Advanced Quantum Chemistry III: Part 3 Haruyuki Nakano Kyushu University 2013 Winter Term
1. Hartree-Fock theory Density Functional Theory 2. Hohenberg-Kohn theorem 3. Kohn-Sham method 4. Exchange-correlation correlation functionals
Independent Particle Model 1. Hartree-Fock theory 2-electron operator is approximately replaced by a 1-electron operator N N 1 rij u( ri ) ij i1 1 2 Z u() () () 2 R r r r r Does u(r) exist that gives exact energy (and electron density) keeping the independent particle model? Yes! Density functional theory Kohn-Sham method
2. Hohenberg-Kohn theorem Definition of electron density Integration of probability amplitude over (N 1) spatial and all spin coordinates gives electron density 2 () r N d1dr2d2d r N d N 1st Hohenberg-Kohn theorem Electron density ρ(r) determines external potential v(r) This means that ρ(r) determines Ψ and E (via v(r)) E[ ] V [ ] T[ ] V [ ] ne ( r) v( r)d rt[ ] Vee[ ] 1 1 J[ ] ( ) ( )d d r1 r2 r1 r2 2 r 12 ee
2nd Hohenberg-Kohn theorem variational theorem () r : approximate electron density ()()d r v( r) r T [ ] V [ ] E [ ] E [ ] ee Hence E[ ] ( )d N 0 r r μ: Lagrange multiplier T [ ] V [ ] ee ()()d r v r r dr d ()d 0 r r r T[ ] Vee[ ] v() r 0 If T[ρ] and V ee [ρ] are known, exact energy is obtained
3. Kohn-Sham method Non-interacting system: N electrons and orbitals {φ i } 1 N 2 TS[ ] i i i 2 N () r i () r These are exact for non-interacting systems i 2 Orbitals {φ i } follows 1-electron Schrödinger equation 1 2 vs () i() i() 2 r r r and the energy is given by E[ ] T [ ] v( r) ( r)d r (i) S
Interacting system E[ ] v( r) ( r)d rt[ ] V [ ] ee J v () r ()d r r TS [ ] J [ ] T [ ] TS [ ] Vee [ ] J [ ] v() r ()d r rt [ ] J[ ] E [ ] (ii) S xc known functional unknown functional Exchange-correlation functional E [ ] T[ ] T [ ] V [ ] J[ ] xc S ee Exchange-correlation potential v xc Exc [ ] [ ]
From (i), for non-interacting systems, the 2nd Hohenberg-Kohn theorem gives TS[ ] vs () r Hence, (ii) gives ( r) TS[ ] v () r d r vxc () r r r () r v eff TS[ ] () r () r Under the external potential, v S (r) = v eff (r), equation for the interacting system is same as that for the non-interacting system Thus, 1-electron equation (Kohn-Sham equation) about v eff (r) becomes 1 2 veff () i() i() 2 r r r and the density becomesn () r i () r i 2
4. Exchange-correlation functional Exc[ ] Ex[ ] Ec[ ] Exchange Correlation Local density approximation (LDA) Approximation: Electron density can be locally treated as a uniform electron gas (cf. LSDA local spin density approximation) Exchange functional LDA 4 3 Ex [ ] Cx ( )d r r LDA 1 3 x [ ] Cx Correlation functional Dirac formula Correlation energy estimated by Monte-Carlo method is fitted to analytical function Vosko, Wilk, and Nusair (VWN) ε c VWN Perdew and Wang (PW91) LSDA underestimates exchange energy by about 10%
Generalized gradient approximation (GGA) Electron density is treated as a non-uniform electron gas [Correlation energy depends on ρ Word generalized means GGA is not simply a Taylor expansion] Exchange functional Perdew and Wang (PW86) ε x PW86 Becke (B or B88) ε x B88 Becke and Roussel (BR) ε BR x Perdew and Wang (PW91) ε x PW91 Correlation eato functional cto Lee, Yang, and Parr (LYP) ε c LYP Perdew (P86) ε c P86 Perdew and Wang (PW91 or P91) ε c pw91 Becke (B95) ε c B95
Hybrid functional (Hyper-GGA) Mixture of several exchange-correlation functionals Half-and-half E H+H xc 1 exact 1 LSDA 2 E 2 E E LSDA x x c E xc for the case that v xc does not exist: Hartree-Fock exchange E xc for the case that v xc does exist: Approximated by LSDA Beck 3 parameter a functional (B3) E (1 a) E ae be E ce B3 LSDA exact B88 B88 GGA xc x x x c c a 0.2, b 0.7, c 0.8
Combination of exchange and correlation functionals BLYP B88 + LYP BPW91 B88 + PW91 B3LYP B3 + LYP B3PW91 B3 + PW91 Perdew classification of exchange-correlation correlation functionals Level Name Variables Examples 1 Local density ρ LDA, LSDA, X α 2 GGA ρ, ρ BLYP, OPTX, OLYP, PW86, PW91, PBE, HCTH 3 Meta-GGA ρ, ρ, ρ or τ BR, B95, VSXC, PKZB, TPSS, τ-hcth 4 Hyper-GGA ρ, ρ, ρ or τ HF exchange H+H, ACM, B3LYP, B3PW91, O3LYP, PBE0, TPSSh, τ-hcth-hybrid 5 Generalized RPA ρ, ρ, ρ or τ HF exchange virtual orbitals OEP2 occ 1 Note : τ: orbital kinetic energy density ( () r ()) i r 2 i