcubicstructureisstableabove330K.ofaninorganicframework.ThisclearlyindicatesthattheMAcationrotationshouldbeIntheenergyrangebetween−0.5and−2.5eV,theDOStakenintoaccountfortheaccurateevaluationofthecrystalandbandstructureobtainedbyDFT/3D-RISMaredifferentstructureofMAPbI3atfinitetemperature.TheimaginaryfromthoseobtainedbyDFT.ThemostdistinctdifferenceismodedoesnotappearintheresultscalculatedbyDFT/3D-anexistenceofapeakintheDOSatabout−2.3eVbyDFT/RISMwiththeRISMtemperatureof330K,indicatingthatthe3D-RISM.TheelectronicstatesobtainedbyDFT/3D-RISMcubicstructureisstableatthistargettemperature.WhenthearedegenerateatpointsΓandR,whiletheyarenotbyDFT.MAcationistreatedasacontinuummodel,thedistributionofThisisduetotheloweredtriclinicsymmetryintheDFTMAcationsbecomesisotropicalongthex,y,andzaxes,model.TheDOSandbandstructureofcubicCsPbI3obtainedresultinginthePbI3inorganicframeworkbeingabletobyDFTareshowninFigureS5.BecausetheAsiteofCsPbI3maintaincubicsymmetry.ThesameDFT/3D-RISMcalcu-isanisotropicCsatominsteadofananisotropicorganiclationforPbI−(perovskitestructurewithouttheMAcationat36605https://doi.org/10.1021/acs.jpcc.1c01171J.Phys.Chem.C2021,125,6601−6610

5TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleFigure5.(a)RISMchargedensitydistributionsonthe(001)planeincubicMA(1−x)FAxPbI3obtainedbyDFT/3D-RISMatFAcationratios(x)of0.0,0.25,0.5,0.75,and1.0.(b)DependenceofthelatticeconstantontheFAcationratioincubicMA1−xFAxPbI3.Circleandcrossmarks65indicatetheresultsobtainedbyDFT/3D-RISMandtheexperiments,respectively.(c)DependenceoftheGibbsenergychangeontheFAcationratioincubicMA1−xFAxPbI3.TheredlinesaretheresultsataRISMtemperature(T)of300K.Thedashedlineistheresultwithouttheentropychangeofmixing.(d)TemperaturedependenceofthebandgapsofMAPbI3andMA0.13FA0.87PbI3obtainedbyDFT/3D-RISMand59,66experiments.theAsite)isshowninFigureS6.Animaginarymodeofthetemperature.ThelatticeparameteroptimizedbyDFT/3D-phononsisalsoobserved,aswithMAPbI3obtainedbytheRISMislargerthanthatoptimizedbyDFTandincreasesasconventionalDFTcalculation.ThisresultsuggeststhatthetheRISMtemperatureincreases.ThisresultreflectsthepresenceofapositivelychargedMAcationstabilizestheincreaseinthethermalrotationoftheMAcationandthestructureofthecubiccell.Therefore,whenthethermalexpansionoftheRISMdistributionoftheMAcationwithbehavioroftheorganicmoleculesisaveragedwiththeRISM,itincreasingtemperature.ispossibletoevaluatethethermodynamicalstabilityoftheThetemperaturedependenceofthelatticeconstantofcubicOIHMsatfinitetemperature,whichisimpossiblewiththeMAPbI3andFAPbI3obtainedbyDFT/3D-RISMandthe59conventionalDFTapproach.experimentisshowninFigure4c,d.ThelatticeconstantsInpreviousstudies,thethermalpropertiesevaluatedbyobtainedbyDFT/3D-RISMincreasewithincreasingtemper-conventionalDFTunderthequasi-harmonicapproximationature,whichagreeswiththeexperimentaltrend.Thelinear56−58(QHA)havebeenreported.However,thepropertiesthermalexpansioncoefficientsofMAPbI3forT>300KobtainedbythisapproachonlyincludetheeffectoftheobtainedbyDFT/3D-RISMandtheexperimentsare2.7×inorganicframeworkandnottheeffectofthermalrotationof10−5/Kand4.8×10−5/K,59,60respectively.Similarly,thoseoftheorganicmolecules.Moreover,whenimaginaryphononsFAPbIbyDFT/3D-RISMandtheexperimentare3.5×10−5/3appearasinthepreviousresults,itisdifficulttoaccuratelyKand3.6×10−5/K,61respectively.Itisseenthat,thecloserpredictthethermalpropertiesunderQHA.Incontrast,thethecalculatedvaluesforthelatticeconstantaretotheDFT/3D-RISMcansimulatetheeffectofthermalrotationofexperimentalvalues,thebetterthelinearthermalexpansiontheorganicmoleculesonthethermalpropertiesatdifferentcoefficientisinagreementbetweenthem.Thisshouldbethetemperatures.TheRISMchargedistributionsincubicMAPbI3casewherethethermalexpansionofthesystemisdominantlycalculatedbyDFT/3D-RISMwiththeRISMtemperaturedeterminedbythethermalmotionoftheorganicmoleculesfrom200to500KareshowninFigure4a.TheseresultsratherthantheanharmonicpotentialoftheinorganicindicatethattheRISMchargedistributionbroadenswithframework.increasingtemperature,reflectinganincreaseinthethermalFormaterialssuchasMAPbI3andFAPbI3,inwhichthemotionoftheMAcation.ThedependenceofthetotalenergyorganicmoleculesarenotstronglyhybridizedneartheVBMorofcubicMAPbI3onthelatticeconstantatdifferentRISMCBM,DFT/3D-RISMcanaccuratelypredictthethermophys-temperaturesisshowninFigure4b.TheresultobtainedbyicalpropertiesatlowcostwithoutusingtheMDapproach.OnDFT,whichcorrespondstothestateatT=0K,isalsoshown.theotherhand,DFT/3D-RISMcannotbeappliedtotheIneachcase,thedependenceofthetotalenergyonthelatticeOIHMswheretheinteractionsbetweentheorganicmoleculesconstantshowsadownwardconvexcurve,ofwhichthelowestandtheinorganicframeworkareverystrong(i.e.,wherethepointcorrespondstothemoststablelatticeconstantatagivenorganicmoleculesdonotrotate).Additionally,itisnot6606https://doi.org/10.1021/acs.jpcc.1c01171J.Phys.Chem.C2021,125,6601−6610

6TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticlestraightforwardtoevaluatealow-symmetrystructurebyDFT/whereΔHmixistheenthalpychangeofmixingobtainedby3D-RISMbecausetheautomaticschemeofthelatticeDFT/3D-RISM.Whenaregularsolidsolutionisassumed,relaxationisnotyetimplementedinQuantumESPRESSO.ΔSmixistheentropychangeofmixingbetweentwosolutionsTherefore,thefuturechallengewillbetocalculatethelower-obtainedbythefollowingequation:temperaturephasesoflow-symmetryMAPbI3andFAPbI3byDFT/3D-RISMwiththeappropriatelatticerelaxationsteps.ΔSkmix=−{Bxlnxxx−(1−)ln(1−})Here,onthebasisofourongoingwork,wewouldliketopointoutthatourDFT/3D-RISMapproachcanreproducethewherexistheratiooftheFAcationandkBisBoltzmann’sexperimentalresultsoftheOIHMswithmoderateinteractions.constant.InFigure5c,theenthalpytermat300KisalmostAsshowninFigureS7,thedistortionofthePbI3octahedroninzero,andthefreeenergyofmixingbecomesnegativeowingtothetetragonalMAPbI3from300to100K,duetothereducedthecontributionoftheentropyofmixing.Therefore,itis(enhanced)molecularrotation(interactions)withdecreasingsuggestedthattheMAandFAcationsareisomorphousatatemperature,hasbeensuccessfullysimulatedbyDFT/3D-finitetemperature,whichisconsistentwiththeexperimental19,20RISM.Theextendedcalculationssuchasanisotropiclatticeresults.expansion,bandgap,phononproperties,andthermodynamicDFT/3D-RISMcanbeusedtoevaluatethethermophysicalpropertieshavebeeninprogress,whichwillbereportedinpropertiesforanycompositionoforganicmoleculesatlowdetailseparatelyelsewhere.cost.Asanexample,thetemperaturedependenceoftheIntheexperiments,compositionalengineeringtechniquesbandgapofexperimentallyreportedcubicMA0.13FA0.87PbI3at66havebeendevelopedtoimprovethepowerconversionhightemperaturewascalculatedusingDFT/3D-RISMand62,63plottedinFigure5d.Forcomparison,theexperimentalefficiencyandstabilityofsolarcelldevices.However,59whencalculatingamixedmodelwithseveraltypesoforganicreportandcalculatedresultofthetemperaturedependencemoleculesbyconventionalDFT,itisnecessarytomakeofthebandgapofcubicMAPbI3arealsoshown.appropriatesupercellstoconsidertheorientationandsitesofMA0.13FA0.87PbI3hasalargerbandgapthanMAPbI3.Thethemolecules,resultinginhugecomputationalcost.InbandgapsofbothMAPbI3andMA0.13FA0.87PbI3linearlycontrast,usingDFT/3D-RISM,becausetheorganicmoleculesincreasewithincreasingtemperature.Thisisbecausethearetreatedasacontinuummodel,aunitcellmodelcanbeinteractionbetweenthePbandIatomsisweakenedbythecalculatedatanarbitrarymixingratioregardlessofthenumberthermalexpansionofthelattice.Thistrendisconsistentwith59,66oftypesofmixedmolecules.Todemonstratethis,DFT/3D-experimentalresultsoncubicperovskitestructures.RISMcalculationswereperformedoncubicMA1−xFAxPbI3,inAsdemonstrated,DFT/3D-RISMisaveryeffectivewhichtheMAandFAcationsaremixed.TheRISMcalculationmethodformixedsystemsbecausethesimulationtemperaturewassetto300K.Thechargedensitydistributionsmodelsareunitcells,notsupercells,evenifthenumberofonthe(001)planeforFAratios(x)of0.0,0.25,0.5,0.75,andtypesoforganiccationsincreases.Therefore,finally,wewould1.0areshowninFigure5a.ThechargedistributionofFAPbI3liketoshowhowmuchthecomputationalcostscanbeishigherinthecenterregionthanMAPbI3.ThisdifferencereducedbyusingDFT/3D-RISMascomparedwiththeMDoccursbecausetheFAcationhasapositivechargeattheCsimulation,whichistheconventionalDFTapproachtotakeatomlocatedatthecenterofthetwo−NH2groups(seeintoaccountthethermalmotionoforganicmoleculesatfiniteFigure5a)andislesspolarizablethantheMAcation.Becausetemperatures.ThecomputationalcostsforMA0.25FA0.75PbI3the−HCgroupsoftheFAcationarepositivelycharged,theusingMDandDFT/3D-RISMaresummarizedinTable1.On35−NH2groupsappeartoberelativelynegativelycharged.Thethebasisofapreviousstudy,weassumedthatthetotaltimenegativelychargedregionislocatedneartheIioninthechargefortheMDsimulationtoreachequilibriummotionwas40psdistribution.Theseresultsindicatethatthehydrogenatomsofwithatimestepof0.1fs,and4×4×2cubicsupercellswerethe−NH2groupsareboundtotheIions,whichisingoodusedtorealizetherandomarrangementandorientationoftheagreementwiththeresultsoftheneutrondiffractionorganicmolecules.ThePbandIatomswerefixed.DFT/3D-64experiments.ThechargedensitydistributioncontinuouslychangeswiththechangeofthemixingratioofMAandFATable1.ComputationalCostofMA0.25FA0.75PbI3Usingcations,indicatingthattheDFT/3D-RISMcalculationsConventionalDFT(MD)andDFT/3D-RISMsuccessfullytakeintoaccountthemixingeffectsofdifferenttotaltotalorganiccations.atomsCPUThedependenceofthelatticeconstantofcubicinthetotaltimeaMA1−xFAxPbI3onthemixingratioofMAandFAcationscompoundmethodcellsizecellsteps(h)bfromDFT/3D-RISMcalculationsandtheexperimentalMA0.25FA0.75PbI3DFT4×4×2384400000938265(MD)resultsareshowninFigure5b.ThemolecularradiioftheMAandFAcationsare216and253pm,respectively.SinceDFT/1×1×141213D-theFAcationislargerthanMAcation,thelatticeconstantRISMincreasesastheratiooftheFAcationincreases,whichagrees65awellwiththeexperimentalresults.ThedependenceoftheThecalculationswereperformedinthefollowingenvironment:anGibbsfreeenergyofmixingonthemixingratiooftheMAandIntel(R)Xeon(R)CPUE5-2698v3@2.30GHz,132GBmemory,bFAcationsincubicMA1−xFAxPbI3isshowninFigure5c.and16cores.ThetotalCPUtimettotalrequiredfortheMDsimulationwasdeterminedbythefollowingformula:t=tfirst+MAPbI3andFAPbI3aresetastheenergystandards.Thetotaltsecond+tave×(n−2),wheretfirstandtsecondaretheCPUtimesforGibbsfreeenergychangeofmixing(ΔGmix)wascalculatedbystepavethefirstandsecondSCFloops,respectively,tistheaverageCPUthefollowingequation:timefromthethirdlooptothe50thloop,andnstepisthetotalionicstepforMD.tfirst=317.42s,tsecond=122.47s,tave=84.44s,andnstepΔGTmix()=ΔHTTmix()−ΔSmix=400000stepswereused.6607https://doi.org/10.1021/acs.jpcc.1c01171J.Phys.Chem.C2021,125,6601−6610

7TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleRISMcancalculateaself-consistentloopofrandomorganicLaboratoryforMaterialsandStructures,TokyoInstituteofmoleculemotionsatfinitetemperaturewithoutusingaTechnology,Yokohama,Kanagawa226-8503,Japan;supercell.Thus,usingDFT/3D-RISM,thecomputationalorcid.org/0000-0001-8993-7681;costcanbereducedbyupto500timeswithoutcompromisingEmail:yokoyama.tomoyasu@jp.panasonic.comthecalculationaccuracycomparedwiththeMDsimulation.WhenDFT/3D-RISMwasincorporatedintothecasesofAuthorstheOIHMs,itwasfoundthatthecalculationcostwasSatoruOhuchi−TechnologyDivision,PanasonicsignificantlyreducedwhilemaintainingacertainlevelofCorporation,Moriguchi,Osaka570-8501,Japanaccuracy,asifthefirst-principlescalculationswereswitchedTaisukeMatsui−TechnologyDivision,Panasonicfromtheall-electrontothepseudopotentialmethods.Corporation,Moriguchi,Osaka570-8501,JapanFurthermore,thefactthatthecalculationsoftheOIHMYukihiroKaneko−TechnologyDivision,PanasonicsystemswithamixtureofdifferentorganicmoleculescouldbeCorporation,Moriguchi,Osaka570-8501,Japan;computedlikeavirtualcrystalapproximation(VCA)wasalsoorcid.org/0000-0003-0000-5840reminiscentoftheaboverelationship.Therefore,theDFT/3D-TakaoSasagawa−LaboratoryforMaterialsandStructures,RISMapproachproposedinthisstudyisexpectedtobewidelyTokyoInstituteofTechnology,Yokohama,Kanagawa226-usedforthedevelopmentofnewOIHMssuitablefor8503,Japanoptoelectronicandthermoelectricapplications.Completecontactinformationisavailableat:■https://pubs.acs.org/10.1021/acs.jpcc.1c01171CONCLUSIONInthiswork,weappliedtheDFT/3D-RISMapproach,whichNoteshasbeenusedtosimulatethesolid/liquidinterface,totheTheauthorsdeclarenocompetingfinancialinterest.OIHMsanddemonstratedthepredictionofthethermophys-icalpropertiesofcubicMAPbI3atlowcomputationalcost■comparedwithMD.BecausetheorganicmoleculesrotateACKNOWLEDGMENTSwithintheinorganicframeworkatfinitetemperature,itisWethankM.Otanifordiscussions.ThisworkwassupporteddifficulttocalculatetheOIHMsbytheconventionalDFTbyajointindustry−universityprojectbetweenPanasonicandapproachintermsofcomputationalcost.WeappliedDFT/theTokyoInstituteofTechnology[KY300121]andpartially3D-RISMtocubicMAPbI,whichisoneofthemostfamousbyaCRESTproject[JPMJCR16F2]fromtheJapanScience3OIHMs.MAcationsweretreatedasacontinuumdistributionandTechnologyAgency.(liquid)of3D-RISM,whereasPbandIatomsweretreatedasasolidframework.Wesuccessfullyreproducedtheexperimen-■REFERENCEStallyobserveddistributionoftheMAcationatafinite(1)Poglitsch,A.;Weber,D.DynamicDisorderinMethylammo-temperature.WhentheRISMtemperaturewaschanged,theniumtrihalogenoplumbates(II)ObservedbyMillimeter-WaveSpec-thermalexpansioncoefficientandtemperaturedependenceoftroscopy.J.Chem.Phys.1987,87,6373−6378.thebandgapofMAPbI3werepredicted,andtheresultswere(2)Onoda-Yamamuro,N.;Matsuo,T.;Suga,H.DielectricStudyofingoodagreementwiththeexperimentalreports.Inaddition,CH3NH3PbX3(X=Cl,Br,I).J.Phys.Chem.Solids1992,53,935−becausetheRISMtreatsdifferentcontinuummodelson939.(3)Kawamura,Y.;Mashiyama,H.;Hasebe,K.StructuralStudyonaverage,theaverageelectronicstateofacompoundcontainingCubic−TetragonalTransitionofCH3NH3PbI3.J.Phys.Soc.Jpn.2002,severaltypesoforganicmoleculescanbecalculatedbyDFT/71,1694−1697.3D-RISM.Thelatticeconstant,thebandgap,andfreeenergy(4)Stoumpos,C.C.;Malliakas,C.D.;Kanatzidis,M.G.ofmixingofMA1−xFAxPbI3,inwhichMAandFAcationsareSemiconductingTinandLeadIodidePerovskiteswithOrganicmixed,canbeevaluatedwithoutusingasupercell.ThismethodCations:PhaseTransitions,HighMobilities,andnear-InfraredmakestheevaluationofthermophysicalpropertieswithoutPhotoluminescentProperties.Inorg.Chem.2013,52,9019−9038.high-costcomputingpossible.(5)Baikie,T.;Fang,Y.;Kadro,J.M.;Schreyer,M.;Wei,F.;Mhaisalkar,S.G.;Graetzel,M.;White,T.J.SynthesisandCrystal■ChemistryoftheHybridPerovskite(CH3NH3)PbI3forSolid-StateASSOCIATEDCONTENTSensitisedSolarCellApplications.J.Mater.Chem.A2013,1,5628−*sıSupportingInformation5641.TheSupportingInformationisavailablefreeofchargeat(6)Fabini,D.H.;Siaw,T.A.;Stoumpos,C.C.;Laurita,G.;Olds,https://pubs.acs.org/doi/10.1021/acs.jpcc.1c01171.D.;Page,K.;Hu,J.G.;Kanatzidis,M.G.;Han,S.;Seshadri,R.MOLfilesoftheMAandFAcationsusedinthisworkUniversalDynamicsofMolecularReorientationinHybridLead(ZIP)IodidePerovskites.J.Am.Chem.Soc.2017,139,16875−16884.(7)Dang,Y.;Liu,Y.;Sun,Y.;Yuan,D.;Liu,X.;Lu,W.;Liu,G.;Xia,Detailedsimulationmethod;RISMchargedensityH.;Tao,X.BulkCrystalGrowthofHybridPerovskiteMaterialdistribution;electronicdensityofstates;totalenergyCH3NH3PbI3.CrystEngComm2015,17,665−670.dependences;distributionsofCandN;bandstructures(8)Kojima,A.;Teshima,K.;Shirai,Y.;Miyasaka,T.OrganometalofCsPbIandPbI−;Harmonicphonondispersionof33HalidePerovskitesasVisible-LightSensitizersforPhotovoltaicCells.PbI−;dependenceoftheinorganicframeworkinthe3J.Am.Chem.Soc.2009,131,6050−6051.tetragonalMAPbI3ontheRISMtemperature(PDF)(9)Yang,W.S.;Noh,J.H.;Jeon,N.J.;Kim,Y.C.;Ryu,S.;Seo,J.;Seok,S.Il.High-PerformancePhotovoltaicPerovskiteLayers■FabricatedthroughIntramolecularExchange.S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