lecture03.pdf
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lecture03.pdf
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CASTEPWorkshop,DurhamUniversity,613December2001TheNutsandBoltsofFirst-PrinciplesSimulation3:
DensityFunctionalTheoryCASTEPDevelopersGroupwithsupportfromtheESFkNetworkCASTEPWorkshop,DurhamUniversity,613December2001DensityfunctionaltheoryMikeGillan,UniversityCollegeLondonGround-stateenergeticsofelectronsincondensedmatterEnergyasfunctionalofdensity:
thetwofundamentaltheoremsEquivalenceoftheinteractingelectronsystemtoanon-interactingsysteminaneffectiveexternalpotentialKohn-shamequationLocal-densityapproximationforexchange-correlationenergyCASTEPWorkshop,DurhamUniversity,613December2001TheproblemHamiltonianHforsystemofinteractingelectronsactedonbyelectrostaticfieldofnuclei:
withTkineticenergy,Umutualinteractionenergyofelectrons,Vinteractionenergywithfieldofnuclei.Todeveloptheory,Vwillbeinteractionwithanarbitraryexternalfield:
1()NiiVv=rwithripositionofelectroni.Ground-stateenergyisimpossibletocalculateexactly,becauseofelectroncorrelation.DFTincludescorrelation,butisstilltractablebecauseithastheformofanon-interactingelectrontheory.HTUV=+CASTEPWorkshop,DurhamUniversity,613December2001Energyasfunctionalofdensity:
thefirsttheoremForgivenexternalpotentialv(r),letmany-bodywavefunctionbe.Thenground-stateenergyEgis:
0gEHV=+andtheelectrondensityn(r)by:
()()nn=rrwherethedensityoperatorisdefinedas:
1()()Niin=rrr()nrTheorem1:
Itisimpossiblethattwodifferentpotentialsgiverisetothesameground-statedensitydistributionn(r).Corollary:
n(r)uniquelyspecifiestheexternalpotentialv(r)andhencethemany-bodywavefunction.CASTEPWorkshop,DurhamUniversity,613December2001Convexityoftheenergy
(1)Theorem1expressesconvexityoftheenergyEgasfunctionofexternalpotential.Convexitymeans:
Fortwoexternalpotentialsand,goalonglinearpathbetweenthem;ifisground-stateenergyforthen:
()gE(0;)vr(1;)vr(;)(1(0;)(1;)vvv=)+rrr01,+Proofoffollowsfrom2nd-orderperturbationtheory:
()
(1)(0)
(1)gggEEE+020200/()()()()/20,()()gngnndEdVVdEdEE02=withandwavefnsofgroundandexcitedstates,andtheirenergies,and.()0()n0()E()nE
(1)(0)Vvv=CASTEPWorkshop,DurhamUniversity,613December2001Convexityoftheenergy
(2)Theorem1isequivalenttosayingthatachangeofexternalpotentialcannotgiveavanishingchangeofdensity()vr()nrThisfollowsfromconvexity.Convexityimpliesthatatislessthanat.But,sothat:
/gdEd0=/gdEd1=0/gdEdV0=0000
(1)(0)(0)(0)VVsothat:
()(1,)d()(0,).dvnvnrrrrrrHence:
()()0,dvnrrrwhichdemonstratesthat,andthisisTheorem1.()0nrCASTEPWorkshop,DurhamUniversity,613December2001Sinceground-stateenergyEgisuniquelyspecifiedbyn(r),writeitasEgn(r).Itsusefultoseparateouttheinteractionwiththeexternalfield,andwrite:
WhereFn(r)isground-stateexpectationvalueofH0whendensityisn(r).()()()(),gEndvnFn=+rrrrrDFTvariationalprinciplethesecondtheoremTheorem2(variationalprinciple):
Ground-stateenergyforagivenv(r)isobtainedbyminimisingEgn(r)withrespectton(r)forfixedv(r),andthen(r)thatyieldstheminimumisthedensityinthegroundstate.Wheren(r)isdensityassociatedwith.Thisprovesthetheorem.Theusualassumptionsofnon-degenerategroundstateisneeded.Proof:
Letv(r)andv(r)betwodifferentexternalpotentials,withground-stateenergiesEgandEgandground-statewavefnsand.ByRayleigh-Ritzvariationalprinciple:
0()()(),gEHVdvnFn+=+rrrrCASTEPWorkshop,DurhamUniversity,613December2001TheEulerequationWriteFn(r)as:
whereTniskineticenergyofasystemofnon-interactingelectronswhosedensitydistributionisn(r).Then:
FnTnGn=+()().EndvnTnGn=+rrrVariationalprinciple:
subjecttoconstraint:
0()(),()()TGEdvnnn=+rrrrr()0.dn=rrHandletheconstant-numberconstraintbyLagrangeundeterminedmultiplier,andget:
(),()()TGvnn+=rrrwithundeterminedmultiplierthechemicalpotential.CASTEPWorkshop,DurhamUniversity,613December2001Kohn-ShamequationRewritetheEulerequationforinteractingelectrons:
bydefining,sothat:
()()()TGvnn+=rrreff()()/()vvGn=+rrreff()()Tvn+=rrButthisisEulerequationfornon-interactingelectronsinpotentialveff(r),andmustbeexactlyequivalenttoSchroedingerequation:
withn(r)givenby:
22eff(),2vm+=rh2()2().n=rrThenputn(r)backintoGn(r)togettotalenergy:
tot()()().EndvnTnGn=+rrrrCASTEPWorkshop,DurhamUniversity,613December2001SelfconsistencyHowtodoDFTinpractice?
WedontknowGn(r),andprobablyneverwill,butsupposeweknowanadequateapproximationtoit.Makeaninitialguessatn(r),calculateandhenceforthisinitialn(r).SolvetheKohn-Shamequationwiththisveff(r)togettheKSorbitalsandhencecalculatethenewn(r):
Theoutputn(r)isnotthesameastheinputn(r).Soiteratetoreduceresidual:
Thewholeprocedureiscalledsearchingforselfconsistency./()Gnreff()()/()vvGn=+rrr2()2()|.in=|rr1/22d()().nnn=rrrCASTEPWorkshop,Durham
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