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》由會(huì)員上傳分享，免費(fèi)在線閱讀，更多相關(guān)內(nèi)容在學(xué)術(shù)論文-天天文庫(kù)。

pubs.acs.org/JPCLLetterCrystalSymmetryandStaticElectronCorrelationGreatlyAccelerateNonradiativeDynamicsinLeadHalidePerovskites§§BrendanSmith,MohammadShakiba,andAlexeyV.Akimov*CiteThis:J.Phys.Chem.Lett.2021,12,2444?2453ReadOnlineACCESSMetrics&MoreArticleRecommendations*s?SupportingInformationABSTRACT:Usingarecentlydevelopedmany-bodynonadiabaticmoleculardynamics(NA-MD)frameworkforlargecondensedmattersystems,westudythephonon-drivennonradiativerelaxationofexcesselectronicexcitationenergyincubicandtetragonalphasesoftheleadhalideperovskiteCsPbI3.We?ndthatthemany-bodytreatmentoftheelectronicexcitedstatessigni?cantlychangesthestructureoftheexcitedstates’coupling,promotesastrongernonadiabaticcouplingofstates,andultimatelyacceleratestherelaxationdynamicsrelativetothesingle-particledescriptionofexcitedstates.Theaccelerationofthenonadiabaticdynamicscorrelateswiththedegreeofcon?gurationalmixing,whichiscontrolledbythecrystalsymmetry.Thehigher-symmetrycubicphaseofCsPbI3exhibitsstrongercon?gurationmixingthandoesthetetragonalphaseandsubsequentlyyieldsfasternonradiativedynamics.Overall,usingamany-bodytreatmentofexcitedstatesandaccountingfordecoherencedynamicsareimportantforclosingthegapbetweenthecomputationallyderivedandexperimentallymeasurednonradiativeexcitationenergyrelaxationrates.Nonadiabatic(NA)moleculardynamics(MD)isaformationmaybestronglyfavored.Furthermore,modelingpromisingmethodforrevealingmechanismsandprocessesthatinvolveexcitedstates’interaction,suchas28?31characterizingthedynamicsofNAprocesses.MultipleNA-triplet?tripletannihilationandphotonupconversion,MDstudiesofbulkperovskiteshavebeenundertakentodate,32?3435,36singlet?ssion,andexcimerformation,wouldalso12providinginsightintotheroleofcationandhalideidentity,requiresteppingbeyondthecommonlyadoptedSPapprox-34symmetrybreakingatgrainboundaries,andvacanciesinimationandextendingtheNAmethodologytotheMBdeterminingthekineticsofsuchprocesses.Otherworkshave(multicon?gurational)treatmentoftheelectronicstates.reportedNA-MDstudiesofthenonradiativehotcarrierNA-MDcalculationsthatutilizeahigh-leveldescriptionofrelaxationandelectron?holerecombinationprocessesinelectronicexcitedstates(naturallyincludingMBe?ects)are56?8perovskitenanocrystals,2Dperovskites,andrelatedroutinelypossiblenowadaysdirectlyforrelativelysmallDownloadedviaUNIVOFCONNECTICUTonMay16,2021at05:33:09(UTC).heterostructuresystems.9?11Currently,NA-MDsimulations37?41molecularsystemsorviaQM/MMapproachesforlargerofcomplexsystemssuchascondensedphaseornanoscalesystems.Incontrast,theinclusionofMBe?ectsintheNA-MDmaterials,includingleadhalideperovskites(LHPs),relyontheofnanoscale,periodic,andextendedmolecularsystemscanbeSeehttps://pubs.acs.org/sharingguidelinesforoptionsonhowtolegitimatelysharepublishedarticles.useofasingle-particle(SP)descriptionoftheelectronicprohibitivelyexpensive.WhileanMBdescriptionofelectronic12?18excitedstates.Withinthisdescription,theCoulombandexcitedstatesincondensedmattersystemsispossible,42?48exchangeinteractionsbetweenelectronsandholesaresuchcalculationsareextremelyexpensivefortheirroutineneglected,andtheelectronsandholesareconsideredfreeapplicationsindynamics.Anumberofworkshavereportedparticles.variousapproachestoincorporateMBe?ectsintotheNA-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-con?ned22?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?59reportedrecentlyasana?ordablewayofincorporatingcorrelationofthevalenceelectronsisdescribedbythe60,6166MBe?ectsintoNA-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.Thee?ectsofthecoreelectronsareaccounted72theroleofMBe?ectsintheNA-MDofsuchextendedsystemsforusingGoedecker?Teter?Hutter(GTH)pseudopoten-isofhighimportance.tials.Theplane-wavebasisisdeterminedbythechargedensityInthiswork,wereportourstudiesofMBe?ectsintheNA-cuto?of300Ry.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).Ourexpectationhereisthatthedi?erenceincorrection.Eachsystemis?rstthermalizedto300KusingsymmetriesofthecrystalstructuresofthetwophasescanMDfollowedbyproductionMD.TheproductionMDa?ectthedegeneraciesofelectronicstates,leadingtotrajectoriesarerunfor1.8psandaresampledusingnucleardi?erencesinthemany-bodycompositionoftheexcitedintegrationtimestepsof1fs.Thermale?ectsofthebathareelectronicstatesforthetwosystems.Inthisway,weexaminedescribedbyacanonicalsamplingthroughvelocityrescaling76theroleofcrystalsymmetryontheNAdynamicsincondensed(CSVR)thermostatwithatimeconstantof200fs,aswasmattersystems.Furthermore,weassesstherolethatMBusedinapreviousstudybyUrataniandNakai,whichiswithin49e?ectshaveintheNA-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/?andthemaximumgeometrythetetragonalsystemhastwogroupsofconductionbandchangebecomeslessthan0.002Bohr.statesintheenergywindow2?4eV,whereasthecubichasForbothsystems,thethermallyaveragedelectronicbandonlyone.Thegroupofstatesnear2eVissplitinto3substatesgapsareroughly1.8eV(Figure1),whichisingoodagreementinthecubicsysteminthestaticstructure(FigureS1),butthis78,79withexperiments.Thermalaveragingisdoneoverallthe?nestructureishiddeninthethermallyaveragedpDOScon?gurationssampledbytheMDtrajectories.We?ndthat(Figure1).ThetwobandsofthetetragonalsystemdonotaveragingyieldsaconvergedpDOS(FigureS1).Consideringshowanynotable?nestructureinthestaticcalculations.Thethelevelofelectronicstructurecalculationsusedinthiswork,pDOSstructurecanberationalizedbythesystemsymmetries:suchagoodagreementofthebandgapvaluesfortheseLHPsthe3-foldsplittingofasinglebandinthecubicstructurecanlikelystemsfromtheknownerrorcancellationthatoccursbeattributedtothe3-foldsymmetryofthesystem,whereasthewhenusingpurefunctionalswithoutspin?orbitcouplingpresenceofthetwonotablysplitbandsinthetetragonal(SOC)e?ects.Theprojecteddensityofstates(pDOS)systemcanbeattributedtoanotableanisotropyofitscrystalcalculationsrevealthatforbothsystems,thevalencebandsstructure,withatleast2distinctdirections(e.g.,cvsaorb).areprimarilycomposedofatomicorbitalsoftheiodineatoms,WecomparepropertiesrelevantforNA-MDcalculationswhereastheconductionbandsarecomposedprimarilyofleadcomputedattheSPandMBlevels(Figure2).Aswasshown82,83orbitalswithasmallerfractionofiodineorbitals(Figure1),inearlier,NACsbetweendistinctSlaterdeterminants(SDs)80,81agreementwithprevioustheoreticalworks.ThemaincanbereducedtotheNACsbetweenorbitals.Forthisreason,di?erencesbetweenthecubicandtetragonalphasesisthethebasisofsingleSDexcitationsisconsideredaSPincreasedpDOSinthelatter,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,weconsideredonly?niteTakentogether,weexpectthatboththeshiftoftheNACenergywindowsofexcitedstatestocompute:roughly0.9eVprobabilitydensitytowardlargervaluesandthechangedforthecubicand0.5eVforthetetragonalphases.AttheMBstructureoftheNACmatrixshouldacceleratetheexcited-statelevel,151and76excitedstates?tintotheseenergyspansfordynamics(e.g.,excited-staterelaxation)whencomputedatthethecubicandtetragonalphases,respectively.TheseMBstatesMBlevelrelativetodynamicsattheSPlevel.areformedinthebasisof229and118uniqueSDsfortheItisillustrativetodiscusstheoriginofthedi?erenceinthecubicandtetragonalsystems,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??=theSPexcitedstatesareestimatedviathedi?erencesoforbitali?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,onecanshowthattheaveragemagnitudeofthethattheexcitonice?ects(staticcorrelationandCoulombic?tinteractionofelectron?holepairs)arerelativelysmall.ThisNACsinthetwobasesareequal,asforinstancecouldbequanti?edby∑,|d|2.UsingthefactthatthematrixUisaresultisconsistentwithexperimentalstudiesreportingsmallijijexcitonbindingenergiesinLHPs.94,95Smallexcitonice?ectsinunitarytransformation,onecanshowthatthepresentlystudiedsystemsarealsoexpectedbecauseofthe22∑||=dd∑||.Thus,theaveragemagnitudeij,M∈{B}ijij,S∈{P}ijlackofquantumcon?nementononehandandtheuseofaofthecouplingwouldnotdependonwhethertheSPorMBpuredensityfunctionalontheother.AssuggestedbyIzmaylov9697descriptionofexcitedstatesisused.However,inmostandScuseria,andasalsofollowsfromotherstudies,situations,thetransformationmatrixUistime-dependent,capturingexcitonice?ectsinTD-DFTcalculationsrequiresthebecauseofthetime-dependenceoftheHamiltonianviaitsfunctionalwiththecorrectasymptoticbehavioroftheparametricdependenceonnucleartrajectories.Assuch,theexchange,suchasachievedviatheuseofhybridfunctionals,?especiallythelong-rangecorrectedones.However,suchterm∑aUUia*?tajcannotbeneglected.Itisthistermthatiscalculationsareprohibitivelyexpensive,andweleavethisresponsibleforthedi?erenceintheaverageNACmagnitudesquestionanopenproblem.(asquanti?edbythecentersofgravityintheprobabilityGiventhesimilarityoftheDOSintheMBandSPbases,densitydistributionsshowninFigureS2).Inotherwords,theonemayexpectthattheNACsintheMBandSPexcitationtime-dependenceoftheMBstatescompositionintermsofthebaseswouldbecomparable.However,adetailedanalysisofthecorrespondingSPstatesdeterminesthedi?erenceinaverageNACsbetweenthepairsofMBandSPstatesbreaksthisNACs.Havingsaidthat,eventheconditionexpectation.The?rstdistinctioncomesinthestructureofthe∑||=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,althoughourcorrespondingSDsmaydi?erbymorethanoneelectroncurrentcalculationssuggestfasterdynamicsintheMBbasis,excitation,leadingtozerocouplingbetweensuchpairsofthereisnoreasontoexpectthistobeageneraltrend.Instates.Ontheotherhand,theMBelectronicexcitedstatesare?composedofmultipleSPtransitions,andtwoMBstatesmayprinciple,thereisnolimitationfortheterm∑aUUia*?tajtotakebecomecoupledviathecouplingoftheunderlyingSDs.Asavaluesoppositeinsigntothoseofthe∑U*dSPUterm,thusa,biaabbjresult,theNACmatrixhasamore“?lled-in”structurewhendecreasingthee?ectiveNACsintheMBbasisasopposedtocomputedinthebasisofMBexcitedstates(Figure2,panelsethoseintheSPbasis.andf).Furthermore,we?ndthattheprobabilitydistributionsComparingthecrystalsymmetries,we?ndthatNACsareoftheNACsbetweenMBstatesisshiftedtowardlargervalueslargerinthecubicsystemthaninthetetragonalone(Figures2comparedtotheprobabilitydistributionsoftheNACsandS2).Thisdi?erencecanberationalizedbythelargerbetweenSPstates(FigureS2).Thismeansthatoneismoredegreeofcon?gurationalmixinginthecubicsystem.WelikelytoencounterlargermagnitudesofNACsduringthequantifythedegreeofcon?gurationalmixingbythesquaredcourseofthedynamicsifMBe?ectsareaccountedfor.Theamplitudesofthecon?gurationinteractioncoe?cientsofprobabilityto?ndnear-zeroNACsishigherattheSPleveldominantSDsenteringthecompositionoftheMBstatesthanattheMBlevel.Forbothcubicandtetragonalphases,the(FiguresS3andS4).Ouranalysisshowsthatthe?rstfewprobabilityto?ndNACvaluesgreaterthanca.0.2meVinexcitedstatesaremainlySPinbothsystemsattheiroptimizedabsolutevalueislargerintheMBbasisthantheSPbasisgeometries,whichvalidatesthewidelyusedSPapproximation(FigureS2,panelsaandd).Theprobabilityof?ndingNACsinmodelingNAprocessessuchaselectron?holerecombina-85,98,99withtheabsolutevaluesof5?50meVissmallforbothtion.Inthetetragonalsystem,electronicstatesretainasystems,whichisordersofmagnitudesmallerthantostrongSPcharacterformanyofthelow-lyingelectronicstatesencounterNACsintherange0?0.5meV.However,forall(FiguresS4),whereasinthecubicsystem,allbutthelowestvaluesofNACmagnitudes,theprobabilitydensityisexcitedstatesexhibitsigni?cantcon?gurationalmixing(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.HighTheMBe?ectsontheNAdynamicscanbebestseenbysymmetryleadstohighdegeneracyofelectronicstatesinthecomparingtheexcitationenergyexcessdecaykineticsatthespaceofSPtransitionsandpromotestheirmixingintheMBFSSHlevelforbothsystems.WhenMBe?ectsareaccountedpicture.Atlowertemperatures,whenthermallyinducedatomicfor,therelaxationofexcessexcitationenergyisacceleratedbymotionisreduced,thesymmetryofthecrystalstructureisthefactorof2.6(Figure3).Moreover,theinclusionofMBbetterpreserved.Thisexplainsthestrongercon?gurationale?ectsqualitativelychangesthedynamics.AttheMBlevel,itismixingpresentattheoptimizedgeometries(0K,FigureS3,typicallythecasethattheexponentialcomponentineq1ispanelsaandb;FigureS4,panelsaandb)comparedtothesmallerthanitisintheSPbasis(e.g.,seeTableS1ofthethermallysampledsetofcon?gurationsat300K(FigureS3,SupportingInformation),whichsigni?esthatcoherentpanelscandd;FigureS4,panelscandd).At300K,thedynamicsisprominentintheMBbasis.ThecoherentatomicmotionbreaksthesymmetryandreducesthedegreeofdynamicsintheMBbasisisfavoredbythemoreextensivecon?gurationalmixing.However,thein?uenceofthesystems’couplingofallstatestoeachotherascomparedtotheSPsymmetryisstillpresent.Forthecubicsystem,signi?cantpicture.AttheSPlevel,thedynamicsexhibitslittletonodecaycon?gurationalmixingoftheelectronicstatesisstillpresentatforthe?rst100fsandisfollowedbyaslowdecay(Figure3a).300K,especiallyforhigher-energyexcitations.IntheSPbasis,thedecaykineticsisdominatedbytheTodirectlyassesstheroleofMBe?ectsontheNAexponentialcomponentineq1(e.g.,seeTableS1ofthedynamics,weconductexplicitNA-MDcalculationsusingtheSupportingInformation),whichsigni?esthatthecoherentfewestswitchessurfacehopping(FSSH),100Belyaev?Leb-dynamicsissuppressedorintrinsicallyslowerintheSPbasis.edev?Landau?Zener(BLLZ),101,102andseveraldecoherenceThissuppressionofthecoherentdynamicscanbeexplainedbycorrectionmethodologies,103?105asalsodetailedinthesectionlargerenergygapsbetweenthecoupledstates.Inthisregard,4oftheSupportingInformation.WeemployarecentlyoneshouldnotbemisleadbytheapparentlysimilardensitiesdevelopedinterfaceoftheLibrasoftwareforNA-MDofexcitedstatesintheSPandMBbases(Figure2,panelsacalculations62,63andtheCP2K64code.Thedetailsofourandb).Althoughthedensitiesaresimilar,theydonotre?ectthestructureofthecouplingofthestatestheenergeticallycomputationalsetupsaresummarizedinsection5ofthenearbySPstatesmaybeuncoupled,whereastheaverageSupportingInformation.FurtherdetailsofourNA-MD106energygapsbetweenthecoupledstateswouldbelargerinthisframeworkarediscussedelsewhere.Thedynamicsofexcessbasis.electronicexcitationenergyrelaxationfortheconsideredWecomputethetimescalesofexcessexcitationenergysystemsisquanti?edby?ttingtheaverageexcesselectronicdecayinbothcubicandtetragonalphasesofCsPbI3usingbothexcitationenergyrelaxationdynamicsoverallNA-MDtheSPandMBdescriptionofexcitedstatesandseveralNA-trajectoriestothefollowingfunctionalform:MDmethodologies(Table1,alsoseesection6oftheiiyyiiy2ySupportingInformationformoredetails).OurmainjjjjjjtzzzzzzjjjjjjtzzzzzzobservationisthatthedynamicswiththeMBe?ectsaccountedf(;tE0)=?+??Aexpjjjjzzzz(EA0)expjjjjjzzzzzkkττ1{{jkk2{z{(1)forisgenerallyfasterthanitisinthebasisofSPstates.ThisconclusionisconsistentwiththechangesoftheNACmatrixstructurediscussedabove.TheinclusionoftheMBe?ectsSucha?ttingfunctionhasbeenusedinpastNA-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-foldComputedUsingVariousSurfaceHoppingMethodsataccelerationbecauseofweakexcitonice?ectsseenintheEithertheSPorMBDescriptionoftheElectronicExcitedpresentwork,althoughthetwovaluescannotbecomparedaStatesdirectlyastheyareobtainedfordistinctsystems.AnanalysisofTable1showsthatthecomputedenergy108476,Bretschneideretal.,109,110relaxationtimescalesmaybeinagoodagreementwiththeexperiment,1.4eV,0.8eV～1000Shenetal.reportedexperimentaldatadependingonthecombinationofcubicFSSHID-AmSDMBLLZtheexcitedstates’descriptionlevelandtheTSHmethodologyMB0.8eV48012091187486used.SuchagreementsanddisagreementsshouldbetakenSP0.8eV122617572492444criticallybecausetheymaybeduetofortuitouserrorMB0.4eV67911201645984cancellationsorthelackofknowne?ectsthataretooSP0.4eV111613231858889expensivetoinclude,respectively.AttheMBlevel,theFSSHtetragonalFSSHID-AmSDMBLLZcalculationsforthecubicsystemareinexcellentagreementMB0.4eV1002161718231865withtheexperimentaltimescalesofapproximately476fsofSP0.4eV1655242430901843Bretschneideretal.108AttheSPlevel,thecomputedtimeaE0(eV)istheinitialexcesselectronicexcitationenergyusedinthescalesarenearlytwiceasslow:1.1?1.2psforarangeofinitialcalculationsetupandthe?ttingfunction,eq1.excitationenergylevels.Thesetimescalesareconsistentwiththevaluesreportedinapreviouscomputationalstudythat1fasterdynamicsthanmSDM.IncontrasttoalltheNAC-basedreliedonasimilarSPdescriptionofexcitedstates.Incontrast,109,110TSHmethodsused(FSSH,ID-A,andmSDM),theenergy-Shenetal.reporttimescalesintherangeof1?30psbasedBLLZmethodpredictshighlysimilardynamicsofthedependingonthecarrierdensity,withasubpicosecond(0.8?excitedstatesatboththeMBandSPlevels.MBe?ectscan1.0ps)rangeforlowcarrierdensitiesstudiedintheirin?uencethedynamicsintwoways.Oneisviathewaveexperiment.OnemaythinkthattheFSSHattheSPlevelfunctions(andhencetheNACs),viathemixingofexcitedyieldsareasonableagreementwiththeexperiment.However,SDs,asdiscussedabove,whichwerefertoasweakexcitonictheFSSHdoesnotaccountfordecoherencee?ectspresentine?ects.Theotherisviatheenergiesoftheexcitonicstates,realisticsystems.Thus,oneneedstoshiftattentiontotheID-Awhichwerefertoasstrongexcitonice?ects.AsdiscussedandmSDMresults.AttheMBlevel,thedynamicscomputedpreviously,thestrongexcitonice?ectswouldmanifestusingtheseschemescomesintocloseragreementwiththeca.1109,110themselvesinquantum-con?nedsystemsandrequiretheusepstimescalesofShenetal.TheSPdescriptionwouldofdensityfunctionalswiththeproperasymptoticbehaviorofoverestimatethetimescalesbyapproximatelyafactorof2theexchangeterms97(pragmaticallyspeaking,theuseofrange-comparedtothedataofShenetal.andbyafactorof4?5correctedhybridfunctionals).Inotherwords,thestrongcomparedtothedataofBretschneideretal.Thus,theexcitonice?ectswouldmanifestthemselvesviaanotableinclusionofevenweakexcitonice?ectsiscriticalformergingexcitonbindingenergy.Forthesystemsconsideredinthisthegapbetweencomputedandexperimentallymeasuredtimework,thedensitiesoftheMBandSPexcitedstatesagreewithscalesofexcitationenergyrelaxation.eachother(Figure2,panelsaandb),suggestingtheexcitonTheabovediscussionshouldbetakencritically.OurbindingenergiesaresmallandtheSPpictureworksasfarassimulationsdonotexplicitlyincludeSOCe?ects,whichhave111theenergiesoftheelectronicexcitedstatesareconcerned.Thisbeenshowntosigni?cantlyacceleratedynamicsinLHPs.Atobservationagreeswiththe10meVvalueforcubicCsPbI3thesametime,weuseapuredensityfunctionalinsteadofthereportedbythepriorstudies.94,95Thisenergyissmallerthancomputationallymoreexpensivehybridfunctionals,andithasthermalenergyatroomtemperature,sotheexcitonice?ectsbeendemonstratedbeforethattheuseofhybridfunctionals68arenegligible.Thesimilarityofthedensitiesofexcitedstatesatmayslowdownthedynamics.ItispossiblethatthetwotheSPandMBlevelsexplainstheinsensitivityoftheenergy-approximationsmaycountereachother’se?ect,althoughwebasedBLLZtothelevelofthedescriptionofexcited-statecannottelltowhatextent.Nonetheless,asshowninthiswork,energies.Thisbehaviormayberegardedasaconsequenceofweakexcitonice?ectsmayleadtoanotableaccelerationofthethelackofstrongexcitonice?ectsorinabilitytocaptureit,dynamics.Thise?ectislikelytoholdregardlessofwhatwhichcanbeaconsequenceofthelackofquantumfunctionalisusedandwhethertheSOCe?ectsareincluded,con?nement(3Dsystems)andtheuseofpuredensityalthoughfurtherstudiesonthismatterwouldbehighlyfunctional.Withthe“strongexcitonice?ects”ruledoutinthedesirable.presentstudy,weconcludethataccelerationoftheNA-MDweAnotherpotentiallimitationofthecurrentmethodology,asobserveinperiodicLHPscanbeattributedtothe“weak”wellasofothersimilartechniques,concernsthetreatmentofexcitonice?ects,thatis,themixingofthequasi-degenerateelectronandphononk-points.Ononehand,includingalargerstates,whichinturncanbea?ectedbyasystem’ssymmetry.numberofk-points(or,equivalently,usingalargersupercell)Weanticipatethoughthatapplyingthecurrentapproachtocouldincreasethedensityofelectronicstatesandaccelerate2Dor0Dperovskites/systemsand/orusinghybridfunctionalsthedecaydynamics.Ontheotherhand,describingthe(whichisprohibitivelyexpensiveatthispoint)maychangethetransitionsbetweenthek-pointsrequiresaccountingforcomputedtimescalesandthequalitativecomparisonofthephononquantization(q-points),whichwouldrequireatimescalescomputedwiththeBLLZapproach.Inthisregard,di?erentcomputationalmethodologythatisnotavailablein60recentworkbyLiuetal.demonstratedthataccountingforthecurrentscheme.Suchcalculationsmayalsoinvolvean112strongexcitonice?ectsinquantum-con?nedsystemsliketheextremelylargenumberofk-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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