Conserving Approximations in Time-Dependent Density Functional Theory

UlfvonBarth,1NilsErikDahlen,2RobertvanLeeuwen,2andGianlucaStefanucci1

1

arXiv:cond-mat/0507604v2 [cond-mat.stat-mech] 17 Dec 2005SolidStateTheory,InstituteofPhysics,LundUniversity,S¨olvegatan14A,S-22362Lund,Sweden

2

RijkuniversiteitGroningen,TheoreticalChemistry,MaterialsScienceCenter,

9747AG,Nijenborgh4,Groningen,TheNetherlands

(Dated:February2,2008)Inthepresentworkweproposeatheoryforobtainingsuccessivelybetterapproximationstothelinearresponsefunctionsoftime-dependentdensityorcurrent-densityfunctionaltheory.Thenewtechniqueisbasedonthevariationalapproachtomany-bodyperturbationtheory(MBPT)asdevelopedduringthesixtiesandlaterexpandedbyusinthemidnineties.Duetothisfeaturetheresultingresponsefunctionsobeyalargenumberofconservationlawssuchasparticleandmomentumconservationandsumrules.ThequalityoftheobtainedresultsisgovernedbythephysicalprocessesbuiltinthroughMBPTbutalsobythechoiceofvariationalexpressions.Weherepresentseveralconservingresponsefunctionsofdifferentsophisticationtobeusedinthecalculationoftheopticalresponseofsolidsandnano-scalesystems.

PACSnumbers:

I.INTRODUCTION

Opticalspectraconstituteimportanttoolsforgain-inginformationontheelectronicstructureofsolids,molecules,andnano-systems.Inmanysystemstheparticle-holeinteractionleadstoastrongexcitonicdis-tortionoftheopticalspectrum-particularlyinnano-scaleobjects.Thetheoreticaldescriptionofsuchspectraisrelativelysophisticatedandverycostlyfromacom-putationalpointofview.Sometimeago,itwasreal-izedthatthesespectraarealsowithinreachusingtime-dependent(TD)density-functionaltheory(DFT)-butwithmuchlesscomputationaleffort.FromtheRunge-Grosstheorem1ofTDDFTweknowhowtoconstructtheexactdensityresponsefunctionofanyelectronicsystemintermsofanexchange-correlationkerneldescribingtheparticle-holeinteractions.Andfromrecentworkbysev-eralresearchers2,3wehavearathergoodideaaboutthepropertiesofthiskernelifitistoreproducetheratherac-curateresultsobtainedfromsolvingtheBethe-Salpeterequationofmany-bodyperturbationtheory(MBPT).Thekernel,usuallynamedfxc,hasbeencalculatedintheexchange-onlyapproximationofTDDFTbyseveralpeopleinthepast,seeforinstanceRefs.4,5.Morere-cently,thekernelfxchasbeencalculatedinthesameapproximationbyPetersilka,Gossmann,andGross6fortheheliumatom,byKurthandvonBarthfortheden-sityresponseofthehomogeneouselectrongas,7andbyKimandG¨orling8inthecaseofbulksilicon.Inthecasesofatomicheliumandthehomogeneouselectrongastheresultingresponsefunctionrepresentedasubstantialim-provementonthatoftheRandomPhaseApproximation(RPA).Theexcitationenergiesofheliumweremuchim-provedandthetotalenergiesobtainedfromtheresponsefunctionweremuchsuperiortothoseobtainedfromtheRPAresponsefunctioninbothheliumandthehomoge-neouselectrongas.Unfortunately,thisabinitioapproachdidnotworkverywellinbulksiliconunlessoneratherarbitrarilyintroducessomekindofstaticscreeningofthe

particle-holeinteraction.

Inactualfact,withinTDDFT,nosystematicandre-alisticroutetowardsuccessivelybetterapproximationshas,sofar,beenavailable.Inthepresentworkwehaveconstructedsuchaschemebasedonthevariationalap-proachtomany-bodytheorydevelopedinRef.11.Intermsoftheone-electronGreenfunctionofMBPT,thesefunctionalsgivestationaryexpressionsforthetotalac-tionofthesystemathand-orthetotalenergyinthecaseoftimeindependentproblems.Fromastationaryactionitisratherstraight-forwardtoconstructthetimedepen-dentdensityresponsefunction.BuildingthefunctionalsfromtheΦ-derivabletheoryofBaymandKadanoff,12,13alwaysresultsinresponsefunctionswhichobeyessentialphysicalconstraintslikeparticle,momentumorenergyconservation.

ThesimpleideaofthepresentworkistorestrictthevariationalfreedomofthefunctionalstothedomainofGreenfunctionswhicharenon-interactingandgivenbyalocalone-electronpotential-andvector-potentialincaseofcurrent-DFT.AccordingtotheRunge-Grossthe-oremthisrestrictionimmediatelyresultsinadensity-functionaltheorythequalityofwhichisdeterminedbythesophisticationwhichisbuildintothechoiceofΦderivableapproximationfortheactionfunctional.Thus,toeveryconservingschemewithinMBPTthereisacor-respondinglevelofapproximationwithinTDDFT.ThelatterisdeterminedvariationallyandthereisnolongeraneedforanadhocproceduretoequatecorrespondingquantitiesbetweenTDDFTandMBPT.Apotentiallyin-terestingconsequenceofthetheoryproposedhere,isthattheoftendiscussedlinearizedSham-Schl¨uterequation14fortheexchange-correlationpotentialisnothingbutthestationaryconditionfortheactionfunctional.Inthepar-ticularversionofthevariationalfunctionalsdevelopedinRef.11andnamedΨderivabletheories,alsothescreenedCoulombinteractionbecomesanindependentvariableatonesdisposal.ThisleadstoapproximationswithinTDDFTwhicharepotentiallyasaccurateasthose

ofmoreelaborateschemeswithinMBPTbutwhicharecomparativelyeasiertoimplement-especiallyinnano-systemsandcomplexsolids.

II.

VARIATIONALAPPROACHTOTDDFT

Letusconsiderasystemofinteractingfermionsex-posedtoanexternal,possiblytime-dependentfieldw(rt).Thefullmany-bodyHamiltonianreads

H

ˆ=Tˆ+Uˆ+Wˆ,(1)

where

Tˆ=−12

󰀂

d3rd3r′ψ†(r)ψ†(r′)v(r,r′)ψ(r′)ψ(r),

istheinteractionoperator(v(r,r′)=1/|r−r′|).The

couplingtotheexternalfieldisgivenby

Wˆ=󰀂

d3rw(rt)nˆ(r),wherenˆ(r)=ψ†(r)ψ(r)isthedensityoperator.The

GreenfunctionGobeysDyson’sequation

G=GH+GHΣG

whereGHistheHartreeGreenfunctionandΣistheexchange-correlationpartoftheelectronicself-energy.Diagrammaticperturbationtheoryprovidesatoolforgeneratingapproximateself-energiesand,inturn,ap-proximateGreenfunctions.Exceptforphysicalintu-ition,thediagrammatictechniquesrelysolelyonthevalidityofWick’stheorem.15,16Thus,atypicalcontri-butiontotheself-energyisrepresentedbyadiagramcontainingnon-interactingpropagatorsandinteractionlines.However,anyapproximationwhichcontainsonlyafinitenumberofthesediagramsviolatesmanyconser-vationlaws.Conservingapproximationsrequireaproperchoiceofaninfinitesetofdiagrams.TheconservingapproachbyBaym12wasbasedonsuchchoices.Alsothevariational11schemebyAlmbladh,vonBarthandvanLeeuwen(ABL)wasdesignedwiththesameobjectiveinmind.TheformerapproachisreferredtoasaΦ-derivableschemebecauseitscentralquantityisauniver-salfunctional,calledΦ,oftheone-electronGreenfunc-tionGandthebareCoulombpotentialv.Itiscon-structedsuchthatitsfunctionalderivativewithrespecttoGgivestheexchange-correlationpartoftheelectronicself-energyΣwhereasthefunctionalderivativewithre-specttotheCoulombinteractionvessentiallygivesthereduciblepolarizabilityχofthesystem,

Σ(1,2)=

δΦ

δv(2,1)

.

(2)

2

(Hereandinthefollowingweusetheshort-handnota-tion1=(r1,t1),2=(r2,t2)andsoon).Notice,how-ever,thatthereisnoreferencetoanactualsystemintheΦfunctional.ItacquiresameaningonlywhenitisevaluatedataGreenfunctionofanactualsystem.IntheapproachofABL,thecentralquantityisinsteadthefunc-tionalΨhavingtheGreenfunctionGandthescreenedCoulombinteractionWasindependentvariables.Itisconstructedsoastogivetheself-energywhenitisdif-ferentiatedwithrespecttoGandtheirreduciblepolar-izabilltyPwhenfunctionallydifferentiatedwithrespecttoW.Again,thereisnoreferencetotheactualsys-temcontainedinthefunctionalΨ.ByaddingfunctionalpiecestotheΦortheΨfunctionalrespectively,pieceswhichdocontainclearconnectionstothesystemunderstudy(like,e.g.,theexternallyappliedpotentialw),oneconstructsfunctionalsforthetotalenergy-ortheac-tioninthecaseoftimedependentproblems-which,asfunctionalsofG,havetheirstationarypointattheGreenfunctionGwhichisthesolutiontoDyson’sequation.InthecaseoftheΨ-basedfunctionalstheyarealsostation-arywhenthescreenedinteractionWobeysthesocalledreducedBethe-Salpeterequationtobediscussedlater.Thefirstvariationalfunctional17ofthiskindwascon-structedbyLuttingerandWard(LW).ItisaΦfunc-tionalandithastheappearance

iYLW[G]=Φ[G]−Tr󰀈ΣG+ln(Σ−G−H

1

)−

󰀋−iUH[G].(3)

InEq.(3),thefunctionalUH[G]=i

δD

󰀇

=0,

D=0

oneobtainsanewvariationalfunctionalhavingthesamestationarypointandthesamevalueatthestationarypoint.Itmight,however,bedesignedtogiveasecondderivativewhichalsovanishesatthestationarypoint-

somethingthatwouldbeofutmostpracticalvalue.Suchpossibilitiescouldopenupawholenewfieldofresearch.ChoosingtoaddF[D]totheLWfunctional,where

F[D]=Tr{−D+ln(D+1)}

(5)

obviouslyhasthedesiredproperties,leadstothefunc-tional

iYK[G]=Φ[G]−Tr󰀈GG−H

1−1+ln(−G−1

)󰀋−iUH[G].ThisfunctionalwasfirstwrittendownbyKlein,21andcouldthusbecalledtheKleinfunctionalinordertodistinguishitfromtheLWfunctionalabove.Unfortu-nately,thisfunctionalislessstable(largesecondderiva-tive)atthe22,23,24,25stationarypointascomparedtotheLWfunctional.SincetheconstructionofresponsefunctionsforTDDFTfromthevariationalfunctionalsinvolveevaluatingthematnon-interactingKohn-ShamGreenfunctions,onemightexpectalessstablefunctionaltogiverisetoinferiorresponsefunctions.Andthisissomethingwhichhastobethoroughlyinvestigated.ButitisclearthattheKleinfunctionalismucheasiertoevaluateandmanipulateascomparedto,e.g.,theLWfunctional.

AlltheΦfunctionalsleadtoaDysonequationwhichhastobesolvedself-consistentlyforG.Thisis,ingen-eral,averydemandingtaskbecauseofthecomplicatedsatellitestructureinherenttoanyinteractingGreenfunc-tion.Thisseverecomplicationis,however,circumventedbyswitchingtoTDDFT.

OurapproximationswithinTDDFTarejustspecialcasesofthevariationalfunctionalsinwhichwere-strictthevariationaldomainoftheGreenfunctiontobeallGreenfunctionsobtainablefromaone-electronSchr¨odingerequationwithalocalmultiplicativepoten-tial-orvectorpotentialinthecaseofcurrent-DFT.Weremarkthatthisrestrictiononthevariationalfreedomrenders1,20,26allthevariationalfunctionalsdensityfunctionals.Givenadensitythereisalocalpoten-tialwhichinanon-interactingsystemproducesthatden-sity.Thispotentialproducesthenon-interactingGreenfunctionwhichweusetoevaluateourfunctionals.Thus,thevariationalapproachnaturallygeneratesdifferentap-proximationswithinDFTforstaticproblemsandwithinTDDFTfortime-dependentproblemsorfortheresponsefunctionsofstationaryproblems.Asweshallsee,theexchange-correlationquantitiesdependonthechoiceoftheactionfunctionalsothattoeveryapproximateBaymfunctionalΦcorresponddifferentapproximateexchange-correlationpotentialsandkernels.

Below,wediscussTDDFTandTDcurrent-DFT(TD-CDFT)approximationsintheframeworkoftheKleinfunctionalandoftheLWfunctional.WealsogeneralizethetheorytoΨfunctionalsandgivesomeexamplesofapproximationswhichwebelievetobequitefeasibletoapplytorealisticsystemstakingdueaccountofthefullelectronicstructureofone-bodyorigin.

3

III.

TDDFTFROMTHEKLEINFUNCTIONAL

LetGsbetheGreenfunctionofanon-interactingsys-temofelectronsexposedtotheexternal,possiblytime-dependent,potentialV(rt).TheKleinfunctionalevalu-atedatGscanthenberegardedasafunctionalofV:

iYK[V]=Φ[Gs]−Tr󰀈GsG−H1−1+ln(−G−s1

)󰀋−iUH[Gs].WecouldnowdirectlyusethestationarypropertyoftheKleinfunctionalwithrespecttovariationsintheun-knownone-bodypotentialVinordertoobtainanequa-tionforthatpotential.Becauseofthesimplicityofanon-interactingGreenfunction,however,thefunctionalYKcanfirstbemanipulatedtoacquireaphysicallyappeal-ingform.Thiscan,mosteasily,beseeninthestaticcaseelaboratedbelow.Thefollowingequationsarestillvalidinthecaseoftimedependentproblemsand/orproblemsatelevatedtemperatures.This,however,requiressomereinterpretationsofstandardDFTquantitieslike,e.g.,TsorUH,whichthenbecomefunctionalsontheKeldyshcontour.27Fornon-interactingGreenfunctionstheloga-rithmoftheinverseofGsisjustthesumoftheoccupied

eigenvaluescontainedinGs.17AndthetraceofGsG−theintegraloftheparticledensitymultipliedH1

−1isjustbythepotentialsV−w−VH.Expressing2theeigenvaluesoftheone-electronHamiltonian−∇/2+Vasexpectationvaluesthenleadsto,

YK[V]=−iΦ[Gs]+Ts[n]+󰀂

wn+UH.(6)Here,thequantityTs[n]isthewellknownfunctionalfor

thekineticenergyofnon-interactingelectronsinthepo-tentialV-whichproducesthedensityn.ComparingnowwithstandardDFTweseethattheΦ-functionalpre-ciselyplaystheroleoftheexchange-correlationenergy.ThismeansthatwemayreusestandardDFTresultsandrealizethattheKleinfunctionalisstationarywhen

V=w+VH−i

δΦ

δn

=−i

󰀂

δΦ

δV

δV

4wherewehavedefinedageneralizednon-interactingre-sponsefunctionΛaccordingto

A.

Theexchange-onlyapproximationLetusconsider,forinstance,thesimplestapproxima-iΛ(2,3;1)≡

δGs(2,3)

δn(2)

.

ThekernelfxccannowbeobtainedfromonefurthervariationwithrespecttothetotalpotentialV.Thevari-ationofvxcwithrespecttoVcanbeexpressedintermsoftheexchange-correlationkernelfxcas

δvxc(1)

(34)

+󰀂

δV(2)

Λ(4,3;1)dΛ(1,3;2)∆(3,4)Gs(4,1)d(34)+󰀂

Gs(1,3)∆(3,4)Λ(4,1;2)d(34),

(10)

where

∆(1,2)=Σs(1,2)−δ(1,2)vxc(1).

WhenthepotentialvxchasbeenobtainedfromEq.(9),theright-handsideofEq.(10)isacalculableexpressionforanygivenapproximateΦandnoself-consistencyisrequired.Asanadditionalbonus,alloccurringGreenfunctionsarenon-interactingasopposedtointeractingasonewouldhaveinmostiterativeschemesbasedonMBPT.(Consider,e.g.,theresponsefunctionofthetime-dependentHartree-Fockapproximation.)

tionforΦ,namelytheHartree-Fockapproximation:Φix=δG(2,1)=iv(1,2)Gs(1,2).Inthiscase

δΣx(1,2)fxxcivxc=+ ++= -i Wxivxc++ + FIG.2:Exchange-correlationkernelintheGWapproxima-tionAllGreenfunctionsareKohn-ShamGreenfunctionsandallinteractionsareRPAscreenedinteractions.Thisresponsefunctionforwhichthe”time-dependentGW(TDGW)response”wouldbeadescriptivename,ispresentlytoodifficulttocomputeinrealsystems.Gel-dartandTaylorusedittoinvestigatetheeffectsofthestaticscreeningpropertieson29theelectrongas.28ItwasusedbyLangrethandPerdewinthestaticlongwave-lengthlimitinordertoextractgradient30approximationsforDFT.RichardsonandAshcrofthavepublishedanapproximationtotheTDGWresponseoftheelectrongasbutonlyatimaginaryfrequencies.AnotherapplicationoftheTDGWresponseisduetoLangrethetal.31anddealswithVanderWaalsforces.TheTDGWresponseisgenerallybelievedtobeveryaccuratebutthecompu-tationofthescreenedinteractionisknowntobeabottleneckinGWcalculationsonrealsolids.Unfortunately,theTDGWresponsecontainstwosuchcomplicatedfac-tors(screenedinteractions).C.TDCDFTfromtheKleinFunctionalInTDCDFTthedensitynandthephysicalcurrentdensityjareuniquelyfixedbytheexternalvectorpo-tentialAextandthescalarpotentialw.32,33ThecouplingtotheexternalfieldsisgivenbyJˆ=󰀂d3r[Aext(rt)·jp(rt)+w˜(rt)n(rt)],wherew˜=w+A2ext/2andjpistheparamagneticcurrentoperator.AccordingtoourprescriptionwerendertheKleinfunctionalafunctionalofjµ=(n,j)byrestrictingthevariationalfreedomoftheGreenfunctionstobeallthoseGs’swhicharenon-interactingandgivenbyalocalscalarpotentialandavector-potential,Aµ=(V,A).ItisconvenientwhereV˜toconsiderthefour-vectorA˜µ=(V˜,A),=V+A2/2,astheindependentvariablessincethefour-vectordensityjp,µ=(n,jp)istoA˜coupledlinearlyµ.5Asinthecaseofonlydensityvariations,thesimplic-ityofanon-interactingGreenfunctionagainallowstheKleinfunctionaltobewritteninamuchmoreconvenientform.UsingsimilarmanipulationsasinthebeginningofSec.III,wearriveattheexpressionYK=Ts[n,j]+UH+󰀂A˜µjp,µ−iΦ,(11)wherewehaveusedthenormalconventiontosumoverrepeatedindices.Here,thefunctionalTsforthenon-interactingkineticenergyalsodependsonthephysicalcurrentdensityjandnotonlyonthedensityn.Asbefore,theΦ-functionalplaystheroleoftheexchange-correlationenergy.WethenrealizethatthefunctionalYKisstationarywhenV˜=w˜+VδΦH+vxcwherevxc=−iδj.(13)Letusnowfocusonthosesystemwithavanishingex-ternalvectorpotential.FollowingthesamestepsasledtoEq.(9),i.e.,thechainrulefordifferentiation,weobtainthe”linearized”Sham-Schl¨uterequationofTDCDFT,34,35󰀂Σs(2,3)Λµ(3,2;1)d(23)=󰀂χs,µν(1,2)Axc,ν(2)d2.(14)(NoticethatAxc,µ=(vxc,Axc)innormalfour-vectorno-tation.)ThegeneralizedresponsefunctionΛµappearingaboveisdefinedaccordingtoiΛδGo(2,3;1)≡s(2,3)1δA(1)=δA.(17)ν(2)Themany-bodyresponsefunctionχδjµν(1,2)=µ(1)6

wherefxc,µν=δAxc,µ/δjν.InourvariationalschemetheequationforfxcisobtainedfromonefurthervariationofEq.(14)withrespecttotheKohn-ShampotentialAµ.Thecorrespondingresponsefunctionχµνobeysthef-sumruleandWardidentities19sinceunderagaugetransformationthescalarpotentialVandvectorpotentialAchangeasintheexactCDFT,namelyV→V+df/dtandA→A+∇f.Inordertoprovethispropertywechangetheexternalfieldsaccordingtow→w+df/dt,Aext→Aext+∇fandweaskthequestionhowthescalarpotentialVandvectorpotentialAchangeatthestationarypoint.FromEqs.(12-13),itisstraightforwardtorealizethatV→V+df/dtandA→A+∇fprovidedtheexchange-correlationpotentialschangeaccordingtovxc→vxc+Axc·∇fandAxc→Axc.TakingintoaccountthatunderthisgaugetransformationGs(1,2)→e−if(1)Gs(1,2)eif(2),itisamatterofverysimplealgebratoshowthatthelinearized

SSequation(14)isgaugeinvariantforanyΦ-derivableself-energy.

D.TheEXOwithinTDCDFT

Letusconsider,forinstance,theexchange-onlyap-proximationforthehomogeneouselectrongas.Extract-ingthetime-orderedcomponentofEq.(18)andtakingadvantageofthetranslationalinvarianceofthehomoge-neouselectrongas,wefind34

χs,µρ(q,ω)fx,ρσ(q,ω)χs,σν(q,ω)=Vµν(q,ω)+Sµν(q,ω)whereallquantitiesaretime-orderedandwhereVµνandSµν,atzerotemperature,aregivenby

Vµν(q,ω)=

󰀂

d3pd3kpµv(|p−k|)kν×

×󰀆

󰀆

¯p−q/2θp+q/2θ

ω−εp+q/2+εp−q/2+iη

¯θk+q/2θk−q/2

ω−εk+q/2+εk−q/2+iη

󰀉

󰀆

󰀉

󰀉

,(19)

Sµν(q,ω)=

󰀂

d3ppµpν×

¯p+q/2θp−q/2θ

(ω−εp+q/2+εp−q/2−iη)2

×{Σx(p+q/2)−Σx(p−q/2)}.(20)

Here,wehavedenotedbypµ,kµthefour-dimensional

vectorsofcomponents(1,p),(1,k),whiletheHeavisidestepfunctions

θq=θ(εF−εq)

and

¯q=1−θqθ

containtheFermienergyεF.

InthelargeωlimitthesumVoo+Soogoeslike1/ω4andthereforeχoo=χs,oo+O(1/ω4).Sincetheresidueofthesecond-orderpoleinχs,ooonlydependsonthedensity,theapproximatedresponsefunctionχooobeysthef-sumrule,asitshould.

E.

Conservationlaws

Asmentionedseveraltimes,thevariationalandΦ-derivableapproachtoTDDFTleadstodensity-functionalapproximationswhichpreservemanyphysi-calpropertieswhenthesystemissubjecttoexternalperturbations.Ofcourse,TDDFTbeingaone-electronliketheorywithamultiplicativepotentialtriviallyobeysthecontinuityequationandthusparticleconservation

foranyapproximationtoexchangeandcorrelation.Theconservationofotherquantitieswillhoweverdependonthechoiceofsuchapproximations.

Inthissubsectionwewill,asanexample,showhowmomentumconservationfollowsfromthegeneralfor-malism.Intheone-electronliketheoryofTDDFT,thechange󰀅oftotalmomentumperunittimeissimplygivenbyn∇(w+VH+vxc).Theapproximationtoexchangeandcorrelationismomentumconservingprovidedvxcsatisfiesthezeroforcetheorem.36Designingexchange-correlationpotentialsthatfulfillsuchaconstraintisnontrivial,37andseveralwell-knownapproximationsareac-tuallynotconserving.38,39Below,weshowthatanyap-proximatevxcgeneratedbyourvariationalapproachisfullyconserving.

FromSec.III,weknowthatthechangeδΦintheΦ-functionalisjust

󰀂

δΦ=ivxc(1)δn(1)d1(21)whenwechangetheone-bodypotentialfromVtoV+δV.

InthevariationalapproachalaKlein,Eq.(21)playsa

similarroleastheBaymconstructionδΦ=Tr[ΣδG].InordertoprovetheconservationofthetotalmomentumwehavetoshownthatvxcdoesnotexertanyforceontheKohn-Shamsystem.Letusshiftallcoordinatesbythesametimedependentinfinitesimalvectorδ(t).ThefunctionalΦdoesnotchangesincetheinteractionpo-tentialisinvariantundertranslations.Thisimpliesthat

0=δΦ=i

󰀂

vxc(1)δ(t1)·∇1n(1)d1.

(22)

Onepartialintegrationandthefactthatthevectorδ(t)isarbitraryandindependentofpositiongives

󰀂

n(rt)∇vxc(rt)d3r=0.(23)Thismeansthatthereisnocontributionfromexchange

andcorrelationtothetotalforceappliedtotheclassicalexpressionF=−should.

Theproofofmomentumconservationinthepresence󰀅systemwhichisgivenbythen∇w,asitofvectorpotentialsandcurrentsfollowsinasimilarwayfromthecorrespondingresult

δΦ=i󰀂

Axc,µ(1)δjµ(1)d1,(24)whichweobtainedfromtheKleinfunctional.

IV.LWFUNCTIONAL

LetusnowdiscussthevariationalfunctionalofLut-tingerandWard.FromEq.(3)wefindiδYLW=Tr

󰀎󰀄

1

G−1=GH+GHΣsG,

˜H

−Σs

i.e.,G

˜representsthefirstiterationtowardthefullself-consistentmany-bodyGreenfunctionstartingfromtheKohn-ShamGreenfunctionGs.Writingthetotalpoten-tialVas

V=w+VH+vxc

andeliminating˜Gs+G˜GHbetweenGsandG

,oneobtainsG˜=[Σs−vxc]Gs,andthusiδYLW

δV(1)

+

δVH

7thoseproducedbylocalpotentials(TDDFT)theLWandKleinfunctionalsgiverisetodifferentresponsefunctionsatthesamelevelofmany-bodyperturbationtheory.

V.ΨFUNCTIONALS

ThemainadvantageoftheΨfunctionalsisthattheygivethepossibilityofusingphysicalmodelsforthescreening,thecalculationofwhichisactuallyabottle-neckinpracticalapplications.Awordofcautionis,how-ever,appropriateinthiscontext.Withmodelscreenedinteractions(W’s)thereisusuallynoself-consistencywithrespecttoW,afactthatmightcompromisetheconservingpropertyofthetheory.TheΨfunctionalshavetwoindependentarguments(GandW)resultingintermslinearinthedeviationoftheactualGreenfunctionfromtheself-consistentonewhenWisawayfromthevaluewhichrendersthefunctionalstationary.WhentheΨfunctionalsareusedtoconstructresponsefunctionsofTDDFTthetheoryis,however,variationalwithrespecttotheone-bodypotentialgeneratingthenon-interactingGreenfunction-evenwhenamodelWisused.Thisfactactuallyrestoresseveralconservingpropertiesalthoughthishastobeverifiedfromcasetocase.Forinstance,choosingmodelW:swhich,likethebareCoulombin-teraction,areinstantaneousandtranslationallyinvariantwillclearlynotspoiltheconservingproperties.

The11firstΨfunctionalwasconstructedbyABLin1996.Ithastheappearance

iYABL[G,W]=Ψ[G,W]−Tr󰀈ΣG−ln󰀁Σ−G−H

1

+

1󰀍󰀋(2,1)

,

P(1,2)=−2

δΨ

δGδQ

[0]=0.

WethenobtainanewΨfunctionalwiththesamesta-tionarypointandthesamevalueatthestationarypoint.

8

Anexampleofasimplefunctionalobtainedinthiswayis

iYLWS[G,W]=Ψ[G,W]−Tr󰀈ΣG−ln󰀁Σ−G−H

1

+

1󰀍󰀋2

Tr{WP+ln(1−vP)}−iUH[G].

Again,duetothesimplicityofthe”Klein”expression,wecanhereusethesamemanipulationsasweappliedtotheoriginalKleinfunctionalinordertoarriveatEq.(6).Thus,insertingthenon-interactingGreenfunctionGsintothefunctionalYABLK,YABLK[V]=Ts[n]+󰀂

wethenobtain

wn+UH+Exc[n],(27)where

Exc[n]=−iΨ[Gs,W]

−

iδn

.(28)

Infact,allfunctionals,betheyoftheΦortheΨvari-ety,havingthe“Klein”formfortheirdependenceontheexternalpotentialwhavethenicepropertythattheopti-mizingpotentialconsistsoftheexternalpotentialw,theHartreepotentialVH,andthefunctionalderivativeoftheexchange-correlationenergywithrespecttothedensityn.InEq.(28),thelastderivativeiscalculatedfromthechainrulefordifferentiationgivingtheOPM-likeequa-tion

󰀂

χvδEs(1,2)xc(2)d2=

xc

2

󰀂

∆W(4,5)δP(5,4)

Thequantity∆WisW−W

˜=W−v/(1−vP)andweremindthereaderthatwearehereallowedtouseany

modelforW.Inparticular,wecouldchooseWtobeW

˜,inwhichcaseourequationfortheexchange-correlationpotentialvxcreducestothesameexpressionasobtainedfromthe“Kleinversion”oftheΦformalismdescribedinSec.III.Furthermore,itiseasilyseendirectlyfromitsdefinitionthatthefunctionalYABLKbecomesindepen-dentofthechoiceofmodelWattheleveloftheRPA.Thus,atthatlevel,thisfunctionaldoesnotaddany-thingtothepreviouslydiscussedΦ-derivableschemeatthesamelevel(RPA).Being,forthemoment,contentwiththatlevelwewillherenotpursuetheYABLKanyfurther.

Finally,byaddinganappropriatechoiceforthefunc-tionalK[Q],asdiscussedabove,tothefunctionalYABLK,weobtainthesimplestfunctionalYKKofthosediscussedinthepresentwork.Wehave

iYKK=Ψ−Tr󰀈GG−H

1−1+ln+

1󰀁−G−1

󰀍󰀋2

Tr{GsGsWo}.

Consequently,inthisapproximation,weobtaintheGWoself-energy

ΣδΨ

s=

910

leadingtothesamequalityofapproximation.OurmethodforimprovedapproximationswithinTDDFThasthesamefeature.Differentfunctionalshavedifferentvariationalaccuracymeaningdifferentsizesofthesec-ondordererrors.Inthepresentpaperwehavedis-cussedmainlytwofunctionals-thatduetoLuttingerandWard(LW)andthatduetoKlein(K).Theformerhasprovedtobemorestableascomparedtothelat-terasfarasconcernsthecalculationoftotalenergiesofavarietyofsystemsrangingfromthosewithverylo-calizedelectronstothosewithitinerantelectrons.ThiswouldsuggestthattheLWfunctionaloughttobeusedalsofortheconstructionofresponsefunctionswithinTDDFT.Inthepresentwork,wehavegiventheformulasfortheexchange-correlationkernelofTDDFTresultingfrombothfunctionals.Sadlyenough,wejudgethatofthesupposedlybetterLWfunctionaltobebeyondourpresentcomputationalfacilities-evenataratherlowlevelofapproximationwithinMBPT.Inordertodemon-stratethispoint,wehavegiventhediagramsrepresentingthedensityresponsefunctionresultingfromtheLWfor-mulationwithintheexchange-onlyapproximation.Wewouldstillliketodrawthereadersattentiontothefactthattheambiguityinthechoiceoffunctionalscanmostlikelybeusedtoouradvantage.Butmuchmoreresearchisneededinordertoseehowthisshouldbedone.

AveryimportantfeatureofourvariationalapproachtoTDDFTisthefactitreliesontheΦorΨderiv-abilityoftheunderlyingapproximationwithinMBPT.Combinedwiththevariationalpropertyofthechosenfunctional,thisleadstothepreservationofmanyphys-icallyimportantconservationlawsandsumrules.Andthisistrueregardlessoftheactualchosenlevelofap-proximationwithinMBPT.Thishighlydesirablefeatureisnotguaranteedinotheravailableapproachesbasedonstraight-forwarddiagrammaticexpansions,iterativetechniques,ordecouplingschemes.Forinstance,inRef.42onedevelopsadiagrammaticrepresentationfortheparticularmany-bodyperturbationschemewhichstartsfromazero-thorderHamiltonianwhichalreadygivesthecorrectdensity.43Unfortunately,thistechniquesuf-fersfromthesamebasicdrawbacksasordinaryMBPT-itis,inprinciple,divergent,summationsmustbecarriedtoinfiniteorder,andthereisnoguaranteeforobtain-ingapproximationswhichhavecertaindesirablephysi-calpropertiesautomaticallyincluded.ThesameholdstrueforexpansionswhicharebasedoniteratingthesocalledHedinequations44usingthescreenedinteractionasthe“small”parameter.45Asanexample,wehave,inthepresentwork,demonstratedhowthevariationalap-proachleadstomomentumconservationinthecaseoftheKleinfunctional.

Itisworthwhileobservingthatthesocalled”lin-earized”Sham-Schl¨uterequationactuallyturnsouttobearesultofourvariationalapproachstartingfromtheKleinfunctional.Butthisisonlytrueiftheself-energyinvolvedisaΦ-orΨ-derivableone.Inthatcase,theresultingapproximationfortheresponsefunctionis,ofcourse,conserving.

Wealsoremarkthatthesocalledoptimizedpotentialmethod(OPM)andmanygeneralizationsthereofreadilyfollowsfromthetheorypresentedhere.Asanexample,wehavegiventheexplicitformulasforthecurrentdensityresponseofahomogeneoussystemwithintheexchange-onlyapproximation.

Eventhoughwenowhaveasystematicwayofobtain-ingbetterresponsefunctionswithinTDDFTtheexpres-sionsquicklybecometoocomplicatedtobeimplementedinlow-symmetrysystems,especiallywhenwewanttoin-cludeallphysicallyrelevantprocesses.Inthiscontext,weadvocatetheuseoftheΨ-derivabletheorieswhichallowsfortheuseofmodelscreenedinteractionswithoutloosingtheimportantconservingproperties.Inthisway,impor-tantphysicaleffectslike,e.g.,astrongparticle-holeinter-actioncanbeincorporatedwithoutanexcessiveincreaseinthecomputationaleffort.Oneshould,however,keepinmindthatmodelsforthescreenedinteractionmustpossesscertainsymmetriesrelatedtotheactualsysteminorderfortheconservingpropertiestobepreserved.Wehavediscussedtheimplementationofthetheo-riespresentedherewithotherresearchgroups.OneparticularlypromisingapproachisthatwhichisbasedontheKleinfunctionalandtheΨ-formulationusingamodelscreenedinteractionlike,e.g.,astaticallyscreenedCoulombinteractionasoftenusedintheBethe-Salpeterapproach,orasimpleplasmon-poleapproximation.To-getherwithourcollaborators,40,41wehopetobeabletopresentsomenumericalresultswithinthenearfuture.

ACKNOWLEDGEMENTS

WewouldliketothankCarl-OlofAlmbladhforfruitfuldiscussionsduringthecourseofthiswork.

ThisworkwassupportedbytheEuropeanCummunity6:thframeworkNetworkofExcellenceNANOQUANTA(NMP4-CT-2004-500198).

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