Accurate and systematically improvable density functional theory embed - PDF document

Accurate and systematically improvable density functional theory embedding forcorrelated wavefunctions Citation: 140, 18A507 (2014); doi: 10.1063/1.4864040 View online: View Table of Contents: Published by the Articles you may be interested in J. Chem. Phys. 137, 224113 (2012); 10.1063/1.4770226 J. Chem. Phys. 136, 044105 (2012); 10.1063/1.3678180 J. Chem. Phys. 132, 164115 (2010); 10.1063/1.3380834 J. Chem. Phys. 127, 214302 (2007); 10.1063/1.2814157 J. Chem. Phys. 125, 014105 (2006); 10.1063/1.2209688 THEJOURNALOFCHEMICALPHYSICS,18A507(2014)AccurateandsystematicallyimprovabledensityfunctionaltheoryembeddingforcorrelatedwavefunctionsJasonD.Goodpaster,TaylorA.Barnes,FrederickR.Manby,andThomasF.MillerIIIDivisionofChemistryandChemicalEngineering,CaliforniaInstituteofTechnology,Pasadena,California91125,USA Electronicmail:fred.manby@bristol.ac.uk.Presentaddress:CaliforniaInstituteofTechnology,Pasadena,California,USA.Electronicmail:tfm@caltech.eduoftheinteractionbetweensubsystemsisalsohandledusingDFTtheory,whichintroduceserrorsintoboththeembeddingpotentialoftheWFTsubsystem,andthenonadditiveenergybetweensubsystems.WeanalyzeWFT-in-DFTembeddingbydecomposingtheerrorintothesethreecontributions,andusetheresultstosuggestfurtherimprovementstoprojector-basedembedding.Theanalysisisperformedthroughcarefulcom-parisonwithlocalcoupled-clustercalculations. 18A507-2Goodpasteretal.J.Chem.Phys.,18A507(2014),whichcorrespondtosubsystemsAandB,respectively.Thesetwosetsoforbitalsareusedtoformthedensityma-tricesofsubsystemsAandBintheatomicorbitalbasis,Next,thesubsystemFockmatrixisformedfortheem-beddingcalculation,suchthatAinBnBA,B]+g[˜A],(1)wheretheembeddedcoreHamiltonianisAinBnBA,B]=h+g[A+B]Šg[A]+µPB.(2)Here,histhestandardone-electroncoreHamiltonian,cludesallthetwo-electronterms,isaprojectionoperator,isalevel-shiftparameter;˜isthedensitymatrixas-sociatedwiththeMOeigenstatesof.Theprojectionoperatorisgivenbylabeltheatomicorbitalbasisfunctions.Inthelimitof,theMOsinareconstrainedtobemu-tuallyorthogonalwiththeMOsofsubsystemB;ifinad-ditionthesamedensityfunctionalisusedforallcalculations,theMOscoincidewiththeoriginalorbitalsAself-consistenteldoptimization,usingtheFockma-,isperformedtoobtain˜,andthenalDFT-in-DFTenergyisisA;A,B]=EDFT[˜A]+EDFT[B]+EnadDFT[A,B]+tr[(˜AinBnBA,B]Šh)],(4)whereEDFTisthestandardDFTenergy(evaluatedwithcore-)andandA,B]isthenonadditiveenergybetweenthesubsystemdensities.Thelasttermisarst-ordercorrectiontothedifferencebetweenbetweenA,B]andandA,B].31InthelimitofandtheDFT-in-DFTembeddingenergyisidenticaltotheenergyfromthecorrespondingKScalculationperformedoverthefullsystem;asaresult,theprojector-basedapproachisnu-mericallyexactforDFT-in-DFTembeddingcalculations.practice,alargenitevalueofisused,andanadditionalperturbativecorrectiontotheenergycanbeperformed;forappropriatevaluesofthiscorrectionistypicallyfarsmallerthantheenergydifferencesdiscussedinthispaperandisthusneglectedthroughout.Furthermore,ashasbeenpreviouslyemphasized,thisembeddingschemeisexactforanyself-consistenteldmethod,suchasHartree-Fock(HF)theory.Thenonadditivecontributiontotheenergy,,A,B],canbedecomposedintoelectrostaticandexchange-correlationcontributions A,B]=Exc[A+B]ŠExc[A]ŠExc[B].(7)Theelectrostaticterm,,iseasilyevaluated,andalthoughtheexactformofisunknown,approxi-matefunctionalsarewellestablished.Sincetheembed-dedMOsareorthogonaltothoseinsubsystemB,thereisnononadditivityinthekineticenergy.Thisremovestherequirementofperformingoptimizedeffectivepotentialorusingapproximatenonadditivekineticenergyfunctionals.Theprojector-basedformalismeasilyallowsforWFT-in-DFTembedding,inwhichsubsystemAistreatedusingaWFT-leveldescriptionandsubsystemBisdescribedattheDFTlevel.Thisisachievedbyreplacingthestandardone-electroncoreHamiltonianwiththeembeddedcoreHamilto-nianofEq..TheelectronicenergyfromtheWFT-in-DFTembeddingapproachisisA;A,B]=A|ˆHAinBnBA,B]|AŠtr[A(hAinBnBA,B]Šh)]+EDFT[B]+EnadDFT[A,B],(8)where|AistheembeddedwavefunctionfromtheWFTmethod,andAinBnBA,B]istheWFTHamiltonianresultingfromreplacingthestandardone-electroncoreHamiltonianwiththeembeddedcoreHamiltonian.Thetermtr[˜AinBnBA,B]Šh)]isin-cludedinthersttermofEq.andthusdoesnotshowupintherst-ordercorrectionterm,asitdidinEq.III.RESULTSI:SOURCESOFERRORINWFT-IN-DFTEMBEDDINGA.Term-by-termcomparisonwithLCSSD(T)WenowformulateanapproachtocomparetheindividualtermsintheenergyexpressionofaCCSD(T)-in-DFTembed-dingcalculationwiththecorrespondingvaluescalculatedattheCCSD(T)level.Todothis,werstrecognizethatthelo-cal(L)CCSD(T)methodbySchützandWernerexactlyequivalenttothecanonicalCCSD(T)methodwhenallorbitalpairsarecorrelatedandallexcitationdomainsaresettothefullvirtualbasis.ThetermsintheLCCSD(T)energyexpression,inturn,canbeorganizedinawaythatenablesdirectcomparisontothetermsintheCCSD(T)-in-DFTem-beddingenergyexpression.TheLCCSD(T)energycanbedecomposedasafunctionoftheamplitudesandtheatomic-orbitaldensitymatricesas 18A507-3Goodpasteretal.J.Chem.Phys.,18A507(2014)istheHFenergyandandA,B]isthesameasEq.,exceptwiththecorrespondingexchangetermsre-e-A,B].Whenthefullvirtualspaceisincluded,thesinglesareadditiveandthusthereisnononadditivecom-ponent.Thenonadditivecorrelationforthedouble-excitationtermsissimplyandlikewiseforthetriple-excitationcorrelationenergy.ThecorrelationenergyfromthesingleexcitationswithinsubsystemAisgivenbywherethesummationspanstheoccupiedorbitalsofsubsys-temA,istheinternal-externalpartoftheFockmatrixinvectorform,andarethesingleexcitationamplitudesinvec-torform.ThecorrelationenergyfromthedoubleexcitationswithinsubsystemAisgivenbyi,jjLijCij],(12)wherethesummationspanstheoccupiedorbitalsofsub-systemA,andaretheinternalcoulombandexchangematrices.Thematrixelementsofaregivenby,wherearethedoubleandsingleexcitationamplitudes,respectively.Finally,thecorrelationenergyfromthetripleexcitationswithinsubsystemAisgivenbyi,j,krstrijkrstrstsijkrstrsttijkrstrstijkrstijkrstwheretherstsummationspanstheoccupiedorbitalsofsub-systemA,theindicesrepresentoccupiedorbitals,andtheindicesrepresentunoccupiedorbitals.isanelementoftheoverlapmatrixoftheprojectedatomicor-bitals,()aretwo-electronintegrals,andarethesingleijkrstisdenedasijkrstijkrstijkrtsijktsrijksrtijktrsijkstrijkrstarethetriplesamplitudes.Thetensorelementijkrstcontainsthedouble-excitationam-B.CalculationdetailsAllgeometryoptimizationsareperformedusingandareprovidedinthesupplementaryAllothercalculationsareperformedinMolproInallcalculationstheorbitalsarelocalizedusingPipek-Mezeylocalization.TheatomsassociatedwithsubsystemAforeachreactionaregiveninthesupplementaryAnylocalizedorbitalwithaLöwdinchargeof0.4onanatomassociatedwithsubsystemAisincludedinthesetoforbitalsassociatedwithsubsystemA.Allcalculationsemployalevelshiftparameter,whichissetto10a.u.AllKS-DFTcalculationsemployalargegridfortheexchange-correlationfunctionalevaluation,achievedbyspecifyingtheMolprooptionGRID.Forcom-putationalefciency,allLCCSD(T)calculationsemploydensitytting(DF),andthetriplesareapproximatedusingthenoniterative(T0)procedure.WeemphasizethattheT0procedureisonlyusedforcalculationsinvolving,andwhichariseinourerroranalysis;thisnoniterativeprocedureisnotusedforanycalculationsoutsideofSec.IIICToenabletherigorouscomparisonoftermsfromtheLCCSD(T)calculationandtheembeddingcalculation,somecaremustbetaken.First,allorbitalpairsarecorrelatedtorecovertheenergyfromcanonicalCCSD(T).Second,thechoiceoforbitalsmustbeconsistentbetweentheLCCSD(T)andCCSD(T)-in-DFTembeddingcalculations.IntheWFT-in-DFTembeddingmethod,subsystemBcomprisesKSMOs,andthusevaluationoftheerrorsresult-ingfromusingtheDFTenergyofsubsystemBrequirestheuseofKSMOsasthereferenceMOs.Thedifferencebe-tweencanonicalCCSD(T)usingtheHFreferenceandDF-LCCSD(T0)usingtheKSreferenceiswithin0.3mforallreactionsdiscussedinSec.IIIC,whichissmallerthantheothersourcesoferrorthatareanalyzed;therefore,through-outSec.IIIC,wewillsimplyrefertotermscalculatedfromDF-LCCSD(T0)asCCSD(T).Likewise,consistentevaluationoftheerrorarisingfromtheembeddingpotentialrequiresthatthereferenceMOsfortheembeddedCCSD(T)calculationonsubsystemAbeob-tainedfromthecorrespondingDFTcalculation.ThischoiceofreferenceMOsisonlyusedinSec.IIIC.Inallothersec-tions,thereferenceMOsoftheembeddedCCSD(T)calcula-tionarechosentobethesetofMOsresultingfromanem-beddedHFcalculation.WenotethatthedifferencebetweenCCSD(T)-in-DFTembeddingwheretheMOsforsubsystemAareobtainedfromanembeddedDFTcalculationcomparedtoanembeddedHFcalculationiswithin0.2mforthere-actionsconsideredinSec.IIICBelowweanalyzethecontributionstotheembeddinger-rorforasetofsixenergiesassociatedwithdifferentreac-tions.Allofthechosenreactionsarenotonlylargeenoughtoinvolvepartitioningacrossacovalentbond,butalsosmallenoughtoallowforcalculationoftheCCSD(T)referenceen-ergyforthefullsystem.ThereactionsconsideredaregiveninTableThedatasetconsistsofthefollowingreactions:(1)ac-tivationenergyforthesymmetricS2reactionofClpropylchloride;(2)acidhydrolysisofdimethylethertoformmethanol;(3)deprotonationofthephenolhydroxylgroup;(4)ring-closingisomerizationof3-methylene-1-heptenetoformbutylcylobutane;(5)theDiels-Alderreactionof2-methoxy-1,3-butadienewithmethylvinylketone;and(6)theactivationenergyfortheDiels-Alderreaction.Thegeometriesareprovidedinthesupplementarymaterial. 18A507-4Goodpasteretal.J.Chem.Phys.,18A507(2014)TABLEI.CCSD(T)reactionenergiesandbarriersinthetestsetobtainedusingcc-pVTZwithaug-cc-pV(Td)ZonCl,andaug-cc-pVTZforallatomsforreactions2–4.Foreaseoferroranalysis,weadoptasignconventioninwhichallreactionsoractivationprocessesarepositiveinenergy.Geome-trieswereobtainedusingB3LYPwith6-311G*(reaction1),def2-TZVP(reactions2–4),or6-31G*(reactions5,6). Eh 2activationbarrier7.82Acidhydrolysis177.83Phenoldeprotonation568.84Ringclosing10.65Diels-Alderreaction63.16Diels-Alderbarrier34.0 C.SourcesoferrorinWFT-in-DFTembedding1.ErrorfromtheembeddingpotentialNowwediscusshowcomparisonoftermsintheenergyexpressionsforCCSD(T)andCCSD(T)-in-DFTembeddingcanbeusedtodeterminetheerrorarisingfromtheembeddingpotential.TheenergyofsubsystemAfromtheCCSD(T)cal-culationisthesumoftheHFenergy(usingtheKSdensity)andthecorrelationenergyofsubsystemA,,A]+EA(S)+EA(D)+EA(T).(14)ThetotalenergyofsubsystemAfromaCCSD(T)-in-DFTembeddingcalculationisAinBnBA,B]|AŠtr[A(hAinBnBA,B]Šh)].(15)ForanembeddingpotentialthatincludesalloftheCCSD(T)many-bodyeffects,theenergyofwouldbeidentical;therefore,theerrorarisingfromtheembeddingpotentialiscalculatedasTheerrorinthereactionenergiesarisingfromtheem-beddingpotentialisthereforethechangeinproductsandreactants,ThebluesquaresinFigureshowthevalueofforthedataset,comparedtothetotalCCSD(T)-in-B3LYPembeddingerrorshownintheblackcircles.Fornosystemistheerrorlargerthan1.5m,withtheaverageerrorbeing0.8m.Thisdemonstratesakeyinsightofthispaper,whichisthattheembeddingpotentialcalculatedusingWFT-in-DFTembeddingisveryaccurate.2.ErrorfromuseofDFTforsubsystemBNext,wequantifytheWFT-in-DFTembeddingerrorre-sultingfromtreatingsubsystemBusingDFT.ThiserrorisobtainedbycomputingcomputingB]ŠEHF[B]+EB(S)+EB(D)+EB(T),(17)whichallowsforadirectcomparisonoftheDFTandCCSD(T)energiesofsubsystemB. 2 3 4 5 6 -5 0 5 10 15 ReactionError (mEh) FIG.1.Theerrorarisingfromtheembeddingpotential(bluesquares),theDFTenergyofsubsystemB(violettriangles),andthenonadditiveexchange-correlationenergy(greendiamonds)comparedtothetotalCCSD(T)-in-B3LYPembeddingerror(blackcircles).CCSD(T)calculationsperformedonthefullsystemareusedasthereference.Thelargestsourceoferroristhenonadditiveexchange-correlationenergyfunctional.ThevaluescalculatedforareshowninFigureasviolettriangles.Thelargesterrorinthisdatasetis2.5mandtheaverageerroris1.5m.Theseerrorsarelargerthanthoseresultingfromtheembeddingpotential,butarestillrel-ativelysmallcomparedtothetotalWFT-in-DFTembeddingerror.Therefore,forthisdataset,DFTdoesanadequatejobdescribingtheenergychangelocalizedwithintheenviron-mentandisnotthedominatesourceoferror.3.Errorfromthenonadditiveexchange-correlationenergyFinally,weanalyzetheerrorthatarisesfromevaluationofthenonadditiveexchange-correlationenergywithanap-proximatefunctional.TheerrorisobtainedbycomputingcomputingA,B]ŠEnadHF[A,B]+Enad(D)corr+Enad(T)corr,(18)whichallowsforthedirectcomparisonoftheapproximatedensityfunctionaltotheenergyobtainedattheCCSD(T)level.ThevaluesforaregiveninFigureasgreendiamonds.ThistermdominatestheWFT-in-DFTembeddingerror,withthelargestvalueofbeing14.2mandtheaveragevaluebeing7.2m.ItisthusthistermthatisresponsibleforintroducingthelargesterrorintheWFT-in-DFTembeddingmethodology.Thesumof,andallofthediscrepancybetweentheCCSD(T)-in-DFTcalcula-tionsandtheCCSD(T)calculationsperformedoverthefullsystem.DuetotheuseofdensityttingandthenoniterativetriplesapproximationusedintheCCSD(T)calculation,thesumoftheseerrorsisoffbyanaverageof0.4mtothetotalCCSD(T)-in-B3LYPembeddingerror;thismakesnodifferenceintheinterpretationofthedata.Toconrmthatourresultsarenotsensitivetotheap-proximateexchange-correlationfunctional,werepeatedtheanalysisusingbothPBEandM06(notshown).Theseconclusionsarerobustwithrespecttotheapproximate 18A507-5Goodpasteretal.J.Chem.Phys.,18A507(2014)exchange-correlationfunctional.Thenonadditiveexchange-correlationenergyremainsthelargestsourceoferror,fol-lowedbytheDFTenergyofsubsystemB.Again,wendthatDFT,forallofthefunctionalstested,providesveryaccurateembeddingpotentials.D.Improvementofthenonadditiveexchange-correlationenergyHavingdeterminedthenonadditiveexchange-correlationenergytobethedominatesourceoferror,newalgorithmscanbeproposedtocalculatethistermmoreaccurately.Oneapproachwouldbetoevaluatethenonadditiveexchangeex-actlyandtouseacomputationallycheapWFTmethod,suchasMP2,toevaluatethenonadditivecorrelation.There-sultingcorrectiontotheWFT-in-DFTembeddingenergyisisA,B]Štr ˜AHFŠA(hAinBnBA,B]Šh),(19)where˜istheHFembeddeddensityofsubsystemA,istheMP2amplitude,andaretheexchangetwoelectronintegrals.FortheMP2calculation,theorbitalsareused,whichallowsforthedirectcalculationoftheMP2correlationbetweentheHForbitalsforAandtheKSorbitalsofB.comparestheCCSD(T)-in-B3LYPembeddingerror(black)totheMP2-correctedCCSD(T)-in-B3LYPem-beddingerror(red).TheaverageerrorofWFT-in-DFTem-beddingis4.6m,whichdropsto1.2mwhentheMP2correctionisapplied.Alternatively,insteadofcalculatingthefullMP2energyinEq.,onecouldonlycalculatethescaledoppositespin(SOS)-MP2correlationenergy.ingtheoppositespinMP2correlationbytheusualempiricalfactorof1.3leadstotheSOS-MP2-correctedCCSD(T)-in-B3LYPembeddingerrorshowninblueinFigure.Apply-ingtheSOS-MP2correctionresultsinanaverageerrorof1.1m,whichisessentiallythesameerrorasthatofthefull 0 2 4 6 8 10 12 1 2 3 4 5 6 Energy (mEh)Reaction FIG.2.BargraphoftheerrorintheenergyobtainedfromCCSD(T)-in-B3LYPembedding(black),MP2-correctedCCSD(T)-in-B3LYPembedding(red),andSOS-MP2-correctedCCSD(T)-in-B3LYPembedding.CCSD(T)calculationsperformedonthefullsystemareusedasthereference.MP2correction,andonlyrequirescomputationsthatscalecomparedtoforthefullMP2energy.TheaverageerrorofstandardMP2calculationsonthesesystemsis6.3mrelativetoCCSD(T);itisthusclearthateffectivenessoftheMP2correctiondoesnotrelyontheMP2energybeingparticularlyaccurateforthedescriptionofthefullsystem.Instead,weobservethatMP2theoryaccuratelyrepresentsthecorrelationenergybetweensubsystemsAandB,whilenotnecessarilyrepresentingothercorrelationtermsaccurately.Thisisconsistentwithotherlocalcoupled-clustermethodsthattreatdistantpairsattheMP2level.IV.RESULTSII:CONTINUITY,CONVERGENCE,ANDCONJUGATIONINWFT-IN-DFTEMBEDDINGA.PotentialenergysurfacesNext,weexaminethepotentialenergysurfaceforhet-erolyticbondcleavage.Localcorrelationmethodsshowdis-continuitiesinthepotentialenergysurfacefortheheterolyticbondcleavageofCOdissociationinketene.Here,westudyarelatedsystem,COdissociationin1-penten-1-one.showspotentialenergycurvescalculatedusingCCSD(T),B3LYP,andCCSD(T)-in-B3LYPembed-ding.Thecc-pVDZbasiswasusedforallcalculations.Here,B3LYPperformsrelativelywellnearequilibrium,butover-estimatestheenergybyupto16mneardissociation.TheCCSD(T)-in-B3LYPcalculationsareveryaccuratenear (b)(c)(d) FIG.3.(a)PotentialenergycurvesforthedissociationoftheC-Cbondinsinglet1-penten-1-oneobtainedusingCCSD(T)(green),KS-DFTwithB3LYP(blue),andCCSD(T)-in-B3LYPembedding(black).ThestructurewasreoptimizedattheHF/cc-pVDZleveloftheoryforeachvalueoftheC-Cbonddistance.TheOCH–moietywastreatedattheCCSD(T)levelfortheCCSD(T)-in-B3LYPembeddingcalculations.(b)–(d)Theer-rorinCCSD(T)-in-B3LYPembedding(black)andMP2-correctedCCSD(T)-in-B3LYPembedding(red)asafunctionofdistancebetweenthecarbon-carbondoublebond.Theresultsareshownforthreepartitioningsofthemolecule,withsubsystemAcorrespondingto(b)CH–(18electrons),(c)OCH–(22electrons),or(d)O–(30electrons). 18A507-6Goodpasteretal.J.Chem.Phys.,18A507(2014)equilibriumandslightlyunderestimatetheenergynearthedissociationlimit.MP2-correctedCCSD(T)-in-B3LYPwerealsoperformedforthissystem;theresultsarenotshowninpanelAofFigure,becausetheyaregraphicallyin-distinguishablefromtheuncorrectedCCSD(T)-in-B3LYPshowtheerrorinCCSD(T)-in-B3LYPembeddingandMP2-correctedCCSD(T)-in-B3LYPem-beddingforthreedifferentsubsystempartitioningsofthemolecule.Theerrorandthechangeoftheslopeatthederiva-tivediscontinuityaround1.5ÅdecreasesbytreatingmoreofthesystemattheCCSD(T)level.Energydiscontinuitiesof50areseenatshortdistances,asshowninFigureofthesupplementarymaterial.Likeotherlocalcorrelationmethods,abruptchangesinthelocalizedorbitalsfordifferentnuclearcongurationsleadtodiscontinuitiesintheWFT-in-DFTembeddingenergyanditsderivatives.Here,thesede-fectsaresmallandcanbesystematicallycontrolledbyin-creasingthesizeofsubsystemA.B.WFT-in-DFTembeddingofconjugatedsystemsAdemandingcaseforanyembeddingmethodologyisthepartitioningofa-conjugatedsystem.TheapplicabilityofWFT-in-DFTembeddingtotreatsuchsystemsistestedandcomparedtosystemswithoutconjugation.First,weconsiderthedissociationofauorideanionfrombothanalkanechain(1-uorodecane)andanalkenechain(1-uoro-1,3,5,7,9-decapentaene).ThegeometriesforbothcompoundsandtheirdissociatedproductswereobtainedusingB3LYP/def2-TZVP.AllCCSD(T)andembeddingcal-culationswereperformedusingthecc-pVDZbasis,withaug-cc-pVDZforuorine.showstheCCSD(T)-in-B3LYPwithandwithouttheMP2correctionforuorideaniondissociationfromthealkanechain.Resultsareprovidedforanumberofdifferentchoicesofthesubsystempartitioning,andtheerrorofbothmethodscanbeseentorapidlyvanishasmoreatomsareincludedintheWFTsubsystem.TheindividualsourcesoferrorintheCCSD(T)-in-B3LYPembeddingcalculations,computedinthesamewayasinSec.IIIC,areshowninFigure.Again,itisobservedthattheerrorarisingfromtheembeddingpotentialissmall,accountingforonlyasmallportionofthetotalerror.Unlikepreviousresults,theerrorarisingfromtreatingsubsystemBattheDFTlevelisofsimilarmagnitudeasthenonadditiveexchange-correlationenergyerror.Astheseerrorsareofop-positesign,evaluatingthenonadditiveexchange-correlationenergyusingDFTleadstoafavorablecancelationoferror.TheMP2correctiononlyincreasestheaccuracyofthesub-systeminteractionenergy,andcannotbeexpectedtocorrectlargeerrorsassociatedwiththeDFTenergyofsubsystemB.showstheMullikenpopulationofthedensityassociatedwithsubsystemBontheatomsassociatedwithsubsystemA.Inthedissociatedproduct,thedensityasso-ciatedwithsubsystemBdistributesontotheatomsofsub-systemAtostabilizethepositivecharge.Wendthatwhenthedifferenceofthisquantityislargebetweentwocong- 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9 B charge on ANumber of Carbons in Subsystem A(a)(b)(c) -10 0 10 20 Error (mEh) -10 -5 0 5 Error (mEh) FIG.4.(a)TheerrorinCCSD(T)-in-B3LYPembedding(blackopencircles)andMP2-correctedCCSD(T)-in-B3LYPembedding(redlledcircles)asafunctionofthenumberofcarbonsincludedinsubsystemAforthedisso-ciationofthealkane.TheB3LYPenergyisgivenbytheblackdottedline.(b)ContributionstotheWFT-in-DFTerror:embeddingpotential(blueopencircles),DFTforsubsystemB(violetlledcircles),andDFTfornonadditiveexchange-correlationenergy(greensquares).(c)DFTMullikenpopulationofthedensityassociatedwithsubsystemBontheatomsinsubsystemA,shownfor1-uorodecane(blackopencircles)andthedissociatedalkanechain(redlledcircles).urations,thereistypicallyafavorablecancelationoferrorbetweentheerrorarisingfromtreatingsubsystemBusingDFTandtheerrorarisingfromevaluatingthenonadditiveexchange-correlationenergyusingDFT.Ingeneral,wenotethatifthenonadditiveexchange-correlationisnotthedomi-nantsourceoferror,theMP2correctioncannotsignicantlyimprovetheaccuracyoftheembeddingcalculation.Afterdissociationoftheuorideanionfrom1-uoro-1,3,5,7,9-decapentaene,thesubsequentgeometryoptimiza-tionleadstoanisomerizationwheretheprotononthesec-ondcarbonmovestotherst.Therefore,theanalysisforthisreactionbeginsatthesecondcarbon.FigureshowstheerrorinCCSD(T)-in-B3LYPembedding(blackopencircles)andMP2-correctedCCSD(T)-in-B3LYPembedding(redlledcircles)asafunctionofthenumberofcarbonsin-cludedinsubsystemAforuorideaniondissociationfrom1-uoro-1,3,5,7,9-decapentaene.Unlikethealkanecase,the 18A507-7Goodpasteretal.J.Chem.Phys.,18A507(2014) 0.1 0.2 0.3 0.4 0.5 2 3 4 5 6 7 8 9 B charge on ANumber of Carbons in Subsystem A(a)(b)(c) -10 0 10 20 Error (mEh) -20 -15 -10 -5 0 Error (mEh) FIG.5.(a)TheerrorinCCSD(T)-in-B3LYPembedding(blackopencir-cles)andMP2-correctedCCSD(T)-in-B3LYPembedding(redlledcircles)asafunctionofthenumberofcarbonsincludedinsubsystemAforthedissociationofthealkene.TheB3LYPenergyisgivenbytheblackdottedline.(b)ContributionstotheWFT-in-DFTerror:embeddingpotential(blueopencircles),useofDFTforsubsystemB(violetlledcircles),nonadditiveexchange-correlationenergy(greensquares).(c)DFTMullikenpopulationofthedensityassociatedwithsubsystemBontheatomsinsubsystemA,shownfor1-uoro-1,3,5,7,9-decapentaene(blackopencircles)andthedissociatedalkenechain(redlledcircles).alkenecaseexhibitslargeerrorswhichslowlydecreaseoncethemajorityofthesystemistreatedattheCCSD(T)level.showsthedecompositionofthecontributionstotheerrorinCCSD(T)-in-B3LYPembedding.Inthiscalcu-lation,theerrorarisingfromtreatingsubsystemBusingDFTisthedominatesourceoferror.ThisexplainswhytheerrorremainslargeuntilthemajorityofthesystemistreatedattheCCSD(T)level,andwhytheMP2-correctionisinsufcienttoreducetheerror.showstheMullikenpopulationoftheden-sityassociatedwithsubsystemBontheatomsassociatedwithsubsystemAforthealkenecase.Aswiththealkanecase,alargedifferenceinthisquantityisseenbetweentheuorinatedanddeuorinatedcompounds.Thisobserva-tionprovidesinsightintowhytheerrorfromtheDFTen-ergyofBcontributesstronglytotheerroroftheembeddingTABLEII.ThemagnitudeofthechangeinthedipolemomentbetweenproductsandreactantsforthedissociationofFfromthealkaneandalkenechains,aswellasthecorrespondingmagnitudesfortheH-Fexchangereac-tion.Valuesarereportedinatomicunits. MethodDissociationExchange AlkaneB3LYP7.3380.781CCSD7.5390.802AlkeneB3LYP1.7020.551CCSD3.0340.630 Themagnitudeofthechangeinthedipolemomentbe-tweentheuorinatedanddeuorinatedcompoundsisshowninTableforKS-DFTwithB3LYPandCCSD.Inthealkanedissociation,thechangeinthedipolemomentislarge,demonstratingasmallpolarizability,andthereisgoodagree-mentbetweenKS-DFTandCCSD.Inthealkenedissocia-tion,thechangeindipolemomentisconsiderablysmallerthanthealkanecase,demonstratingthatthedensitypolar-izestostabilizecharge.Forthealkene,thereislargedis-agreementbetweenKS-DFTandCCSD,demonstratingtheknownfailureofDFTtoaccuratelytreatpolarizabilitythough-conjugatedsystem.Therefore,whentherearelargeer-rorsassociatedwithKS-DFT,theselargeerrorswillaffecttheDFTenergyofsubsystemB,causinglargeWFT-in-DFTem-beddingerrors.WeemphasizethatforcasesinwhichDFTdoescorrectlydescribethepolarizationoftheenvironment,thislargesourceoferrordoesnotarise.ThefailureofWFT-in-DFTembeddinginFigureisnotafailureofembeddingitself,butratherafailureofDFTtoaccuratelytreatthepolar-izabilityof-conjugatedsystems.Finally,weconsiderthereactionofexchangingtheuorideanionfrom1-uorodecaneand1-uoro-1,3,5,7,9-decapentaenewithahydride(Figure).ThechangeindipolemomentforthesereactionsisprovidedinTable.Thesereac-tionsexhibitamoderatechangeindipolemoment,andthereisgoodagreementbetweenCCSDandKS-DFT.plottheerrorintheCCSD(T)-in-B3LYPembeddingandMP2-correctedCCSD(T)-in-B3LYPembeddingenergiesforthehydrideexchangereactionsfromalkaneandalkenechains,respectively,asafunctionofthenumberofcarbonsincludedinsubsystemA.Foreveryparti-tion,theerrorsaresmall.Forthesmallestdivision,theMP2correctionprovidesasignicantimprovementintheaccuracyoftheCCSD(T)-in-B3LYPembeddingenergy;forlargerdi-visions,theeffectoftheMP2correctionismuchsmaller.Un-likeinthecaseofuorideaniondissociation,DFTappliedtothehydrideexchangereactionaccuratelyrepresentsthechangeindipole.AstherearenolargeerrorsarisingfromtheDFTenergyofsubsystemB,WFT-in-DFTembeddingper-formsaccuratelyandtheMP2correctionfurtherimprovestheenergetics.TheimportantobservationfromthesecalculationsisthatwhenthereisalargeerrorintheDFTcalculationontheenvironment,therewillbecorrespondinglylargeerrorsintheWFT-in-DFTembeddingenergy.Importantly,thisfail-ureisassociatedwitherrorsintrinsictotheDFTfunc-tionals,anddoesnotariseduetoerrorsintheembedding 18A507-8Goodpasteretal.J.Chem.Phys.,18A507(2014) -1 0 1 1 2 3 4 5 6 7 8 9 Error (mEh)Number of Carbons in Ssystem A(a)(b) -1 0 1 2 Error (mEh) FIG.6.TheerrorinCCSD(T)-in-B3LYPembedding(blackopencircles)andMP2-correctedCCSD(T)-in-B3LYPembedding(redlledcircles)asafunctionofthenumberofcarbonsincludedinsubsystemAfortheexchangeofuoridetoahydridein(a)1-uorodecaneand(b)1-uoro-1,3,5,7,9-potential.WhenachemicalprocessinvolvesalargechangeintheMullikenpopulationofsubsystemBlocatedonthesub-systemAatoms,itislikelythattheembeddingerrorwillbedominatedbyerrorsarisingfromtheDFT-leveltreatmentofsubsystemB;errorsofthissortcannotbereducedbytheMP2V.CONCLUSIONSProjector-basedquantumembeddingprovidesaschemeformultiscaledescriptionswiththeexactnesspropertythatDFT-in-DFTisequivalenttoDFTonthewholesystem.Inmanytestsandapplications,wendtheaccuracyoftheschemetobeexcellent,allowingforaggressivepartitioningacrosscovalentbondsclosetothereactivecenterofthesys-temofinterest.However,forsomeapplications,theerrorsin-troducedbyembeddingarelargerthanwouldtypicallybeac-ceptable,andtheprincipalaimsofthispaperhavebeentounderstandandtakestepstowardsresolvingtheerrorsinsuchCarefulcomparisonofCCSD(T)-in-DFTembeddingcal-culationswithCCSD(T)calculationsperformedoverthefullsystemhasledtokeyinsightsregardingthesourcesoferrorintheembeddingcalculations.First,theembeddingpotentialobtainedusingapproximatedensityfunctionalsisfoundtobeaccurateforallofthecaseswehaveinvestigated,makingacontributiontotheoverallerroroftheembeddingcalculationthatisnegligiblecomparedtoothersourcesoferror.Itwasnotimmediatelyobviousthatthiswouldbethecase,becausefunctionals(particularlyincaseswheretheyareparameter-ized)aredesignedwithaccurateenergiesinmind.Andsecond,itisfoundthatinmanycases,theprimarysourceoferrorinCCSD(T)-in-DFTembeddingisthetreat-mentofnonadditiveexchange-correlationeffectswithanap-proximatedensityfunctional.Thisisimportantbecauseitistheonetermintheerrorforwhichsimplecorrectionscanbedevelopedthatconservetheefciencyoftheoriginalmethod.Here,wefoundthatuseofMP2orSOS-MP2correctionsforthistermtypicallyimprovedtheaccuracyoftheenergeticsforchemicalreactions,reducingtheaverageerrorfrom4.6mto1.2mwithrespecttoCCSD(T)calculationsperformedoverthefullsystem.ToinvestigatetheconvergencewithrespecttothesizeofsubsystemA,westudieddissociationandexchangeeventsattheterminusof10-carbonalkylandconjugatedchains.FortheremovalofF,theresultsoftheCCSD(T)-in-DFTem-beddingcalculationfortheconjugatedsystemarenoticeablyworsethanforthealkane,anditisfoundthattheMP2cor-rectiondoesnotreducethiserrorinthecomputedreactionenergy.Ouranalysisshows,however,thattheseresultsfollowfromthefactthatDFTprovidesapoordescriptionofthepo-larizationofthechargedalkenefragmentandthattheuncor-rectedCCSD(T)-in-DFTresultsbenetfromacancellationoferrorsintheDFTtreatmentofsubsystemBandintheDFTtreatmentofnonadditiveexchange-correlation.TheMP2cor-rectionimprovesthedescriptionofnonadditiveenergyterm,butitdoesnotcompensatefortheinaccuraciesintheDFTdescriptionofsubsystemB.Forahydrideexchangereactionattheterminusofthealkylandconjugatedchains,theCCSD(T)-in-DFTem-beddingresultsconvergesmoothlyandrapidlytoreferenceCCSD(T)calculationsperformedoverthefullsystem,regard-lessofinclusionoftheMP2correctionandregardlessofcon-jugationinthechain.TheseresultsdemonstratethatintheregimewhereDFTisadequateforthetreatmentoftheen-vironment,ourprojector-basedembeddingschemecaneffec-tivelypartitionthesystem,eveninconjugatedmolecules.Thecurrentworkdemonstratesthatprojection-basedem-beddingprovidesbotharigorousandpracticalapproachtoembeddingcorrelatedwavefunctionsinaDFTdescriptionoftheenvironment.Althoughtheresultspresentedhereutilizecoupled-clustermethodsfordescribingthecorrelatedwave-fuction,weemphasizethatprojection-basedembeddingcanbecombinedjustaseasilywithmulti-referenceelectronicstructuremethods,aswellasanymean-elddescriptionoftheenvironment.Theembeddingmethodisstraightforwardtoemploy—requiringonlythespecicationofwhichatomsaretobetreatedattheWFTandDFTlevelsoftheory—anditisfullyimplementedandavailableintheMOLPROchemistrypackage.ACKNOWLEDGMENTSThisworkissupportedbytheU.S.ArmyResearchLab-oratoryandtheU.S.ArmyResearchOfce(USARO)un-derGrantNo.W911NF-10-1-0202(J.D.G.),bytheAirForceOfceofScienticResearch(USAFOSR)underGrantNo.FA9550-11-1-0288(T.A.B.),andbythe(U.S.)DepartmentofEnergy(DOE)underGrantNo.DE-SC0006598(J.D.G.).T.F.M.acknowledgessupportfromaCamilleandHenryDreyfusFoundationTeacher-ScholarAwardandanAlfredP.SloanFoundationResearchFellowship.F.R.M.wasvis-itingCaltechwhilemostoftheresearchwasperformed.Hegratefullyacknowledgessupportforthesabbaticalthrough 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