The correlation energy as an explicit functional o

arXiv:physics/0506186v2[physics.chem-ph]28Jun2005Thecorrelationenergyasanexplicitfunctionaloftheone-particledensitymatrixfromadeterminantalreferencestateJamesP.Finley∗DepartmentofPhysicalSciences,EasternNewMexicoUniversity,Station#33,Portales,NM88130andDepartmentofAppliedChemistry,GraduateSchoolofEngineering,TheUniversityofTokyo,7-3-1Hongo,Bunkyo-ku,Tokyo,113-8656Japan(Dated:February2,2008)AbstractUsinganapproachbasedonmanybodyperturbationtheory,thecorrelationenergyEcoisex-pressedasanexplicitfunctionalofρ1,v,andvs,whereρ1istheone-particledensitymatrixfromthenoninteracting,orreference,determinantal-state;vistheexternalpotentialfromtheinteracting,ortarget,state;vsisthe(kernelofthe)externalpotentialfromthenoninteractingdeterminantal-state.InotherwordswehaveEco[ρ1,v,vs].Antherpossibilityisthefollowingexplicitfunctional:Eco[ρ1,vco,vs],wherevcoisthe(kernelofthe)correlationpotentialfromthenoninteractingHamil-tonian.Theproposedmethodcan,inprinciple,beusedtocomputeEcoinaveryaccurateandefficientmanner,since,liketheKohn–Shamapproach,therearenovirtualorbitalstoconsider.However,incontrasttotheKohn–Shamapproach,Ecoisaknown,explicitfunctionalthatcanbeapproximatedinasystematicmanner.Forsimplicity,weonlyconsidernoninteractingclosed-shellstatesandtargetstatesthatarenondegenerate,singletground-states;so,inthatcase,ρ1denotesthespin-lessone-particledensitymatrixfromthedeterminantalreferencestate.1I.INTRODUCTIONTheKohn-Shamversionofdensityfunctionaltheoryhasbeenverysuccessfulinthedescriptionofelectronicstructureforquantumchemistryandcondensedmatterphysics.1,2,3,4,5,6,7Unlikepuredensityfunctionalapproaches,1,2,8theKohn–Shammethodusesasetofoccupiedorbitalsfromanoninteractingstate,wherethisdeterminantalstatesharesitselectrondensityρwiththetarget,orinteracting,state|Ψi.IntheKohn–Shamap-proach,thekineticenergy(throughthe)firstorderisnotanexplicitfunctionalofρ,but,in-stead,thisfunctionaldependsontheone-particledensitymatrixρ1fromthenoninteracting,determinantalstate.TheKohn-Shamapproachrequiresasinputtheexchange-correlationfunctionalEKSxc—orequivalentlytheexchangeExandcorrelation-energyEKScofunctionals—whereEKSxcisrequiredtobeanexplicitfunctionalofρ.Unfortunately,EKSxcisanunknownfunctional,andthereisnosystematicmethodtoimproveapproximations.Theoptimizedpotentialmethod9,10,11,12,13,14,15isadensityfunctionalapproachthatcanconvertanonlocaloperatorintoalocalone,wheretheexchange-correlationfunctionalscandependonboththeoccupiedandvirtualorbitals.Unfortunately,thismethodlackstheefficiencyofotherKohn–Shamapproaches.Furthermore,theoptimizedpotentialmethodintroducesfunctionalthat—incontrasttomanywavefunctionmethods—arenotinvarianttoaunitarytransformationofeithertheoccupiedorvirtualorbitals;thelocalpotentialsalsodependontheorbitalenergies.Highlevelsofapproximations—beyondtheKohn–Shamapproaches—canbeobtainedbywavefunctionmethods,16,17,18,19includingthecoupledclustermethod,many-bodypertur-bationtheory,andconfigurationinteraction.Often,however,thesemethodsaremuchlessefficientthantheKohnShamapproaches,where,typically,wavefunctionmethodsconsideralargenumberof2-electronmolecularintegrals,dependingonboththeoccupiedandvirtualorbitals,andtheseintegralsmustbecomputedandutilizedincalculationsinvolvinglargeatomic-orbitalbasissets.Inordertoimprovetheefficiencyofthewavefunctionmethods,especiallyinregardstotheirscalingwithmolecularsize,perturbativemethodshavebeendevelopedbasedonlocalizedmolecularorbitals.20,21,22,23,24,25,26AnalternativeapproachusesaLaplacetransformtoremovetheenergydenominatorsinperturbationtheory,yieldingapproachesinvolvingcorrelationenergyexpressionsthatdependexplicitlyontheatomic-orbitalbasisset.27,282Thismethodhasalsobeenusedwiththecoupledclustertheory.29OurresearchinterestisinthedevelopmentofmethodsthatbridgethegapbetweenwavefunctionapproachesandKohn–Shamdensityfunctionaltheory.However,thesetwodifferentapproachesalreadysharesomecommonfeatures.Forexample,asmentionedabove,Kohn–Shamintroducesadeterminantalwavefunctionandtreatsthefirst-orderkineticenergyinthesamemannerasinwavefunctionmethods,i.e.,asafunctionaloftheone-bodydensitymatrixfromadeterminant.Furthermore,hybriddensityfunctionals,30,31,32,33includingB3LYP,30,34introduceacomponentoftheexactexchange—afunctionalofρ1—eventhoughtheseapproachesviolatetheHohenberg–Kohntheorem8byusingthenonlocalexchangeoperator.AsimplegeneralizationoftheKohn–ShamfunctionalsinvolvesusingtheexactexchangeenergyEx(withitscorrespondingnon-localoperator)and,inaddition,acorrelation-energyfunctionalthatalsodependsonρ1.SuchanapproachcouldprobablybederivedwithinavariantoftheKohn–Shanscheme.However,itisalsoreasonabletobasesuchaformalismonwavefunctionmethods,since,forexample,thecorrelation-energy,sayEco,fromwavefunctionmethodsisanimplicitfunctionalofρ1,anddoesnotimplicitlydependontheorbitals,eventhoughthereisanexplicitdependence.(ThisiseasilyprovenbynotingthatwehaveEco=E−E1[ρ1],whereEistheexactelectronenergy,andE1isthefirstorderenergythatisdeterminedbyρ1.)Furthermore,suchaformalismcanbebasedonanyreasonableorbitals:Hartree–Fock,Brueckn