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Ab Initio Density Functional Theory

University of Florida Institutional Repository
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TABLEOFCONTENTS page LISTOFTABLES ................................. v LISTOFFIGURES ................................ vi ABSTRACT .................................... vii CHAPTER 1INTRODUCTION .............................. 1 1.1Ab-InitioWavefunction-BasedMethods ............... 3 1.2Kohn-ShamDensityFunctionalTheory ............... 9 1.3ProblemswithConventionalFunctionals .............. 12 1.4Orbital-DependentFunctionals .................... 14 1.5AbinitioDensityFunctionalTheory ................ 15 2EXACTORBITAL-DEPENDENTEXCHANGEFUNCTIONAL .... 17 2.1ExactExchangeFunctional ...................... 17 2.2OptimizedEectivePotentialMethod ................ 19 2.3PerformanceoftheAuxiliary-BasisEXXMethod ......... 24 3CORRELATIONFUNCTIONALSFROMSECOND-ORDER PERTURBATIONTHEORY ....................... 31 3.1CorrelationFunctionalfromSecond-OrderPerturbationTheory 31 3.2CorrelationfunctionalfromSecond-OrderPerturbationTheory withPartialInnite-OrderResummation ............. 33 3.3ImplementationofthePT2andPT2SCFunctionals ........ 39 3.4NumericalTestsforAbinitioFunctionals .............. 40 4OTHERTHEORETICALANDNUMERICALRESULTS ........ 48 4.1ConnectionbetweenEnergy,Density,andPotential ........ 48 4.2DiagrammaticDerivationoftheOptimizedEectivePotential Equation .............................. 49 4.3MixingExactNonlocalandLocalExchange ............ 55 4.4Second-OrderPotentialwithinCommonEnergyDenominator Approximation ........................... 59 5CONCLUSIONS ............................... 65 iii

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APPENDIX AFUNCTIONALDERIVATIVEVIATHECHAINRULE ......... 69 BSINGULARVALUEDECOMPOSITION ................. 73 CDERIVATIVEOFTHESECOND-ORDERCORRELATIONENERGIES 75 REFERENCES ................................... 79 BIOGRAPHICALSKETCH ............................ 81 iv

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Table page 2{1EectofbasissetontheperformanceoftheEXXmethod. ...... 26 2{2EectoftheexplicitasymptotictermontheperformanceoftheEXXmethod .................................. 27 2{3EectoftheSingularValueDecompositionthresholdontheperformanceoftheEXXmethod .................... 28 2{4PerformanceoftheEXXmethodsforthe35closed-shellmoleculesoftheG1testset .............................. 30 3{1Performanceofabinitioandconventionalcorrelationfunctionalsinthehigh-densitylimit .......................... 42 3{2Performanceofabinitiocorrelationfunctionalsforclosed-shellatoms 42 3{3DensitymomentsofNecalculatedwithabinitioDFT,abinitiowavefunctionandconventionalDFTmethods ............. 46 4{1PerformanceofthehybridabinitiofunctionalEXX-PT2hwithoptimizedfractionofnonlocalexchange ................ 57 v

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Figure page 2{1ExplicitasymptotictermsforNeandthecorrespondingEXXpotentials ................................ 28 3{1PerformanceofabinitioDFTandabinitiowavefunctionmethodsintotalenergycalculationsfortheG1testset. ............. 43 3{2PerformanceofabinitioDFT,abinitiowavefunction,andconventionalDFTmethodsincalculationsofthetotalenergyasafunctionofthebondlengths ..................... 45 3{3PerformanceofabinitioDFT,abinitiowavefunction,andconventionalDFTmethodsindipolemomentcalculationsfortheG1testset ............................... 47 4{1ThetotalenergyandrstdensitymomentofBecalculatedwiththeEXX-PT2hfunctionalwithvariousfractionsofthenonlocalexchangeoperator ........................... 58 vi

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vii

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viii

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Theunderlyingphysicallawsnecessaryforthemathematicaltheoryofalargepartofphysicsandthewholeofchemistryarethuscompletelyknown,andthedicultyisonlythattheexactapplicationoftheselawsleadstoequationsmuchtoocomplicatedtobesoluble.P.A.M.Dirac,Proc.Roy.Soc.London,p.174,1929Atthemicroscopiclevel,achemicalreactionisthetransitionfromonestableconglomerateofnucleiandelectrons(reagent)toanotherone(product).Giventheinitialcongurationofthesystem,thetransitionpropertiesandthenalstatearedeterminedbytheinteractionsoftheparticleswitheachotherandwiththeenvironment.Sincethenatureoftheseinteractionisknown,itisthenthetaskoftheoreticalchemistrytopredicttheoutcomeofthereactionbysolvingthefundamentalequationdescribingtheseparticles.Dirac'sfamouswordsstatetheultimategoaloftheoreticalchemistry|thecompletesubstitutionoftheexperimentbyatheoreticalcalculation|andwarnabouttheultimatediculty|theimmensecomplexityoftheproblem.Evennow,givenallthecomputationalpoweratourdisposal,thenear-exactsolutionsoftheelectronicproblemarestilllimitedtofew-electronsystems.Facingtheintractabilityoftheexactsolution,onemustrelyonapproximations.Although,forsystemsbeyondseveralthousandsparticlesonehasnochoicebuttorelyonclassicalmechanics,mostchemicalphenomenarequireaquantum-mechanicaldescriptiontoobtainatleastqualitativeresemblancewithreality.Inquantumtheory,achemicalsystemisdescribedbythemolecular 1

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Hamiltonian(neglectingmagneticandrelativisticeectsforsimplicity) ^H=elec:Xi=11 2r2inucl:XA=11 2r2Aelec:Xinucl:XAZA ^H(t;x1;::;xN)=i@ @t(t;x1;::;xN)(1-2)Thus,quantumchemistry,thecollectionofquantum-mechanicalmethodstosolveEq. 1-2 ,playsthemajorroleintheoreticalchemistry.AbinitioDensityFunctionalTheorybelongstotheclassofquantum-chemicalmethodscalledelectronicstructuremethods.ThesemethodsfurthersimplifytheHamiltonianofEq. 1-1 usingtheBorn-Oppenheimerapproximation.Inthisapproximationoneneglectsthecouplingbetweentheelectronicandnucleardegreesoffreedom,whichallowsonetofactorizethecorrespondingvariablesandconcentrateontheelectronicpartofthewavefunction.Also,ifoneisinterestedinstationarysolutions,thetimevariablecanalsobefactorized.Thus,thetime-independentelectronicstructureproblemistondthesolutionofthetime-independentSchrodingerequation ^H(x1;::;xN)=E(x1;::;xN)(1-3)denedbythenonrelativistic,Born-OppenheimerelectronicHamiltonian ^H=elec:Xi1 2r2i+elec:Xinucl:XAZA 1-3 .Itsformalismisbasedontwofundamentalelectronicstructureapproaches:abinitiowavefunctionmethodsandDensityFunctionalTheory.

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1.1 Ab-InitioWavefunction-BasedMethods Hartree-FockMethod.Thesimplestapproximatewavefunctionthatretainsthecorrectfermionsymmetryisgivenbytheantisymmetricproductofsingle-electronwavefunctions (x1;::;xN)=(N!)1=2^A1(x1)::N(xN);(1-5)where ^A=XP(1)P^P(1-6)ensuresthatisantisymmetricwithrespecttoapermutationofthelabelsofanypairofelectrons.Thistypeofwavefunctioncanbeconvenientlywrittenasadeterminant HF=1(r1):::1(rN)::::::N(r1):::N(rN)(1-7)andisoftencalledaSlaterdeterminantorsingle-determinantwavefunction.IntheHartree-Fockmethod,thesingle-electronwavefunctions(ororbitals)aredeterminedbytheconditionthatthecorrespondingdeterminantminimizestheexpectationvalueofthetruemany-electronHamiltonian[ 1 ] 1-5 intothisexpectationvalue,oneobtainstheexpressionfortheHartree-Fockenergyintermsoftheorbitals 2r2+vextp+1 2elec:Xijijij(1-9)

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whereijijistheDiracnotationforthetwo-electronintegralsdenedbyEq. 1-10 .ijij=ijijijji=Zdrdr0i(r)j(r0)i(r)j(r0) (1 2r2+vext+vH+^vnlx)p=Xqpqq(1-11)where 1-11 isnotlimitedtothenumberofelectrons.ThelowestNsolutionsarereferredtoasoccupiedorbitals(Nbeingthenumberofelectrons)andtheremainingsolutionsarereferredtoasvirtualorbitals.

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UsingthefactthattheFockoperator ^f=1 2r2+vext+vH+^vnlx(1-16)isinvariantwithrespecttoanyunitarytransformationoftheoccupiedorbitals,onecantransformEq. 1-11 toitscanonicalform, ^fp=pp;(1-17)whichistheeigenvalueproblemfortheFockoperator.SincetheFockoperatordependsonfigthroughthevHand^vnlx,Eq. 1-11 isanintegro-dierentialequationthatcanbesolvediteratively,untilself-consistencyisreached.Therefore,theHartree-FockapproximationbelongstotheclassofSelf-ConsistentField(SCF)approximations.OnecansolvetheHartree-Fockequationsnumerically.However,amorepracticalapproachistouseanitebasisset(usuallyatom-centeredGaussian-typefunctions)toexpandtheHForbitals.Astheresult,theHartree-Fockintegro-dierentialequationsaretransformedintoamatrixproblem. Electron-CorrelationMethods.TheHartree-Fockmethodcanrecoverasmuchas99%ofthetotalelectronicenergy.Still,eventheremainingerrorof1%istoolargeonthechemicalscaleandmayleadtoaqualitativelywrongtheoreticalprediction.ThedierencebetweentheSCFandexactsolutionsisduetoelectron-correlationeects.Inabinitioelectron-correlationmethods,onereliesonelaboratemany-bodytechniquestogobeyondtheSCFapproximationandaccountforthesimultaneouselectron-electroninteractions.Thesemethods,incontrasttotherelativelysimpleHartree-Fockapproximation,canbequitechallengingconceptuallyandcomputationally.

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Thecorrelationlimit(i.e.,theexactsolutionofEq. 1-3 inagivenbasisset)canbeobtainedviatheFullCongurationInteractionmethod.Inthismethod,thecorrelationcorrectiontotheHartree-Fockdeterminantisexpandedoverallpossibleexciteddeterminants FCI=HF+occ:Xivirt:XaCaiai+occ:Xi6=jvirt:Xa6=bCijCababij+:::(1-18)whereai;abij,etc.areformedbybysubstitutingseveraloccupiedorbitalsintheHartree-Fockdeterminantbyvirtualorbitals,e.g. FCIFCI(1-20)However,thenumberofpossibleexciteddeterminantsgrowsexponentiallywiththenumberofelectronsandbasisfunctions,therefore,theFullCImethodiscomputationallyintractableforanybutverysmallsystems.Amongtheapproximateelectron-correlationmethods,themostcommonarethetruncatedandmulti-referenceCongurationInteractionmethods,Coupled-Clustermethods[ 2 ],andMany-BodyPerturbationTheory[ 3 ].Forexample,forsystemswherethemulti-referencetreatmentisnotnecessary(i.e.,whentheHartree-FockwavefunctiondominatestheFullCIexpansion),theCoupled-Clustermethodshaveprovedtobethemostsystematicandcomputationallyrobustapproachtothemany-electronproblem.

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Many-BodyPerturbationTheory.Insomecases,perturbationtheorycanprovideanaccuratedescriptionofelectron-correlationeectsatasignicantlylowercostthanrequiredbyCoupled-Clusterormultireferencemethods.Forexample,second-orderRayleigh-Schrodingerperturbationtheoryisthesimplestandleastexpensiveabinitiomethodforelectroncorrelation.Thatiswhyitwaschosenasthebasisfortheabinitiocorrelationfunctional(Chapter3).Insuchperturbationtheory,onendsthesolutionofthemany-bodyproblem(Eq. 1-21 )usinganSCFmodel(Equations 1-22 and 1-23 )asthereference. ^H=E(1-21) (1 2r2+^u)p=pp:(1-22) 2r2+^u=E0;(1-23)whereisthesingle-determinantwavefunctionconstructedfromtheNlowestsolutionstoEq. 1-22 .Theremainingeigenfunctionsof^H0areobtainedbysubstitutingthecorrespondingnumberofoccupiedorbitalsinbythevirtualorbitals.Todothis,thetrueHamiltonianispartitionedintothereferenceHamiltonianandperturbation ^H=H0+^V(1-24)where ^V=^H^H0=elec:Xivext(ri)+elecXi6=j1 1-21 isthenfoundbyintroducingtheperturbationparameterandexpressingthecorrectionstothereferencewavefunctionand

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energyasseriesoftermsofincreasingpowersof (E0H0)(1)=(^VE(1))(1-29) (E0H0)(2)=(^VE(1))(1)E(2)(1-30)andsoforth.Choosingtheperturbativecorrectionstobeorthogonaltothereferencewavefunction,(n)=0,onecanreadilyobtaintheexpressionsfortheorder-by-ordercontributionstotheenergybyprojectingtheEquations 1-29 and 1-30 ontothereferencespace 4 ](theinverseofintegro-dierentialoperatorE0H0intheHilbertsubspace)

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^R0=^Q E0H0;(1-35)and^Q=^1istheprojectorontothecomplementaryspaceof(Hilbertspacewithexcluded.)TheactualexpressionfortheresolventoperatorcanreadilybefoundbyrecognizingthatE0H0isdiagonalintermsofeigenfunctionsofH0 1.2 Kohn-ShamDensityFunctionalTheoryDensityFunctionalTheoryisanalternativeapproachtotheelectronicstructureproblemofEq. 1-3 thatusestheelectronicdensityratherthanthewavefunctionasthebasicvariable.TheformalbasisofDFTisprovidedbytwotheoremsintroducedbyHohenbergandKohn[ 5 ].Thersttheoremestablishestheone-to-onecorrespondencebetweentheelectronicground-statedensityandtheexternalpotential.Sinceitistheexternalpotentialthatdenesaparticular

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molecule,theexistenceofsuchacorrespondenceensuresthattheground-stateelectronicdensityalonecarriesalltheinformationaboutthesystem.Inparticular,theground-stateenergycanbewrittenasafunctionalofthedensity.However,thereisnoequationofmotionfortheelectronicdensity.Instead,onemustrelyonthesecondHohenberg-Kohntheoremthatstatesthattheground-stateenergyasafunctionalofthedensityisminimizedbythetrueground-statedensity.Therefore,giventheenergyfunctional,onecanobtaintheground-statedensityandenergybyvariationalminimizationofthefunctional.However,theformaldenitionofDensityFunctionalTheorydoesnottellhowtoconstructsuchfunctional.Severalapproximateformshavebeensuggested;however,theyarefarfromaccurate.Thekineticenergyofelectronsisparticularlydiculttoapproximateasafunctionalofthedensity.TheideaofKohnandSham[ 6 ]wastouseaSCFmodel(Eq. 1-39 )totransformthevariationalsearchoverthedensityintoasearchovertheSCForbitalsthatintegratetoagiventrialdensity. [1 2r2+vs(r)]p(r)=p(r)(1-39)Suchatransformationdoesnotrestrictthevariationalspace,providedthateveryphysicallymeaningfuldensitycorrespondtoauniquesetofSCForbitals(thev-representabilitycondition).NotonlydoestheuseoftheKohn-ShamSCFmodelensurethatthevariationalsearchbetofermionicdensities,butalsoitprovidesagoodapproximationforthekineticenergy.Indeed,providedthattheorbitalsintegratetotheexactdensity,theso-callednoninteractingkineticenergy 2r2s=occ:Xii1 2r2i(1-40)shouldaccountforalargepartofthetruekineticenergy.

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Theremainingunknowntermsoftheenergyfunctionalaregroupedintotheexchange-correlationfunctional

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sinceitisanSCFmodel,practicalimplementationoftheKSprocedureisverysimilartotheHartree-Fockmethod.Usually,theSCForbitalsareexpandedinaGaussian-typeatom-specicbasis,whichtransformstheKohn-Shamintegro-dierentialequationintoamatrixSCFequation.Afterself-consistencyisreached,theSCForbitalsareguaranteedtoreproducethetruedensityofthemany-electronsystem.Also,thetrueenergycanbefoundbyinsertingthisdensityintotheenergyfunctional.VirtuallyallmodernimplementationsofDFTusetheKohn-Shamscheme.However,thetheorystillleavesopenthequestionofhowtoconstructtheexchange-correlationfunctional.Therefore,theprincipalchallengeforthetheoreticaldevelopmentofDFTremainstheconstructionofaccurateexchange-correlationfunctionals. 1.3 ProblemswithConventionalFunctionalsTheconventionalapproachistoapproximatetheenergyfunctionalasananalyticalexpressionofthedensityanditsgradients.Theeectivepotentialcanthenbeobtainedinanalyticalformaswell,andtheKSequationscanbesolvedreadily.ThisapproachstartedwiththesimplestLocalDensityApproximation(LDA)wheretheenergyisgiventhroughanintegralofalocalfunctionalofthedensity.Thenext-level,GeneralizedGradientApproximation(GGA)functionals,improvedontheLDAfunctionalformbyincludingthedependenceonthegradientsofthedensity.Thisextensionprovidedacertainfreedomindeningtheformofthefunctional,andanumberofdierentformshavebeensuggested.Typically,thebasicformofaGGAfunctionalischosentosatisfyasetofconditionsknowntobesatisedbytheexactfunctional.Thebasicformistheneitherparameterizedtoreproduceexperimentaldata(empiricalfunctionals)orfurthermodiedtosatisfyanextendedsetofconditions(non-empiricalfunctionals).

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Withtheconventionalfunctionals,KSDFTsurpassesthequalityoftheHFmethod,andbecomescomparablewiththesimplestabinitiocorrelationmethods.Nevertheless,restrictingthefunctionalformtoanalyticalexpressionsofthedensityimposescertainlimitationsontheenergyfunctional.GGAexchangefunctionalsarenotcapableofcompleteeliminationofthespuriousself-interactioncomponentoftheHartreeenergy.Sincetheexchangepartoftendominatestheexchange-correlationenergy,theself-interactionerrorcanconsiderablyreducetheaccuracyoftheGGAfunctional.Similarly,semilocalcorrelationfunctionalscannotdescribepurenonlocalcomponentsofthecorrelationenergysuchasdispersion.ThisomissiongreatlyreducestheapplicabilityoftheconventionalKSDFTmethodstoweakly-interactingsystems.AnotherproblemisthatwhiletheGGAfunctionalsresultinrelativelyaccurateenergies,thefunctionalderivatives(i.e.,thecorrespondingKSpotentials)arenotnearlyasaccurate,especiallyintheinter-shellandasymptoticregions.Consequently,oneshouldnotexpectthesamelevelofaccuracyforthedensityasfortheenergy.Furthermore,thequalitativelyincorrectpotentialsreducesubstantiallytheusefulnessoftheKSorbitalsandorbitalenergies,whichareoftenusedtocalculatecertainground-statepropertiesorasthebasisforresponseandtime-dependentKSDFTcalculations.Someoftheseproblemscanbeaddressedwithoutextendingthefunctionalform.Severalpost-SCFcorrectionshavebeensuggestedtopartiallyremovetheself-interactionerror.Forexample,aftertheKSequationshavebeensolved,onecanintroducecorrectionstotheenergytoincludedispersionorensurethecorrectasymptoticbehavioroftheKSpotential.However,thesecorrectionsarespecictotheparticularfunctionalandclassofsystemsandtheylikelyareincompatiblewitheachother.Clearly,oneneedstogobeyondtheGGAfunctionalformtoresolvetheseproblemsinaconsistentanduniversalfashion.

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1.4 Orbital-DependentFunctionalsIthasnowbeenfullyrecognizedthatKSorbitalscanprovideextrainformationaboutthesystemthatcannotbe\extracted"easilyfromthedensityoritsgradients.Thenext-generationfunctionals(hybrid-andmeta-GGA)augmenttheGGAfunctionalformwithtermsthatdependexplicitlyontheorbitalratherthanthedensity.Analternativeapproachistodismisscompletelytheconventionalhierarchyofapproximationsandconstructthefunctionalusingsolelytheorbitals.Incontrasttotheconventionalfunctionals,orbital-dependentfunctionalsareanalyticalexpressionsoftheorbitals(andorbitaleigenvalues).Theystillareimplicitfunctionalsofthedensity,however.Indeed,thecentralassumptionofKSDFTisthatthereexistsone-to-onemappingbetweentheexactdensityandsomealocalpotential.Therefore,agivendensityuniquelydenesthepotential,which,inturn,uniquelydenestheorbitalsthroughtheKSSCFequations.Therefore,theorbitalsandexplicitlyorbital-dependentfunctionalareimplicitfunctionalsofthedensity.OnecanthinkoftheKSorbitalsastheintermediatestepinthemappingfromthedensitytotheenergy.Themostsignicantdierencebetweentheorbital-dependentandconventionalfunctionalsishowthecorrespondingpotential(i.e.,thefunctionalderivativewithrespecttothedensity)isdetermined.Theconventionalfunctionalsaregivenasanalyticalexpressionsintermsofthedensity.Therefore,onecantakethefunctionalderivativestraightforwardlytoobtainananalyticalexpressionforthepotential.Theorbital-dependentfunctionalsareanalyticalexpressionsintermsoftheorbitals,whosedependenceonthedensityisgiventhroughtheeectivepotentialandKSintegro-dierentialequation.Therefore,theanalyticalexpressionforthefunctionalderivative(hence,potential)cannotbeobtaineddirectly.Instead,onemustrelyonthechainruletoobtainanintegralequationforthepotential

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(Chapter2).ThisintegralequationisidenticaltotheoneusedintheOptimizedEectivePotential(OEP)method.TheOEPmethodis,therefore,thecornerstoneofDFTwithorbital-dependentfunctionals.Theimmediateadvantageoftheorbital-basedapproachisthattheexactexchangefunctionalisknownintermoforbitals.OnecanthinkoftheEXXmethodasanextensiontotheideaofKohnandSham,whereSCForbitalsareusedtocalculateboththelargerpartofthekineticenergyanda(presumablylarger)partoftheexchange-correlationenergy. 1.5 AbinitioDensityFunctionalTheoryWhiletheEXXfunctionalprovidestheexactdescriptionoftheexchangeinteractions,itisjustarststeptowardstheexactexchange-correlationfunctional.ItistheeectivedescriptionofelectroncorrelationeectsthatmakesKSDFTapowerfulalternativetotheabinitiowavefunctionmethods.Thus,oneneedsacorrelationfunctionalthatcanbecombinedwiththeEXXfunctional.Conventional(GGAorhigher-level)correlationfunctionalsaredevelopedincombinationwiththecorrespondingapproximateexchangefunctionalsandoftencompensatethedecienciesofthelatter.Forexample,theGGAcorrelationfunctionalsusuallyresultincorrelationpotentialsthathavetheoppositesigntotheexactone.Thetermscorrectingtheapproximateexchangeare\hidden"inthecorrelationfunctionalsandinseparablefromthe\true"correlationterms.Thus,itisnotsurprisingthatsubstitutingtheapproximateexchangebyitsexactcounterpartdestroysthebalancebetweentheapproximateexchangeandapproximatecorrelationcomponentsandresultsinafunctionalinferiortotheexchange-onlyapproximation.Inotherwords,theconventionalcorrelationfunctionalsarenotcompatiblewiththeEXXfunctional.Thus,theprimarychallengeintheorbital-basedapproachtoexchange-correlationfunctionalis

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todevelopanorbital-dependentcorrelationfunctionalthatcanbecombinedseamlesslywiththeexactexchangefunctional.AbinitioDFTsolvestheproblemofconstructinganorbital-dependentcorrelationfunctionalbyreferringtoabinitiowavefunctionmethods.Theideaissimple:thegoalofabinitiomethodsistocalculatethecorrectiontotheexactexchangeapproximation(i.e.,correlationenergy)intermsoftheSCForbitals.Thus,suchanenergyexpressiontreatedastheorbital-dependentfunctionalresultsinacorrelationfunctionalthatcanbeseamlesslyaddedtotheexactexchangefunctional.AbinitioDFTmakesaplethoraofwavefunction-basedapproximationsavailableasthecorrelationfunctionals.Unliketheconventionalones,abinitiofunctionalsaresystematicallyimprovable,sinceonecanalwaysuseahigher-levelapproximationtoobtainamoreaccuratefunctional.Theyalsohaveawell-denedexactlimitrepresentedbytheFCImethod.Thenexttwochaptersdescribetheformaldevelopment,implementation,andsometestapplicationsfortheexactexchangefunctionalandthecorrelationfunctionalbasedonsecond-orderperturbationtheory.Chapter4discussespossibleextensionsoftheabinitioDFTapproach.TheresultsofthestudyaresummarizedinChapter5.

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2.1 ExactExchangeFunctionalTheimmediateadvantageofconstructingtheenergyfunctionalintermsoforbitalsisthatsincetheexchangeenergyisdenedintermsoforbitals,theexactorbital-dependentexchangefunctionalisknown.Indeed,sincetheexchangecomponentoftheexchange-correlationenergyisdenedas 2occ:Xi;jijji(2-1)treatingitasanimplicitfunctionalofthedensityresultsintheexactexchange(EXX)functional.ThemostimportantfeatureoftheEXXfunctionalisthat,unlikeanyoftheconventionalfunctionals,itcompletelyeliminatesthespuriousself-interactioncomponentoftheHartreeenergy.Similarly,thecorrespondingEXXpotentialcancelstheself-interactioncomponentoftheHartreepotential.Thus,usingtheEXXfunctionalandpotentialwillavoidmanypathologicalproblemscausedbytheself-interactionerrorinconventionalDFTapproximations,bothattheenergyanddensitylevels.Theexplicitdependenceontheorbitalsamountstoonecomplication,however.SincetheEXXfunctionaldoesnotdependonthedensityexplicitly(i.e.,itisnotananalyticalexpressionofthedensity),thefunctionalderivativecannotbetakendirectly.Instead,onemustrelyonthechainrule,whichaccountsfortheimplicitdependenceexpressingthederivativeofinterestthroughtheproductofknownderivatives. 17

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Thechainruleplaysacentralroleintheorbital-basedapproachbecauseitallowstotakethefunctionalderivativeofanexpressionintermsoforbitalswithrespecttothedensity.Todothatoneneedstodetermineforwhichderivativestheexpressionsareknown,andthenexpressthederivativeofinterestintermsoftheknownderivatives.First,onerecognizesthattheKSpotentialisthemostconvenientvariable.Indeed,theresponseoftheorbitalsandorbitalenergiestoainnitesimallysmallchangeinthepotentialisreadilyavailablethroughthelinearresponseKSequations(Appendix A ).Andsoistheresponseofthedensity.Thus,thefunctionalderivativesoftheorbital,orbitalenergies,anddensitywithrespecttothepotentialareknown.Second,sincetheexchangeenergyisgivenasananalyticalexpressionintermsoforbitals,itsderivativewithrespecttotheorbitalscanbeobtaineddirectly.Thus,startingwiththedenitionfortheexchangepotential

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and,asshowninAppendix A 7 ],longbeforethefoundationoftheDFT.TheirgoalwastondalocalapproximationtotheHFexchangeoperator.Theydenedtheoptimizedpotentialastheonethatmakesthesingle-determinantexpectationvalueofthetrueHamiltonianstationary[ 8 ].MuchlaterwasitrealizedthattheOEPmethodresultsintheexactexchangepotentialintheKSDFTcontext.Becauseofthisequivalence,theterms\EXXmethod"and\OEPmethod"areoftenusedinterchangeably.However,theapplicationofthechainruleisnotlimitedtotheexchangefunctional.Inthenextchapter,thechainrulewillbeappliedtotheorbital-dependentcorrelationfunctional.Thus,itispreferrablytousetheterm\OEPmethod"todenotethewaytodeterminethelocalpotentialforagivenorbital-dependentfunctional.Consequently,onereferstotheEXXmethodastheKSDFTmethodwiththeexactorbital-dependentexchangefunctionalandcorrespondingpotentialobtainedviatheOEPmethod. 2.2 OptimizedEectivePotentialMethodTheOEPequationisaFredholmintegralequationoftherstkind.Itsintegralkernelaswellastheright-handsidedependontheSCForbitalsandorbitaleigenvalues,therefore,itmustbesolvedsimultaneouslywiththeSCFequations.Hirataetal.[ 9 ]analyzedtheintegralkernelandshowedthatitdenesthepotentialuniquelyuptoanirrelevantconstantiftheSCForbitalsformacomplete

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basisset.However,virtuallyallpracticalimplementationsoftheSCFprocedureemployanitebasistorepresenttheorbitals.Inthiscase,theSCForbitalsdonotformthecompletesetandtheintegralkernel,ingeneral,hassingulareigenfunctions.This,meansthatthepotentialisdeneduptoalinearcombinationofthesingulareigenfunctions.Thus,inapracticalimplementationoftheOEPmethod,onemustexcludethesubspacespannedbythesingulareigenfunctionsofthekernelfromthesolution.TheincompletenessoftheorbitalbasiscanalsoleadtotheOEPintegralkernelthatdoesnotsamplecertainregionsoftherealspace.Forexample,Gaussian-typeorbitals,whicharetypicallyusedastheorbitalbasis,fallotoorapidlywithincreasingr.Theydecayaser2,whiletheexactexchangepotentialisknowntohavethe1=rasymptoticbehavior.Asaresult,theOEPkerneldecaystoorapidlyanddoesnotsamplethesolutionintheasymptoticregion.Consequently,thepotentialobtainedfromtheOEPequationintheniteorbitalbasiscandeviatearbitrarilyfromtheexactsolutionintheasymptoticregion.TherstimplementationoftheSCFprocedurewiththeexchangepotentialgivenbytheOEPequationwasreportedbyTalmanandShadwick[ 8 ].Theyusedaexpansionoveraspatialgridtosolvetheintegralequation.However,suchgrid-basedimplementationisinevitablylimitedtoatoms,forwhichthesphericalsymmetrypermitsexcludingtheangularpointsfromconsideration.Inthecaseofapolyatomicmolecule,thenumberofgridpointsnecessaryforanadequaterepresentationoftheEXXpotentialissignicantlylargerandagrid-basedOEPmethodbecomescomputationallyintractable.Kriegeretal.[ 10 ]suggestedneglectingtheorbitalstructureoftheintegralkerneltoavoidthesolutionsoftheintegralequation,theso-calledKLIapproximation.TheOEPintegralequationintheKLIapproximationreducestoaverysimplenonlinearequationfortheapproximateKLIexchangepotential.

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TheKLIequationcaneasilybesolvedself-consistentlyeveninthecaseofpolyatomicmolecules.However,theresultingpotentialdoesnotreproducethecharacteristic\bumps"oftheEXXpotentialintheinter-shellregion.Recently,amoreelaboratedapproximationtotheOEPequationhasbeensuggested,knownasCommonEnergyDenominatorApproximation[ 11 ]orLocalizedHartree-Fock[ 12 ].TheexchangepotentialinthisapproximationmoreaccuratelyreproducesthestructureoftheEXXpotential.However,theerrorintroducedbythisapproximationstillcannotbemeasuredaprioriorcontrolled.Theauxiliary-basisapproach[ 13 14 ]presentsanattractivealternativetogrid-basedandapproximateOEPmethods.Inthismethod,thepotentialisexpandedinaniteauxiliarybasissetandtheOEPintegralequationistransformedintoalinearmatrixproblem.Thisisverysimilartohowtheintegro-dierentialSCFequationsaresolvedintheLCAOapproximation.Theauxiliary-basisapproachtothesolutionoftheOEPequationdoesnotrequireanespatialgridnordoesitintroduceanyapproximationtothekernel.Theerrorintroducedbythenite-basisexpansionalwayscanbereducedbyincreasingthesizeoftheauxiliarybasisset.Inthisapproach,theOEPintegralequation

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2-8 .Asithasbeenalreadymentioned,inaniteorbitalbasistheoccupied-virtualorbitalproductsdonotspantheentirespaceand,therefore,theintegralkernelmayhavenontrivialeigenfunctionswithzeroeigenvalue.Consequently,theauxiliary-basisrepresentationofthekernel,X,mayhavesingulareigenvaluesandnotbeinvertible.Inthiscase,onecanusetheSingularValueDecomposition(SVD)procedure.TheSingularValueDecomposition(Appendix B )withagivenSVDthresholdwillprovidetheapproximatesolutiontoEq. 2-8 ~u=(XSVD)w(2-13)thatminimizestheerrorinaleast-squaressense

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thequalityoftheapproximatesolution.Therefore,thereisnoaprioripreferredvalueforthethresholdanditsactualchoiceissubjecttoinvestigation.AnotherproblemmentionedinthebeginningofthissectionconcernswiththeasymptoticbehavioroftheEXXpotential.Theexactexchangepotentialmustdecayas1=ratlarger.However,iftheOEPkernelisobtainedwithGaussian-typeSCForbitalsitdecaystoorapidlyanddoesnotsamplethepotentialintheasymptoticregion.Thesolutiontothisproblem,consideredbymanyauthors[ 15 16 17 ],istouseanumericalpotentialthathastheasymptoticbehavioroftheexactexchangepotential.Thereareseveralchoicesforsuchapotential:theFermi-AmaldiscaledCoulombpotential

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sectiondiscussesthetypicalchoicesfortheseparametersandtheireectontheperformanceoftheEXXmethod. 2.3 PerformanceoftheAuxiliary-BasisEXXMethodIntheconventionalDFTapproximations,thequalityofagivenexchange-onlyapproximationissolelydeterminedbythequalityoftheapproximatefunctional.IntheEXXmethodthefunctionalisknownandthequalityofthemethodisdeterminedbytheimplementationoftheOEPmethodforthecorrespondingpotential.Inthisstudy,theauxiliary-basisimplementationischosenbecauseitallowsapplicationstogeneralpolyatomicmoleculesanddoesnotintroduceanysimplicationstotheintegralkernelstructure.Theonlyerrorisintroducedbytheincompletenessoftheorbitalandauxiliarybases,butthiserrorcanbecontrolledbyincreasingthesizesofthebasissets.Also,giventheinevitableincompletenessofthesebases,thequalityofthepotentialisaectedbytheexplicitasymptotictermusedtoensurethecorrectlongrangebehaviorandtheSVDthreshold.ThetaskofndingtheoptimalcombinationofthenumericalparametersisgreatlyfacilitatedbythefactthatthereferencefortheEXXmethodisgivenbytheHartree-Fockmethodand,thus,isreadilyavailable.Indeed,theonlypurposeoftheEXXmethodastherststeptowardtheexchange-correlationfunctionalistoaccuratelyincludetheexchangeinteractionwithintheDFTframework.SincetheHFresultsrepresenttheexactexchangelimitinagivenbasisset,animplementationoftheEXXmethodmustbeassessedbyhowwellitreproducestheHFresults.Onemustunderstand,however,thatthesetwomethodsarenotidentical.First,bothmethodsresultinsingle-determinantsthatminimizestheexpectationvalueofthetrueHamiltonian.However,intheHartree-Fockcase,theSCFoperatorisnotconstrainedtobelocal.Therefore,theHartree-FockenergyisalwayslowerthantheEXXone.Second,thetwodensitiesmustbeveryclose,but

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notexactlyequal,sincetheOEPconditionensuresthatthedierencebetweenthetwodensitiesiszeroonlythroughrstorder.TherealsoisavirialconditionontheexactlocalexchangepotentialthatprovidetheequivalencebetweentheHOMOenergiesintheHFandEXXSCFmodels.Butthisequivalenceisaectedbytheincompletenessoftheorbitalbasisset.Thus,thereisnoexactequivalenceontheenergy,density,orHOMOenergy;however,oneexpectsthesequantitiestobeverycloseintheHFandEXXmethods.Thus,inthiswork,allthreequantities|energy,density,andHOMOenergies|werecomparedtoprovideanexhaustiveassessmentoftheEXXimplementation.Thereareexactlyfourparametersthataecttheperformanceoftheauxiliary-basisOEPmethod:thesizeoftheorbitalbasisset,thesizeoftheauxiliarybasisset,theexplicitasymptoticterm,andtheSVDthreshold.Themostfundamentaleectcomesfromtheincompletenessofthebasissetusedtoexpandthemolecularorbitals.Thesecondmostimportanteectshouldcomefromthesizeoftheauxiliarybasis.InthecurrentimplementationthesameGaussian-typeatomicbasiswaschosenasboththeorbitalandauxiliarybases.Conventionalatomicbasesmaynotbethebestchoicetoexpandthepotentialbecauseofthedierentphysicalnatureoftheorbitalsandpotential.However,theuseofconventionalbasissetssignicantlyfacilitatestheimplementationandalsoremovesanecessityofdevelopingandtestingauxiliarybases.Table 2{1 reportsthedeviationsoftheEXXtotalenergies,highest-occupiedorbitalenergiesanddensitiesfromtheHartree-FockvaluesasafunctionoftheGaussian-typebases.Sinceitisdiculttomeaningfullycomparetheatomicormoleculardensitiesdirectly,thedensitymomentswerecomparedinstead.InallthesecalculationstheexchangeenergydensitywasusedastheexplicitasymptotictermandSVDthresholdof105wasused.Asonecanseefromtheseresultsitisdiculttodenetheoptimalbasis;however,theuncontractedRoosaugmented

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doublezetaresultsinadequateerrorsforallthequantities.Unlikethesingle-zetabases(6-311Gand6-311G),itisextensiveenoughtodescribethecorrelationeects(whichwillbecomeimportantinthenextchapter)andnotaslargeastriple-andquadruple-zetabasessothecalculationsremainaordable. Table2{1: EectofbasissetontheperformanceoftheEXXmethod.Shownaredeviationsoftotalenergy(inmilliHartree),HOMOenergy(ineV)anddensitymoments(ina.u.)fromtheHFvalues.\(u)"indicateuncontractedbasissets. BasisENDensitymoments Ne H2O Table 2{2 comparestheperformanceoftheEXXmethodwithandwithouttheexplicitasymptoticterms(EAT).Asonecansee,usingtheFermi-Amaldipotentialasexplicitasymptotictermonlyslightlyimprovestheenergyanddensity,but

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dramaticallychangesthevalueoftheHOMOenergy.UsingtheexchangeenergydensityfurtherimprovestheperformanceoftheEXXmethodasthispotentialismuchbetterapproximationtotheEXXpotentialand,therefore,isbettersuitedtoensurethecorrectlongrangebehavior. Table2{2: EectoftheexplicitasymptotictermontheperformanceoftheEXXmethod.UncontractedRoosaugmenteddouble-zetabasissetandSVDthresholdof105. Ne H2O Figure 2{1 showstheFermi-Amaldipotential,exchangeenergydensity,andEXXpotentialswithorwithouttheexplicitasymptoticterms.NotethattheEXXpotentialwithouttheEATwasshiftedtofacilitatethecomparison.Asonecansee,theuseoftheEATdoesnotaecttheshapeofthepotentialbutensuresthecorrectasymptoticbehavior.Table 2{3 showstheresultsoftheEXXcalculationsforNeandH2OwithgradualincreaseoftheSVDthreshold.Ascanbeseenuptoabout105106atomicunits,theSVDthresholdhadsmalleecttheenergyordensity.However,usingthethresholdslessthanthesevaluesaectsthequalityoftheHOMOenergy.Basedontheseresults,weconcludethattheoptimalcongurationfortheauxiliary-basisOEPimplementationwasachievedwhenweuseduncontractedRoosaugmenteddoublezeta,theexchangeenergydensityastheexplicitasymptoticterm,andtheSVDthresholdof105or106.AsthenaltestfortheEXXmethod,theHFandEXXcalculationswereperformedforthe35closed-shellmoleculeswithsingletgroundstatechosenfrom

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ExplicitasymptotictermsforNeandthecorrespondingEXXpotentials.UncontractedRoosaugmenteddoublezetaANObasisset. Table2{3: EectoftheSingularValueDecompositionthresholdontheperformanceoftheEXXmethod.UncontractedRoosaugmenteddouble-zetabasisset. Ne H2O

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theG1testset.TheG1setwastherstonefromtheseriesofsets[ 18 ]developedtoteststandardelectronicstructuremethods.Althoughnotexhaustive,thissetaimsatrepresentingdierenttypesofmoleculesandchemicalbonding.Alsotheexperimental(ratherthancomputed)structureswereusedtoavoidtheambiguityincomparison.Theexperimentalvaluesforthebondlengthandanglesarereadilyavailableonline(ComputationalChemistryComparisonandBenchmarkDataBase,http://srdata.nist.gov/cccbdb).Table 2{4 reportstheHFenergies,theabsoluteandrelativedierencesbetweenHFandEXXenergies,theHFdipolemomentsandtheabsoluteandrelativedierencebetweentheHFandEXXdipolemoments.Asonecansee,theEXXenergiesareveryclosetotheHFones,withthelargestdeviationsofabout10milliHartreeforSO2,whichislessthan2%oftheMP2correlationenergyof631milli-Hartree.

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Table2{4: PerformanceoftheEXXmethodsforthe35closed-shellmoleculesoftheG1testset.UncontractedRoosaugmenteddoublezetabasisset.EnergiesareinmilliHartreeanddipolemomentsareinDebye. SystemEnergyDipole LiH0.20.0025%0.0060.10%CH41.50.0037%NH31.20.0021%0.0271.68%H2O0.90.0011%0.0201.02%HF0.60.0006%0.0090.47%SiH44.50.0016%PH33.90.0011%0.0395.78%H2S3.30.0008%0.0302.79%HCl2.30.0005%0.0312.64%Li20.70.0046%LiF0.70.0006%0.0040.06%C2H21.70.0022%H2C=CH23.00.0038%H3C-CH33.90.0050%HCN2.40.0026%0.0371.12%CO3.00.0027%H2C=O3.70.0032%0.0401.41%CH3-OH3.70.0032%0.0372.05%N22.70.0025%HO-OH4.00.0027%0.0221.31%F24.70.0024%CO25.50.0029%Na20.50.0002%P24.60.0007%Cl27.80.0008%NaCl2.80.0004%0.0030.03%SiO4.00.0011%0.0240.65%CS6.00.0014%0.0452.77%ClF5.80.0010%0.0110.98%H3Si-SiH38.40.0015%CH3Cl5.60.0011%0.0341.65%H3C-SH5.90.0013%0.0362.11%HOCl5.80.0011%0.0120.74%SO29.20.0017%0.0422.14% Average3.70.0019%0.0251.57%Maximum9.20.0050%0.0455.78%

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3.1 CorrelationfunctionalfromSecond-OrderPerturbationTheoryTheideaofusingperturbationtheorytoapproachtheexactcorrelationfunctionalbelongstoGorlingandLevy[ 19 ].TheydemonstratedhowtheexactcorrelationfunctionalcanbeformallyconstructedfromtheperturbationexpansionwithintheAdiabaticConnectionformalism.TruncatedinsecondordertheGorling-Levyperturbationtheorygivetherstapproximationtothecorrelationfunctional 4occ:Xi;jvirt:Xa;bijab2 31

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SimilarlytotheEXXfunctional,thecorrespondingfunctionalderivative(i.e.,thecorrelationpotential)canbeobtainedusingthechainrule 20 ]werersttosolvethisequationbasedonthegrid-basedOEPalgorithm.However,theydidnotincludethepotentialintotheSCFiterationsanddidnottakeintoaccountthersttermofEq. 3-1 ,whichdescribesthecontributionofthesingleexcitationstothesecond-orderenergy.Also,intheirnumericalsolutionoftheOEPequation,certaintermsweretreatedseparatelywhichapparentlyleadtoanumericalsingularity.Basedonthissingularity,theyconcludedthatthecorrelationpotentialfromthesecond-orderperturbationseriesdoesnotvanishatlarger,astheexactoneshould.Thisraisedthequestionwhetherthesecond-orderperturbationtheorycanleadtoameaningfulcorrelationpotential.Niquetetal.[ 21 ]disputedthisconclusionandarguedthatitisthenumericalprocedureusedtocalculatethepotentialthatisthesourceofthisproblem.Recently,theyshowedthatthecorrelationpotentialfromthesecond-orderperturbationtheoryhasthecorrectC=r4asymptote,atleastforclosed-shellsystemswithsphericalsymmetry.Asithasbeendiscussedinthepreviouschapter,anygrid-basedOEPimplementationislimitedtosmallsystems.Grabowskietal.[ 22 ]rederivedtheexpressionforthesecond-orderpotentialthatincludedthecontributionsofsingleexcitationsandimplementeditbasedontheauxiliary-basisOEPmethod.TheyhaveshownthatthePT2functionalresultsinaccuratecorrelationenergiesof

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two-electronsystems,quicklyapproachingtheexactcorrelationenergyinthehigh-densitylimitandresultsinqualitativelycorrectcorrelationpotentialsforatoms. 3.2 CorrelationFunctionalfromSecond-OrderPerturbationTheorywith PartialInnite-OrderResummationMorerecently,thesameauthors[ 23 ]foundthattheiterativesolutionoftheOEPequationdivergesfortheBeatomandthePT2functionalsignicantlyoverestimatesthecorrelationenergiesforsmallmolecules.Thispoorperformanceofthesecond-orderfunctionalisnotsurprising.ItisknownthattheKSreferenceisusuallyabadreferenceforperturbationexpansions.Forexample,Warken[ 24 ]analyzedtheperturbationseriesbasedontheKSorbitalsandshowedthatitusuallyhasradiusofconvergencesmallerthan1.Therefore,theKS-basedperturbationseriesoftendivergesformolecularsystems.Thispresentsanevenbiggerproblemforthedeterminationofthepotential.Indeed,truncatingadivergentseriesatsomeniteorderstillresultsinaniteenergy.Moreover,itisknownthatsomeasymptoticallydivergentseriesmaygivedecentapproximationsinlowerorders.However,theconvergenceoftheseriesiscrucialtoobtainevenalower-orderpotential.BecausetheOEPequationforthepotentialissolvediteratively,thelargetermswillaccumulateandtheiterativesolutionwilldiverge.Inouropinion,therearetwoprimarycauseforthedivergenceoftheseries.First,theKSmodelfeaturesalocalSCFpotential,andastheresult,thevirtualorbitalsliemuchlowerthen,forexample,Hartree-Fockones.Astheresult,theoccupied-virtualenergydierencearesmallerandtheresultingseriesfeaturessmalldenominators.Second,unliketheHartree-Fockcase,theKSreferenceHamiltonianisnotequaltotheone-bodypartofthetrueHamiltonian.Thus,theperturbationcontainsaone-bodypartthatcanbelargeandultimatelyleadtothe

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divergenceoftheseries.Althoughnotimmediatelyobvious,thesetwoproblemsarecloselyrelated.Indeed,asitwillbeshownbelow,thelargeone-bodytermscanberemovedfromtheseriesbyresummingthemtoallorders.Astheresults,thedenominatorsareformedbydiagonalmatrixelementsoftheFockoperator,whichcorrespondtolargedierences.Toseewherethesetermsarise,considerthetrueHamiltonianinsecond-quantizedform^H=allXp;qhpq^ayp^aq+1 4Xp;q;r;spqsr^ayp^ayq^as^ar=allXp;qhpq+occ:Xipiqi^ayp^aq+1 40Xp;q;r;spqsr^ayp^ayq^as^ar ^V=^H^Hs=allXp;qh+vH+vnlxhspq^ayp^aq+^W=allXp;qvnlxvxcpq^ayp^aq+^W(3-6)where^Wstandsforthetwo-bodyterms.Theseone-bodytermscanbelargeandpotentiallyleadtoadivergentperturbationseries.IncaseofabinitioDFT,thesituationisslightlyimproved,becausetheuseoftheexactexchangepotentialreducesthesizeofcertainone-body

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terms.Indeed,theOEPequationfortheEXXpotential

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^QE0H0=0(3-13)where^Q=^1istheprojectoronthecomplementaryspacespannedbyallbutground-stateeigenfunctionsofH0.ProjectingEq. 3-12 onthecomplementaryspaceandaddingEq. 3-13 oneobtains ^QE0H0^QVE=0;(3-14)or E0H0VE:(3-15)Applyingthisrelationiteratively,oneobtainstheseriesforthetruewavefunction=+=+^Q E0H0VE=+^Q E0H0VE+"^Q E0H0VE#2=1Xn=0^Q E0H0VEn

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Notethatintheexpression,thetermsofthesumabovedonotcorrespondtoanyorderofperturbationtheorybecauseEhastermsofallorders.Toobtaintheorder-by-orderexpansiononewouldhavetosubstitutetheorder-by-orderexpressionforE.This,however,isnotnecessaryforthecurrentdiscussion.Instead,oneproceedsbyidentifyingtheproblematictermsV00andreorderingthem=+1Xn=0^Q E0H0V00E00n+1Xn=0^Q E0H0V00E00nV00E00n^Q E0H0V0E0+::: ^QV00E00=0(3-18)onecanperformthesummationtoobtain=1Xn=01Xm=0^Q E0H0V00E00m^Q E0H0V0E0n=1Xn=0^R0V0E0n E0H0V00E00m^Q E0H0=^Q E0+E00H0V00

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operator^R00.ItisbasedonthenewreferenceHamiltonian^H00=H0+^V00=all:Xpphs+up^ayp^ap+occ:Xi;jivH+^vnlxuj^ayi^aj+virt:Xa;bavH+^vnlxub^aya^ab=occ:Xi;jfij^ayi^aj+occ:Xa;bfab^aya^ab 1-36 .ThecasewhenthereferenceHamiltonianisnotdiagonalintermsoftheSCFeigenfunctionsistypicalforthegeneralizedMany-BodyPerturbationTheory[ 2 ].Twopossiblesolutionstothisproblemaretondtheinverse(i.e.,theresolventoperator)iterativelyortondaunitarytransformationthatmakestheoccupied-occupiedandvirtual-virtualblocksofthereferenceHamiltoniandiagonal.Whilethesetwosolutionsareformallyequivalent,theunitarytransformationislesscomputationallyexpensivesinceitinvolvesonlyoperationswithtwo-indexquantities.Thus,ifateachSCFiteration,onetransformstheoccupiedorbitalssothatfij=ijfiiandthevirtualorbitalssothatfab=abfaathen^H00becomediagonalinthisbasis.Sincethistransformationdoesnotmixtheoccupiedandvirtualorbitals,allthephysicallyrelevantquantitiessuchasenergyanddensityarenotaectedbythistransformation.ThisnewsetoforbitalsisoneofthepossiblenoncanonicalrepresentationsoftheKSorbitalsandiscalledsemicanonicalbecauseitdiagonalizetheoccupied-occupiedandvirtual-virtualblockoftheFockoperator.Thisis

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whythenewperturbationseriesandcorrespondingfunctionalsarereferredtoas\semicanonical"(SC). 3.3 ImplementationofthePT2andPT2SCFunctionalsAsinthecaseoftheEXXfunctional,thecentralelementofimplementationoforbital-dependentcorrelationfunctionalsistheOEPmethodforthecorrespondingpotential.AsonecanseefromEq. 3-3 ,theonlydierencebetweentheintegralequationsfortheEXXandPT2(orPT2SC)potentialsisontheright-handside.Thus,ifonehastheOEPmethodimplementedfortheEXXpotential,itsextensionforthesecond-orderpotentialisatedious,butstraightforwardtask.AsfortheEXXpotential,theimplementationbasedontheauxiliary-basisOEPmethodwasused.InthismethodtheintegralOEPequationistransformedintoalinearmatrixproblembyprojectingthereal-spacequantitiesontoanauxiliarybasis.Theonlydierenceisthatthecalculationoftheright-handsiderequiresanumberofcontractionsofmatrixelementsofauxiliarybasisfunctionswiththeone-andtwo-electronintegrals.Thus,atypicalSCFiterationwithboththeEXXandPT2potentialsproceedsasfollows

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potentialintheauxiliary-basisrepresentation.Calculateitsmatrixelementswithrespecttoatomicorbitals. 3.4 NumericalTestsforAbinitioFunctionals CorrelationEnergyintheHigh-DensityLimit.Thersttestwasfortheperformanceoftheabinitiocorrelationfunctionalsinthehigh-densitylimit.Itisknown[ 19 ]thatthecontributionsoforderhigherthansecondscaleasnegativepowersofthescalingparameterand,therefore,

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vanishasthescalingparameterapproachesinnity(i.e.,inthehigh-densitylimit).Therefore,thesecond-orderenergyexpressionistheexactlimitforthecorrelationfunctionalatinnitelylargescalingparameterandthecombinationoftheexactexchangeandsecond-ordercorrelationfunctionalsbecomestheexactexchange-correlationfunctionalinthehigh-densitylimit.ThePT2SCfunctionalisbasedontheenergyexpressionthatdoesnotscalehomogeneouslyduetothepresenceoftheFockoperatorinthedenominators.Nevertheless,itisequivalenttoainnite-orderseries,wherethehigher-ordertermsagainscaleasthenegativepowersofthescalingparameterandvanishasinthehigh-densitylimit.Therefore,theEXX-PT2SCfunctionalmustapproachtheexactexchange-correlationfunctionalaswell.Totestthepropertiesofabinitiofunctionalsinthehigh-densitylimit,wecalculatedthecorrelationenergiesoftheseriesoftwo-electronatomicionswithincreasingnuclearchargeZ(Table 3{1 .Two-electronsystemsareparticularlyconvenientbecausetheexactexchangepotentialisjusthalfoftheHartreepotential,hence,thereisnoerrorassociatedwiththeauxiliary-basisimplementationofEXXpotential.Moreover,thefullCIenergyisreadilyavailableasonlysingleanddoubleexcitationscontributetothecorrelatedwavefunction(theCoupled-Clustermethodwithsingleanddoubleexcitations[CCSD]wasusedtoobtainthefullCIenergy.)TheresultsdemonstratethatthePT2energyindeedrapidlyapproachesthefullCIvalue.ThisisinagreementwiththeresultsreportedbyGrabowskietal.[ 22 ].NotethattheGGAcorrelationfunctionalssuchasPBEorLYPdonothavethecorrectscalingandresultinnonvanishingerror. CorrelationEnergiesofClosed-ShellAtoms.Thenexttestsetconsistedoftherstsixclosed-shellatomswithsingletgroundstates.Table 3{2 reportsthedeviationofthePT2andPT2SCcorrelation

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Table3{1: Performanceofabinitioandconventionalcorrelationfunctionalsinthehigh-densitylimit.TherstcolumngivesfullCIcorrelationenergiesandtheremainingcolumnsgivethedierencesbetweenthesevaluesandcorrelationenergycalculatedwithabinitioandconventionalcorrelationfunctionals.UncontractedRoosaugmenteddoublezetabasisset.AllvaluesareinmilliHartree. ZIonFCIPT2PT2SCPBEBLYP 2He0+38:634:02(10%)6:41(17%)2:27(6%)5:05(13%)4Be2+39:781:84(5%)3:42(9%)5:75(14%)9:48(24%)10Ne8+40:190:65(2%)1:47(4%)7:52(19%)10:21(25%)12Mg10+40:370:54(1%)1:24(3%)7:52(19%)9:97(25%)18Ar16+40:510:35(1%)0:83(2%)7:61(19%)9:54(24%)20Ca18+40:480:08(0%)0:99(2%)7:67(19%)9:55(24%) energiesfromCCSD(T)values.TheCoupledClustermethodwithsingleanddoubleexcitations,andnoniterativeinclusionoftripleexcitationsprovidesveryaccurateenergiesforclosed-shellatomsandmoleculesattheequilibriumgeometriesandwillberegardedasthecorrelationlimitforthegivenbasisset.Theenergiesobtainedwithsecond-orderMller-PlessetperturbationtheoryandCCSDmethodaregivenforcomparison. Table3{2: Performanceofabinitiocorrelationfunctionalsforclosed-shellatoms.TherstcolumngivetheCCSD(T)correlationenergiesandtheremainingcolumnsgiveabsoluteandrelativedeviationsfromthesevalues.TheMP2andCCSDvaluesaregivenforcomparison.Roosaugmenteddoublezetabasisset.AllvaluesareinmilliHartree. AtomCCSD(T)PT2PT2SCMP2CCSD He37:13:3(9%)6:9(19%)6:9(19%)0:0(0%)Be53:4N/C18:4(34%)18:5(35%)0:2(0%)Ne267:580:2(30%)3:7(1%)4:5(2%)4:1(2%)Mg51:224:5(48%)11:7(23%)11:9(23%)0:7(1%)Ar228:587:7(38%)15:6(7%)15:8(7%)4:1(2%)Ca84:942:8(50%)11:6(14%)12:0(14%)2:0(2%) Asonecansee,thePT2functionalresultsinaccuratecorrelationenergyforHe,butsignicantlyoverestimatesthecorrelationenergyforlargeratoms.AlsotheiterativesolutionforthePT2potentialdidnotconvergeinthecaseofBe.Onthecontrary,thePT2SCenergyisslightlyworseforHe,butgivesmuchbetter

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estimationofcorrelationenergiesforotheratoms.Also,theiterativesolutionsforthePT2SCpotentialconvergedineverycase.NotethatthePT2SCfunctionalperformsslightlybetterthanMP2. CorrelationEnergiesofMolecules.BasedontheresultsforatomiccorrelationenergiesonecanconcludethatthePT2functionalsignicantlyoverestimatesthecorrelationenergies,whilethePT2SCfunctionaloersamoreadequatedescriptionofthecorrelationeects.Tofurtherverifythisconclusionaseriesofcalculationswasperformedforthesamesetof35closed-shellmoleculesthatwasusedinChapter2.Figure 3{1 reportstherelativedeviationsofthecorrelationenergiesfromtheCCSD(T)valuesaveragedover35molecules (method)=1 PerformanceofabinitioDFTandabinitiowavefunctionmethodsintotalenergycalculationsfortheG1testset.ShownareaveragerelativedeviationfromtheCCSD(T)values.Roosaugmenteddoublezetabasisset.

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Theresultsareverysimilartothoseforatoms.First,theiterativesolutionofOEPequationforthePT2potentialdivergedfor19molecules(includingLiH,NH3,N2,CO,andothers).Fortheremaining16molecules,thePT2functionaloverestimatedthecorrelationenergyonaverageby40%.Onthecontrary,theiterativesolutionforthePT2SCpotentialconvergedforall35molecules.ThePT2SCfunctionalledtoanaverageerrorof11.7%,slightlybetterthantheMP2valueof12.3%.Notethatforthesesystems,theHF-basedperturbationtheoryindeedprovidesaseriesthatsystematicallyconvergestotheexactanswer.Includinghigher-ordercorrectionsreducestheerrorfromtheMP2valueof12.3%,totheMP3valueof8.0%totheMP4valueof3.8%.ThissupportsthepromiseofabinitioDFTtoprovideaseriesofsystematicallyimprovingapproximationstothecorrelationfunctional.Ofcourse,theCoupledClustermethodprovidesamorerapidlyconvergingseriesresultingintheaverageerrorof6.1%alreadyattheCCSDlevel.ItshouldbeemphasizedthatitistheabinitiocharacterofthePT2andPT2SCfunctionalsthatallowsustocomparetheabsolutevaluesofthecorrelationenergy.Onthecontrary,onecannotdirectlycomparetheGGAenergiestoabinitioresults.Itisawell-knownfactthattheabsolutevaluesoftheDFTenergiescanbeverydierentfromthewavefunctioncorrelationlimit.Instead,onehastocomparerelativequantitieslikeatomizationenergiestoassessthequalityofconventionalfunctionals. TotalEnergyasaFunctionoftheBondLength.Nexttest(Figure 3{2 )assessedtheperformanceofabinitioDFTfunctionalsindescriptionofthepotentialenergysurfaces(i.e.,thetotalenergyofamoleculeasafunctionofthebondlength.)Fourmoleculeswerechosentorepresentdierenttypesofchemicalbonds:theionicbond(HF),symmetricsinglecovalentbond(F2),doublebond(H2Owherethehydrogenatomsweresimultaneouslypulled

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awayfromtheoxygenatom),andatriplebond(N2).Toremovetheambiguitywithrespecttothecurves'absoluteposition,thecurveswereshiftedverticallysothatallcurvescrosstheCCSDTcurveattheexperimentalbondlengths.Suchashiftnotonlyfacilitatesthecomparisonoftheshapes,butalsoallowsadirectcomparisonbetweentheabinitioandconventionalDFT(PBEinthiscase)methods. PerformanceofabinitioDFT,abinitiowavefunction,andconventionalDFTmethodsincalculationsofthetotalenergyasafunctionofthebondlengths.Roosaugmenteddoublezetabasisset. Asonecanseefromthegures,theEXX-PT2functionalfailedtoreproduceameaningfulenergycurveforanyofthefourmolecules.ForallthesystemsexceptN2theEXX-PT2SCcurvesliesclosertotheCCSDTonethatMP2.TheN2plotclearlydemonstratesthatoneshouldnotapplyaperturbativemethodtoamultiple-bondbreakingproblem.

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Atomicandmoleculardensities.ThequalityofaDFTfunctionalmustalsobereectedintheconvergeddensity,i.e.,thedensitythatminimizesthefunctional.Sinceitisdiculttocompareatomicormoleculardensitiesdirectly,thedensitymomentswerecomparedinstead.Table 3{3 reportsthedensitymomentsforNecalculatedwithabinitioDFT,abinitiowavefunction,andconventionalDFTmethods.ThedensityobtainedwiththeCCSD(T)methodwasusedasthereference.Also,unlikeinthecaseoftotalenergies,theGGAdensity(obtainedwithPBEfunctional)canbedirectlycomparedtotheabinitioresults.Asevidentfromthetable,theoverestimationofcorrelationeectsbythePT2functionalleadstothesituationwheretheexchange-only(EXX)densityisbetterthanwhenthecorrelationeectsareincluded(EXX-PT2).However,theEXX-PT2SCfunctionalresultsinadensitythatisclosertoCCSD(T)thananyothermethod,intheregionsbothclosetothenucleus(assampledbytheaveragevaluesofthenegativepowersofr)andawayfromthenucleus(assampledbytheaveragevaluesofthepositivepowersofr). Table3{3: DensitymomentsofNecalculatedwithabinitioDFT,abinitiowavefunctionandconventionalDFTmethods.UncontractedRoosaugmenteddoublezetabasisset. Methodr2r1rr2r3r4r5 Asthetestfortheelectronicdensityofmolecules,themoleculardipolemomentswerecalculated.Figure 3{3 showstheaverage(over22systemswithnonzerodipolemoments)deviationofthecomputeddipolemomentsfromtheexperimentalvalues.For9(outof22)systemswherethesolutionforthePT2potentialconverged,thedeviationfromtheexperimentaldipolemomentisnoless

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than0.13Debye(HF)withthelargesterrorof0.56DebyeforH2O.Thisresultsinanaverageerrorgreaterthanthatofexchange-onlymethods.Again,EXX-PT2SCresultsindipolemomentsthatimprovesupontheMP2values.InthecaseofEXX-PT2SC,thelargesterrorsinthecomputeddipolemomentareforSiO:0.45Debye(CCSD:0.14,PBE:0.22)andSO2:0.22Debye(CCSD:0.10,PBE:0.16).InthecaseofPBE,forNaCl:0.40Debye(OEP2(SC):0.10,CCSD:0.22)andLiH:0.27(OEP(SC):0.04,CCSD:0.01).IncaseofCCSD,thelargesterrorsareforNaClandSiO. PerformanceofabinitioDFT,abinitiowavefunction,andconventionalDFTmethodsindipolemomentcalculationsfortheG1testset.Shownaretheaverageabsolutedeviationfromtheexperimentalvalues.Roosaugmenteddoublezetabasisset.

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4.1 ConnectionbetweenEnergy,Density,andPotentialTheeectivepotentialoftheKohn-Shammodelisdenedasthederivativeoftheenergyfunctionalwithrespecttothedensity.Thisdenitionfollowsfromthevariationalconditionontheenergyfunctional.ItensuresthatiftheKohn-ShamSCFmodelgeneratesthedensitythatminimizestheenergyfunctionaland,thus,isequaltotheexactground-statedensity.InthecontextofabinitioDFT,thisshouldmeanthatthepotentialdenedthroughthefunctionalderivativegeneratestheSCFdensitythatminimizestheabinitioenergyexpression,uponwhichthefunctionalisbased.ItalsoshouldmeanthattheSCFdensityisequaltothecorrespondingabinitiomany-bodydensity.Forexample,ifthefunctionalisbasedontheenergyexpressionthroughsecond-orderthendeningthepotentialas 4-1 andthestationaryconditionofEq. 4-2 .Applyingthechainrule(Appendix 48

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A ),wetransformthefunctionalderivativeintothefollowingintegralrelation vs(r)Zdrs(r)vs(r)=s(r)s(r)Zdr0vs(r0)s(r0) 4-4 becomesthestationaryconditionontheenergy vs(r)=0(4-7) 4.2 DiagrammaticDerivationoftheOptimizedEectivePotentialEquationInChapter2wedenedthecorrelationpotentialthroughthefunctionalderivative.Toobtaintheactualexpression,wehavetondthederivativeofthesecond-orderenergywithrespecttothepotential.Totakethisderivativealgebraicallyisquiteanexercise(Appendix C ).Here,wedescribeadiagrammatictechniquethatsignicantlysimpliesthederivationoftheseexpressions.ThistechniqueissimilartotheoneusedbyBartlettetal.[ 23 ]butoperatesdirectlywiththeenergyfunctionalratherthanthedensity.Wewillusetheoperatorsinsecond-quantizedformsandantisymmetric

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Goldstone(ASG)diagramstorepresenttheperturbativecorrectionstotheenergyandenergyderivative.Aproperintroductionofthesecondquantizationandthediagrammatictechniquescanbefoundinbooksonmany-bodytheory(see,forexample,Ref.[ 3 ]).Inthediagrammaticrepresentation,thetrueHamiltoniancanbewrittenas^H=allXp;qhpq^ayp^aq+1 4Xp;q;r;spqsr^ayp^ayq^as^ar= ^Hc=elec:Xih(ri)=elec:Xi[1 2r2i+vext(ri)];(4-9)andtheremainingdiagramsrepresentthetwo-electronterms ^W=elec:Xi6=j1

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whereudenotestheeectiveelectron-electronpartoftheSCFpotential

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parts).Thus,theperturbationseriesfortheenergyisrepresentedbythefollowingdiagramsE0+E(1)+E(2)+:::= 3 ]resultsinthefamiliarexpressionfortheenergyseriesE0+E(1)+E(2)=occ:Xii1 2r2+vext+ui+occ:Xii^vnlxui+occ:Xiocc:Xaa^vnlxui E[u+u]=E[u]+E(1)+:::=E[u]+ZdrE u(r)u(r)+:::(4-18)SinceE[u]isgiventhroughtheperturbationtheoryaswell,thelinearresponseoftheorder-by-ordercorrectionstotheenergyisgiventhroughthedouble-perturbationexpansion,whereoneperturbationisthemany-bodyperturbationandthesecondoneisu.

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Thus,theorder-by-ordercontributionstothestationaryconditionareobtaineddiagrammaticallybydrawingallpossibleclosed,linkeddiagramsthatinvolveone-vertexandthecorrespondingnumberofthemany-bodyvertexesE0+E(1)+E(2)+::: 4-19 isanonlinear,inhomogeneousequationfortheeectiveelectron-electronpartoftheKSpotential,u.Tobeconsistentwiththeperturbativetreatmentoftheenergy,onehastosolvethisnonlinearequationiteratively,byneglectingthecontributionofthehigher-ordertermsateveryiterations.Thus,ontherstiterationoneneglectsallbutlinearterms,andsolvestheresultinglinearequationfortherstapproximationtou(r) 4-21 ).Then,oneproceedsbyincludinghigherandhighertermsuntilthenth-orderapproximationfortheoptimizedpotentialisobtained.Withthenewvalueforthepotential,one

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constructstheOEPHamiltoniananddiagonalizesittoobtainanewsetofOEPorbitalsandorbitalenergies.ThisconcludesoneiterationoftheSCFprocedure,whichmustberepeateduntilself-consistencyisreachedforboththeorbitalsandoptimizedpotential. 2occ:Xi;j;kvirt:Xa;bk(r)a(r)ijkbabij 2occ:Xi;jvirt:Xa;b;ci(r)c(r)cjababij 2occ:Xi;j;kvirt:Xa;bi(r)k(r)kjababij 2occ:Xi;jvirt:Xa;b;cc(r)a(r)ijcbabij (4-21)

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4.3 MixingExactNonlocalandLocalExchangeAninterestingextensionofHohenberg-Kohn-Shammodelistousethenonlocalexchangeoperatorinsteadof(orinadditionto)thelocalexchangepotential.Inparticular,usingtheadiabaticintegrationformula,Becke[ 25 ]showedthataGGAexchange-correlationfunctionalcombinedinequalproportionswiththeHartree-Fockexchangefunctional,so-calledBeckehalf-and-halffunctional,shouldbeabetterapproximationtotheexactfunctional.Semiempiricalparametrizationtoexperimentaldatashowedthattheinclusionofabout1=5ofHartree-FockexchangeintheBLYPfunctional(B3LYPfunctional)actuallyleadstothemostsignicantimprovementintheperformance.SuchhybridfunctionalsarecurrentlythemostsuccessfulextensionoftheGGAformandthemostaccuratefunctionalsintheoverallhierarchyofconventionalfunctionals.Allabinitioexchange-correlationfunctionalsincludetheexactexchangefunctional,butwecantreatapartofitasanimplicitfunctionalofthedensityandtheotherpartasafunctionaloforbitals.ThisistheideaofbehindtheHartree-Fock-Kohn-Shammodel[ 26 ],whichtreatsapartoftheenergyasafunctionaloforbitalsandtheremainingpartasafunctionalofthedensity.ThecorrespondingconstrainedminimizationprocedureresultsinEuler-Lagrangeequationsthatmixtheorbitalderivative(likeinHartree-Focktheory)andthedensityderivative(likeinKohn-Shamtheory).WecanalsofurtherextendtheHFKSmodelandtreatonlyafractionoftheexactexchangefunctionalasthefunctionaloforbitals.Thus,thehybridabinitiofunctionalcanbewrittenas

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wheredenesthefractionoftheexactexchangeenergytreatedasthefunctionaloforbitalsratherthanimplicitfunctionalofthedensity.Besidejusttheoreticalcuriosity,thereisasimplepracticalreasonwhysuchacombinationisworthinvestigating.Thepresenceofnonlocalexchangepartiallyremovestheself-interactioncomponentfromtheoccupiedorbitalenergies.Italsoresultsinmorediusevirtualorbitalsthatcorrespondtoelectronattachment(ratherthanexcitation)states.Althoughitmaynotappearadvantageousfromaformalpointofview,itdoesimprovetheSCFmodelforpurposeoftheperturbationexpansion.Indeed,aswediscussedinChapter3,alocalSCFpotentialleadstosmalldierencesbetweentheoccupiedandvirtualorbitalandastheresult,theperturbationseriesfeaturessmalldenominatorsandoftendiverges.Presenceofthenonlocalexchangeshiftsthevirtualenergiesupand,therefore,shouldresultinanimprovedperturbationseries.WeimplementedthehybridabinitiofunctionalinourabinitioDFTcode.TheimplementationrequiredonlyminormodicationsoftheschemedescribedinChapter2.First,whenthesecond-orderpotentialandenergyarecalculated,theone-bodypartoftheperturbationisscaledby1(beingthefractionofthenonlocalexchangeoperator).Second,theSCFHamiltonianisnowthesumofthecoreHamiltonian,thenonlocalexchangeoperator,localexchanges,andlocalcorrelationpotentials.AswediscussedinChapter3,inthecaseofBe,theiterativesolutionforthePT2potentialcannotbeobtainedbecausetheperturbationseriesdiverges.However,already20percentofnonlocalexchangeresultedinaconvergentseriesandwecouldobtainthecorrespondingcorrelationpotential.Forotheratoms(Table 3{2 ),thePT2energyexpression(smallerdenominators)tendstooverestimatethecorrelationenergies,whilethePT2SCandMP2expressions(largerdenominators)tendtounderestimatethecorrelation

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energy.Therefore,includingafractionofnonlocalexchangecanleadtothecorrelationenergythatisveryclosetotheexactone.ThisisexactlywhathappenedinthecaseofBe(Figure 4{1 ).Atabout40%ofnonlocalexchange,theEXX-PT2henergyandthedensitymomentcrossedtheCCSD(T)values.Thus,inthecaseofBe,about2=5ofthenonlocalexchangeoperatorresultsintheoptimalreferenceforthesecond-orderfunctionalandpotential.Wewereabletondsuchoptimalfractionsforeachsystemoutoftheseveralclosed-shellatomsandmoleculesconsidered(Table 4{1 ). Table4{1: PerformanceofthehybridabinitiofunctionalEXX-PT2hwithoptimizedfractionofnonlocalexchange.Shownaretotalenergies(milliHartree)anddensitymoments(atomicunits)deviationsfromtheCCSD(T)values.TheEXX-PT2SCandCCSDvaluesareshownforcomparison.UncontractedRoosaugmenteddoublezetabasis. Atom EXX-PT2h EXX-PT2SC CCSD 310:00:00 6:40:00 0:00:00Be 360:10:01 20:70:05 0:60:00Ne 970:10:03 4:00:02 5:10:01Mg 770:00:01 12:60:00 3:10:01Ar 850:40:00 12:90:01 5:70:01Ca 770:20:06 12:40:04 3:40:01 Molecule EXX-PT2h EXX-PT2SC CCSD 990:10:02 4:50:03 5:70:02H2O 920:00:04 9:70:02 6:70:02NH3 800:10:05 18:00:00 6:50:02CH4 650:30:00 25:80:00 5:20:00CO 950:10:30 11:80:18 15:00:05N2 970:40:00 9:60:00 15:90:00 However,thevaluesoftheoptimizedfractionisdierentforeverysystem.Inparticular,wenoticedthatforsystemswherethedynamicalcorrelationprevailsandthePT2SCenergyisclosetoCCSD(T)one(Ne,H2O,N2),theoptimalfractiontendtobecloseto100%.Onthecontrary,ifthePT2SCerrorismuchlargerthantheCCSDone(Be,CH4),thentheoptimalfractionissmaller.Also,

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ThetotalenergyandrstdensitymomentofBecalculatedwiththeEXX-PT2hfunctionalwithvariousfractionsofthenonlocalexchangeoperator.UncontractedRoosaugmenteddoublezetabasis.

PAGE 67

thefractionthatresultsinanalmostexactcorrelationenergydoesnotnecessarilyleadtoanimproveddensity.Forexample,inthecaseofCO,95%ofnonlocalexchangeresultsintheerrorof0.1milliHartreeintheenergy,but0.3Debyeinthedipolemoment.Thus,thepresentresultsdonotsupporttheutilityofthehybridabinitiofunctional.Apossibleroutetoapracticalmethodistondthefractionofnonlocalexchangethatminimizestheaverageerrorinthetotalenergiesofalargesetofsystems. 4.4 Second-OrderPotentialwithinCommonEnergyDenominatorApproximationAswehavementionedinSection2,analternativetonumericalornite-basis-basedsolutionoftheOEPintegralequationistointroduceanapproximationtotheOEPkernel.Forexample,theKLIapproximationofKriegeretal.[ 10 ]neglectsthecontributionofallbutdiagonalmatrixelementstothekernel.Asaresult,theKLIpotentialisgivenviaasimplerecursiverelationandcanreadilybefounditeratively.Arelatedapproximation,theCommonEnergyDenominatorApproximation(CEDA)proposedbyGruningetal.[ 11 ]retainsmorefeaturesoftheexactkernel,butliketheKLIapproximation,allowsonetocalculatetheapproximatepotentialviaasimpleiterativeprocedure.Anequivalentscheme,LocalizedHartree-Fock(LHF)method,wasindependentlyintroducedbyDellaSalaandGorling[ 27 ].Theseapproximationscanbeusedtoobtainanapproximatesecond-orderpotentialaswell.InwhatfollowswerstintroducethemathematicsoftheseapproximationsandthenapplythemtotheOEPequationforthesecond-orderpotential.TheCommonEnergyDenominatorapproximationconsistoftwosteps.TherststepistoapproximatealloftheenergydenominatorsintheOEPequationby

PAGE 68

aconstantocc:XivirtXai(r)a(r) occ:XivirtXai(r)a(r)Zdr0a(r0)vexx(r0)i(r0)+c:c:=1 occ:XivirtXai(r)a(r)Zdr0a(r0)^vnlx(r0)i(r0)+c:c: 2Zdr(r)x(r)(4-30)

PAGE 69

IntheKLIapproximation,thisexpressionisfurthersimpliedbyneglectingallbutdiagonalelementsfromthesummation 2{1 ,x(r)isacloseapproximationtotheEXXpotentialintheinnerandasymptoticregions.However,itlacksthestructureoftheEXXpotentialintheintershellregion.TheKLIapproximationand,subsequently,CEDAaddmoreorbital-dependenttermstotheapproximationpotentialthatwillcorrectx(r)towardsthetrueEXXpotential.TheideaproposedinthissectionistoapplytheCommonEnergyDenominatorApproximationtothesecond-ordercorrelationpotential.ThiswillallowonetosubstitutetheOEPintegralequationbyarecursiverelationforthecorrelationpotential.Moreover,applyingCEDAwillsignicantlysimplifytheexpressionfortheright-handsideoftheequationresultinginareducedcomputationcostofthesecond-orderscheme.IntroducingCEDAtotheequationforthesecond-orderpotentialocc:Xivirt:Xai(r)a(r)av2ci (4-32)weobtain 2(r)E(2) C-8 )tothisderivativeas

PAGE 70

2occ:Xi;jvirt:Xaa(r)j(r)tiaZdr0j(r0)i(r0) 2occ:Xi;jvirt:Xaa(r)j(r)j^vnlxvexxitiac:c:3 2occ:Xi;jvirt:Xaj(r)i(r)a^vnlxvexxjtiac:c:+2occ:Xi;j;kvirt:Xaj(r)k(r)kaijtia+c:c:

PAGE 71

Similarly,thedouble-excitationcontribution(Eq. C-9 )canbewrittenasE(2)D 2occ:Xi;jvirt:Xa;b;cc(r)i(r)abcjtijab 2occ:Xi;j;kvirt:Xa;ba(r)k(r)kbijtijab 2occ:Xi;j;kvirt:Xa;bk(r)i(r)abkjtijab 2occ:Xi;jvirt:Xa;b;ca(r)c(r)cbijtijab 4occ:Xi;jvirt:Xa;ba(r)i(r)Zdr0b(r0)j(r0) 4occ:Xi;j;kvirt:Xa;bk(r)i(r)abkjtijabc:c:5 4occ:Xi;j;kvirt:Xa;ba(r)k(r)kbijtijabc:c:

PAGE 72

second-orderpotentialrequiresexactlythesameamountofmemoryandoperationsasanMP2calculation.Weexpectthattheadvantagesoftreatingthesecond-orderenergyexpressionasafunctionalwillberetainedevenifthepotentialisfoundapproximately.Therefore,theEXX-PT2SCfunctionalwiththepotentialfoundwithintheCEDAwillbeapracticalalternativetotheMP2method.

PAGE 73

65

PAGE 74

Thenewfunctionaloersamorebalanceddescriptionofcorrelationeects,aswasdemonstratedinapplicationstoanumberofclosed-shellatomsandmolecules.Thecombinationoftheexactexchangefunctionalandmodiedcorrelationfunctional,EXX-PT2SC,resultsintotalenergiesthatimproveupontheMP2values,bothattheequilibriumgeometriesandasafunctionofthebondlength.EXX-PT2SCresultsindensitiesthatareslightlybetterthantheMP2orGGAones.Also,theEXX-PT2SCpotentialqualitativelyreproducestheshapeoftheexactexchange-correlationpotential.Thecorrectshapeofthepotentialleadstoameaningfulone-electronenergyspectrumwiththeHOMOenergyapproximatingtheprincipalionizationpotential.Thus,abinitioDFTalreadywiththeEXX-PT2SCexchange-correlationfunctionalleadstoaveryaccurateDFTmethodsuperiortothecurrentconventionalfunctionals.ThecurrentimplementationcanbeextendedtohigherordersofperturbationtheoryorCoupled-Clusteransatz.Forexample,onecanstraightforwardlyderiveacorrelationpotentialfromthelinearizedCoupledClusterenergyexpressions.ThiswillfullltheoriginalpromiseofabinitioDFTtoprovideahierarchyofexchange-correlationfunctionalsofmonotonicallyincreasingaccuracy.Forpracticalpurposes,abinitioDFTmustbecomparedtotheabinitiowavefunctionmethodsbecauseitscomputationalcostissignicantlyhigherthenconventionalDFT.TheresultsofthisstudyshowedthatEXX-PT2SCfunctionalimprovesontheconventionalperturbationtheory,MP2.However,thecomputationalcostofobtainingtheEXX-PT2SCpotentialisthatofMP2timesthenumberofiterations.Formanysystemsconsideredinthetests,theseimprovementsarenotverysignicantgiventheincreasedcomputationalcost.FurthertestsarenecessarytoestablishwhetherabinitioDFTcanbecomea

PAGE 75

practicalalternativetoMP2methodswhenCoupled-Clusterormultireferencemethodsarenotaccessible.Besidesimprovementsintheenergyanddensity,abinitioDFTalsooerstheadvantageoftransformingtheabinitioqualitydescriptionofthecorrelationeectsintoasimpleSCFmodel.Theresultingmolecularorbitalsincludetheexactexchangeinteractionandmostofthecorrelationeects.TheseorbitalsaresuperiortobothHF(nocorrelationeects)andconventionalDFT(poorexchange-correlationpotentials)andcanbeusedtodescribethechemicalpropertiesofthesystem.Also,theSCFmodelofabinitioDFTshouldprovideanexcellentstartingpointforlinear-responseandtime-dependentmethods.Infact,theextensionoftheEXXmethodtolinear-responseproblemshasbeenalreadyformulated[ 28 ]andimplemented[ 16 ]andtheextensionofthesecond-ordermethodisunderway.TomakeabinitioDFTmorepractical,onecanreducethecostoftheEXX-PT2SCschemebyintroducingapproximationstothefunctionalorpotential.ThemostdirectrouteistosimplifytheOEPprocedureforthepotential.Theexpressionforthesecond-orderpotentialwithintheCommonEnergyDenominatorApproximationwasderived.Usingthisapproximationwilldecreasetheoverallcomputationalcostandeliminatetheintegralequation.Giventhatthecorrelationcontributiontotheorbitalsissmall,onecanuseexchange-onlyorbitalstocalculatetheCEDApotentialratherthansolveforititerativelyuntilfullself-consistency.TheresultingschemewillrequireexactlythesametimeandcomputationresourcesastheMP2method.IfusingthecorrelationpotentialstillimprovestheMP2results,suchanapproximateschemewillbeanalternativetotheMP2method.Notethatanyapproximationstothepotentialdoesnotreducethecostassociatedwiththesecond-orderenergyexpression.Thus,toreducethecostbeyondMP2,oneneedstodevelopanapproximationtothefunctional.Suchanapproximation

PAGE 76

mustmakeuseofthevariationaloptimizationtocompensateforthesimplicationsintheenergyexpression.

PAGE 77

(^t+vs)p=pp(A-1)theirfunctionalderivativewithrespecttothepotentialisdenedthroughthelinearresponseKSequations.Considerainnitesimalchangeinthepotential,vs!vs+vs (vsp)p=(p^tvs)p(A-3)Byprojectingthisequationonpandusingtheorbitalorthonormality 69

PAGE 78

Byprojectionontheqsuchthatp6=q,forthelinearreponseoftheeigenfunctionsonehas

PAGE 79

therefore,thesummationindexcanberestrictedtorunovervirtualorbitalsonly (A-18)Forexample,theexpressionfortheright-handsideoftheOEPequationfortheEXXpotentialisgivenbythefunctionalderivativeoftheEXXfunctional 2occ:Xi;jijji(A-19)withrespecttothepotential.SinceEEXXdoesnotdependonthevirtualorbitalsnororbitalenergies,theonlycontributioncomesfromthetermsofthechainrule

PAGE 80

involvingtheoccupiedorbitalsEEXX 2occ:Xj;k i(r)Zdr0r00j(r0)k(r00)k(r0)j(r00)

PAGE 81

73

PAGE 82

where

PAGE 83

A-18 ,oneobtainsthefollowingexpressionforthederivativeofatwo-electronintegralpqrs (C-3) 75

PAGE 84

Thus,p^vnlxvexxq vs(r)1 p1 (ia)(ia)+a(r)a(r) (ia)(ia) (C-5)Thus,thederivativeofthesingle-excitationcontributiontothesecond-orderenergyisE(2)S vs(r)occ:Xivirt:Xai^vnlxvexxaa^vnlxvexxi

PAGE 85

thefourthtermoftheaboveexpression,thedenominatorisequaltothedierenceofthetwovirtualorbitalenergies.Ifthesetwoorbitalenergiesareaccidentiallyveryclosetoeachother(physicallyacceptablesituation)thenthisdenominatorwillleadtoanarbitrarilylargepotential.However,thefollowingtransformationdemonstratesthattermsinvolvingsuchdenominatorsalwayscanceleachotherandarenotpresentintheactualexpression.Givensomematrixelementsgabsuchthatgab=gba,onecanuseasimplerearrangementofindicestoshowthatocc:Xivirt:Xa;b6=agab (C-7)

PAGE 86

ApplyingthistransformationtoEq. C-6 oneobtainsthenalexpressionforthederivativeofthesingle-excitationcontributiontothesecond-orderenergyE(2)S (C-8)Similarly,thederivativeofthedouble-excitationcontributionisE(2)D 2occ:Xi;jvirt:Xa;b;cc(r)i(r)ijababcj 2occ:Xi;j;kvirt:Xa;bk(r)a(r)ijkbabij 2occ:Xi;j;kvirt:Xa;bk(r)i(r)ijababkj 2occ:Xi;jvirt:Xa;b;cc(r)a(r)ijcbabij (C-9)Notethatthelasttwotermsdonothavethecomplexconjugatecounterparts.

PAGE 87

[1] A.SzaboandN.S.Ostlund,ModernQuantumChemistry,IntroductiontoAdvancedElectronicStructureTheory(McGraw-HillInc.,NewYork,1990). [2] R.J.Bartlett,inModernElectronicStructureTheory,PartI,editedbyD.R.Yarkony(WorldScienticPublishingCo.,Singapore,1995). [3] F.E.Harris,H.J.Monkhorst,andD.L.Freeman,AlgebraicandDiagrammaticMethodsinMany-FermionTheory(OxfordUniversityPress,NewYork,1992). [4] P.-O.Lowdin,J.Chem.Phys.19,1396(1951). [5] P.HohenbergandW.Kohn,Phys.Rev.B136,B864(1964). [6] W.KohnandL.J.Sham,Phys.Rev.140,1133(1965). [7] R.T.SharpandG.K.Horton,Phys.Rev.90,317(1953). [8] J.D.TalmanandW.F.Shadwick,Phys.Rev.A14,36(1976). [9] S.Hirataetal.,J.Chem.Phys.115,1635(2001). [10] J.B.Krieger,Y.Li,andG.J.Iafrate,Phys.Lett.A148,470(1990). [11] M.Gruning,O.V.Gritsenko,andE.J.Baerends,J.Chem.Phys.116,6435(2002). [12] F.DellaSalaandA.Gorling,J.Chem.Phys.115,5718(2001). [13] S.Ivanov,S.Hirata,andR.J.Bartlett,Phys.Rev.Lett.83,5455(1999). [14] A.Gorling,Phys.Rev.Lett.83,5459(1999). [15] R.ColleandR.K.Nesbet,J.Phys.B-At.Mol.Opt.Phys.34,2475(2001). [16] S.Hirata,S.Ivanov,I.Grabowski,andR.J.Bartlett,J.Chem.Phys.116,6468(2002). [17] Q.Wu,P.W.Ayers,andW.T.Yang,J.Chem.Phys.119,2978(2003). [18] L.Curtiss,K.Raghavachari,G.W.Trucks,andJ.A.Pople,J.Chem.Phys.94,7221(1991). [19] A.GorlingandM.Levy,Phys.Rev.A50,196(1994). 79

PAGE 88

[20] E.Engeletal.,Phys.Rev.A58,964(1998). [21] Y.M.Niquet,M.Fuchs,andX.Gonze,Phys.Rev.A68,(2003). [22] I.Grabowski,S.Hirata,S.Ivanov,andR.J.Bartlett,J.Chem.Phys.116,4415(2002). [23] R.Bartlett,I.Grabowski,S.Hirata,andS.Ivanov,J.Chem.Phys.122,034104(2005). [24] M.Warken,Chem.Phys.Lett.237,256(1995). [25] A.Becke,J.Chem.Phys.98,5648(1993). [26] R.G.ParrandW.Yang,Density-FunctionalTheoryofAtomsandMolecules(OxfordUniversityPress,NewYork,1989). [27] F.DellaSalaandA.Gorling,J.Chem.Phys.118,10439(2003). [28] A.Gorling,Phys.Rev.A57,3433(1998).

PAGE 89

IwasbornandraisedinNovosibirsk,thethirdmostpopulouscityinRussia,locatedinthesouth-westcornerofthevastlandofSiberia.Bothmyparentsaredevotedphysicists,well-knownintheireld.IwasbornwhentheywerestillworkingontheirdegreesattheNovosibirskStateUniversity,soIspentmyinfancyinthePhysicsdorm,surroundedbylecturenotesandLandau'sbooks.Givingthatbackground,Ineverdoubtedmyscienticdestiny,andin1995enteredthePhysicsprogramatNovosibirskStateUniversity.In1999,Ireceivedthebachelor'sdegreeinPhysics(specializinginChemicalPhysics).ThenextyearIspentconductingresearchwithProf.GeorgiyM.ZhidomirovandDr.IgorL.ZilberbergattheLaboratoryofQuantumChemistryintheBoreskovInstituteofCatalysis,Novosibirsk.In2000,Itookmyparents'adviceandsearchedforagraduateschoolabroad.Lookingforfundamentalelectronicstructuretheory,onecannotoverlookProf.RodneyJ.Bartlett'sgroup,andIlookednofurther.Inthefallof2000,IenteredthePh.D.programofDeparmentofChemistryattheUniversityofFloridaandjoinedProf.Bartlett'sgroupattheQuantumTheoryProject. 81


Permanent Link: http://ufdc.ufl.edu/UFE0011614/00001

Material Information

Title: Ab Initio Density Functional Theory
Physical Description: Mixed Material
Language: English
Creator: Schweigert, Igor Vitalyevich ( Dissertant )
Bartlett, Rodney J. ( Thesis advisor )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2005
Copyright Date: 2005

Subjects

Subjects / Keywords: Chemistry thesis, Ph. D.
Dissertations, Academic -- UF -- Chemistry

Notes

Abstract: Ab initio Density Functional Theory (DFT) is a new approach to the electronic structure problem that combines elements of both density-functional and wavefunction-based approaches. It avoids the limitations of conventional DFT methods by using orbital-dependent functionals based on the systematic approximations of wavefunction theory. The starting point of ab initio DFT is the exact exchange functional. This functional was implemented with the auxiliary-basis Optimized Effective Potential method. The effect of numerical parameters on the performance of the method was also examined. It has been suggested in the literature to use perturbation theory to construct the correlation counterpart of the exact exchange functional. In this study, an ab initio correlation functional from second-order perturbation theory was implemented. However, numerical tests showed that this functional fails to provide an adequate description of correlation effects in molecules. This problem was attributed to the poor convergence of the perturbation series based on the Kohn-Sham determinant and a partial infinite-order resummation of one-body terms was proposed as a solution. The new functional offers a more balanced description of correlation effects, as was demonstrated in applications to a number of closed-shell atoms and molecules. It resulted in energies and densities superior to conventional (Møller-Plesset) second-order perturbation theory or DFT methods, accurately reproduced potential energy surfaces, and led to qualitatively correct effective potentials and single-electron spectra. An extension of the method based on mixing exact local and nonlocal exchange and an approximate second-order correlation potential were also examined.
Subject: density, electronic, exchange, Kohn, optimized, perturbation, quantum, wavefunction
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 89 pages.
General Note: Includes vita.
Thesis: Thesis (Ph. D.)--University of Florida, 2005.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

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Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
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Permanent Link: http://ufdc.ufl.edu/UFE0011614/00001

Material Information

Title: Ab Initio Density Functional Theory
Physical Description: Mixed Material
Language: English
Creator: Schweigert, Igor Vitalyevich ( Dissertant )
Bartlett, Rodney J. ( Thesis advisor )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2005
Copyright Date: 2005

Subjects

Subjects / Keywords: Chemistry thesis, Ph. D.
Dissertations, Academic -- UF -- Chemistry

Notes

Abstract: Ab initio Density Functional Theory (DFT) is a new approach to the electronic structure problem that combines elements of both density-functional and wavefunction-based approaches. It avoids the limitations of conventional DFT methods by using orbital-dependent functionals based on the systematic approximations of wavefunction theory. The starting point of ab initio DFT is the exact exchange functional. This functional was implemented with the auxiliary-basis Optimized Effective Potential method. The effect of numerical parameters on the performance of the method was also examined. It has been suggested in the literature to use perturbation theory to construct the correlation counterpart of the exact exchange functional. In this study, an ab initio correlation functional from second-order perturbation theory was implemented. However, numerical tests showed that this functional fails to provide an adequate description of correlation effects in molecules. This problem was attributed to the poor convergence of the perturbation series based on the Kohn-Sham determinant and a partial infinite-order resummation of one-body terms was proposed as a solution. The new functional offers a more balanced description of correlation effects, as was demonstrated in applications to a number of closed-shell atoms and molecules. It resulted in energies and densities superior to conventional (Møller-Plesset) second-order perturbation theory or DFT methods, accurately reproduced potential energy surfaces, and led to qualitatively correct effective potentials and single-electron spectra. An extension of the method based on mixing exact local and nonlocal exchange and an approximate second-order correlation potential were also examined.
Subject: density, electronic, exchange, Kohn, optimized, perturbation, quantum, wavefunction
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 89 pages.
General Note: Includes vita.
Thesis: Thesis (Ph. D.)--University of Florida, 2005.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0011614:00001


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AB INITIO DENSITY FUNCTIONAL THEORY


By

IGOR VITALYEVICH SCHWEIGERT















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

Igor Vitalyevich Schweigert















TABLE OF CONTENTS
page

LIST OF TABLES ........ ........................ v

LIST OF FIGURES .. ... ....................... vi

ABSTRACT ................... .............. vii

CHAPTER

1 INTRODUCTION .................... ....... 1

1.1 Ab-Initio Wavefunction-Based Methods ....... ........ 3
1.2 Kohn-Sham Density Functional Theory ...... ......... 9
1.3 Problems with Conventional Functionals . . ...... 12
1.4 Orbital-Dependent Functionals ............. .. .. 14
1.5 Ab initio Density Functional Theory ............... .. 15

2 EXACT ORBITAL-DEPENDENT EXCHANGE FUNCTIONAL .... 17

2.1 Exact Exchange Functional .................. ..... 17
2.2 Optimized Effective Potential Method . . ...... 19
2.3 Performance of the Auxiliary-Basis EXX Method . ... 24

3 CORRELATION FUNCTIONALS FROM SECOND-ORDER
PERTURBATION THEORY ........... ..... ... 31

3.1 Correlation Functional from Second-Order Perturbation Theory 31
3.2 Correlation functional from Second-Order Perturbation Theory
with Partial Infinite-Order Resummation . . 33
3.3 Implementation of the PT2 and PT2SC Functionals . ... 39
3.4 Numerical Tests for Ab initio Functionals . . ..... 40

4 OTHER THEORETICAL AND NUMERICAL RESULTS ........ 48

4.1 Connection between Energy, Density, and Potential . ... 48
4.2 Diagrammatic Derivation of the Optimized Effective Potential
Equation .............. . . .... 49
4.3 Mixing Exact Nonlocal and Local Exchange . . ... 55
4.4 Second-Order Potential within Common Energy Denominator
Approximation .................. ........ .. 59

5 CONCLUSIONS .................. ........... .. 65









APPENDIX

A FUNCTIONAL DERIVATIVE VIA THE CHAIN RULE ........ 69

B SINGULAR VALUE DECOMPOSITION ................ 73

C DERIVATIVE OF THE SECOND-ORDER CORRELATION ENERGIES 75

REFERENCES ............................. ..... 79

BIOGRAPHICAL SKETCH ............................ 81















LIST OF TABLES
Table page

2-1 Effect of basis set on the performance of the EXX method. . 26

2-2 Effect of the explicit .,-i-i!,l, .' ic term on the performance of the EXX
method ...... ............ ............... .. 27

2-3 Effect of the Singular Value Decomposition threshold on the
performance of the EXX method ............. .. .. .. 28

2-4 Performance of the EXX methods for the 35 closed-shell molecules of
the G 1 test set .. ... .. .. .. .. ... .. .. .. ....... 30

3-1 Performance of ab initio and conventional correlation functionals in
the high-density limit .................. ..... .. .. 42

3-2 Performance of ab initio correlation functionals for closed-shell atoms 42

3-3 Density moments of Ne calculated with ab initio DFT, ab initio
wavefunction and conventional DFT methods . . 46

4-1 Performance of the hybrid ab initio functional EXX-PT2h with
optimized fraction of nonlocal exchange . . ...... 57















LIST OF FIGURES
Figure page

2-1 Explicit .,-i~i!.ll. ic terms for Ne and the corresponding EXX
potentials . . . . . . ... .. 28

3-1 Performance of ab initio DFT and ab initio wavefunction methods in
total energy calculations for the G1 test set. ............ ..43

3-2 Performance of ab initio DFT, ab initio wavefunction, and
conventional DFT methods in calculations of the total energy as
a function of the bond lengths .................. .. 45

3-3 Performance of ab initio DFT, ab initio wavefunction, and
conventional DFT methods in dipole moment calculations for the
G1 test set .................. ............ .. 47

4-1 The total energy and first density moment of Be calculated with
the EXX-PT2h functional with various fractions of the nonlocal
exchange operator .................. ........ .. 58















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

AB INITIO DENSITY FUNCTIONAL THEORY

By

Igor Vitalyevich Schweigert

August 2005

C'!I ir: Rodney J. Bartlett
Major Department: C('! i ii-I ry

Ab initio Density Functional Theory (DFT) is a new approach to the

electronic structure problem that combines elements of both density-functional

and wavefunction-based approaches. It avoids the limitations of conventional

DFT methods by using orbital-dependent functionals based on the systematic

approximations of wavefunction theory.

The starting point of ab initio DFT is the exact exchange functional. This

functional was implemented with the auxiliary-basis Optimized Effective Potential

method. The effect of numerical parameters on the performance of the method was

also examined.

It has been -I-.: -1. 1 in the literature to use perturbation theory to construct

the correlation counterpart of the exact exchange functional. In this study,

an ab initio correlation functional from second-order perturbation theory was

implemented. However, numerical tests showed that this functional fails to provide

an adequate description of correlation effects in molecules. This problem was

attributed to the poor convergence of the perturbation series based on the Kohn-

Sham determinant and a partial infinite-order resummation of one-body terms was









proposed as a solution. The new functional offers a more balanced description of

correlation effects, as was demonstrated in applications to a number of closed-shell

atoms and molecules. It resulted in energies and densities superior to conventional

(\ !.. r-Plesset) second-order perturbation theory or DFT methods, accurately

reproduced potential energy surfaces, and led to qualitatively correct effective

potentials and single-electron spectra.

An extension of the method based on mixing exact local and nonlocal

exchange and an approximate second-order correlation potential were also

examined.















CHAPTER 1
INTRODUCTION


The underlying physical laws necessary for the mathematical theory of
a large part of physics and the whole of chemistry are thus completely
known, and the difficulty is only that the exact application of these laws
leads to equations much too complicated to be soluble.
P. A. M. Dirac, Proc. Roy. Soc. London, p. 174, 1929

At the microscopic level, a chemical reaction is the transition from one stable

conglomerate of nuclei and electrons (reagent) to another one (product). Given

the initial configuration of the system, the transition properties and the final

state are determined by the interactions of the particles with each other and with

the environment. Since the nature of these interaction is known, it is then the

task of theoretical chemistry to predict the outcome of the reaction by solving

the fundamental equation describing these particles. Dirac's famous words state

the ultimate goal of theoretical chemistry -the complete substitution of the

experiment by a theoretical calculation -and warn about the ultimate difficulty

-the immense complexity of the problem. Even now, given all the computational

power at our disposal, the near-exact solutions of the electronic problem are still

limited to few-electron systems.

Facing the intractability of the exact solution, one must rely on

approximations. Although, for systems beyond several thousands particles one

has no choice but to rely on classical mechanics, most chemical phenomena require

a quantum-mechanical description to obtain at least qualitative resemblance

with reality. In quantum theory, a chemical system is described by the molecular









Hamiltonian (neglecting magnetic and relativistic effects for simplicity)

elec. ucl. lec. nucl. lec. nucl.
1 12 1 ZA + ZAZB
S22 IAj RA i= A=l i A i r A (1-1)
The solution of the Shridinger equation information about the system evolution.


Hf (t; X1,.., XN) =i (t; X, .. XN) (1-2)


Thus, quantum 1,' ,ii;li the collection of quantum-mechanical methods to solve

Eq. 1-2, p]1 i the 1i. ii" role in theoretical chemistry.

Ab initio Density Functional Theory belongs to the class of quantum-chemical

methods called electronic structure methods. These methods further simplify

the Hamiltonian of Eq. 1-1 using the Born-Oppenheimer approximation. In this

approximation one neglects the coupling between the electronic and nuclear

degrees of freedom, which allows one to factorize the corresponding variables and

concentrate on the electronic part of the wavefunction. Also, if one is interested

in stationary solutions, the time variable can also be factorized. Thus, the time-

independent electronic structure problem is to find the solution of the time-

independent Schridinger equation


HX(X, .., XN) = EI(, .., XN) (1-3)


defined by the nonrelativistic, Born-Oppenheimer electronic Hamiltonian
elec. elec. nucl. elec.
2v 7 ZA 1 (1-4)
2 RA i rj
i i A ri R i
Ab initio DFT is a new method for obtaining approximate solutions to Eq. 1-3.

Its formalism is based on two fundamental electronic structure approaches : ab

initio wavefunction methods and Density Functional Theory.









1.1 Ab-Initio Wavefunction-Based Methods

Hartree-Fock Method.

The simplest approximate wavefunction that retains the correct fermion

symmetry is given by the antisymmetric product of single-electron wavefunctions


X(xI, .., N) =(N!)1/2 A[1(1).N(xN)], (1-5)


where

A = (-1)P (1-6)
P
ensures that + is antisymmetric with respect to a permutation of the labels of

any pair of electrons. This type of wavefunction can be conveniently written as a

determinant
d1(ri) ... Q1(rN)

#HF ..... (-7)

QN(ri) ... N(rNw)
and is often called a Slater determinant or single-determinant wavefunction.

In the Hartree-Fock method, the single-electron wavefunctions (or orbitals)

are determined by the condition that the corresponding determinant minimizes the

expectation value of the true many-electron Hamiltonian [1]


EHF K HF H HF = min H ), (1-8)


subject to the constraint that the orbitals remain orthonormal, ()p ) = pq.

Inserting the expression for K from Eq. 1-5 into this expectation value, one

obtains the expression for the Hartree-Fock energy in terms of the orbitals
dec. 2 + e dec.
2 1
EHF = HF + HF)= C (i -9 2 + Veyt p) + i < |> (1-9)
i ij









where (ijij) is the Dirac notation for the two-electron integral defined by
Eq. 1-10.


{ij ij} = {ij ij) {ij ji

= drdr' --- / drr' (1-10)

Requiring that EHF be stationary with respect to an arbitrary variation of

{f4} one obtains the Hartree-Fock equations

/1 V2 + V"' + VH + OP ) / ,c-
^2+ *,> (1-11)
q
where
nucl.
t() ZRA (1-12)
r R~
is the external Coulomb field created by the nuclei and

vH(r)= dr' p(r') (1-13)

is the Hartree potential (i.e., the Coulomb field created by the total electron

density),
elec.
p(r) = J(r) {(r), (1-14)

Oni is the nonlocal exchange operator,

(r pdr' (r)4 (r) (1-15)
i J
and Cpq are the Lagrange multipliers that ensure the orthonormality of the Hartree-

Fock orbitals.

Note that the number of solutions of Eq. 1-11 is not limited to the number of

electrons. The lowest N solutions are referred to as occupied orbitals (N being the

number of electrons) and the remaining solutions are referred to as virtual orbitals.









Using the fact that the Fock operator


2
=-2 + Vgt + VH + Unlx (1-16)

is invariant with respect to any unitary transformation of the occupied orbitals, one

can transform Eq. 1-11 to its canonical form,


f ip) = c p ,) (1-17)

which is the eigenvalue problem for the Fock operator.

Since the Fock operator depends on {fi} through the VH and vx, Eq. 1-11 is

an integro-differential equation that can be solved iteratively, until self-consistency

is reached. Therefore, the Hartree-Fock approximation belongs to the class of

Self-Consistent Field (SCF) approximations.

One can solve the Hartree-Fock equations numerically. However, a more

practical approach is to use a finite basis set (usually atom-centered Gaussian-type

functions) to expand the HF orbitals. As the result, the Hartree-Fock integro-

differential equations are transformed into a matrix problem.

Electron-Correlation Methods.

The Hartree-Fock method can recover as much as 9' I. of the total electronic

energy. Still, even the remaining error of 1 is too large on the chemical scale and

may lead to a qualitatively wrong theoretical prediction.

The difference between the SCF and exact solutions is due to electron-

correlation effects. In ab initio electron-correlation methods, one relies on elaborate

I ii ,iv-1 ody techniques to go beyond the SCF approximation and account for the

simultaneous electron-electron interactions. These methods, in contrast to the

relatively simple Hartree-Fock approximation, can be quite challenging conceptually

and computationally.









The correlation limit (i.e., the exact solution of Eq. 1-3 in a given basis set)

can be obtained via the Full Configuration Interaction method. In this method, the

correlation correction to the Hartree-Fock determinant is expanded over all possible

excited determinants
occ. virt. occ. virt.
^FCI = HF + K cyjC + >1K CUCab +... (1-18)
i a iHj acb

where ab, etc. are formed by by substituting several occupied orbitals in the

Hartree-Fock determinant by virtual orbitals, e.g.


< = (]VN!)1/2A[ 1(1**..Qa(Xi)N(ZN)]. (1-19)

The expansion coefficients are found from the variational condition on the

expectation value of the true Hamiltonian


EFCI = min tc H CI) (1-20)
CT, CT".... {q^FCI 9 qFCl)

However, the number of possible excited determinants grows exponentially

with the number of electrons and basis functions, therefore, the Full CI method

is computationally intractable for any but very small systems. Among the

approximate electron-correlation methods, the most common are the truncated

and multi-reference Configuration Interaction methods, Coupled-Cluster methods

[2], and A! m:-Body Perturbation Theory [3]. For example, for systems where

the multi-reference treatment is not necessary (i.e., when the Hartree-Fock

wavefunction dominates the Full CI expansion), the Coupled-Cluster methods

have proved to be the most systematic and computationally robust approach to the

ir ini -electron problem.









Many-Body Perturbation Theory.

In some cases, perturbation theory can provide an accurate description of

electron-correlation effects at a significantly lower cost than required by Coupled-

Cluster or multireference methods. For example, second-order Rayleigh-Schr6dinger

perturbation theory is the simplest and least expensive ab initio method for

electron correlation. That is why it was chosen as the basis for the ab initio

correlation functional (C!i plter 3).

In such perturbation theory, one finds the solution of the many-body problem

(Eq.1-21) using an SCF model (Equations 1-22 and 1-23) as the reference.


HI ) = E ) (1-21)



1
(- 22 + u) Qp) CP ). (1-22)


elec.
o (-V2 + u) ) = E=o, (1-23)

where K is the single-determinant wavefunction constructed from the N lowest

solutions to Eq. 1-22. The remaining eigenfunctions of Ho are obtained by

substituting the corresponding number of occupied orbitals in KP by the virtual

orbitals.

To do this, the true Hamiltonian is partitioned into the reference Hamiltonian

and perturbation

H = Ho + V (1-24)

where
elec. elec elec.
V H- Ho = vt(r)+) + Y (1-25)
i i' j i
The solution to Eq. 1-21 is then found by introducing the perturbation

parameter A and expressing the corrections to the reference wavefunction and








energy as series of terms of increasing powers of A

H = Ho + AV (1-26)



T1) = + A ~()) + A2 (2)) +... (1-27)


E = Eo + AE(1) + A2E(2) + ... (1-28)

These order-by-order corrections can be found by neglecting all higher terms
from the Schr6dinger equation

(Eo- Ho) () =(V- E (1) )) (1-29)



(Eo H0) q(2)) (V- E(1)) (1)) E(2) l) (1-30)

and so forth.
C'I ...!-ig the perturbative corrections to be orthogonal to the reference
wavefunction, (K1")l I = 0, one can readily obtain the expressions for the order-
by-order contributions to the energy by projecting the Equations 1-29 and 1-30
onto the reference space
E(1) (= V ) (1-31)



E(2) ( | V | (1)) (1-32)

The order-by-order contributions to the wavefunction can be written in terms
of the resolvent operator [4] (the inverse of integro-differential operator Eo Ho in
the Hilbert subspace)


P(l)) IoV 4


(1-33)











|p(2)) 7Z(V ) T(1)) 7Zo(V E- I)7o >), (1-34)

where

Ro =- (1-35)
Eo Ho'

and = 1 )( is the projector onto the complementary space of +) (Hilbert

space with () excluded.)

The actual expression for the resolvent operator can readily be found by

recognizing that Eo Ho is diagonal in terms of eigenfunctions of Ho


RY Eo E,
n 0

Note that the Hartree-Fock SCF model presents a special case as the reference

for the perturbation expansion. First, the HF energy is correct through first order


Eo + E') = ( Ho0 ) + (4 V ) =- (4I HK H} = EHF (1-37)


Second, the HF SCF Hamiltonian cancels all the effective one-body terms of the

true Hamiltonian, leaving only two-body terms in the perturbation, so that only

double-excited determinants contribute to the second-order energy
occ. virt. 2
E CC ab) (1-38)
~HF ~ @ (j -Y Ya +b
i,j a,b


1.2 Kohn-Sham Density Functional Theory

Density Functional Theory is an alternative approach to the electronic

structure problem of Eq. 1-3 that uses the electronic density rather than the

wavefunction as the basic variable. The formal basis of DFT is provided by two

theorems introduced by Hohenberg and Kohn [5]. The first theorem establishes

the one-to-one correspondence between the electronic ground-state density and

the external potential. Since it is the external potential that defines a particular









molecule, the existence of such a correspondence ensures that the ground-state

electronic density alone carries all the information about the system. In particular,

the ground-state energy can be written as a functional of the density. However,

there is no equation of motion for the electronic density. Instead, one must rely on

the second Hohenberg-Kohn theorem that states that the ground-state energy as a

functional of the density is minimized by the true ground-state density. Therefore,

given the energy functional, one can obtain the ground-state density and energy by

variational minimization of the functional.

However, the formal definition of Density Functional Theory does not tell

how to construct such functional. Several approximate forms have been -.-. -i -1 I 1.

however, they are far from accurate. The kinetic energy of electrons is particularly

difficult to approximate as a functional of the density.

The idea of Kohn and Sham [6] was to use a SCF model (Eq. 1-39) to

transform the variational search over the density into a search over the SCF

orbitals that integrate to a given trial density.

1
[--V + V (r)O(r) = e(r) (1-39)
2

Such a transformation does not restrict the variational space, provided that every

physically meaningful density correspond to a unique set of SCF orbitals (the

v-representability condition). Not only does the use of the Kohn-Sham SCF model

ensure that the variational search be to fermionic densities, but also it provides

a good approximation for the kinetic energy. Indeed, provided that the orbitals

integrate to the exact density, the so-called noninteracting kinetic energy
elec. occ.
Ts = (s V2 ) i -_ 2 i) (1-40)
2 2


should account for a large part of the true kinetic energy.









The remaining unknown terms of the energy functional are grouped into the

exchange-correlation functional


Exc[p] = E[p] T E.t EH (1-41)

where EH is the Hartree energy, which (as well as T,) can be readily calculated

for a given set of SCF orbitals. Since T, should reproduce a large part of T,

this procedure eliminates the necessity to model the entire kinetic energy as a

functional of the density. Thus, it is expected that E,, is easier to approximate as a

functional of the density than E.

Note that according to the definition of the exchange interaction in

wavefunction theory, the exchange component of the exchange-correlation

functional is defined as

E ( Vee > EH (1-42)

and the correlation component is the remaining part


Ec = E, E, (1-43)


The Kohn-Sham SCF orbitals are defined by the effective potential vs.

Transforming the variational condition on the energy functional into the condition

for the constrained search over the orbitals, one can obtain

v, (r) [E[p] T,] [Et + EH + E]r-44
Mp(r) 6p(r)

where the exchange-correlation potential is defined as the functional derivative of

the exchange-correlation functional


Vc(r) -= E (1-45)
6p(r)

Thus, given an exchange-correlation functional, one defines the exchange-

correlation potential and then solves the Kohn-Sham SCF equations. Note that









since it is an SCF model, practical implementation of the KS procedure is very

similar to the Hartree-Fock method. Usually, the SCF orbitals are expanded in

a Gaussian-type atom-specific basis, which transforms the Kohn-Sham integro-

differential equation into a matrix SCF equation. After self-consistency is reached,

the SCF orbitals are guaranteed to reproduce the true density of the many-electron

system. Also, the true energy can be found by inserting this density into the energy

functional.

Virtually all modern implementations of DFT use the Kohn-Sham scheme.

However, the theory still leaves open the question of how to construct the

exchange-correlation functional. Therefore, the principal challenge for the

theoretical development of DFT remains the construction of accurate exchange-

correlation functionals.

1.3 Problems with Conventional Functionals

The conventional approach is to approximate the energy functional as an

analytical expression of the density and its gradients. The effective potential can

then be obtained in analytical form as well, and the KS equations can be solved

readily. This approach started with the simplest Local Density Approximation

(LDA) where the energy is given through an integral of a local functional

of the density. The next-level, Generalized Gradient Approximation (GGA)

functionals, improved on the LDA functional form by including the dependence

on the gradients of the density. This extension provided a certain freedom in

defining the form of the functional, and a number of different forms have been

siu---- -. .1 Typically, the basic form of a GGA functional is chosen to satisfy

a set of conditions known to be satisfied by the exact functional. The basic

form is then either parameterized to reproduce experimental data (empirical

functionals) or further modified to satisfy an extended set of conditions (non-

empirical functionals).









With the conventional functionals, KS DFT surpasses the quality of the HF

method, and becomes comparable with the simplest ab initio correlation methods.

Nevertheless, restricting the functional form to analytical expressions of the density

imposes certain limitations on the energy functional. GGA exchange functionals

are not capable of complete elimination of the spurious self-interaction component

of the Hartree energy. Since the exchange part often dominates the exchange-

correlation energy, the self-interaction error can considerably reduce the accuracy

of the GGA functional. Similarly, semilocal correlation functionals cannot describe

pure nonlocal components of the correlation energy such as dispersion. This

omission greatly reduces the applicability of the conventional KS DFT methods to

weakly-interacting systems.

Another problem is that while the GGA functionals result in relatively

accurate energies, the functional derivatives (i.e., the corresponding KS potentials)

are not nearly as accurate, especially in the inter-shell and .,-i-!!,I, ,i ir regions.

Consequently, one should not expect the same level of accuracy for the density

as for the energy. Furthermore, the qualitatively incorrect potentials reduce

substantially the usefulness of the KS orbitals and orbital energies, which are often

used to calculate certain ground-state properties or as the basis for response and

time-dependent KS DFT calculations.

Some of these problems can be addressed without extending the functional

form. Several post-SCF corrections have been -i-i:. -1. I1 to partially remove the

self-interaction error. For example, after the KS equations have been solved, one

can introduce corrections to the energy to include dispersion or ensure the correct

.,-vmptotic behavior of the KS potential. However, these corrections are specific to

the particular functional and class of systems and they likely are incompatible with

each other. Clearly, one needs to go beyond the GGA functional form to resolve

these problems in a consistent and universal fashion.









1.4 Orbital-Dependent Functionals

It has now been fully recognized that KS orbitals can provide extra

information about the system that cannot be ::i I I" easily from the density or

its gradients. The next-generation functionals (hybrid- and meta-GGA) augment

the GGA functional form with terms that depend explicitly on the orbital rather

than the density.

An alternative approach is to dismiss completely the conventional hierarchy of

approximations and construct the functional using solely the orbitals. In contrast

to the conventional functionals, orbital-dependent functionals are analytical

expressions of the orbitals (and orbital eigenvalues). They still are implicit

functionals of the density, however. Indeed, the central assumption of KS DFT

is that there exists one-to-one mapping between the exact density and some a local

potential. Therefore, a given density uniquely defines the potential, which, in turn,

uniquely defines the orbitals through the KS SCF equations. Therefore, the orbitals

and explicitly orbital-dependent functional are implicit functionals of the density.

One can think of the KS orbitals as the intermediate step in the mapping from the

density to the energy.

The most significant difference between the orbital-dependent and conventional

functionals is how the corresponding potential (i.e., the functional derivative with

respect to the density) is determined. The conventional functionals are given

as analytical expressions in terms of the density. Therefore, one can take the

functional derivative straightforwardly to obtain an analytical expression for the

potential. The orbital-dependent functionals are analytical expressions in terms

of the orbitals, whose dependence on the density is given through the effective

potential and KS integro-differential equation. Therefore, the analytical expression

for the functional derivative (hence, potential) cannot be obtained directly. Instead,

one must rely on the chain rule to obtain an integral equation for the potential









(C'!i jpter 2). This integral equation is identical to the one used in the Optimized

Effective Potential (OEP) method. The OEP method is, therefore, the cornerstone

of DFT with orbital-dependent functionals.

The immediate advantage of the orbital-based approach is that the exact

exchange functional is known in term of orbitals. One can think of the EXX

method as an extension to the idea of Kohn and Sham, where SCF orbitals are

used to calculate both the larger part of the kinetic energy and a (presumably

larger) part of the exchange-correlation energy.

1.5 Ab initio Density Functional Theory

While the EXX functional provides the exact description of the exchange

interactions, it is just a first step towards the exact exchange-correlation functional.

It is the effective description of electron correlation effects that makes KS DFT

a powerful alternative to the ab initio wavefunction methods. Thus, one needs a

correlation functional that can be combined with the EXX functional.

Conventional (GGA or higher-level) correlation functionals are developed

in combination with the corresponding approximate exchange functionals and

often compensate the deficiencies of the latter. For example, the GGA correlation

functionals usually result in correlation potentials that have the opposite sign to

the exact one. The terms correcting the approximate exchange are "hidden" in the

correlation functionals and inseparable from the "true" correlation terms.

Thus, it is not surprising that substituting the approximate exchange by

its exact counterpart destroys the balance between the approximate exchange

and approximate correlation components and results in a functional inferior to

the exchange-only approximation. In other words, the conventional correlation

functionals are not compatible with the EXX functional. Thus, the primary

challenge in the orbital-based approach to exchange-correlation functional is









to develop an orbital-dependent correlation functional that can be combined

seamlessly with the exact exchange functional.

Ab initio DFT solves the problem of constructing an orbital-dependent

correlation functional by referring to ab initio wavefunction methods. The idea

is simple: the goal of ab initio methods is to calculate the correction to the exact

exchange approximation (i.e., correlation energy) in terms of the SCF orbitals.

Thus, such an energy expression treated as the orbital-dependent functional results

in a correlation functional that can be seamlessly added to the exact exchange

functional.

Ab initio DFT makes a plethora of wavefunction-based approximations

available as the correlation functionals. Unlike the conventional ones, ab initio

functionals are systematically improvable, since one can ..1- ,iv-; use a higher-level

approximation to obtain a more accurate functional. They also have a well-defined

exact limit represented by the FCI method.

The next two chapters describe the formal development, implementation,

and some test applications for the exact exchange functional and the correlation

functional based on second-order perturbation theory. C'! lpter 4 discusses possible

extensions of the ab initio DFT approach. The results of the study are summarized

in C'!i pter 5.















CHAPTER 2
EXACT ORBITAL-DEPENDENT EXCHANGE FUNCTIONAL

2.1 Exact Exchange Functional

The immediate advantage of constructing the energy functional in terms

of orbitals is that since the exchange energy is defined in terms of orbitals, the

exact orbital-dependent exchange functional is known. Indeed, since the exchange

component of the exchange-correlation energy is defined as
OCC.
E, = (4 V, ) EH -= iI j*i) (2-1)
i,j

treating it as an implicit functional of the density results in the exact exchange

(EXX) functional.

The most important feature of the EXX functional is that, unlike any of the

conventional functionals, it completely eliminates the spurious self-interaction

component of the Hartree energy. Similarly, the corresponding EXX potential

cancels the self-interaction component of the Hartree potential. Thus, using the

EXX functional and potential will avoid many pathological problems caused by the

self-interaction error in conventional DFT approximations, both at the energy and

density levels.

The explicit dependence on the orbitals amounts to one complication, however.

Since the EXX functional does not depend on the density explicitly (i.e., it is not

an analytical expression of the density), the functional derivative cannot be taken

directly. Instead, one must rely on the chain rule, which accounts for the implicit

dependence expressing the derivative of interest through the product of known

derivatives.









The chain rule pl a central role in the orbital-based approach because it
allows to take the functional derivative of an expression in terms of orbitals with
respect to the density. To do that one needs to determine for which derivatives the
expressions are known, and then express the derivative of interest in terms of the
known derivatives.
First, one recognizes that the KS potential is the most convenient variable.
Indeed, the response of the orbitals and orbital energies to a infinitesimally
small change in the potential is readily available through the linear response
KS equations (Appendix A). And so is the response of the density. Thus, the
functional derivatives of the orbital, orbital energies, and density with respect
to the potential are known. Second, since the exchange energy is given as an
analytical expression in terms of orbitals, its derivative with respect to the orbitals
can be obtained directly.
Thus, starting with the definition for the exchange potential

6EExx / s Isr')6I EEXX
VEX(r) EEx dr' 6v ) () (2-2)
6p(r) 6p(r) 6v,(r')

and recognizing that it is 6p(r)/6vs(r') that is known in analytical form, one
obtains
f Ip(r') 5EEXX
dr'EXX(r)p() 6E(EX (2-3)
J 6v,(r) 6v,(r)
Thus, the EXX potential is given through an integral equation

Sdr'X(r, r')ve,(r) = w(r) (2-4)

where
W ( r' ) occ. ,it.
X(r,r') P ( YY +. C. (2-5)
6v, (r) i a-C









and, as shown in Appendix A


w(r) 6EEXX OCC. i(r') EEXX
w(r) dr
6v,(r) 6v,(r) 6p (r')
occ. virt. .* ( r) i r)\ I i \
CC rt + (2-6)
i a

This integral equation was first written in the context of the Optimized

Effective Potential (OEP) method. The OEP method was originally introduced

by Sharp and Horton [7], long before the foundation of the DFT. Their goal was

to find a local approximation to the HF exchange operator. They defined the

optimized potential as the one that makes the single-determinant expectation value

of the true Hamiltonian stationary [8]. Much later was it realized that the OEP

method results in the exact exchange potential in the KS DFT context.

Because of this equivalence, the terms "EXX method" and "OEP method"

are often used interchangeably. However, the application of the chain rule is not

limited to the exchange functional. In the next chapter, the chain rule will be

applied to the orbital-dependent correlation functional. Thus, it is preferably to

use the term "OEP method" to denote the way to determine the local potential for

a given orbital-dependent functional. Consequently, one refers to the EXX method

as the KS DFT method with the exact orbital-dependent exchange functional and

corresponding potential obtained via the OEP method.

2.2 Optimized Effective Potential Method

The OEP equation is a Fredholm integral equation of the first kind. Its

integral kernel as well as the right-hand side depend on the SCF orbitals and

orbital eigenvalues, therefore, it must be solved simultaneously with the SCF

equations.

Hirata et al.[9] analyzed the integral kernel and showed that it defines the

potential uniquely up to an irrelevant constant if the SCF orbitals form a complete









basis set. However, virtually all practical implementations of the SCF procedure

employ a finite basis to represent the orbitals. In this case, the SCF orbitals

do not form the complete set and the integral kernel, in general, has singular

eigenfunctions. This, means that the potential is defined up to a linear combination

of the singular eigenfunctions. Thus, in a practical implementation of the OEP

method, one must exclude the subspace spanned by the singular eigenfunctions of

the kernel from the solution.

The incompleteness of the orbital basis can also lead to the OEP integral

kernel that does not sample certain regions of the real space. For example,

Gaussian-type orbitals, which are typically used as the orbital basis, fall off too

rapidly with increasing r. They decay as e- while the exact exchange potential

is known to have the -1/r .i-vmptotic behavior. As a result, the OEP kernel

decays too rapidly and does not sample the solution in the .,-,ill l ic region.

Consequently, the potential obtained from the OEP equation in the finite orbital

basis can deviate arbitrarily from the exact solution in the .,-i-, !!ii i. region .

The first implementation of the SCF procedure with the exchange potential

given by the OEP equation was reported by Talman and Shadwick [8]. They

used a expansion over a spatial grid to solve the integral equation. However, such

grid-based implementation is inevitably limited to atoms, for which the spherical

symmetry permits excluding the angular points from consideration. In the case

of a polyatomic molecule, the number of grid points necessary for an adequate

representation of the EXX potential is significantly larger and a grid-based OEP

method becomes computationally intractable.

Krieger et al.[10] -1i--.- -1. I neglecting the orbital structure of the integral

kernel to avoid the solutions of the integral equation, the so-called KLI

approximation. The OEP integral equation in the KLI approximation reduces

to a very simple nonlinear equation for the approximate KLI exchange potential.









The KLI equation can easily be solved self-consistently even in the case of

polyatomic molecules. However, the resulting potential does not reproduce the

characteristic "bumps" of the EXX potential in the inter-shell region. Recently, a

more elaborated approximation to the OEP equation has been -ii-. -1. I known

as Common Energy Denominator Approximation [11] or Localized Hartree-Fock

[12]. The exchange potential in this approximation more accurately reproduces

the structure of the EXX potential. However, the error introduced by this

approximation still cannot be measured a priori or controlled.

The auxiliary-basis approach [13, 14] presents an attractive alternative to grid-

based and approximate OEP methods. In this method, the potential is expanded

in a finite auxiliary basis set and the OEP integral equation is transformed into

a linear matrix problem. This is very similar to how the integro-differential SCF

equations are solved in the LCAO approximation. The auxiliary-basis approach to

the solution of the OEP equation does not require a fine spatial grid nor does it

introduce any approximation to the kernel. The error introduced by the finite-basis

expansion alv-, can be reduced by increasing the size of the auxiliary basis set.

In this approach, the OEP integral equation


Sdr'X(r, r')vex(r) = w(r) (2-7)

is transformed into a linear matrix problem
aux.
-X"u'" J' (2-8)


by projecting the equation onto a finite basis set:
aux.
X(r, r') Z= XX (')(r (r') (2-9)


aux.
W(r}) C WX(r) (2-10)










aux.
uW(r') > ur'), (2-11)

where the expansion coefficients over the orthonormal set of auxiliary basis

functions

I drx(r)xv(r) = 6, (2-12)

are found by solving the linear matrix problem of Eq. 2-8.

As it has been already mentioned, in a finite orbital basis the occupied-

virtual orbital products do not span the entire space and, therefore, the integral

kernel may have nontrivial eigenfunctions with zero eigenvalue. Consequently, the

auxiliary-basis representation of the kernel, XE, may have singular eigenvalues

and not be invertible. In this case, one can use the Singular Value Decomposition

(SVD) procedure.

The Singular Value Decomposition (Appendix B) with a given SVD threshold

will provide the approximate solution to Eq. 2-8


il = (XSVD)nw (2-13)


that minimizes the error in a least-squares sense
aux.
svD J wX~ n? (2-14)


where the actual value of the residual error, eSVD, depends both on the matrix X

and the SVD threshold. In the hypothetical case of the complete orbital basis, one

would have to set the threshold at the hardware-specific numerical precision. In the

case of a finite orbital (and finite auxiliary) basis set, retaining very small singular

values of X may introduce instabilities into the solution and ultimately lead to the

divergence of the iterative solution. Conversely, a large SVD threshold decreases









the quality of the approximate solution. Therefore, there is no a priori preferred

value for the threshold and its actual choice is subject to investigation.

Another problem mentioned in the beginning of this section concerns with

the .,-vmptotic behavior of the EXX potential. The exact exchange potential must

decay as -1/r at large r. However, if the OEP kernel is obtained with Gaussian-

type SCF orbitals it decays too rapidly and does not sample the potential in the

.,-vmptotic region. The solution to this problem, considered by many authors

[15, 16, 17], is to use a numerical potential that has the .,-i- i!'Ill .ic behavior of

the exact exchange potential. There are several choices for such a potential: the

Fermi-Amaldi scaled Coulomb potential

N -, 1 p(r')
vfa(r) N dr' (2-15)
N r r'

the local exchange energy density, defined so that


c (r) =C (r)(r) Jdr' i + c.c. (2-16)
p(r) J r r

and others. These potentials depend explicitly on the SCF orbitals and are

guaranteed to have the correct .,-i: ii!.ll Iic behavior. One can then use them as the

explicit ,i /,,/.1.: ': term in the EXX potential


vexx(r) = veat(r) + ezxx(r) (2-17)

and solve the OEP equation for vexx(r)

J dr'X (r, r')xx(r) w (r) dr'X(r, r') veat(r) (2-18)

Now choosing a larger SVD threshold will ensure that v~ex decays rapidly and vex

becomes Veat at large r and thus has the correct .,-i-,i! ill ic.

Thus, a practical implementation of the OEP method has several parameters

that will affect the quality of the corresponding EXX potential. The next









section discusses the typical choices for these parameters and their effect on

the performance of the EXX method.

2.3 Performance of the Auxiliary-Basis EXX Method

In the conventional DFT approximations, the quality of a given exchange-only

approximation is solely determined by the quality of the approximate functional.

In the EXX method the functional is known and the quality of the method is

determined by the implementation of the OEP method for the corresponding

potential. In this study, the auxiliary-basis implementation is chosen because it

allows applications to general polyatomic molecules and does not introduce any

simplifications to the integral kernel structure. The only error is introduced by the

incompleteness of the orbital and auxiliary bases, but this error can be controlled

by increasing the sizes of the basis sets. Also, given the inevitable incompleteness of

these bases, the quality of the potential is affected by the explicit .,-,iii ltic term

used to ensure the correct long range behavior and the SVD threshold.

The task of finding the optimal combination of the numerical parameters is

greatly facilitated by the fact that the reference for the EXX method is given by

the Hartree-Fock method and, thus, is readily available. Indeed, the only purpose

of the EXX method as the first step toward the exchange-correlation functional

is to accurately include the exchange interaction within the DFT framework.

Since the HF results represent the exact exchange limit in a given basis set, an

implementation of the EXX method must be assessed by how well it reproduces the

HF results.

One must understand, however, that these two methods are not identical.

First, both methods result in single-determinants that minimizes the expectation

value of the true Hamiltonian. However, in the Hartree-Fock case, the SCF

operator is not constrained to be local. Therefore, the Hartree-Fock energy is

alv--i,- lower than the EXX one. Second, the two densities must be very close, but









not exactly equal, since the OEP condition ensures that the difference between

the two densities is zero only through first order. There also is a virial condition

on the exact local exchange potential that provide the equivalence between the

HOMO energies in the HF and EXX SCF models. But this equivalence is affected

by the incompleteness of the orbital basis set. Thus, there is no exact equivalence

on the energy, density, or HOMO energy; however, one expects these quantities to

be very close in the HF and EXX methods. Thus, in this work, all three quantities

-energy, density, and HOMO energies -were compared to provide an exhaustive

assessment of the EXX implementation.

There are exactly four parameters that affect the performance of the auxiliary-

basis OEP method: the size of the orbital basis set, the size of the auxiliary basis

set, the explicit .,-vmptotic term, and the SVD threshold. The most fundamental

effect comes from the incompleteness of the basis set used to expand the molecular

orbitals. The second most important effect should come from the size of the

auxiliary basis. In the current implementation the same Gaussian-type atomic

basis was chosen as both the orbital and auxiliary bases. Conventional atomic

bases may not be the best choice to expand the potential because of the different

physical nature of the orbitals and potential. However, the use of conventional basis

sets significantly facilitates the implementation and also removes a necessity of

developing and testing auxiliary bases.

Table 2-1 reports the deviations of the EXX total energies, highest-occupied

orbital energies and densities from the Hartree-Fock values as a function of the

Gaussian-type bases. Since it is difficult to meaningfully compare the atomic or

molecular densities directly, the density moments were compared instead. In all

these calculations the exchange energy density was used as the explicit .i-iv!!ll] i. ic

term and SVD threshold of 10-5 was used. As one can see from these results it is

difficult to define the optimal basis; however, the uncontracted Roos augmented









double zeta results in adequate errors for all the quantities. Unlike the single-zeta

bases (6-311G and 6-311G**), it is extensive enough to describe the correlation

effects (which will become important in the next chapter) and not as large as

triple- and quadruple-zeta bases so the calculations remain affordable.

Table 2-1: Effect of basis set on the performance of the EXX method. Shown are
deviations of total energy (in milliHartree), HOMO energy (in eV) and density
moments (in a.u.) from the HF values. "(u)" indicate uncontracted basis sets.

Basis E CN Density moments
Ne < r> < -1> < -2
DZP 0.6 0.54 0.000747 0.003584 0.002573 0.189502
Roos-ADZP 0.2 0.50 0.000001 0.000151 0.000261 0.020306
Roos-ATZP 0.9 1.19 0.000963 0.005382 0.001295 0.080455
cc-pVTZ 0.5 7.35 0.000524 0.001968 0.001261 0.053778
cc-pVQZ 0.9 1.18 0.000025 0.000270 0.000326 0.062478
cc-pV5Z 1.8 0.65 0.000244 0.001795 0.001736 0.1- !I;
6-311G(u) 1.4 1.57 0.000032 0.000037 0.000173 0.008471
6-311G**(u) 1.4 0.84 0.000030 0.000098 0.000186 0.007592
DZP(u) 1.5 0.97 0.000031 0.000069 0.000118 0.005479
Roos-ADZP(u) 1.6 0.38 0.000017 0.000117 0.000151 0.001379
Roos-ATZP(u) 1.6 0.24 0.000007 0.000182 0.000141 0.001171
cc-pVTZ(u) 1.5 0.76 0.000050 0.000040 0.000246 0.000281
cc-pVQZ(u) 1.4 0.56 0.000010 0.000015 0.000109 0.008050
H20 < z > < 2 > <2 > < 2 >
DZP 1.2 0.16 0.0017 0.0035 0.0011 0.0003
Roos-ADZP 0.9 3.38 0.0193 0.0064 0.0008 0.0017
Roos-ATZP 1.5 0.59 0.0170 0.0051 0.0038 0.0018
cc-pVTZ 2.0 0.36 0.0027 0.0057 0.0127 0.0010
cc-pVQZ 1.9 0.56 0.0164 0.0027 0.0073 0.0021
6-311G(u) 1.9 0.14 0.0073 0.0120 0.0144 0.0150
6-311G**(u) 2.0 3.58 0.0047 0.0006 0.0048 0.0040
DZP(u) 2.0 1.47 0.0100 0.0005 0.0032 0.0010
Roos-ADZP(u) 2.1 0.13 0.0161 0.0011 0.0043 0.0001
Roos-ATZP(u) 2.3 0.16 0.0156 0.0040 0.0059 0.0022
cc-pVTZ(u) 2.3 0.36 0.0139 0.0050 0.0111 0.0025
cc-pVQZ(u) 2.2 0.42 0.0155 0.0032 0.0065 0.0025


Table 2-2 compares the performance of the EXX method with and without the

explicit .,-vmptotic terms (EAT). As one can see, using the Fermi-Amaldi potential

as explicit .,-vmptotic term only slightly improves the energy and density, but









dramatically changes the value of the HOMO energy. Using the exchange energy

density further improves the performance of the EXX method as this potential is

much better approximation to the EXX potential and, therefore, is better suited to

ensure the correct long range behavior.
Table 2-2: Effect of the explicit .,-i-~:i !ll ic term on the performance of the EXX
method. Uncontracted Roos augmented double-zeta basis set and SVD threshold of
10-5.

E CN AC Density moments
Ne < r > < r-1> < r-2>
None -128.545041 -7.85 18.52 7.891344 9.372690 31.112819 414.831247
F-A -128.545041 -0.50 18.50 7.891344 9.372689 31.112820 414.831243
CeXX -128.545041 0.37 18.49 7.891345 9.372692 31.112818 414.831234
H20 < z > <2 < 2 > < 2 >
None -76.063395 -6.71 7.95 1.987344 5.639106 7.237257 6.509336
F-A -76.063409 -0.74 8.37 1 ,*il,'_' 5.640095 7.2 ;i:.-; 6.508710
CeeX -76.063437 0.15 8.42 1.986697 5.640234 7.2- :. 124 6.508589


Figure 2-1 shows the Fermi-Amaldi potential, exchange energy density, and

EXX potentials with or without the explicit .,-vmptotic terms. Note that the EXX

potential without the EAT was shifted to facilitate the comparison. As one can

see, the use of the EAT does not affect the shape of the potential but ensures the

correct .,-i 'iii ll ic behavior.

Table 2-3 shows the results of the EXX calculations for Ne and H20 with

gradual increase of the SVD threshold. As can be seen up to about 10-5 10-6

atomic units, the SVD threshold had small effect the energy or density. However,

using the thresholds less than these values affects the quality of the HOMO energy.

Based on these results, we conclude that the optimal configuration for the

auxiliary-basis OEP implementation was achieved when we used uncontracted Roos

augmented double zeta, the exchange energy density as the explicit .-i-~:!,iiil i,

term, and the SVD threshold of 10-5 or 10-6

As the final test for the EXX method, the HF and EXX calculations were

performed for the 35 closed-shell molecules with singlet ground state chosen from















S_------------

--21-


-3

I 0.5 1

-1/r --
-4 Fermi-Amaldi ------
EXX (shifted)-
EXX (FA)
EXX(x) ......
-5 1 1 1
0 1 2 3 4 5 6 7 8
Distance from the nucleus, Angstrom

Figure 2-1: Explicit .,-vmptotic terms for Ne and the corresponding EXX
potentials. Uncontracted Roos augmented double zeta ANO basis set.

Table 2-3: Effect of the Singular Value Decomposition threshold on the
performance of the EXX method. Uncontracted Roos augmented double-zeta basis
set.

CSVD E CN Density moments
Ne < r > < r2 > < -2 >
10-1 0.0041 1.37 0.030887 0.121066 0.010612 0.217301
10-2 0.0017 1.00 0.006959 0.034878 0.001004 0.004677
10-3 0.0016 0.48 0.000239 0.001822 0.000122 0.001399
10-4 0.0016 0.26 0.000045 0.000105 0.000152 0.001378
10-5 0.0016 0.38 0.000017 0.000117 0.000151 0.001379
10-6 0.0016 0.38 0.000017 0.000117 0.000151 0.001379
H20 < z > < 2 2 > < z2 >
10-1 0.0062 1.12 0.0531 0.1112 0.0947 0.1061
10-2 0.0023 0.81 0.0044 0.0336 0.0140 0.0225
10-3 0.0022 0.61 0.0112 0.0068 0.0042 0.0071
10-4 0.0021 0.46 0.0140 0.0017 0.0025 0.0029
10-5 0.0021 0.13 0.0161 0.0011 0.0043 0.0001
10-6 0.0021 0.59 0.0147 0.0025 0.0055 0.0016
10-7 0.0021 1.28 0.0148 0.0026 0.0056 0.0017
10-8 0.0021 1.15 0.0148 0.0026 0.0056 0.0017
10-9 0.0021 1.15 0.0148 0.0026 0.0056 0.0017









the G1 test set. The G1 set was the first one from the series of sets [18] developed

to test standard electronic structure methods. Although not exhaustive, this set

aims at representing different types of molecules and chemical bonding. Also the

experimental (rather than computed) structures were used to avoid the ambiguity

in comparison. The experimental values for the bond length and angles are readily

available online (Computational C('!i ii-I ry Comparison and Benchmark DataBase,

http://srdata.nist.gov/cccbdb).

Table 2-4 reports the HF energies, the absolute and relative differences

between HF and EXX energies, the HF dipole moments and the absolute and

relative difference between the HF and EXX dipole moments.

As one can see, the EXX energies are very close to the HF ones, with the

largest deviations of about 10 milliHartree for SO2, which is less than 2' of the

MP2 correlation energy of 631 milli-Hartree.












Table 2-4: P i. ::.. : : of the EXX methods for the 35 closed-shell molecules of
the C1 test set. Uncontracted Roos augmented double zeta basis set. Energies are
in : : :TT : tree and (i: )ie moments are in D i


Di-
0.0 .. .

0.027
0 .i


1.
0.1

1:.
1.0-,.


0.009 0.47%


0.::
0.0 .'
0.031


5.78%
2..


System
LilH
C i :
NHs3
H20
HF
SiH4
PH3
H2S

L, C
Li2



Li I
C212

H3(C-CH3
HCN
Co
CO-
H2C= O

HO-OH

F2
CO0

P2
C12

SiO
C'
CIF

C1i Cl
SC-SI1
HOC1
SO2
Average
Maximum


i : :.gy
0 : :: '




o.0:::
0.0- '.
0.- 11 (c
0.C'"






0.C .




0.0011
0 -::

0.0:.
0 o- -

0 .0 :: ..
0

O., .C: .
0 ( : *







0. (:

0.0:: '

0.'"". '
0.0:: .:1
0.- 11'


O.C il


0.001 i
0 1'




0 ::: 1
0.01,. '
0:11'"
0.- (H
:1'ir


0.0
0.-. ,
2.77%
0.1:

1 .,
2.11
0.7
2.1
1.5
5..


0.004 0.0.




0.037 1.1-

0.040 1.41-
0.037 2.05%X

0.022 1.31


0.003
0.024
0.045
0.011




0. -
0.t : 1


0.025
0.045















CHAPTER 3
CORRELATION FUNCTIONALS FROM SECOND-ORDER
PERTURBATION THEORY

While the EXX functional provides the exact description of the exchange

interaction, one still needs a functional to account for electron-correlation

effects. Unlike the exchange case, there is no closed expression for the correlation

energy. Thus, the principal challenge is to find an adequate approximation for the

correlation functional.

Ab initio Density Functional Theory solves this problem by using the energy

expressions from ab initio wavefunction methods. Most ab initio methods calculate

the correlation energy in terms of the SCF orbitals. Treated as orbital-dependent

functionals, these expressions become the correlation functionals in the ab initio

DFT context.

The simplest ab initio approximation for the correlation energy comes from the

second-order perturbation theory. That is why the second-order energy expression

was chosen as the basis for the initial implementation of ab initio DFT.

3.1 Correlation functional from Second-Order Perturbation Theory

The idea of using perturbation theory to approach the exact correlation

functional belongs to Girling and Levy [19]. They demonstrated how the exact

correlation functional can be formally constructed from the perturbation expansion

within the Adiabatic Connection formalism. Truncated in second order the

Gbrling-Levy perturbation theory give the first approximation to the correlation

functional
occ. virt. vl a) 2 occ. virt. ab) 2
E(2) y y^ (3-1)
z^ a-^ Ci-Ca 4 ij a-b Ci + Cj Ca Qb
i a 2,] a~b









Similarly to the EXX functional, the corresponding functional derivative (i.e.,

the correlation potential) can be obtained using the chain rule

6E(2) f ts(r') JE(2)
(2r) -- dr' () (3-2)
c 6p(r) 6p(r) 6vb(r')

which results in the integral equation for the potential
/ 6E(2)
dr (r) 6( (3-3)
]dr%8(rr')v(2) 6v (r')

To derive the algebraic expression for this equation one must take the

functional derivative of the right-hand side. Engel et al. [20] were first to solve

this equation based on the grid-based OEP algorithm. However, they did not

include the potential into the SCF iterations and did not take into account the first

term of Eq. 3-1, which describes the contribution of the single excitations to the

second-order energy. Also, in their numerical solution of the OEP equation, certain

terms were treated separately which apparently lead to a numerical singularity.

Based on this singularity, they concluded that the correlation potential from

the second-order perturbation series does not vanish at large r, as the exact one

should. This raised the question whether the second-order perturbation theory can

lead to a meaningful correlation potential. Niquet et al.[21] disputed this conclusion

and argued that it is the numerical procedure used to calculate the potential that

is the source of this problem. Recently, they showed that the correlation potential

from the second-order perturbation theory has the correct -C/r4 .i ,pi- I at

least for closed-shell systems with spherical symmetry.

As it has been discussed in the previous chapter, any grid-based OEP

implementation is limited to small systems. Grabowski et al.[22] rederived the

expression for the second-order potential that included the contributions of single

excitations and implemented it based on the auxiliary-basis OEP method. They

have shown that the PT2 functional results in accurate correlation energies of









two-electron systems, quickly approaching the exact correlation energy in the

high-density limit and results in qualitatively correct correlation potentials for

atoms.

3.2 Correlation Functional from Second-Order Perturbation Theory with
Partial Infinite-Order Resummation

More recently, the same authors [23] found that the iterative solution of

the OEP equation diverges for the Be atom and the PT2 functional significantly

overestimates the correlation energies for small molecules.

This poor performance of the second-order functional is not surprising. It is

known that the KS reference is usually a bad reference for perturbation expansions.

For example, Warken [24] analyzed the perturbation series based on the KS orbitals

and showed that it usually has radius of convergence smaller than 1. Therefore, the

KS-based perturbation series often diverges for molecular systems.

This presents an even bigger problem for the determination of the potential.

Indeed, truncating a divergent series at some finite order still results in a finite

energy. Moreover, it is known that some .,-i~, ill ically divergent series may give

decent approximations in lower orders. However, the convergence of the series is

crucial to obtain even a lower-order potential. Because the OEP equation for the

potential is solved iteratively, the large terms will accumulate and the iterative

solution will diverge.

In our opinion, there are two primary cause for the divergence of the series.

First, the KS model features a local SCF potential, and as the result, the virtual

orbitals lie much lower then, for example, Hartree-Fock ones. As the result, the

occupied-virtual energy difference are smaller and the resulting series features

small denominators. Second, unlike the Hartree-Fock case, the KS reference

Hamiltonian is not equal to the one-body part of the true Hamiltonian. Thus, the

perturbation contains a one-body part that can be large and ultimately lead to the









divergence of the series. Although not immediately obvious, these two problems

are closely related. Indeed, as it will be shown below, the large one-body terms can

be removed from the series by resumming them to all orders. As the results, the

denominators are formed by diagonal matrix elements of the Fock operator, which

correspond to large differences.

To see where these terms arise, consider the true Hamiltonian in second-

quantized form

all
H > hpqcaq + 1 > pq s r)a>4a-a-
p,q p,q,r,s
all occ. I
=Y [hpq + '.i )caq + > pq\sra]a)aar (3-4)
p,q i p,q,r,s

Thus, the effective one-body part of the Hamiltonian -the Fock operator -consists

of the core Hamiltonian and two-electron terms of one-body character
OCC.
VH +nlx ( '. (3-5)


Note that the Hartree-Fock model uses the Fock operator as the SCF

Hamiltonian, therefore all the one-body terms are included in the reference

Hamiltonian. The KS Hamiltonian is based on a different potential, therefore the

one-body terms remain in the perturbation

all all
V = H H = h+ H + l hs] pqt q +W [unlx V pq + W (3-6)
P,q P,q

where W stands for the two-body terms.

These one-body terms can be large and potentially lead to a divergent

perturbation series. In case of ab initio DFT, the situation is slightly improved,

because the use of the exact exchange potential reduces the size of certain one-body









terms. Indeed, the OEP equation for the EXX potential
OCC. Virt. i / i / \/ *
cC v- z Q(r)Qa(r){a veg1 Unix i)
.( (r) E aL vL. + c.c. 0 (3-7)
i a Ca

can be viewed as a fitting procedure for a local vexx that minimizes the difference

of (a vnx ve i) with the given weights. Since the correlation contribution,

(a v i), corresponds to the higher orders of perturbation theory (hence, is small),

one should expect that the use of the EXX potential as the reference makes the

(a, i) part of the perturbation small.
However, this is not true for occupied-occupied or virtual-virtual elements

of the one-body part of the perturbation. Nothing in the definition of the vexx

indicates that terms (il nx Vexx j) or (a n, vexx b) should be small. The

presence of these terms in the perturbation may ultimately lead to the divergence

of the series.

A usual method to avoid the divergence of a perturbation series is to perform

an infinite-order resummation. In the case of the series under investigation, one is

particularly interested in resummations of the occupied-occupied and virtual-virtual

one-body terms. As we will see below, these terms are particularly easy to resum

because they do not mix the reference and complementary spaces. The infinite-

order resummation of these terms can be effectively performed by redefining the

reference Hamiltonian.

Let us demonstrate how the resummation of these terms can be performed.

First, recall that using the ab initio DFT Hamiltonian leads to the following

perturbation


V = V + V" (3-8)


occ. virt.
V/ > K vnix vxc a)aa + c.c. + W (3-9)
i a








OCC.
V// I V ( ^a Vc x )d a


(3-10)


71,3

where V" indicates the problematic terms.

To find the solution of the Schr6dinger equation using perturbation theory, one

partitions the equation


(E- H)T)- 0


(3-11)


into


(Eo Ho) ) (V 6E) ) 0


(3-12)


Since is the ground-state eigenfunction of Ho, the following is true


(Eo- Ho)0) 1) 0


(3-13)


where Q = 1 V)( is the projector on the complementary space spanned by all

but ground-state eigenfunctions of Ho.

Projecting Eq. 3-12 on the complementary space and adding Eq. 3-13 one

obtains


S(Eo- Ho) }) (V -AE) ) =0,

or

TH- o (V- AE) I).
Eo Ho
Applying this relation iteratively, one obtains the series for the true

wavefunction


qJ) =) + qJ-4)


E o- Ho (

= ) + (V-AE) + (V- AE)


S EoHo (V- AE)]" o-H
=00
ZE[0(V AE)]"4<


(3-14)



(3-15)


(3-16)









Note that in the expression, the terms of the sum above do not correspond to

any order of perturbation theory because AE has terms of all orders. To obtain

the order-by-order expansion one would have to substitute the order-by-order

expression for AE. This, however, is not necessary for the current discussion.

Instead, one proceeds by identifying the problematic terms V" and reordering

them




+ Eo[ o (v" -AE")] ))


+ Q o(V/V-AE")](V" -AE")]Eo Ho(V -AE') 4
n= 0o
+ ... (3-17)


However, since V" is such that


S[V" AE"] ) = 0 (3-18)

one can perform the summation to obtain


III)

n (v'- E') o)
oo0 [7 o Ho -Eo-Ho

& {r'(V'-AE') |) (3-19)
no0
where


i- Eo-- HEo Eo- Ho Eo + E" -Ho V
m=
Thus, the new series where the occupied-occupied and virtual-virtual one-body

terms have been resumed is the perturbation series featuring a new resolvent









operator R7. It is based on the new reference Hamiltonian


H' = Ho + V"
all. occ. virt.
=1p hs+u p)aap + i VH+ Unix ji)aajZ (a VH + Unix u b) alab
p i,j a,b
occ. occ.

i,j a,b

where the fact that (p Ak + u q) = 6pqp h, + q) and


hs + u + VH + unlz h8 + VH + nl = f (3-22)


is the one-particle Fock operator were used.

Note that since H' is no longer diagonal in the basis of eigenfunctions of Ho,

its inverse is no longer given by Eq. 1-36.

The case when the reference Hamiltonian is not diagonal in terms of the SCF

eigenfunctions is typical for the generalized Many-Body Perturbation Theory

[2]. Two possible solutions to this problem are to find the inverse (i.e., the

resolvent operator) iteratively or to find a unitary transformation that makes the

occupied-occupied and virtual-virtual blocks of the reference Hamiltonian diagonal.

While these two solutions are formally equivalent, the unitary transformation is

less computationally expensive since it involves only operations with two-index

quantities.

Thus, if at each SCF iteration, one transforms the occupied orbitals so that

fij = ijfii and the virtual orbitals so that fab -abfaa then H' become diagonal in

this basis. Since this transformation does not mix the occupied and virtual orbitals,

all the physically relevant quantities such as energy and density are not affected

by this transformation. This new set of orbitals is one of the possible noncanonical

representations of the KS orbitals and is called semicanonical because it diagonalize

the occupied-occupied and virtual-virtual block of the Fock operator. This is









why the new perturbation series and corresponding functionals are referred to as

- ,,iii, ,,,,,, ,1 (SC ).

3.3 Implementation of the PT2 and PT2SC Functionals

As in the case of the EXX functional, the central element of implementation of

orbital-dependent correlation functionals is the OEP method for the corresponding

potential. As one can see from Eq. 3-3, the only difference between the integral

equations for the EXX and PT2 (or PT2SC) potentials is on the right-hand side.

Thus, if one has the OEP method implemented for the EXX potential, its extension

for the second-order potential is a tedious, but straightforward task.

As for the EXX potential, the implementation based on the auxiliary-basis

OEP method was used. In this method the integral OEP equation is transformed

into a linear matrix problem by projecting the real-space quantities onto an

auxiliary basis. The only difference is that the calculation of the right-hand side

requires a number of contractions of matrix elements of auxiliary basis functions

with the one- and two-electron integrals.

Thus, a typical SCF iteration with both the EXX and PT2 potentials proceeds

as follows
Read in the one-electron integrals and construct the matrix elements of the
core Hamiltonian.
If the PT2SC is used, read the two-electron integrals, construct the Fock
matrix, and diagonalize its occupied-occupied and virtual-virtual block to
obtain the SCF coefficients in the semicanonical representation.
Read in the two-electron integrals and matrix elements of the auxiliary
basis functions. Transform them using the original or semicanonical SCF
coefficients.
Construct the OEP integral kernel in the auxiliary-basis representation and
find its inverse using the SVD procedure.
Construct the right-hand side of the OEP equation for the EXX potential.
Contract it with the inverse of the kernel to obtain the EXX potential in the
auxiliary-basis representation. Calculate its matrix elements with respect to
atomic and molecular orbitals.
Construct the right-hand side of the OEP equation for the PT2 potential
using the one- and two-electron integrals and matrix elements of the EXX
potential. Contract it with the inverse of the kernel to obtain the PT2









potential in the auxiliary-basis representation. Calculate its matrix elements
with respect to atomic orbitals.
Add the matrix elements of exchange and correlation potential to the core
Hamiltonian. Diagonalize it to obtain new SCF coefficients, check the
convergence, and proceed to the next SCF iteration unless the convergence is
reached.

The most expensive step in this procedure is the transformation of the two-

electron integrals. As in case of conventional MP2 energy calculation, the first

step in this transformation requires the loop over one occupied and four atomic

indeces. Therefore, the computational cost of the SCF iteration with the EXX

and PT2 potential is similar to the MP2 energy calculation and scales as N,,,ccNl,

where Noc is the number of occupied orbitals and Naii is total number of orbitals.

Therefore, the overall cost of a EXX-PT2 calculation is the cost of MP2 times the

number of SCF iterations.

Since the incompleteness of the orbital basis set leads to the singularities

in the integral kernel, one has to use the SVD procedure to find an approximate

solution to the OEP matrix problem. In all the calculations reported below, the

SVD threshold was fixed at 10-6 atomic units. For example, in the case of Roos

augmented double zeta basis set, which was used as orbital and auxiliary bases in

all molecular calculations, the 10-6 threshold results in 0 or 1 singularities removed

in the 1 I i i i ly of atoms and molecules considered below. The largest number of 3

singular eigenvalues (out of total 58 orbitals) was neglected for NaC1.

In the case of the exchange, the same basis set (contracted or uncontracted

Roos augmented double zeta) was used to expand the orbitals and potentials.

3.4 Numerical Tests for Ab initio Functionals

Correlation Energy in the High-Density Limit.

The first test was for the performance of the ab initio correlation functionals

in the high-density limit. It is known [19] that the contributions of order higher

than second scale as negative powers of the scaling parameter and, therefore,









vanish as the scaling parameter approaches infinity (i.e., in the high-density limit).

Therefore, the second-order energy expression is the exact limit for the correlation

functional at infinitely large scaling parameter and the combination of the exact

exchange and second-order correlation functionals becomes the exact exchange-

correlation functional in the high-density limit.

The PT2SC functional is based on the energy expression that does not scale

homogeneously due to the presence of the Fock operator in the denominators.

Nevertheless, it is equivalent to a infinite-order series, where the higher-order terms

again scale as the negative powers of the scaling parameter and vanish as in the

high-density limit. Therefore, the EXX-PT2SC functional must approach the exact

exchange-correlation functional as well.

To test the properties of ab initio functionals in the high-density limit,

we calculated the correlation energies of the series of two-electron atomic

ions with increasing nuclear charge Z (Table 3-1. Two-electron systems are

particularly convenient because the exact exchange potential is just half of the

Hartree potential, hence, there is no error associated with the auxiliary-basis

implementation of EXX potential. Moreover, the full CI energy is readily available

as only single and double excitations contribute to the correlated wavefunction (the

Coupled-Cluster method with single and double excitations [CCSD] was used to

obtain the full CI energy.)

The results demonstrate that the PT2 energy indeed rapidly approaches the

full CI value. This is in agreement with the results reported by Grabowski et al.

[22]. Note that the GGA correlation functionals such as PBE or LYP do not have

the correct scaling and result in nonvanishing error.

Correlation Energies of Closed-Shell Atoms.

The next test set consisted of the first six closed-shell atoms with singlet

ground states. Table 3-2 reports the deviation of the PT2 and PT2SC correlation









Table 3 1: Performance of ab initio and conventional correlation functionals in the
high-density limit. Ti: first column gives full CI correlation energies and the
remaining columns give the <: :', ... between these values and correlation < ...
calculated with ab initio and conventional correlation functional. Uncontracted
Roos .. .... ,1 double zeta basis set. All values are in milliHartree.

Z Ion FCI PT2 PT C PBE BLYP

4 Be"' -39.78 -1.84(E ) 3.42(' ) -5. .1 ) -9. 2' )
10 Ne 19 -0. .) 1.47( ) -7. (1 ) -10.21( .)
12 MI -40.37 -0.54(1) 1.24( ) -7.52(1 -9 '7(25')
18 Ar' 11 -;0.35(1 .) 3(' ) -7.61(1 -9.54(2 .)
: Ca1 40. -0 ) 9( ) -7.67(1 : -9.55(2 )


energies from CCSD(T) values. The Coupled Cluster method with single and

double excitations, and noniterative inclusion of triple excitations provides very

accurate energies for closed-shell atoms and molecules at the equilibrium geometries

and will be regarded as the correlation limit for the given basis set. The energies

obtained with second-order Meller-Plesset perturbation theory and CCSD method

are given for comparison.

Table 3-2: Performance of ab initio correlation functional for closed-shell atoms.
Ti.. first column give the C- .:)(T) correlation energies and the remaining columns
give absolute and relative deviations from these values. T: MP2 and CCSD values
are given for comparison. Roos ..*.... *..1. double zeta basis set. All values are in
milliHartree.

Atom CC' D(T) PT2 PT A Ai i'2 CC .i)
He -37.1 -3.3 ('. ) 6.9 ( ) G.9 ( .) 0.0 (0)
Be -53.4 N/C 18.4 ( ) 18.5 (3E' ) 0.2 ( )
Ne '.7.5 : (. ) 3.7 ( 1 ) 4.5 ( ') 4.1 ( )
Mg -51.2 -24.5 ( ) 11.7( ) 11.9 ( ) 0.7 (1 )
Ar -7.7. ) .6( ) 5. ( 4.1( )
Ca -84.9 -42.8 ( ) 11. ( ) 12.0 (1 .) 2.0 ( )


As one can see, the PT2 functional results in accurate correlation energy for

He, but significantly overestimates the correlation energy for larger atoms. Also

the iterative solution for the PT2 potential did not converge in the case of Be. On

the contrary, the PT2SC energy is slightly worse for He, but gives much better










estimation of correlation energies for other atoms. Also, the iterative solutions for

the PT2SC potential converged in every case. Note that the PT2SC functional

performs slightly better than MP2.

Correlation Energies of Molecules.

Based on the results for atomic correlation energies one can conclude that

the PT2 functional significantly overestimates the correlation energies, while the

PT2SC functional offers a more adequate description of the correlation effects. To

further verify this conclusion a series of calculations was performed for the same set

of 35 closed-shell molecules that was used in C'! lpter 2.

Figure 3-1 reports the relative deviations of the correlation energies from the

CCSD(T) values averaged over 35 molecules


1A( hod) E[method] E,[CCSD(T)]
A(method) = (3-23)
System E[CCSD(T)1
system



oHF EXX 14
100
n12
// MP2
S- ///// 10
S///// PT2SC
| 80 8
6
Ht v\\ / 4 CCSD
S60 -// ---
S/ Mp IP4
I 40 1 1^ ^^ PT2 0 ,
40


20
-a 20
S/PT2SC MP2
U ..P3 CCSD
0/ i -7 _--_ F M P4


Figure 3-1: Performance of ab initio DFT and ab initio wavefunction methods in
total energy calculations for the G1 test set. Shown are average relative deviation
from the CCSD(T) values. Roos augmented double zeta basis set.









The results are very similar to those for atoms. First, the iterative solution

of OEP equation for the PT2 potential diverged for 19 molecules (including LiH,

NH3, N2, CO, and others). For the remaining 16 molecules, the PT2 functional

overestimated the correlation energy on average by !1I' On the contrary, the

iterative solution for the PT2SC potential converged for all 35 molecules. The

PT2SC functional led to an average error of 11.7'. slightly better than the MP2

value of 12.;:' .

Note that for these systems, the HF-based perturbation theory indeed provides

a series that systematically converges to the exact answer. Including higher-order

corrections reduces the error from the MP2 value of 12.;:'. to the MP3 value of

8.0'. to the MP4 value of 3.>' This supports the promise of ab initio DFT to

provide a series of systematically improving approximations to the correlation

functional. Of course, the Coupled Cluster method provides a more rapidly

converging series resulting in the average error of 6.1 already at the CCSD level.

It should be emphasized that it is the ab initio character of the PT2 and

PT2SC functionals that allows us to compare the absolute values of the correlation

energy. On the contrary, one cannot directly compare the GGA energies to ab

initio results. It is a well-known fact that the absolute values of the DFT energies

can be very different from the wavefunction correlation limit. Instead, one has

to compare relative quantities like atomization energies to assess the quality of

conventional functionals.

Total Energy as a Function of the Bond Length.

Next test (Figure 3-2) assessed the performance of ab initio DFT functionals

in description of the potential energy surfaces (i.e., the total energy of a molecule

as a function of the bond length.) Four molecules were chosen to represent different

types of chemical bonds: the ionic bond (HF), symmetric single covalent bond

(F2), double bond (H20 where the hydrogen atoms were simultaneously pulled














away from the oxygen atom), and a triple bond (N2). To remove the ambiguity


with respect to the curves' absolute position, the curves were shifted vertically


so that all curves cross the CCSDT curve at the experimental bond lengths.


Such a shift not only facilitates the comparison of the shapes, but also allows a


direct comparison between the ab initio and conventional DFT (PBE in this case)


methods.


-199 2

-100 15 HF -F
-19922 -

-1002 P-

S-199 24
-100 25 -

-19926
-100 3 -
EXX-PT2 +0 100H O EXX-PT2 +0210H-
EXX+PT2SC 0 00H--- EXX-PT2SC 0 007H-
MP2 007H 199 28 MP2 0 020H
PBE + 0067H ---PBE + 0 147H ---
-100 35 CCSDT CCSDT -
06 08 1 12 14 16 18 2 12 14 16 18
H-F bond length, Angstrom F-F bond length, Angstrom
-109 1
76 H20 ,.. N2
-109 15
76 05

-76 1 -1092

-76 15
-109 25
-76 2 / /

-76 25 -109 3 /
EXX-PT2 + 0 110H EXX-PT2 + 0 170H
EXX-PT2SC 010H EXX-PT2SC 0 009H -
-76 3 MP2 0 012H --- // MP2 0 016H ---- -
PBE + 0 062H ------ -109 35 PBE + 0 095H ---
CCSDT CCSDT
76 35
06 08 1 12 14 16 18 2 08 1 12 14 16 18
H-0 bond length, Angstrom N-N bond length, Angstrom

Figure 3 2: Performance of ab initio DFT, ab initio wavefunction, and

conventional DFT methods in calculations of the total energy as a function of the

bond lengths. Roos augmented double zeta basis set.




As one can see from the figures, the EXX-PT2 functional failed to reproduce


a meaningful energy curve for any of the four molecules. For all the systems except


N2 the EXX-PT2SC curves lies closer to the CCSDT one that MP2. The N2


plot clearly demonstrates that one should not apply a perturbative method to a


multiple-bond breaking problem.









Atomic and molecular densities.

The quality of a DFT functional must also be reflected in the converged

density, i.e., the density that minimizes the functional. Since it is difficult

to compare atomic or molecular densities directly, the density moments were

compared instead. Table 3-3 reports the density moments for Ne calculated

with ab initio DFT, ab initio wavefunction, and conventional DFT methods. The

density obtained with the CCSD(T) method was used as the reference. Also, unlike

in the case of total energies, the GGA density (obtained with PBE functional)

can be directly compared to the ab initio results. As evident from the table, the

overestimation of correlation effects by the PT2 functional leads to the situation

where the exchange-only (EXX) density is better than when the correlation effects

are included (EXX-PT2). However, the EXX-PT2SC functional results in a density

that is closer to CCSD(T) than any other method, in the regions both close to the

nucleus (as sampled by the average values of the negative powers of r) and away

from the nucleus (as sampled by the average values of the positive powers of r).

Table 3-3: Density moments of Ne calculated with ab initio DFT, ab initio
wavefunction and conventional DFT methods. Uncontracted Roos augmented
double zeta basis set.

Method (r-2) (r-1) r) (2) (r3) (r4) r
PBE 415.180 31.089 8.004 9.833 15.972 32.714 82.036
EXX 414.831 31.113 7.891 9.373 14.386 27.200 61.613
MP2 414.819 31.104 7.913 9.447 14.599 27.811 63.472
EXX-PT2 414.892 31.061 8.075 10.092 16.788 35.351 91.162
EXX-PT2SC 414.878 31.097 7.972 9.686 15.398 30.486 72.924
CCSD(T) 414.879 31.101 7.955 9.615 15.155 29.650 69.861


As the test for the electronic density of molecules, the molecular dipole

moments were calculated. Figure 3-3 shows the average (over 22 systems with

nonzero dipole moments) deviation of the computed dipole moments from the

experimental values. For 9 (out of 22) systems where the solution for the PT2

potential converged, the deviation from the experimental dipole moment is no less










than 0.13 Debye (HF) with the largest error of 0.56 Debye for H20. This results in

an average error greater than that of exchange-only methods. Again, EXX-PT2SC

results in dipole moments that improves upon the MP2 values. In the case of EXX-

PT2SC, the largest errors in the computed dipole moment are for SiO: 0.45 Debye

(CCSD:0.14, PBE: 0.22) and SO2: 0.22 Debye (CCSD: 0.10, PBE: 0.16). In the

case of PBE, for NaCl: 0.40 Debye (OEP2(SC): 0.10, CCSD: 0.22) and LiH: 0.27

(OEP(SC): 0.04, CCSD: 0.01). In case of CCSD, the largest errors are for NaCl

and SiO.


Figure 3 3: F .:::: : e of ab initio DFT, ab initio wavefunction, and
conventional DFT methods in dipole moment calculations .. the G I test set.
Shown are the average absolute deviation from the experimental values. Roos
augmentted double zeta basis set.


V I,
3




I c I


^^ ^ a p3
1 1 p i )

D/I'
ii
I> 'Fl ^ .














CHAPTER 4
OTHER THEORETICAL AND NUMERICAL RESULTS

4.1 Connection between Energy, Density, and Potential

The effective potential of the Kohn-Sham model is defined as the derivative of

the energy functional with respect to the density. This definition follows from the

variational condition on the energy functional. It ensures that if the Kohn-Sham

SCF model generates the density that minimizes the energy functional and, thus, is

equal to the exact ground-state density.

In the context of ab initio DFT, this should mean that the potential defined

through the functional derivative generates the SCF density that minimizes the ab

initio energy expression, upon which the functional is based. It also should mean

that the SCF density is equal to the corresponding ab initio mi: iiv- body density.

For example, if the functional is based on the energy expression through

second-order then defining the potential as

S[E + E() + E(2)] (4-
vs r = (4-1)
6p(r)

should ensure that
a [Eo + E(1) + E(2)]
=)O (4-2)
6ps,()
and

ps(T) ps(T) + p )(r) + p(2 ) (4-3)

We examined the correspondence between these relations, and explicitly

showed that their equivalence holds true in every order of perturbation theory.

We first examined the correspondence between the functional derivative of

Eq. 4-1 and the stationary condition of Eq. 4-2. Applying the chain rule (Appendix









A), we transform the functional derivative into the following integral relation

dr'v (r () (4-4)
6v (r) 6,(vsr')

The derivative of the noninteracting kinetic energy can be easily found using the
fact that
elec.
T, ) ( H v(r) ) = E, drp,(r)v,(r). (4-5)

Consequently,

6T, 6E, drp(r)v(r)
6v, (r) 6v, (r) 6v, (r)

=p'(r) p'(r) dr'v(r')6p(r')
S, (r)

dr'v (r') P ) (4-6)
6v1, (r) *

Thus, the derivative of the noninteracting kinetic energy cancels the left-hand

side of the integral equation for the effective potential and Eq. 4-4 becomes the

stationary condition on the energy

5E
S-0 (4-7)
6v,(r)

4.2 Diagrammatic Derivation of the Optimized Effective Potential Equation

In C'! lpter 2 we defined the correlation potential through the functional

derivative. To obtain the actual expression, we have to find the derivative of

the second-order energy with respect to the potential. To take this derivative

algebraically is quite an exercise (Appendix C).

Here, we describe a diagrammatic technique that significantly simplifies

the derivation of these expressions. This technique is similar to the one used by

Bartlett et al.[23] but operates directly with the energy functional rather than the

density. We will use the operators in second-quantized forms and antisymmetric









Goldstone (ASG) diagrams to represent the perturbative corrections to the energy

and energy derivative. A proper introduction of the second quantization and the

diagrammatic techniques can be found in books on rn ,iv- body theory (see, for

example, Ref. [3]).

In the diagrammatic representation, the true Hamiltonian can be written as
all
H hpqP a 4 S q I aaasa
p,q p,q,r,s
-0

----h + + + ---0 + +

-A -0



+ ---- + V+ + (4-8)



where the first three diagrams represent the "core" Hamiltonian
elec. elec. 1
He => h(ri) [ V, + Vxt(ri)], (4-9)
i i

and the remaining diagrams represent the two-electron terms
elec.
W=- (4-10)


Similarly, the reference SCF Hamiltonian is represented by the following

diagrams
all
Ho = [hpq + upql]aaq
P,q


S--- + + + -u + + (4-11)

Sv\- _U









where u denotes the effective electron-electron part of the SCF potential


u(r) = h, h = VH(r) + v'(r)


With this reference Hamiltonian, the perturbation becomes

V = H Ho


+,


+


0


+


where we used a combined symbol for the one-body part of the perturbation


----U


Note that the one-body part of the perturbation contains only the difference
between the nonlocal exchange operator and local exchange-correlation potentials:


all occ.

p,q i


all all
= (p pVH + Vnlx -H vc qaaq = p l vYc %aq. (4-15)
p,q p,q
Once the perturbation is defined in terms of diagrams, the diagrams
representing the perturbative corrections to the energy are constructed by simply
combining the corresponding number of perturbation diagrams. All of the energy
diagrams must be closed (no open lines) and linked (no disconnected closed


(4-12)


(4-13)


(4-14)


--_@









parts). Thus, the perturbation series for the energy is represented by the following

diagrams


Eo + E(1) + E(2) +...



O---h + O----u + 0-- + + +... (4-16)
0--@ 0-

Interpreting these diagrams according to the standard rules [3] results in the

familiar expression for the energy series


Eo + E(1) + E(2)
OCC. OCC.
1 K V2 + 1t + UI)+nx U i
i i
occ. occ. OI ^ i) occ. occ. i b)\ 2
I 0C 0CCa (4-17)
Z^ Z^ FCi Ca Y- FCi+ Cj C Cb
i a ij a,b

Thus, one can see that the diagrammatic technique is a powerful tool for

handling the perturbation theory expressions for the energy. Furthermore, using

the stationary condition on the potential and double-perturbation theory, we

applied this technique to obtain the corresponding potential.

If we consider a infinitesimal change in the SCF potential, 6u, as a

perturbation, then the perturbative corrections to the energy correspond to the

Taylor series of the energy change due to 6u


j 6u~r)
E[u + u ] E lE[] I+ E) +. E" l +[] + dr uQ() + .... (4-18)

Since E[u] is given through the perturbation theory as well, the linear response

of the order-by-order corrections to the energy is given through the double-

perturbation expansion, where one perturbation is the ]riin i,-body perturbation and

the second one is 6u.









Thus, the order-by-order contributions to the stationary condition are obtained
diagrammatically by drawing all possible closed, linked diagrams that involve one

6-vertex and the corresponding number of the many-body vertexes





6 ---


0 0- -6 0
[E0 + ---- + -E(2) ..
(r) --



+ -o + o + + ---_o + -- + (4-19)



Since each one-body vertex involves a matrix element of u, Eq. 4-19 is a

nonlinear, inhomogeneous equation for the effective electron-electron part of the

KS potential, u. To be consistent with the perturbative treatment of the energy,

one has to solve this nonlinear equation iteratively, by neglecting the contribution

of the higher-order terms at every iterations. Thus, on the first iteration one

neglects all but linear terms, and solves the resulting linear equation for the first

approximation to u(r)
occ. virt. \(r)Q i) ui a) occ. virt. i (r) i i ) (420)
SCi Ca i (a
Sa a
On the next step one keeps the unknown in the linear terms, substitutes u(r)

by its first-order approximation, and neglects the higher-order terms to obtain

a linear equation for the second approximation to u(r) (Eq. 4-21). Then, one

proceeds by including higher and higher terms until the nth-order approximation

for the optimized potential is obtained. With the new value for the potential, one











constructs the OEP Hamiltonian and diagonalizes it to obtain a new set of OEP

orbitals and orbital energies. This concludes one iteration of the SCF procedure,

which must be repeated until self-consistency is reached for both the orbitals and

optimized potential.


O~cc. virt ii;i\


a (






U( (
i a~b
Occ. vcirl.





Es
a
c~c virt.


i~j ar,b
(-)Cc. virt.


ij L,b
OC'Z. virt



OCC. virt.


i a,b



i,j,k a,
occ. vir

2
ijJ u~b
occ. vir

2

occ. vir

2
ijJ u~b


I C.C


(2' ii
I C.C
Ci Ca



(Ci Cb)(Ci C,)





((j C a j)







( (C Cb)(C I C)(j LyL, a)

(r') Ob(r) Ii Ab)
+ ccx




Ca)((Ci (Ca
(b)b






(r i ( b) ((i (a)
b ((C k C1,)(( C (a I (b) Ca ()


,c C



'C ( j(i ~ (r)Kj (a K(b) .
b ((,k ( a b)(( C(a (b)



'C ((i (c ( b: C)((i (a (b), -h


(4-21)









4.3 Mixing Exact Nonlocal and Local Exchange

An interesting extension of Hohenberg-Kohn-Sham model is to use the

nonlocal exchange operator instead of (or in addition to) the local exchange

potential. In particular, using the adiabatic integration formula, Becke [25] showed

that a GGA exchange-correlation functional combined in equal proportions with

the Hartree-Fock exchange functional, so-called Becke half-and-half functional,

should be a better approximation to the exact functional. Semiempirical

parametrization to experimental data showed that the inclusion of about 1/5 of

Hartree-Fock exchange in the BLYP functional (B3LYP functional) actually leads

to the most significant improvement in the performance. Such hybrid functionals

are currently the most successful extension of the GGA form and the most accurate

functionals in the overall hierarchy of conventional functionals.

All ab initio exchange-correlation functionals include the exact exchange

functional, but we can treat a part of it as an implicit functional of the density and

the other part as a functional of orbitals. This is the idea of behind the Hartree-

Fock-Kohn-Sham model [26], which treats a part of the energy as a functional of

orbitals and the remaining part as a functional of the density. The corresponding

constrained minimization procedure results in Euler-Lagrange equations that mix

the orbital derivative (like in Hartree-Fock theory) and the density derivative (like

in Kohn-Sham theory). We can also further extend the HFKS model and treat only

a fraction of the exact exchange functional as the functional of orbitals. Thus, the

hybrid ab initio functional can be written as


EH = T + E,,, + EH + Io [',] + (1 a)E, [p] + E,[p], (4-22)


and the corresponding SCF potential as


V,(r) = VUet(+T + H av)+ + (1 a)Vex(rT) + Vc(r),


(4-23)









where a defines the fraction of the exact exchange energy treated as the functional

of orbitals rather than implicit functional of the density.

Beside just theoretical curiosity, there is a simple practical reason why such

a combination is worth investigating. The presence of nonlocal exchange partially

removes the self-interaction component from the occupied orbital energies. It also

results in more diffuse virtual orbitals that correspond to electron attachment

(rather than excitation) states. Although it may not appear advantageous

from a formal point of view, it does improve the SCF model for purpose of

the perturbation expansion. Indeed, as we discussed in C'! lpter 3, a local SCF

potential leads to small differences between the occupied and virtual orbital and as

the result, the perturbation series features small denominators and often diverges.

Presence of the nonlocal exchange shifts the virtual energies up and, therefore,

should result in an improved perturbation series.

We implemented the hybrid ab initio functional in our ab initio DFT code.

The implementation required only minor modifications of the scheme described

in C'! lpter 2. First, when the second-order potential and energy are calculated,

the one-body part of the perturbation is scaled by 1 a (a being the fraction of

the nonlocal exchange operator). Second, the SCF Hamiltonian is now the sum of

the core Hamiltonian, the nonlocal exchange operator, local exchanges, and local

correlation potentials.

As we discussed in C(i lpter 3, in the case of Be, the iterative solution for

the PT2 potential cannot be obtained because the perturbation series diverges.

However, already 20 percent of nonlocal exchange resulted in a convergent series

and we could obtain the corresponding correlation potential.

For other atoms (Table 3-2), the PT2 energy expression (smaller

denominators) tends to overestimate the correlation energies, while the PT2SC

and MP2 expressions (larger denominators) tend to underestimate the correlation









energy. Therefore, including a fraction of nonlocal exchange can lead to the

correlation energy that is very close to the exact one. This is exactly what

happened in the case of Be (Figure 4-1). At about 40'1 of nonlocal exchange,

the EXX-PT2h energy and the density moment crossed the CCSD(T) values.

Thus, in the case of Be, about 2/5 of the nonlocal exchange operator results in

the optimal reference for the second-order functional and potential. We were able

to find such optimal fractions for each system out of the several closed-shell atoms

and molecules considered (Table 4-1).

Table 4-1: Performance of the hybrid ab initio functional EXX-PT2h with
optimized fraction of nonlocal exchange. Shown are total energies (milliHartree)
and density moments (atomic units) deviations from the CCSD(T) values. The
EXX-PT2SC and CCSD values are shown for comparison. Uncontracted Roos
augmented double zeta basis.


Atom EXX-PT2h EXX-PT2SC CCSD
~a, E (r) E (r) E (r)
He 31 -0.0 0.00 6.4 -0.00 0.0 0.00
Be 36 0.1 -0.01 20.7 0.05 0.6 0.00
Ne 97 -0.1 0.03 4.0 0.02 5.1 -0.01
Mg 77 -0.0 -0.01 12.6 0.00 3.1 0.01
Ar 85 -0.4 -0.00 12.9 -0.01 5.7 -0.01
Ca 77 -0.2 -0.06 12.4 -0.04 3.4 0.01
Molecule EXX-PT2h EXX-PT2SC CCSD
ao, E (z) E (z) E W()
HF 99 0.1 -0.02 4.5 -0.03 5.7 0.02
H20 92 0.0 -0.04 9.7 -0.02 6.7 0.02
NH3 80 0.1 -0.05 18.0 0.00 6.5 0.02
CH4 65 0.3 0.00 25.8 0.00 5.2 0.00
CO 95 -0.1 0.30 11.8 0.18 15.0 -0.05
N2 97 -0.4 0.00 9.6 0.00 15.9 0.00


However, the values of the optimized fraction is different for every system. In

particular, we noticed that for systems where the dynamical correlation prevails

and the PT2SC energy is close to CCSD(T) one (Ne, H20, N2), the optimal

fraction tend to be close to 1011' On the contrary, if the PT2SC error is much

larger than the CCSD one (Be, CH4), then the optimal fraction is smaller. Also,







58



8 0.025 I I I--
H 0.025

0 0.0215 -


I 0.015

0.01

o 0.005

D0

S -0.005 -
a / EXX-PT2h -1--
/^ EXX-PT2SCh ---
H -0.01
0 20 40 60 80 100
Fraction of nonlocal exchange operator, %
0.06 I I I

j 0.05 ------- ---*-- ----

S 0.04

0.03

U 0.02


0.01 -
0

-0.01 -

S -0.02

S -0.03 EXX-PT2h -
EXX-PT2SCh --
-0.04
0 20 40 60 80 100
Fraction of nonlocal exchange operator, %

Figure 4-1: The total energy and first density moment of Be calculated with the
EXX-PT2h functional with various fractions of the nonlocal exchange operator.
Uncontracted Roos augmented double zeta basis.









the fraction that results in an almost exact correlation energy does not necessarily

lead to an improved density. For example, in the case of CO, 95'. of nonlocal

exchange results in the error of 0.1 milliHartree in the energy, but 0.3 Debye in the

dipole moment.

Thus, the present results do not support the utility of the hybrid ab initio

functional. A possible route to a practical method is to find the fraction of nonlocal

exchange that minimizes the average error in the total energies of a large set of

systems.

4.4 Second-Order Potential within Common Energy Denominator Approximation

As we have mentioned in Section 2, an alternative to numerical or finite-basis-

based solution of the OEP integral equation is to introduce an approximation

to the OEP kernel. For example, the KLI approximation of Krieger et al.[10]

neglects the contribution of all but diagonal matrix elements to the kernel. As a

result, the KLI potential is given via a simple recursive relation and can readily

be found iteratively. A related approximation, the Common Energy Denominator

Approximation (CEDA) proposed by Gruning et al. [11] retains more features

of the exact kernel, but like the KLI approximation, allows one to calculate the

approximate potential via a simple iterative procedure. An equivalent scheme,

Localized Hartree-Fock (LHF) method, was independently introduced by Della Sala

and G6rling [27].

These approximations can be used to obtain an approximate second-order

potential as well. In what follows we first introduce the mathematics of these

approximations and then apply them to the OEP equation for the second-order

potential.

The Common Energy Denominator approximation consist of two steps. The

first step is to approximate all of the energy denominators in the OEP equation by










a constant

occ. virt
dr' (r') v (r') (r') + c.c.
i a
occ. virt
= r) dr' (r(r'))(rr')4 (r) + c.c. (4-24)


occ. virt
ai-Ca





occ. virt
SS (r) a(r) dr' (r') alZ(r') S(r') + c.c. (4-25)
a a

Note that since the right- and left-hand sides involve the same denominators, the

constant A conveniently disappears.

The second step is to apply the resolution of identity

occ. virt.
r (r) r ((r') + Y,(r)O ) ((4-26)
i a

to exclude the virtual orbitals from the summation

occ. virt
5 ir(r) dr' [6(r r') j(r)(r')] O (r')(r') + c.c.
i a
occ. virt
= (r) dr' [6(r r') j(r)0*(r')] v(r)0i(r') + c.c. (4-27)
i a

This, the CEDA exchange potential is given by the following iterative relation


Vx,ceda(r) C e(r) + (r) Vx,ced Vl i) (4-28)
ij (r)

where


,() ) dr'(r ') (r') dr' )(r) r) (4-29)
X p(r) J r P(rr'

is the exchange energy density defined by


EEXX- drp(r)c(r) (4-30)
2 x









In the KLI approximation, this expression is further simplified by neglecting all

but diagonal elements from the summation


Vx,ceda(r) C(r) + 01 (r) (r) iz Vx,ceda n ix (4-31)
p(r)

Thus, we have a series of approximations to the EXX potential: ex(r)

Vx,KLI(r) = Vx,CEDA(r). As we have seen on Figure 2-1, ex(r) is a close

approximation to the EXX potential in the inner and i,-iv!ll ic regions. However,

it lacks the structure of the EXX potential in the intershell region. The KLI

approximation and, subsequently, CEDA add more orbital-dependent terms to the

approximation potential that will correct e,(r) towards the true EXX potential.

The idea proposed in this section is to apply the Common Energy

Denominator Approximation to the second-order correlation potential. This

will allow one to substitute the OEP integral equation by a recursive relation for

the correlation potential. Moreover, applying CEDA will significantly simplify

the expression for the right-hand side of the equation resulting in a reduced

computation cost of the second-order scheme.

Introducing CEDA to the equation for the second-order potential
occ. virt. i \(r) v(r)a vl / I 6E(2)
l Y a C + -c.c. ( (4-32)
i a "

we obtain
(2) _o r)(2. A 6E(2)
v2)(r) /\( ) -(r) (2 0*) I E + (4-33)
cj p(r) 2p(r)6vs(r)'
where the derivative of the second-order energy must be calculated with the CEDA

as well.

To do that let us rewrite the expression for the single-excitation contribution

(Eq. C-8) to this derivative as













6E 2) cc. virt. t.(r)(r) l e i
6V, (r) Ca
+c.c.
ij a
occ. vUirt. ,) K \ \/n"1
i bC.C.
i a,b


z ab C6 Cb
occ. virt.
0C ct *(r) i(r) aj bli+C.C.
i,j a,b b
occ. virt. (r) (r) a j v, | \


Y, a Cj Ca
ij a,b
occ. virt. \ i i

+ Y irt. (r)b(r)ab Vx Vexx J)tia

a (434)
0i cb
i a,b C Cb

where

tia = -ez- a) (4-35)
Ci Ca

Approximating all the explicitly shown denominators by a constant and using

the resolution of identity to remove one virtual orbital, after some algebra, we

obtain

E s(2 ) + o cc v ir t. (r + .
A Y+ s '(r)tia + c.c.
V,(r) 2 -
Socc. virt.

,111 a


~2 E(r) ij(r)Ka V ej )otia C.C.
3 oCC. crft.

ij a
occ. virt.
+ 2Y (r (r)Kka ij)tia + c.c.
i,j,k a


(4-36)










Similarly, the double-excitation contribution (Eq. C-9) can be written as

5ED 1 0CC t r O(r)}i(r)Kab cj)tijab + .
o c.c.
6vS(r) 2 e E
1 occ. virt. i\ (r)(/I i)tijab

2 Ck Ca- a
occ. virt. 7r)/(r)(ab kj)tijab
2 >1>1

i,j,k a,b
Socc. virt. .
+1 t Y r)(ic(r){cb\ \jz)tijab (437)
ij abc + j Cc (- Cb
ij a,b,c

where
(ij \ab)
tijab =-- ab- (4-38)
Ci + Cj -Ca Cb

Approximating all the explicitly shown denominators by a constant and using

the resolution of identity to remove one virtual orbital, we obtain


A6 () 4 (r (r) dr' r,')' jj' + ..
ij a,b
5 occ. virt.
S- / (r)/i(r)K ab kj)tijab C.C-
i,j,k a,b
5 occ. virt.

i,j,k a,b

(4-39)


Thus, using the CEDA to calculate the second-order potential has several

advantages. First, the potential can be found directly as a function of r via an

iterative procedure. Second, the calculation of the right-hand side within the

CEDA involves only (ij ab)-type of two-electron integrals as opposed to (ia lbc)

integrals needed in the case of original OEP equation. The (ij ab) integrals

are need for the MP2 energy calculation as well, therefore the calculation of the







64

second-order potential requires exactly the same amount of memory and operations

as an MP2 calculation.

We expect that the advantages of treating the second-order energy expression

as a functional will be retained even if the potential is found approximately.

Therefore, the EXX-PT2SC functional with the potential found within the CEDA

will be a practical alternative to the MP2 method.















CHAPTER 5
CONCLUSIONS

Ab initio Density Functional Theory offers an alternative approach to the

exchange-correlation functional. The functional is written as an analytical

expression of KS orbitals rather than the density. Ab initio DFT features the

exact orbital-dependent exchange functional and provides rigorous and systematic

approximations for the correlation functionals based on ab initio wavefunction

methods.

The exact exchange functional was implemented with the auxiliary-basis OEP

method for the corresponding potential. The implementation featured standard

Gaussian bases to represent the potential, explicit .,-vmptotic term to ensure the

correct .,-i~!,iiill ic behavior of the potentials, and Singular Value Decomposition

procedure to solve the integral equation in the auxiliary-basis representation. Test

calculations showed that the EXX method accurately reproduces the Hartree-Fock

values with deviations an order of magnitude smaller than correlation effects. Only

the HOMO energy was found to be very sensitive to the choice of the basis set.

A future study should address the question whether using different bases for the

orbitals and potential can improve the orbital energies.

As the correlation counterpart of the exact exchange functional, the correlation

functional from second-order perturbation theory (PT2) was implemented. The

numerical test showed, however, that it does not provide an adequate description

of correlation effects in molecules. The problem was identified with the presence

of the large one-body terms in the perturbation, which caused the perturbation

series to diverge. It was demonstrated that these terms can be resumed into the

reference Hamiltonian, which resulted in the new correlation functional, PT2SC.









The new functional offers a more balanced description of correlation effects, as was

demonstrated in applications to a number of closed-shell atoms and molecules. The

combination of the exact exchange functional and modified correlation functional,

EXX-PT2SC, results in total energies that improve upon the MP2 values, both

at the equilibrium geometries and as a function of the bond length. EXX-PT2SC

results in densities that are slightly better than the MP2 or GGA ones. Also, the

EXX-PT2SC potential qualitatively reproduces the shape of the exact exchange-

correlation potential. The correct shape of the potential leads to a meaningful

one-electron energy spectrum with the HOMO energy approximating the principal

ionization potential.

Thus, ab initio DFT already with the EXX-PT2SC exchange-correlation

functional leads to a very accurate DFT method superior to the current

conventional functionals. The current implementation can be extended to higher

orders of perturbation theory or Coupled-Cluster ansatz. For example, one can

straightforwardly derive a correlation potential from the linearized Coupled Cluster

energy expressions. This will fulfill the original promise of ab initio DFT to

provide a hierarchy of exchange-correlation functionals of monotonically increasing

accuracy.

For practical purposes, ab initio DFT must be compared to the ab initio

wavefunction methods because its computational cost is significantly higher

then conventional DFT. The results of this study showed that EXX-PT2SC

functional improves on the conventional perturbation theory, MP2. However,

the computational cost of obtaining the EXX-PT2SC potential is that of MP2

times the number of iterations. For many systems considered in the tests, these

improvements are not very significant given the increased computational cost.

Further tests are necessary to establish whether ab initio DFT can become a









practical alternative to MP2 methods when Coupled-Cluster or multireference

methods are not accessible.

Besides improvements in the energy and density, ab initio DFT also offers

the advantage of transforming the ab initio quality description of the correlation

effects into a simple SCF model. The resulting molecular orbitals include the

exact exchange interaction and most of the correlation effects. These orbitals are

superior to both HF (no correlation effects) and conventional DFT (poor exchange-

correlation potentials) and can be used to describe the chemical properties of the

system. Also, the SCF model of ab initio DFT should provide an excellent starting

point for linear-response and time-dependent methods. In fact, the extension of

the EXX method to linear-response problems has been already formulated [28] and

implemented [16] and the extension of the second-order method is under way.

To make ab initio DFT more practical, one can reduce the cost of the EXX-

PT2SC scheme by introducing approximations to the functional or potential.

The most direct route is to simplify the OEP procedure for the potential. The

expression for the second-order potential within the Common Energy Denominator

Approximation was derived. Using this approximation will decrease the overall

computational cost and eliminate the integral equation. Given that the correlation

contribution to the orbitals is small, one can use exchange-only orbitals to calculate

the CEDA potential rather than solve for it iteratively until full self-consistency.

The resulting scheme will require exactly the same time and computation resources

as the MP2 method. If using the correlation potential still improves the MP2

results, such an approximate scheme will be an alternative to the MP2 method.

Note that any approximations to the potential does not reduce the cost associated

with the second-order energy expression. Thus, to reduce the cost beyond MP2,

one needs to develop an approximation to the functional. Such an approximation









must make use of the variational optimization to compensate for the simplifications

in the energy expression.














APPENDIX A
FUNCTIONAL DERIVATIVE VIA THE CHAIN RULE

The functional derivative of A with respect B to is, by definition, the linear

term coefficient in the Taylor expansion of A response to the infinitesimely small

change in B.

Since the KS orbitals and orbital energies depend on the potential through the

KS equations

(i+ v p) ) ,/, ) (A-1)

their functional derivative with respect to the potential is defined through the

linear response KS equations. Consider a infinitesimal change in the potential,

vs -+ Vs + 6Vs

(t + vs + 6vs) \p + 60)p = (C, + 6c,) p + 6p) (A-2)

Subtracting the unperturbed equation and neglecting the second-order quantities

one obtains

(vs o) ) (C -- tV s) 60p) (A-3)

By projecting this equation on ((p and using the orbital orthonormality


K()p = pq, (A-4)

for the linear response of the eigenvalues one has


h) ) (A-5)

hence,


6cp ( qq 6', ..,-).


(A-6)









By projection on the (Qbq such that p / q, for the linear response of the

eigenfunctions one has


(qhene, tp((p) = Cp(pQ6q(p (pq Vs \6p) = (Cp q)(bq 6b p, (A-7)

hence,


Finallyq usin k -- Op)n
Cp Cq

Finally, using the resolution of identity


(A-8)


>,)(Q 1 (A-9)

and the fact that the linear response of (p can be chosen to be orhtogonal to (p


(p (fP) = 0 (A-10)


one obtains
all all O

q qp p- Cq

Also, for the linear response of the density one has
OCC. OCC.
6p(r) = 6 (r)O (r)) = 6(r)0 (r) + c.c.
i i
occ. allc.c.
i i -- ( p
i p7 ir

Note that the contribution of the occupied orbitals to the sum above is zero

because
oCC. oCC. 0*(Or (r) j( v i) i ) () O k /iJ)
y5i -- Cj +r{i -6 }].
i j
OCC. OCC.
oCC >z c (()Yr\ J /()bv r O sk i) + (i(J)s YkJ)
SCi ij Ci cj
j i


(A-ll)








(A-12)










(A-13)









therefore, the summation index can be restricted to run over virtual orbitals only
occ. Virt. \ \)
6P(r) = Y a /as + C.c. (A-14)
i a

Thus, the corresponding functional derivatives are


6 (r) (r) (A-15)
bP, (r) (') (')



(A-16)
6v5(r') p Cp- Cq
qP
and
occ. rirt.
6r) 1 r)a(r (r') + c.c. (A-17)
(Vs(r') >>'i -- Ca

As shown in C'!I Ilter 2, to obtain the right-hand side of the OEP equation for

the exchange or correlation potential, one needs the functional derivative of the

corresponding functional with respect to the total potential. The expressions for

the derivatives of orbitals and orbital energies with respect to the potential are

available, one can use the chain rule

SE[o, C] all c, SE[, C] all ,6J (r') 6E[0, cc
S+ dr + c.c.
6v (r) 6vs(r) ep, 6v J(r) J6(r')
P P
all aE[ l] I a[ (A-18)
Y p Jp ()Op +y Y dr'Oq(r') r
P "C p,qlp ()

For example, the expression for the right-hand side of the OEP equation for

the EXX potential is given by the functional derivative of the EXX functional
OCC.
EEXX = Y( j (A-19)
i,j

with respect to the potential. Since EEXX does not depend on the virtual orbitals

nor orbital energies, the only contribution comes from the terms of the chain rule







72

involving the occupied orbitals

6EExx 1 oCC dr6r (k, ( r')
j,k
Si(r') 2 i(r) r- r()

= dr' = ( ivz r), (A-20)


and

6EEXX occ. all EEXX




i q3i
occ all i iq
Yzz fl2 (A-21)
q i Cq

Also, as in the case of the density, one can show that when the summation index q

is restricted to occupied orbitals, the sum is antisymmetic with respect to renaming

indices i and q and, therefore, is zero. Thus, q can be restricted to virtual orbitals,

which gives the final expression for the right-hand side of the integral equation on

the EXX potential

EEX 7 "i'tx*/~occ. virt. i
EEXX a(r)Vr)( lx\ (A-22)
6V, (r) Ci C(















APPENDIX B
SINGULAR VALUE DECOMPOSITION

Singular Value Decomposition of a rectangular m x n matrix A (m > n) is the

product of an m x n column-orthogonal matrix u, an n x n diagonal matrix w with

positive or zero elements, and the transpose of an n x n orthogonal matrix v:


A = uwvt


(B-l)


where


uut 1



VVt = VtV


(B-2)


(B-3)


and


w = diag{wl,w2,.. ',,} : W W2,.., i',, > n

For a system of linear equations


Ax y


with square (m n) matrix A, the solution is provided by


vw -utAx = vw-uty = x = vw-uy


if A is not singular, i.e., all elements of w are nonzero.

If matrix A is singular, using


(B-4)


(B-5)


(B-6)


(B-7)


w' = diag{w' w',..r',







74

where

S wi (B-8)
0 ,' =0

provides the solution

x' vw'uty (B-9)

that minimizes the residue

r =y Ax' (B-10)

in the least square sense.

Given a finite numerical precision, one must introduce a finite threshold e

(called the SVD threshold) to define which singular values will be discarded:

I w >(
,' wi (B-ll)
0 It <















APPENDIX C
DERIVATIVE OF THE SECOND-ORDER CORRELATION ENERGIES

To take the functional derivative of the second-order energy one needs the

derivatives of two- and one-electron integrals, and energy denominators. Using the

chain rule of Eq. A-18, one obtains the following expression for the derivative of a

two-electron integral


I ) Jlr ) dr'O (r') P 6 rs + C.C.


S+ +c.c.
6v, (r) e c-) t ((r))







a occ a l(r (r)i () dvq r (rl [r) q (r)p onL rr') +c c
all all





0- + c.c.
^r '-- ,o s -- v

The derivative of the one-electron integral is, after some algebra

6{ i vexx q 6Pjq) {+ 6p..q) + C.C.
6v, (r) 6v, (r) 6v, (r)

all vc all~\P3 0 (p~r) q(r) P nlx Vexx V)

v7 j C vq Cq

Note that in the first term one can restrict index v to virtual orbtals only,


(C-l)








(C-2)


since


OCC. OCC. *(r) /, 1 OCC. OCC. (/)r (r) (p jq)

v7j j v7j j
OCC. OtCC. t(r) (r)(pv j) OCC. OCC. (r)o(r)(pv jq)
Cv -- C( ( j CV
j7v v I v7j j


0 (C-3)







76

Thus,


v, (r) vezz-cost.
virt oc. occ. (r) \j(r) p bq all (r) (r) (C-4)
b j Cj -b b Yq Cq C- CV

The derivatives of an energy denominator ise
all
6 1 6a 1
C Y, (r)P (r)5 )r) Ca
6V,(r) Ci Ca P- 6Cp Ci Ca

(+ (C-5)
(C Ca)(Ci Ca) (C -Ca)(Ci C)

Thus, the derivative of the single-excitation contribution to the second-order

energy is

C(2) (occ. virt. Vx \1
6Es o nix -~( a) -a Vnix ) exx b)
6vs(r) 6vs(r) Ci Ca


i a
OCCV Virt. )v'X, a) a Vnlx v;1)
i a (C- Ca)(Ci Ca)
occ. virt. i
Socc. virt. a (rf a nls Vexx a){ x exzz I











+EEE
i a ( Cv )( )
+ c. irt. 0-. ( r- ) \ \*
potentially present a problem. For example, v runs over virtual orbital in
ij a,b (i a \(j -b)
occ. virt. all -
0*\^\^ (Tr) (0 r)( zIx Vexx z a){ nx Vexx V)


i a v C ( -i)


+ c.c. (C-6)


Note that the denominators involving orbital energies of the same type can

potentially present a problem. For example, when v runs over virtual orbitals in







77

the fourth term of the above expression, the denominator is equal to the difference

of the two virtual orbital energies. If these two orbital energies are accidentally

very close to each other (physically acceptable situation) then this denominator will

lead to an arbitrarily large potential.

However, the following transformation demonstrates that terms involving

such denominators ah--,i- cancel each other and are not present in the actual

expression. Given some matrix elements gab such that gb gba, one can use a

simple rearrangement of indices to show that

occ. virt. occ. virt. occ. virt.
(C a gab +a b a z f : gaa
Cb(C -Caa) (Ca Cb) (C Ca) i (Ci Ca)(Ci Ca)
i a,b a i a,b a i a
occ. virt. occ. virt.
Sab gba gaa

S(a,b Ca Cb)( ) (C Ca) (C Ca)
occ. virt. occ. virt.



occ. virt. occ. virt.
g ab gaa
i a,b#a b i a Ca Ca (C- C
occ. virt.
z ( gab
i ,b ( a) a)(C-7)









Applying this transformation to Eq. C-6 one obtains the final expression for

the derivative of the single-excitation contribution to the second-order energy

6E () occ. virt. t ) a vexx a) /i z -\ vez )\
6v(r) c c (C Ca)(Cj Ca)
aj a
oCC. Uirt. (r) ) a(r/a 'i ve x b) b Vnl vx vez i .
+ Y Y --- -------
Sa (C a) (Ci eb)
i a,b
occ. virt. \ f \j(r) 7 ezzi(ijba

i,j a,b ( Ca) Cb)
occ. virt. \
C C (r) j(-r) i vx Vezzi (a +
-- ------ + c c.

occ. virt. ) j / /l *\ ] \
CC. Yrt. (r)/j j(r) a Vnix Vezxx ) i Vnix Vezxx )
S(C- Ca)(Cj -Ca)
o+cc. virt. O* r \O / a / I \v1) I z b)
0CC cut -*),) 'v zv (C-8)
5 V5 (Ci Ca)e(Ci Cb)
i a,b

Similarly, the derivative of the double-excitation contribution is

6E (r)(riabab c) i
6vsr) 2 (C Cc)(Ci + e eCa Cb)
ij2 a,b,c
occ. virt. k a(r) if )/J kb.K) + ...I

i,2 k k Ca) (i Cj Ca C b)

+ 2k
i,j,k a,b

occ. virt. / \/\a(r) ij 7\ /cb .|. ll \
+ r (C-9)
2ij (cg e+ Cc C)(i + Ci ea eb)
i,j a,b,c a
Note that the last two terms do not have the complex conjugate counterparts.














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BIOGRAPHICAL SKETCH

I was born and raised in Novosibirsk, the third most populous city in Russia,

located in the south-west corner of the vast land of Siberia. Both my parents

are devoted physicists, well-known in their field. I was born when they were still

working on their degrees at the Novosibirsk State University, so I spent my infancy

in the Physics dorm, surrounded by lecture notes and Landau's books.

Giving that background, I never doubted my scientific d, -Iii,', and in 1995

entered the Physics program at Novosibirsk State University. In 1999, I received

the bachelor's degree in Physics (specializing in C('! liim I Physics). The next

year I spent conducting research with Prof. Georgiy M. Zhidomirov and Dr. Igor

L. Zilberberg at the Laboratory of Quantum C'! i11i-I ry in the Boreskov Institute of

Catalysis, Novosibirsk.

In 2000, I took my parents' advice and searched for a graduate school

abroad. Looking for fundamental electronic structure theory, one cannot overlook

Prof. Rodney J. Bartlett's group, and I looked no further. In the fall of 2000,

I entered the Ph. D. program of Deparment of C('! i,,,-I ry at the University of

Florida and joined Prof. Bartlett's group at the Quantum Theory Project.