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

APStutorial

What we’ll cover

•  Simplestpossibleexampleofafunctional•  EssentialsofKS‐DFT,andfunctionalzoo•  Importantconditionsnotmetbystandardfunctionals:Self‐interactionandderivativediscontinuity

•  Exactexchange•  Quiz

24APStutorial

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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