Crystal Symmetry and Static Electron Correlation Greatly Accelerate Nonradiative Dynamics in Lead Halide Perovskites - Smith, Shakiba, A

Crystal Symmetry and Static Electron Correlation Greatly Accelerate Nonradiative Dynamics in Lead Halide Perovskites - Smith, Shakiba, A

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pubs.acs.org/JPCLLetterCrystalSymmetryandStaticElectronCorrelationGreatlyAccelerateNonradiativeDynamicsinLeadHalidePerovskites§§BrendanSmith,MohammadShakiba,andAlexeyV.Akimov*CiteThis:J.Phys.Chem.Lett.2021,12,2444−2453ReadOnlineACCESSMetrics&MoreArticleRecommendations*sıSupportingInformationABSTRACT:Usingarecentlydevelopedmany-bodynonadiabaticmoleculardynamics(NA-MD)frameworkforlargecondensedmattersystems,westudythephonon-drivennonradiativerelaxationofexcesselectronicexcitationenergyincubicandtetragonalphasesoftheleadhalideperovskiteCsPbI3.Wefindthatthemany-bodytreatmentoftheelectronicexcitedstatessignificantlychangesthestructureoftheexcitedstates’coupling,promotesastrongernonadiabaticcouplingofstates,andultimatelyacceleratestherelaxationdynamicsrelativetothesingle-particledescriptionofexcitedstates.Theaccelerationofthenonadiabaticdynamicscorrelateswiththedegreeofconfigurationalmixing,whichiscontrolledbythecrystalsymmetry.Thehigher-symmetrycubicphaseofCsPbI3exhibitsstrongerconfigurationmixingthandoesthetetragonalphaseandsubsequentlyyieldsfasternonradiativedynamics.Overall,usingamany-bodytreatmentofexcitedstatesandaccountingfordecoherencedynamicsareimportantforclosingthegapbetweenthecomputationallyderivedandexperimentallymeasurednonradiativeexcitationenergyrelaxationrates.Nonadiabatic(NA)moleculardynamics(MD)isaformationmaybestronglyfavored.Furthermore,modelingpromisingmethodforrevealingmechanismsandprocessesthatinvolveexcitedstates’interaction,suchas28−31characterizingthedynamicsofNAprocesses.MultipleNA-triplet−tripletannihilationandphotonupconversion,MDstudiesofbulkperovskiteshavebeenundertakentodate,32−3435,36singletfission,andexcimerformation,wouldalso12providinginsightintotheroleofcationandhalideidentity,requiresteppingbeyondthecommonlyadoptedSPapprox-34symmetrybreakingatgrainboundaries,andvacanciesinimationandextendingtheNAmethodologytotheMBdeterminingthekineticsofsuchprocesses.Otherworkshave(multiconfigurational)treatmentoftheelectronicstates.reportedNA-MDstudiesofthenonradiativehotcarrierNA-MDcalculationsthatutilizeahigh-leveldescriptionofrelaxationandelectron−holerecombinationprocessesinelectronicexcitedstates(naturallyincludingMBeffects)are56−8perovskitenanocrystals,2Dperovskites,andrelatedroutinelypossiblenowadaysdirectlyforrelativelysmallDownloadedviaUNIVOFCONNECTICUTonMay16,2021at05:33:09(UTC).heterostructuresystems.9−11Currently,NA-MDsimulations37−41molecularsystemsorviaQM/MMapproachesforlargerofcomplexsystemssuchascondensedphaseornanoscalesystems.Incontrast,theinclusionofMBeffectsintheNA-MDmaterials,includingleadhalideperovskites(LHPs),relyontheofnanoscale,periodic,andextendedmolecularsystemscanbeSeehttps://pubs.acs.org/sharingguidelinesforoptionsonhowtolegitimatelysharepublishedarticles.useofasingle-particle(SP)descriptionoftheelectronicprohibitivelyexpensive.WhileanMBdescriptionofelectronic12−18excitedstates.Withinthisdescription,theCoulombandexcitedstatesincondensedmattersystemsispossible,42−48exchangeinteractionsbetweenelectronsandholesaresuchcalculationsareextremelyexpensivefortheirroutineneglected,andtheelectronsandholesareconsideredfreeapplicationsindynamics.Anumberofworkshavereportedparticles.variousapproachestoincorporateMBeffectsintotheNA-MDWhiletheSPtreatmentoftheelectronicstateshasbeen4919inthepast.Nakaiutilizedthetime-dependentdensityshowntobereasonableundercertainconditions,itbreaks50,5120,21functionaltight-binding(TD-DFTB)approachtomodeldowninmanyothercases.Notably,forsystemspossessing52NA-MDinLHPs.Bonafehaverecentlydevelopedansymmetry,electronicstatedegeneraciesbecomeimportant,EhrenfestTD-DFTBapproachtomodelingcoupledelec-suggestingthatthetrueexcitedstatesmaybebestdescribedbythesuperpositionsofsuch(nearly)-degeneratestates.Undersuchconditions,thestaticelectroniccorrelationbecomesReceived:December24,2020importanttoinclude,becausenearlyallexcitedstates,buttheAccepted:March1,2021lowestfew,typicallycontainmultipleSPexcitations(SlaterPublished:March4,2021determinants).Amany-body(MB)descriptionoftheexcitedelectronicstatesisalsonecessaryinquantum-confined22−2526,27systemsandatlowtemperatures,whereexciton©2021AmericanChemicalSocietyhttps://dx.doi.org/10.1021/acs.jpclett.0c037992444J.Phys.Chem.Lett.2021,12,2444−2453

1TheJournalofPhysicalChemistryLetterspubs.acs.org/JPCLLetterFigure1.ThermallyaveragedprojecteddensityofstatesfortheconsideredCsPbI3systemsandschematicsofsomeoftheconsideredelectronicexcitations.(a)pDOSforthecubicphase;(b)schematicofelectronicexcitationresultingin∼0.8eVofexcesselectronicenergyforthecubicphase;(c)pDOSforthetetragonalphase;(d)schematicofelectronicexcitationresultingin∼0.4eVofexcesselectronicenergyforthetetragonalphase.50tron−nucleardynamicswithintheDFTB+packageandcontaining40and160atoms,respectively(Figure1,panelsaappliedittostudytheexcited-statedynamicsinseveralandc,insets).Thegeometryoptimization,molecularexamplesystems.TheTretiakgrouphasdevelopedthedynamics,ground-statedensityfunctionaltheory(DFT),and53,54NEXMDsoftwarepackagethatreliesonthecollectiveTD-DFTcalculationsareperformedusingtheCP2Ksoftware55,5664,65electronoscillator(CEO)approachforNA-MDmodelingpackage.Intheelectronicstructurecalculations,theinextendedorganicsystems.AnumberofschemesbasedonvalenceelectronsforallatomtypesaredescribedusingatheTD-DFTdescriptionofelectronicexcitedstateshavebeenmixedGaussianandplane-wavebasisset.Theexchangeand57−59reportedrecentlyasanaffordablewayofincorporatingcorrelationofthevalenceelectronsisdescribedbythe60,6166MBeffectsintoNA-MD,includingourownimplementa-Perdew−Burke−Ernzerhof(PBE)densityfunctional.tionusingtheneglect-of-back-reactionapproximation(NBRA)Although,thispuredensityfunctionalhasanumberofwell-62,6367−71ofNA-MDwithintheLibrasoftware.Despitetheserecentknownproblems,theuseofgenerallymorereliablehybridadvances,theuseoftheSPdescriptionofelectronicexcitedfunctionalsintheMDcalculationssuchasthoseundertakeninstatesintheNA-MDofnanoscaleandperiodicsystemsisstillthepresentworkisprohibitivelyexpensive.Weanticipatethatprevailing.Todate,littleattentionhasbeenpaidtocriticallythequalitativetrendsdiscussedinthisworkwillholdevenifassessingtheapproximateapproachesinviewofthemorethehybridfunctionalsareused,exceptforthecasesexplicitlyrigorousmethodscurrentlyavailable.Thus,anassessmentofdiscussedlater.Theeffectsofthecoreelectronsareaccounted72theroleofMBeffectsintheNA-MDofsuchextendedsystemsforusingGoedecker−Teter−Hutter(GTH)pseudopoten-isofhighimportance.tials.Theplane-wavebasisisdeterminedbythechargedensityInthiswork,wereportourstudiesofMBeffectsintheNA-cutoffof300Ry.Thedouble-ζ-valence-polarized(DZVP)73MDofperiodiccondensedmattersystemsunderconditionsbasissetisusedastheGaussianbasis.Toensurethefavoringhigh(quasi)degeneraciesoftheelectronicstates,asaccuracyoftheforcecalculations,thek-pointsamplinguses74maybethecaseforLHPs.Inparticular,wefocusonmodelingthe4×4×4and2×2×2Monkhorst−PackgridsfortheexcessexcitationenergyrelaxationintheCsPbI3LHP,whichiscubicandtetragonalsystems,respectively.Dispersioninter-knowntoexistinthecubicandtetragonalphases(amongactionsareaccountedforusingGrimme’sDFT-D3dispersion75others).Ourexpectationhereisthatthedifferenceincorrection.Eachsystemisfirstthermalizedto300KusingsymmetriesofthecrystalstructuresofthetwophasescanMDfollowedbyproductionMD.TheproductionMDaffectthedegeneraciesofelectronicstates,leadingtotrajectoriesarerunfor1.8psandaresampledusingnucleardifferencesinthemany-bodycompositionoftheexcitedintegrationtimestepsof1fs.Thermaleffectsofthebathareelectronicstatesforthetwosystems.Inthisway,weexaminedescribedbyacanonicalsamplingthroughvelocityrescaling76theroleofcrystalsymmetryontheNAdynamicsincondensed(CSVR)thermostatwithatimeconstantof200fs,aswasmattersystems.Furthermore,weassesstherolethatMBusedinapreviousstudybyUrataniandNakai,whichiswithin49effectshaveintheNA-MDbystudyingthedynamicsinthesetherangeofphononmodesforthisperovskite.ForalltwosystemsatboththeMBandSPlevels.geometryoptimizationcalculations,optimizationisperformedWeemployatomisticmodelsofthecubicandtetragonalusingtheBroyden−Fletcher−Goldfarb−Shannon(BFGS)77phasesofCsPbI3composedas2×2×2supercellsandalgorithm.Thegeometryoptimizationprocessiscontinued2445https://dx.doi.org/10.1021/acs.jpclett.0c03799J.Phys.Chem.Lett.2021,12,2444−2453

2TheJournalofPhysicalChemistryLetterspubs.acs.org/JPCLLetterFigure2.Comparisonofdensitiesofexcitedstatesandthermallyaveragednonadiabaticcouplingsforthecubic(a,c,ande)andtetragonal(b,d,andf)phasesofCsPbI3.ThethermallyaveragedNACsarecomputedattheSP(candd)andMB(eandf)levels.foreachstructureuntilthemaximumforceoneachatomexcesselectronicenergyinthetetragonalphase.Inaddition,becomeslessthan15meV/Åandthemaximumgeometrythetetragonalsystemhastwogroupsofconductionbandchangebecomeslessthan0.002Bohr.statesintheenergywindow2−4eV,whereasthecubichasForbothsystems,thethermallyaveragedelectronicbandonlyone.Thegroupofstatesnear2eVissplitinto3substatesgapsareroughly1.8eV(Figure1),whichisingoodagreementinthecubicsysteminthestaticstructure(FigureS1),butthis78,79withexperiments.ThermalaveragingisdoneoverallthefinestructureishiddeninthethermallyaveragedpDOSconfigurationssampledbytheMDtrajectories.Wefindthat(Figure1).ThetwobandsofthetetragonalsystemdonotaveragingyieldsaconvergedpDOS(FigureS1).Consideringshowanynotablefinestructureinthestaticcalculations.Thethelevelofelectronicstructurecalculationsusedinthiswork,pDOSstructurecanberationalizedbythesystemsymmetries:suchagoodagreementofthebandgapvaluesfortheseLHPsthe3-foldsplittingofasinglebandinthecubicstructurecanlikelystemsfromtheknownerrorcancellationthatoccursbeattributedtothe3-foldsymmetryofthesystem,whereasthewhenusingpurefunctionalswithoutspin−orbitcouplingpresenceofthetwonotablysplitbandsinthetetragonal(SOC)effects.Theprojecteddensityofstates(pDOS)systemcanbeattributedtoanotableanisotropyofitscrystalcalculationsrevealthatforbothsystems,thevalencebandsstructure,withatleast2distinctdirections(e.g.,cvsaorb).areprimarilycomposedofatomicorbitalsoftheiodineatoms,WecomparepropertiesrelevantforNA-MDcalculationswhereastheconductionbandsarecomposedprimarilyofleadcomputedattheSPandMBlevels(Figure2).Aswasshown82,83orbitalswithasmallerfractionofiodineorbitals(Figure1),inearlier,NACsbetweendistinctSlaterdeterminants(SDs)80,81agreementwithprevioustheoreticalworks.ThemaincanbereducedtotheNACsbetweenorbitals.Forthisreason,differencesbetweenthecubicandtetragonalphasesisthethebasisofsingleSDexcitationsisconsideredaSPincreasedpDOSinthelatter,whichisaconsequenceofadescription.Incontrast,theMBelectronicstatesaredescribedlargersizeofthecell.Thus,onemayexpectfasterrelaxationofbysuperpositionsoftheSDexcitations.Becausethedensities2446https://dx.doi.org/10.1021/acs.jpclett.0c03799J.Phys.Chem.Lett.2021,12,2444−2453

3TheJournalofPhysicalChemistryLetterspubs.acs.org/JPCLLetterofexcitedstatesareratherhigh,weconsideredonlyfiniteTakentogether,weexpectthatboththeshiftoftheNACenergywindowsofexcitedstatestocompute:roughly0.9eVprobabilitydensitytowardlargervaluesandthechangedforthecubicand0.5eVforthetetragonalphases.AttheMBstructureoftheNACmatrixshouldacceleratetheexcited-statelevel,151and76excitedstatesfitintotheseenergyspansfordynamics(e.g.,excited-staterelaxation)whencomputedatthethecubicandtetragonalphases,respectively.TheseMBstatesMBlevelrelativetodynamicsattheSPlevel.areformedinthebasisof229and118uniqueSDsfortheItisillustrativetodiscusstheoriginofthedifferenceinthecubicandtetragonalsystems,respectively.However,someofNACmagnitudescomputedattheSPandMBlevels.Ateverytheseexcitationsareoutsideoftheenergywindowsconsideredtimeinstant,theMBstates,{Ψi},aregivenbyaunitaryandthereforemaybeexcludedfromtheSP-onlymodelingandtransformation(U)oftheSPexcitations,{Φi}:Ψi=∑jUjiΦj.82TheNACsbetweentheMBstatesaregivenbydMB=calculations.Followingtheearlierapproximationcommonlyij84−93usedinmanySP-basedNA-MDstudies,theenergiesofΨΨ∂=∑UU*ΦΦ∂+∑UU*⟨Φ|Φ⟩∂=theSPexcitedstatesareestimatedviathedifferencesoforbitali∂tjab,iabja∂tbab,iaab∂tbjSP∂energies,neglectingtheCoulombandexchangeintegrals.∑UdU*+∑UU*.Here,weutilizedtheortho-ab,iaabbjaia∂tajSomewhatsurprisingly,thedensitiesoftheexcitedstatesnormalizationoftheSPstates,⟨Φi|Φj⟩=δij.UnderthespecialcomputedatbothSPandMBlevelsforeachsystemarenearlycaseofthetime-independenttransformationmatrixU,thesame(Figure2,panelsaandb).Thissimilarityindicates∂U=0,onecanshowthattheaveragemagnitudeofthethattheexcitoniceffects(staticcorrelationandCoulombic∂tinteractionofelectron−holepairs)arerelativelysmall.ThisNACsinthetwobasesareequal,asforinstancecouldbequantifiedby∑,|d|2.UsingthefactthatthematrixUisaresultisconsistentwithexperimentalstudiesreportingsmallijijexcitonbindingenergiesinLHPs.94,95Smallexcitoniceffectsinunitarytransformation,onecanshowthatthepresentlystudiedsystemsarealsoexpectedbecauseofthe22∑||=dd∑||.Thus,theaveragemagnitudeij,M∈{B}ijij,S∈{P}ijlackofquantumconfinementononehandandtheuseofaofthecouplingwouldnotdependonwhethertheSPorMBpuredensityfunctionalontheother.AssuggestedbyIzmaylov9697descriptionofexcitedstatesisused.However,inmostandScuseria,andasalsofollowsfromotherstudies,situations,thetransformationmatrixUistime-dependent,capturingexcitoniceffectsinTD-DFTcalculationsrequiresthebecauseofthetime-dependenceoftheHamiltonianviaitsfunctionalwiththecorrectasymptoticbehavioroftheparametricdependenceonnucleartrajectories.Assuch,theexchange,suchasachievedviatheuseofhybridfunctionals,∂especiallythelong-rangecorrectedones.However,suchterm∑aUUia*∂tajcannotbeneglected.Itisthistermthatiscalculationsareprohibitivelyexpensive,andweleavethisresponsibleforthedifferenceintheaverageNACmagnitudesquestionanopenproblem.(asquantifiedbythecentersofgravityintheprobabilityGiventhesimilarityoftheDOSintheMBandSPbases,densitydistributionsshowninFigureS2).Inotherwords,theonemayexpectthattheNACsintheMBandSPexcitationtime-dependenceoftheMBstatescompositionintermsofthebaseswouldbecomparable.However,adetailedanalysisofthecorrespondingSPstatesdeterminesthedifferenceinaverageNACsbetweenthepairsofMBandSPstatesbreaksthisNACs.Havingsaidthat,eventheconditionexpectation.Thefirstdistinctioncomesinthestructureofthe∑||=dd2∑||2doesnotimplyasimilarityofij,M∈{B}ijij,S∈{P}ijNACmatrices.AttheSPexcitationlevel,thetime-averagedthedynamicscomputedinthetwobases.TherelativeNACsbetweenelectronicstateshaveascattered-likemagnitudeofthecouplingsbetween“equivalent”states(ifappearance(Figure2,panelscandd).SuchastructurearisessuchamappingoftheSPtoMBbasiscanbemade)maybebecauseinthespaceofSPstatesoftypeHOMO−N→changed,suchthatsomechannelsofthedynamicsmaybeLUMO+M,withvaryingNandM,andorderedbyenergy,thefavoredinonebasisovertheother.Finally,althoughourcorrespondingSDsmaydifferbymorethanoneelectroncurrentcalculationssuggestfasterdynamicsintheMBbasis,excitation,leadingtozerocouplingbetweensuchpairsofthereisnoreasontoexpectthistobeageneraltrend.Instates.Ontheotherhand,theMBelectronicexcitedstatesare∂composedofmultipleSPtransitions,andtwoMBstatesmayprinciple,thereisnolimitationfortheterm∑aUUia*∂tajtotakebecomecoupledviathecouplingoftheunderlyingSDs.Asavaluesoppositeinsigntothoseofthe∑U*dSPUterm,thusa,biaabbjresult,theNACmatrixhasamore“filled-in”structurewhendecreasingtheeffectiveNACsintheMBbasisasopposedtocomputedinthebasisofMBexcitedstates(Figure2,panelsethoseintheSPbasis.andf).Furthermore,wefindthattheprobabilitydistributionsComparingthecrystalsymmetries,wefindthatNACsareoftheNACsbetweenMBstatesisshiftedtowardlargervalueslargerinthecubicsystemthaninthetetragonalone(Figures2comparedtotheprobabilitydistributionsoftheNACsandS2).ThisdifferencecanberationalizedbythelargerbetweenSPstates(FigureS2).Thismeansthatoneismoredegreeofconfigurationalmixinginthecubicsystem.WelikelytoencounterlargermagnitudesofNACsduringthequantifythedegreeofconfigurationalmixingbythesquaredcourseofthedynamicsifMBeffectsareaccountedfor.Theamplitudesoftheconfigurationinteractioncoefficientsofprobabilitytofindnear-zeroNACsishigherattheSPleveldominantSDsenteringthecompositionoftheMBstatesthanattheMBlevel.Forbothcubicandtetragonalphases,the(FiguresS3andS4).OuranalysisshowsthatthefirstfewprobabilitytofindNACvaluesgreaterthanca.0.2meVinexcitedstatesaremainlySPinbothsystemsattheiroptimizedabsolutevalueislargerintheMBbasisthantheSPbasisgeometries,whichvalidatesthewidelyusedSPapproximation(FigureS2,panelsaandd).TheprobabilityoffindingNACsinmodelingNAprocessessuchaselectron−holerecombina-85,98,99withtheabsolutevaluesof5−50meVissmallforbothtion.Inthetetragonalsystem,electronicstatesretainasystems,whichisordersofmagnitudesmallerthantostrongSPcharacterformanyofthelow-lyingelectronicstatesencounterNACsintherange0−0.5meV.However,forall(FiguresS4),whereasinthecubicsystem,allbutthelowestvaluesofNACmagnitudes,theprobabilitydensityisexcitedstatesexhibitsignificantconfigurationalmixing(FigureconsistentlylargerattheMBlevelthatitisattheSPlevel.S3).WeattributesuchapronouncedmixingoftheSPstatesto2447https://dx.doi.org/10.1021/acs.jpclett.0c03799J.Phys.Chem.Lett.2021,12,2444−2453

4TheJournalofPhysicalChemistryLetterspubs.acs.org/JPCLLetterFigure3.ExcesselectronicenergyrelaxationdynamicscomputedwiththeFSSHmethodologyincubicCsPbI3:(a)attheSPleveland(b)attheMBlevel.theincreasedsymmetrypresentinthecubicsystem.HighTheMBeffectsontheNAdynamicscanbebestseenbysymmetryleadstohighdegeneracyofelectronicstatesinthecomparingtheexcitationenergyexcessdecaykineticsatthespaceofSPtransitionsandpromotestheirmixingintheMBFSSHlevelforbothsystems.WhenMBeffectsareaccountedpicture.Atlowertemperatures,whenthermallyinducedatomicfor,therelaxationofexcessexcitationenergyisacceleratedbymotionisreduced,thesymmetryofthecrystalstructureisthefactorof2.6(Figure3).Moreover,theinclusionofMBbetterpreserved.Thisexplainsthestrongerconfigurationaleffectsqualitativelychangesthedynamics.AttheMBlevel,itismixingpresentattheoptimizedgeometries(0K,FigureS3,typicallythecasethattheexponentialcomponentineq1ispanelsaandb;FigureS4,panelsaandb)comparedtothesmallerthanitisintheSPbasis(e.g.,seeTableS1ofthethermallysampledsetofconfigurationsat300K(FigureS3,SupportingInformation),whichsignifiesthatcoherentpanelscandd;FigureS4,panelscandd).At300K,thedynamicsisprominentintheMBbasis.ThecoherentatomicmotionbreaksthesymmetryandreducesthedegreeofdynamicsintheMBbasisisfavoredbythemoreextensiveconfigurationalmixing.However,theinfluenceofthesystems’couplingofallstatestoeachotherascomparedtotheSPsymmetryisstillpresent.Forthecubicsystem,significantpicture.AttheSPlevel,thedynamicsexhibitslittletonodecayconfigurationalmixingoftheelectronicstatesisstillpresentatforthefirst100fsandisfollowedbyaslowdecay(Figure3a).300K,especiallyforhigher-energyexcitations.IntheSPbasis,thedecaykineticsisdominatedbytheTodirectlyassesstheroleofMBeffectsontheNAexponentialcomponentineq1(e.g.,seeTableS1ofthedynamics,weconductexplicitNA-MDcalculationsusingtheSupportingInformation),whichsignifiesthatthecoherentfewestswitchessurfacehopping(FSSH),100Belyaev−Leb-dynamicsissuppressedorintrinsicallyslowerintheSPbasis.edev−Landau−Zener(BLLZ),101,102andseveraldecoherenceThissuppressionofthecoherentdynamicscanbeexplainedbycorrectionmethodologies,103−105asalsodetailedinthesectionlargerenergygapsbetweenthecoupledstates.Inthisregard,4oftheSupportingInformation.WeemployarecentlyoneshouldnotbemisleadbytheapparentlysimilardensitiesdevelopedinterfaceoftheLibrasoftwareforNA-MDofexcitedstatesintheSPandMBbases(Figure2,panelsacalculations62,63andtheCP2K64code.Thedetailsofourandb).Althoughthedensitiesaresimilar,theydonotreflectthestructureofthecouplingofthestatestheenergeticallycomputationalsetupsaresummarizedinsection5ofthenearbySPstatesmaybeuncoupled,whereastheaverageSupportingInformation.FurtherdetailsofourNA-MD106energygapsbetweenthecoupledstateswouldbelargerinthisframeworkarediscussedelsewhere.Thedynamicsofexcessbasis.electronicexcitationenergyrelaxationfortheconsideredWecomputethetimescalesofexcessexcitationenergysystemsisquantifiedbyfittingtheaverageexcesselectronicdecayinbothcubicandtetragonalphasesofCsPbI3usingbothexcitationenergyrelaxationdynamicsoverallNA-MDtheSPandMBdescriptionofexcitedstatesandseveralNA-trajectoriestothefollowingfunctionalform:MDmethodologies(Table1,alsoseesection6oftheiiyyiiy2ySupportingInformationformoredetails).OurmainjjjjjjtzzzzzzjjjjjjtzzzzzzobservationisthatthedynamicswiththeMBeffectsaccountedf(;tE0)=−+−−Aexpjjjjzzzz(EA0)expjjjjjzzzzzkkττ1{{jkk2{z{(1)forisgenerallyfasterthanitisinthebasisofSPstates.ThisconclusionisconsistentwiththechangesoftheNACmatrixstructurediscussedabove.TheinclusionoftheMBeffectsSuchafittingfunctionhasbeenusedinpastNA-MDstudiestoacceleratesthedynamicsmoreinthecubicsystemthanitdoescharacterizethedecayofexcessexcitationenergyincondensed1,13,107inthetetragonalsystem:thetimescalesaredecreasedbythematterandnanoscalesystems.Thisformaccountsforfactorof1.4−2.6inthecubicsystemandbyafactorofonlytheGaussiandecaykineticstypicalforcoherentdynamicsof1.6inthetetragonaloneincomparisontothecorrespondingelectronsindensemanifoldsofexcitedstatesintheshort-timeSP-basedtimescales.SuchtrendsarealsoconsistentwithrangeandtheexponentialdecaykineticstypicalforincoherentslightlylargerNACsinthecubicsystemthaninthetetragonal,dynamicsatthelongertimescalesorinthesparsemanifoldsofasaconsequenceofthecrystalsymmetries(e.g.,seeFiguresexcitedstates.TheoverallrelaxationtimescaleisthenS2−S4).computedaccordingtoAsexpected,accountingforelectronicdecoherence(viaID-104103,105AandmSDMmethods)leadstoslowingtheA()EA0−τ=+ττ12dynamicsdownrelativetoFSSH.Ofthetwodecoherence-E0E0(2)correctedTSHschemestested,theID-Atypicallyyieldsa2448https://dx.doi.org/10.1021/acs.jpclett.0c03799J.Phys.Chem.Lett.2021,12,2444−2453

5TheJournalofPhysicalChemistryLetterspubs.acs.org/JPCLLetterTable1.Excited-StateEnergyDecayTimeConstants,τ(fs),accelerationisnotablylargerthantheapproximately2.6-foldComputedUsingVariousSurfaceHoppingMethodsataccelerationbecauseofweakexcitoniceffectsseenintheEithertheSPorMBDescriptionoftheElectronicExcitedpresentwork,althoughthetwovaluescannotbecomparedaStatesdirectlyastheyareobtainedfordistinctsystems.AnanalysisofTable1showsthatthecomputedenergy108476,Bretschneideretal.,109,110relaxationtimescalesmaybeinagoodagreementwiththeexperiment,1.4eV,0.8eV∼1000Shenetal.reportedexperimentaldatadependingonthecombinationofcubicFSSHID-AmSDMBLLZtheexcitedstates’descriptionlevelandtheTSHmethodologyMB0.8eV48012091187486used.SuchagreementsanddisagreementsshouldbetakenSP0.8eV122617572492444criticallybecausetheymaybeduetofortuitouserrorMB0.4eV67911201645984cancellationsorthelackofknowneffectsthataretooSP0.4eV111613231858889expensivetoinclude,respectively.AttheMBlevel,theFSSHtetragonalFSSHID-AmSDMBLLZcalculationsforthecubicsystemareinexcellentagreementMB0.4eV1002161718231865withtheexperimentaltimescalesofapproximately476fsofSP0.4eV1655242430901843Bretschneideretal.108AttheSPlevel,thecomputedtimeaE0(eV)istheinitialexcesselectronicexcitationenergyusedinthescalesarenearlytwiceasslow:1.1−1.2psforarangeofinitialcalculationsetupandthefittingfunction,eq1.excitationenergylevels.Thesetimescalesareconsistentwiththevaluesreportedinapreviouscomputationalstudythat1fasterdynamicsthanmSDM.IncontrasttoalltheNAC-basedreliedonasimilarSPdescriptionofexcitedstates.Incontrast,109,110TSHmethodsused(FSSH,ID-A,andmSDM),theenergy-Shenetal.reporttimescalesintherangeof1−30psbasedBLLZmethodpredictshighlysimilardynamicsofthedependingonthecarrierdensity,withasubpicosecond(0.8−excitedstatesatboththeMBandSPlevels.MBeffectscan1.0ps)rangeforlowcarrierdensitiesstudiedintheirinfluencethedynamicsintwoways.Oneisviathewaveexperiment.OnemaythinkthattheFSSHattheSPlevelfunctions(andhencetheNACs),viathemixingofexcitedyieldsareasonableagreementwiththeexperiment.However,SDs,asdiscussedabove,whichwerefertoasweakexcitonictheFSSHdoesnotaccountfordecoherenceeffectspresentineffects.Theotherisviatheenergiesoftheexcitonicstates,realisticsystems.Thus,oneneedstoshiftattentiontotheID-Awhichwerefertoasstrongexcitoniceffects.AsdiscussedandmSDMresults.AttheMBlevel,thedynamicscomputedpreviously,thestrongexcitoniceffectswouldmanifestusingtheseschemescomesintocloseragreementwiththeca.1109,110themselvesinquantum-confinedsystemsandrequiretheusepstimescalesofShenetal.TheSPdescriptionwouldofdensityfunctionalswiththeproperasymptoticbehaviorofoverestimatethetimescalesbyapproximatelyafactorof2theexchangeterms97(pragmaticallyspeaking,theuseofrange-comparedtothedataofShenetal.andbyafactorof4−5correctedhybridfunctionals).Inotherwords,thestrongcomparedtothedataofBretschneideretal.Thus,theexcitoniceffectswouldmanifestthemselvesviaanotableinclusionofevenweakexcitoniceffectsiscriticalformergingexcitonbindingenergy.Forthesystemsconsideredinthisthegapbetweencomputedandexperimentallymeasuredtimework,thedensitiesoftheMBandSPexcitedstatesagreewithscalesofexcitationenergyrelaxation.eachother(Figure2,panelsaandb),suggestingtheexcitonTheabovediscussionshouldbetakencritically.OurbindingenergiesaresmallandtheSPpictureworksasfarassimulationsdonotexplicitlyincludeSOCeffects,whichhave111theenergiesoftheelectronicexcitedstatesareconcerned.ThisbeenshowntosignificantlyacceleratedynamicsinLHPs.Atobservationagreeswiththe10meVvalueforcubicCsPbI3thesametime,weuseapuredensityfunctionalinsteadofthereportedbythepriorstudies.94,95Thisenergyissmallerthancomputationallymoreexpensivehybridfunctionals,andithasthermalenergyatroomtemperature,sotheexcitoniceffectsbeendemonstratedbeforethattheuseofhybridfunctionals68arenegligible.Thesimilarityofthedensitiesofexcitedstatesatmayslowdownthedynamics.ItispossiblethatthetwotheSPandMBlevelsexplainstheinsensitivityoftheenergy-approximationsmaycountereachother’seffect,althoughwebasedBLLZtothelevelofthedescriptionofexcited-statecannottelltowhatextent.Nonetheless,asshowninthiswork,energies.Thisbehaviormayberegardedasaconsequenceofweakexcitoniceffectsmayleadtoanotableaccelerationofthethelackofstrongexcitoniceffectsorinabilitytocaptureit,dynamics.ThiseffectislikelytoholdregardlessofwhatwhichcanbeaconsequenceofthelackofquantumfunctionalisusedandwhethertheSOCeffectsareincluded,confinement(3Dsystems)andtheuseofpuredensityalthoughfurtherstudiesonthismatterwouldbehighlyfunctional.Withthe“strongexcitoniceffects”ruledoutinthedesirable.presentstudy,weconcludethataccelerationoftheNA-MDweAnotherpotentiallimitationofthecurrentmethodology,asobserveinperiodicLHPscanbeattributedtothe“weak”wellasofothersimilartechniques,concernsthetreatmentofexcitoniceffects,thatis,themixingofthequasi-degenerateelectronandphononk-points.Ononehand,includingalargerstates,whichinturncanbeaffectedbyasystem’ssymmetry.numberofk-points(or,equivalently,usingalargersupercell)Weanticipatethoughthatapplyingthecurrentapproachtocouldincreasethedensityofelectronicstatesandaccelerate2Dor0Dperovskites/systemsand/orusinghybridfunctionalsthedecaydynamics.Ontheotherhand,describingthe(whichisprohibitivelyexpensiveatthispoint)maychangethetransitionsbetweenthek-pointsrequiresaccountingforcomputedtimescalesandthequalitativecomparisonofthephononquantization(q-points),whichwouldrequireatimescalescomputedwiththeBLLZapproach.Inthisregard,differentcomputationalmethodologythatisnotavailablein60recentworkbyLiuetal.demonstratedthataccountingforthecurrentscheme.Suchcalculationsmayalsoinvolvean112strongexcitoniceffectsinquantum-confinedsystemsliketheextremelylargenumberofk-points,whichisprohibitivelyMoS2/WS2heterojunctionmayleadtouptoa10-foldexpensivefortheatomisticsystemsconsideredhere(especiallyaccelerationofthedynamicscomparedtothecommonlyusedattheTD-DFTlevel).Atthesametime,enablingrelaxationSPKohn−Sham-DFTprescriptionofexcitedstates.Suchanchannelsthatincludemultiplek-pointsmayslowthedecay2449https://dx.doi.org/10.1021/acs.jpclett.0c03799J.Phys.Chem.Lett.2021,12,2444−2453

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