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1、PHYSICALREVIEWVOLUME140 NUMBER4A15NOVEMBER1965 Self ConsistentEquationsIncludingExchangeandCorrelationEffects W KOHNANDL J SHAM UnieersityofCa Bfornia SanDiego laJolta California Received21June1965l FromatheoryofHohenbergandKohn approximationmethodsfortreatinganinhomogeneoussystem ofinteractingelect
2、ronsaredeveloped Thesemethodsareexactforsystemsofslowlyvaryingorhighdensity Forthegroundstate theyleadtoself consistentequationsanalogoustotheHartreeandHartree Fock equations respectively Intheseequationstheexchangeandcorrelationportionsofthechemicalpotential ofauniformelectrongasappearasadditionale
3、ffectivepotentials Theexchangeportionofoureffective potentialdiffersfromthatduetoSlaterbyafactorof Electronicsystemsatfinitetemperaturesandin magneticleldsarealsotreatedbysimilarmethods Anappendixdealswithafurthercorrectionfor systemswithshort wavelengthdensityoscillations I INTRODUCTION N recentyea
4、rsagreatdealofattentionhasbeen giventotheproblemofahomogeneousgasofinter actingelectronsanditspropertieshavebeenestablished withaconsiderabledegreeofconfidenceoverawide rangeofdensities Ofcourse suchahomogeneousgas representsonlyamathematicalmodel sinceinallreal systeins atoms inolecules solids etc
5、theelectronic densityisnonuniform Itisthenamatterofinteresttoseehowproperties ofthehomogeneousgascanbeutilizedintheoretical studiesofinhomogeneoussystems Thewell known methodsof Thomas Fermi andtheSlater exchange holeareinthisspirit Inthepresentpaperweusethe formalismofHohenbergandKohn tocarrythis a
6、pproachfurtherandweobtainasetofself consistent equationswhichinclude inanapproximate way ex changeandcorrelationeffects They requireonlya knowledgeofthetruechemicalpotential tie e ofa homogeneousinteractingelectrongasasafunctionof thedensityn Wederivetwoalternativesetsofequations Eqs 2 8 and 2 22 wh
7、ichareanalogous respec tively totheconventionalHartreeandHartree Fock equations and althoughtheyalsoincludecorrelation effects theyarenomoredifficulttosolve Thelocaleffectivepotentialsintheseequationsare uniqueinasensewhichisdescribedinSec II Inpar ticular weAndthattheSlaterexchange holepotential be
8、sidesitsomissionofcorrelationeffects istoolarge byafactorof Apartfromwork onthecorrelationenergyofthe homogeneouselectrongas mosttheoreticalmany body studieshavebeenconcernedwithelementaryexcita tionsandasaresulttherehasbeenlittlerecentprogress inthetheoryofcohesiveenergies elasticconstants etc ofre
9、al i e inhomogeneous metalsandalloys Themethodsproposedhereofferthehopeofnew progressinthislatterarea Supported inpartbytheU S OfhceofNavalResearch L H Thomas Proc Cambridge Phil Soc 23 542 1927 E Fermi Z Physik48 73 1928 J C Slater Phys Rev 81 385 1951 P HohenbergandW Kohn Phys Rev 136 3864 1964l r
10、eferredtohereafterasHK InSecs IIIandIV wedescribethenecessaryInodid cationstodealwiththefinite temperatureproperties andwiththespinparamagnetismofaninhomogeneous electrongas Ofcourse thesimplemethodswhichareherepro posedingeneralinvolveerrors Theseareoftwogeneral origins4 atoorapidvariationofdensity
11、and for6nite systems boundaryeffects Refinementsaimedatre ducingthe6rsttypeoferrorarebrieQydiscussedin AppendixII II THEGROUNDSTATE A LocalEffectivePotential Ithasbeenshown thattheground stateenergyof an interactinginhomogeneouselectrongasinastaticpo tentialn r canbewrittenintheform 1e r e r Z tt r
12、e r dr drdr G e rr i 2 1 wheree r isthedensityandG e isauniversalfunc tionalofthe density Thisexpression furthermore isa minimumforthecorrectdensityfunctione r Inthis sectionweproposefirstan approximationforG e whichleadstoascheme analogoustoHartree smethod butcontainsthemajorpartoftheeffectsofexcha
13、nge andcorrelation Wefirstwrite G e T e yZ e 2 2 whereT e is thekineticenergyofasystemofnon interactingelectronswithdensitye r andF e is byourdefinition theexchangeand correlationenergy ofaninteractingsystemwithdensitye r Foranarbi trarye r ofcourse onecangivenosimpleexactex pressionforE e However i
14、fe r issufliciently slowlyvarying onecanshow that F e e r e e r dr 2 3 4W Kohn andL J Sham Phys Rev 137 A1697 1965 Forsuch asystemitfollowsfromHKthatthekineticenergy isinfactauniquefunctionalofthedensity 1138 SH MAN thoverwhichd ro a yp sit Int s icallengt seas it z raius lhangeindensty hangeand nap
15、preciabec ndthetrueexc isan canexpan ene gy a F fej a e e dr 2 11 c i e theeffectiveexchangepotential fora stateatthetopoftheFermidistributions Thisis physicallyunderstandablesincedensityadjustments comeaboutbyredistributionoftheelectronsnearthe Fermilevel b Highdertsity Thisregimeischaracterizedbyt
16、he conditionr a 1 whereasistheBohrradius Inthis case theentireexchangeandcorrelationenergy is smallerthanthekineticenergybyafactoroforder r as andhenceourinaccuracyinrepresenting these portionsbecomesnegligible ThereaderwillhavenoticedthatwhileinEq 2 3 weapproximatetheexchangeand correlationenergyby theexpressionvalidforaslowlyvaryingdensity we madenoapproximationforthekinetic energyfunc tionalT LnjofEq 2 2 Thisprocedureisresponsible fortheexactnessofthehigh densitylimit evenwhen thedensityisrap
