Solutions to Holmes - Introduction to Perturbation Methods (Springer-Verlag 1995)
More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al....
Transcript of More basics of DFTPrimer in Density Functional Theory, edited by C. Fiolhais et al....
More basics of DFT
KieronBurkeandfriendsUCIrvinePhysicsandChemistry
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References for ground-state DFT
– ABCofDFT,byKBandRudyMagyar,http://dft.uci.edu
– APrimerinDensityFunctionalTheory,editedbyC.Fiolhaisetal.(Springer‐Verlag,NY,2003)
– DensityFunctionalTheory,DreizlerandGross,(Springer‐Verlag,Berlin,1990)
– DensityFunctionalTheoryofAtomsandMolecules,ParrandYang(Oxford,NewYork,1989)
– AChemist’sGuidetoDensityFunctionalTheory,KochandHolthausen(Wiley‐VCH,Weinheim,2000)
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What we’ll cover
• Simplestpossibleexampleofafunctional• EssentialsofKS‐DFT,andfunctionalzoo• Importantconditionsnotmetbystandardfunctionals:Self‐interactionandderivativediscontinuity
• Exactexchange• Quiz
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Atomic units and particles in box • Inatomicunits,allenergiesareinHartree(1H=
27.2eV)andalldistancesinBohr(1a0=0.529Å)
• Towriteformulasinatomicunits,sete2=Ћ=me=1• E.g.,usualformulaforenergylevelsofinfinitewell
ofwidthL:
• Atomicunits,boxlengthL=1:
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Constructing your very first density functional
• Let’slookatthekineticenergyofspinlessfermionsin1d:
• IstheresomewaytogetTswithoutevaluatingallthosedamnorbitals?Yes!
• Writeitasadensityfunctional,i.e.,anintegraloversomefunctionofn(x).
• Simplestchoice:alocalapprox:
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Particles in box
• Accuracy
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N Ts[0] Ts %err
1 4.112 4.934 -17
2 21.79 24.67 -12
3 62.92 69.09 -9
What we’ve learned
• Densityfunctionalsareapproximationsfortheenergyofmanyparticles
• WorkbestforlargeN,worstforsmallN
• Localapproximationsarecrudelycorrect,butmissdetails
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Essence of Kohn-Sham DFT • EvenwithexactExc[n],onlygetE0andn(r)(andI).Sootherpropertiesmaynotberight.
• Resultsonlyasgoodasfunctionalused.• VastamountofinformationfromE0alone,suchasgeometries,vibrations,bondenergies…
• Well‐fittedfunctionalsareaccurateforlimitedset
• Non‐empiricalfunctionalslessso,butmorereliableforabroaderrange,anderrorsunderstandable
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He atom in Kohn-Sham DFT
Dashed-line:
EXACT KS potential
Everything has (at most) one KS potential
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Functionals in common use • Localdensityapproximation(LDA)– Usesonlyn(r)atapoint.
• Generalizedgradientapprox(GGA)– Usesbothn(r)and|∇n(r)|– Moreaccurate,correctsoverbindingofLDA– ExamplesarePBEandBLYP
• Hybrid:– MixessomefractionofHF– ExamplesareB3LYPandPBE0
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Functional soup
• Good:chooseonefunctionalofeachkindandstickwithit(e.g.,LDAorPBEorB3LYP).
• Bad:Runseveralfunctionals,andpick‘best’answer.
• Ugly:Designyourownfunctionalwith2300parameters.
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Functional Zoology
• Empirical– GGA:BLYP– Hybrid:B3LYP
• Names:– B=B88exchange– LYP=Lee‐Yang‐Parrcorelation
• Non‐empirical– GGA:PBE– Meta‐GGA:TPSS– Hybrid:PBE0
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What we’ll cover
• Simplestpossibleexampleofafunctional• EssentialsofKS‐DFT,andfunctionalzoo• Importantconditionsnotmetbystandardfunctionals:Self‐interactionandderivativediscontinuity
• Exactexchange• Quiz
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What we’ll cover
• Simplestpossibleexampleofafunctional• EssentialsofKS‐DFT,andfunctionalzoo• Importantconditionsnotmetbystandardfunctionals:Self‐interactionandderivativediscontinuity
• Exactexchange• Quiz
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Simple conditions for Coulomb systems
• Asymptoticdecayofthedensity
• LeadstosevereconstraintonKSpotential
• AnddeterminesKSHOMO:
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KS potential for He atom
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Densities
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LDA potential
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Self interaction
• Violatedbymostsemilocalfunctionals(unlessbuiltin)
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Energy as function of N
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FromDreizler+Gross
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Derivative discontinuity
• Whenyouaddatinyfractionofanelectrontoasystem,theKSpotentialshiftsuniformly,sincebefore,εHOMO(N)=‐I,butnow,εHOMO(N+δ)=‐A
• Thusvs(r)mustjumpbyΔxc=(I‐A)‐(εHOMO‐εLUMO)
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Ne Potentials
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Missing derivative discontinuity in LDA
LDAlookslikeexact,shiftedbyaboutI/2
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What we’ll cover
• Simplestpossibleexampleofafunctional• EssentialsofKS‐DFT,andfunctionalzoo• Importantconditionsnotmetbystandardfunctionals:Self‐interactionandderivativediscontinuity
• Exactexchange• Quiz
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What we’ll cover
• Simplestpossibleexampleofafunctional• EssentialsofKS‐DFT,andfunctionalzoo• Importantconditionsnotmetbystandardfunctionals:Self‐interactionandderivativediscontinuity
• Exactexchange• Quiz
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What ever happened to HF?
• WeknowExisjust
• Sowhycan’twejustputthatinKSequations?
• Becausedon’tknowEx[n],somustapproximate
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OEP • Waytohandleorbital‐dependentfunctionalsinKSscheme,i.e.,withsinglemultiplicativeKSpotential
• Stilldensityfunctionals,sinceorbitalsuniquelydeterminedbydensity
• OftencalledOPM• Severalschemestoimplement,allmuchmoreexpensivethanregularKS‐DFT
• Canimproveotherproperties:– Noself‐interactionerror– Potentialsandorbitalenergiesmuchbetter– Approximatesderivativediscontinuity
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SeeRMP,KuemmelandKronik
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HF versus EXX
• HFminimizesEx[{φi}]overallpossiblewavefunctions
• EXXincludesadditionalconstraintofcommonpotential(i.e.,KS)
• Yieldalmostidenticaltotalenergies,withHFaneenstybitlower.
• Occupiedorbitalenergiesverysimilar,butbigdifferenceinunoccupiedorbitals
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A tale of three gaps
• Fundamentalgap:– Δ=I–A=24.6eVforHe
• Kohn‐Shamgap:– Δs=εHOMO‐εLUMO=21.16eV
• Derivativediscontinuity:Δxc=Δ‐Δs
• Lowestopticaltransition:– ωmin=E(1s,2p)‐E(1s2)=21.22eV
• NOTE:Allsameifnon‐interacting,alldifferentwheninteracting
• Of course, εHOMO(LDA)=15.5eVAPStutorial
Quiz
1. Dolocalfunctionalsdobetterfor:A.smallN,B.largeN?
2. Howmanyempiricalparametersaretoomany?A.1;B.10.,C.100+
3. GGA’shavenoself‐interactionerror,Trueorfalse?
4. TheKohn‐Shamgapwouldequalthetruegapifonlywehadtheexactfunctional?
5. WhynotuseExinsmallcalculationstoimprovegeometries,etc.?
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What we’ve learned, maybe • Ground‐statedensitydeterminesallpropertiesofsystem,
inprinciple,butinpractice,onlyreallygetenergyanddensity(whichis90%ofwhatyouwant).
• Localdensityfunctionaltheoriesgiveroughlycorrectanswers,butaretooinaccuratetobehelpfulinquantumchemistry.
• Thecommonly‐usedfunctionalsinchemistryarewell‐foundedandhavefewparameters.
• Thereareknownexactpropertiesofthedensityinrealatoms.
• TherearesubtleandbizarreeffectsintheKSpotentialbecauserealelectronsdointeract.
• Exactexchangeisexpensive,andwedon’thaveacorrelationfunctionaltogowithit,butitimprovessomeproperties.
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