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1、EnsembleDensity-FunctionalTheoryforA/?InitioMolecularDynamicsofMetalsandFinite-TemperatureInsulatorsNicolaMarzari,12DavidVanderbilt,1andM.C.Payne2DepartmentofPhysicsandAstronomy,RutgersUniversity,Piscataway,NewJersey08855-0849-CavendishLaboratory(TCM),UniversityofCambridge,MadingleyRoad,CambridgeCB3
2、OHE,UnitedKingdom(Received7March1997)Anewmetliodispresentedforperformingfirst-principlesmolecular-dynamicssimulationsofsystemswiihvariableoccupancies.Weadoptamatrixrepresentationfbrtheone-particlestatisticaloperatorutointroduceaprojectedfreeenergyfunctionalGthatdependsontheKohn-Shamorbitalsonlyandth
3、atisXiinvariantundertheirunitarytransformations.TheLiouvilleequationr1.Hg-0isalwayssatisfied,guaranteeingaveryefficientandrobustvariationalminimizationalgorithm,thatcanalsobeextendedtononconventionalentropicformulations.S0031-9007(97)03810-6()031-9007y97y79(7)y1337(4)$10.001997TheAmericanPhysicalSoc
4、iety1337PACSnumbers:71.l5.Pd,31.50.+w,65.50.+m,68.35JaInrecentyears,therangeofproblemsthatcanbestudiedwithquantitativeaccuracyusingthemethodsofcomputationalsolidstatephysicshasexpandeddramatically.Itisnowpossibletocalculatemanymaterialspropertieswithaprecisionthatisoftencomparabletothatofexperiments
5、.Thisdegreeofconfidenceisbasedonthefundamentalquantum-mechanicaltreatmentofferedbydensity-functionaltheory(DFT)1,coupledwiththeavailabilityofincreasinglypowerfulcomputersandwiththedevelopmentofalgorithmstunedtowardsoptimalperformance2,3.Theapplicationofthesemethodsandtechniquestometallicsystemshasno
6、nethelessencounteredseveraldifficultiesthathavemadeprogressslowerthanforthecaseofsemiconductorsandinsulators.ThediscontinuousvariationoftheorbitaloccupanciesacrosstheBrillouinzone(BZ)makestheoccupationnumbersratherillconditionedvariables,andtheself-consistentsolutionofthescreeningproblemcansufferfro
7、mseveralinstabilities.TheabsenceofagapintheenergyspectrumandtherequirementofanexactdiagonalizationfortheHamiltonianmatrixeverywhereintheBZ(inordertoassigntheoccupationnumbers)introduceslowfrequenciesintheevolutionoftheorbitalstowardsthegroundstateandprecludethestraightforwardextensiontometalsofalgor
8、ithmswhichperformedwellforinsulators.SmearingtheFermisurfacewithafiniteelectronictemperature4,5allowsforanimprovedBZsampling,butonlypartiallyalleviatestheproblemsalludedtoabove.InthisLetter,weintroduceanewapproachwhichsolvesmanyoftheseproblemsinanaturalway,andwhichprovidesageneralandefficientframewo
9、rkfbrobtainingthegroundstateofaKohn-ShamHamiltonianatafiniteelectronictemperature.ThecontextistheMerminformulationfortheFermi-Diracstatistics5-7,butthemethodalsoapplieswhengeneralizedentropicfunctionalsareintroduced8,asisoftenthecaseformetallicsystems.ThelanguageofensembleDFT9isused,andavariationala
10、lgorithmfortheminimizationtothegroundstateisdevelopedandimplemented.DramaticimprovementsareobtainedintheconvergenceoftheenergiesandespeciallyoftheHellmann-Feynmanforces.WithinensembleDFT,theHelmholtzfreeenergyfunctionalatatemperatureTandforanN-representablechargedensitynsrdin4yfsrdg-F/f/isrUg1RanKxt
11、externalsrd/?srdr/rpotentiaI,whereVFcxitisisthefinile-temperalureMermin-Hohenberg-Kohnfunctional6.ThechargedensityosrdthatminimizesAvistheground-statechargedensity,and/vf?osrdgisthefreeenergyoftheelectronicsystem.AKohnShammappingontononinteractingelectronsleadstoadecompositionofthefunctionalFtintono
12、ninteraciingkinetic-energy,electrostatic,andentropiccontributions,plustheexchange-correlation(XC)functional,forwhichwetakethelocaldensityapproximationI.Akeyassumptionismadebyadoptingamatrixrepresentationfy,inthebasisoftheorbitals,fbrtheone-G人particleeffectivestatisticaloperator,sothatXPsrdcysrd.(1)s
13、rd-fjiCiHerethehc(jareorthonormalsingle-particleKohn-Shamorbitals,(hesumextendsinprincipleoverallthestates,andfijisconstrainedtohavetrf-NandeigenvaluesboundedtoII),lg.ThefunctionalAiobeminimizedisthenAfT;hej,顷jgXA1rcxljc；lIEHxcf昭2仙;顼kc(j7ij(2)theHartreeandXCterms,whichdependonlyonthechargedensity,ha
14、vebeengroupedtogether.TheFermiDiracentropictermisafunctionoftheeigenvaluesoff:Sfh/Jg-trssfd,wheresisfnfIsi2ydlnsl2fA.Typically,Vcuisgeneratedbyanarrayofnonlocalpseudopotcntials.ThefreeenergyfunctionalAisintheformoftracesofoperators,andsoitiscovariantunderachangeofrepresentation(i.e.foraunitaiytransf
15、ormationUoftheorbitalshcj);thiscanbevcrificdPbylettingf!f-UfUyandjc；l!jcyl-mUjp5.ThecovarianceofAallowsforthedefinitionofanewprojectedfunctionalGthatdependsonlyontheorbitalshc；j:GfThejg:-minAfT;hc,j,li/g.响Gisinvariantunderanyunitarytransformationofthehej:thetransformedorbitalscannotleadtoadifferentv
16、alueforG.byvirtueofthecovarianceofA.TheprojectedfunctionalGrepresentsamuchbetterconditionedchoicethantheoriginalfreeenergyAfortheevolutionoftheorbitalstowardsthegroundstate.Thereasonsareseveral,albeitrelated,(i)ThefunctionalGnolongerdependsontheoccupanciesoftheorbitalsorontheirunitarytransformations(rotations)intheoccupiedsubspace.Theseareill-conditioned,nonlocaldegreesoffreedom,withtheaddednonlinearconstraintofchar
