Spring College on Computational Nanoscience May Variational Principles, the Hellmann-Feynman Theorem, Density Functional Theor

Transcription

1 Spring College on Computational Nanoscience May 2010 Variational Principles, the Hellmann-Feynman Theorem, Density Functional Theor Stefano BARONI SISSA & CNR-IOM DEMOCRITOS Simulation Center Trieste Italy

2 variational principles, the Hellmann-Feynman theorem, density-functional theory Stefano Baroni Scuola Internazionale Superiore di Studi Avanzati & CNR-IOM DEMOCRITOS Simulation Center Trieste - Italy lecture given at the Spring College on Computational Nanoscience The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, May 17-28, 2010

3 the saga of time and length scales 10-3 length [m] macro scale = hic sunt leones 10-9 nano scale = 1 time [s]

4 the saga of time and length scales 10-3 length [m] thermodynamics & finite macro elements scale = kinetic Monte hic sunt Carlo leones classical molecular dynamics electronic nano structure scale methods = 1 time [s]

5 size vs. accuracy size/duration classical empirical methods pair potentials force fields shell models quantum empirical methods tight-binding embedded atom quantum self-consistent methods density Functional Theory Hartree-Fock quantum many-body methods quantum Monte Carlo MP2, CCSD(T), CI GW, BSE accuracy

6 size vs. accuracy size/duration classical empirical methods pair potentials force fields shell models quantum empirical methods tight-binding embedded atom quantum self-consistent methods density Functional Theory Hartree-Fock quantum many-body methods quantum Monte Carlo MP2, CCSD(T), CI GW, BSE accuracy

7 ab initio simulations i Φ(r, R; t) t = ( 2 2M 2 R 2 2 2m 2 ) + V (r, R) Φ(r, R; t) r2 The Born-Oppenheimer approximation (M m) ( 2 2m 2 + V (r, R) r2 M R = E(R) ) R Ψ(r R) =E(R)Ψ(r R)

8 density-functional theory V (r, R) = e2 2 Z I Z J R I R J Z Ie 2 r i R I + e2 2 1 r i r j

9 density-functional theory V (r, R) = e2 2 Z I Z J R I R J Z Ie 2 r i R I + e2 2 1 r i r j DFT V (r, R) e2 2 Z I Z J R I R J + v [ρ](r) Kohn-Sham Hamiltonian ( 2 2m ρ(r) = v 2 ) r 2 + v [ρ](r) ψ v (r) =ɛ v ψ v (r) ψ v (r) 2

10 functionals examples: G[f] :{f} R G[f] = f(x 0 ) G[f] = G[f] = b a b a f 2 (x)dx f (x) 2 dx f(x) f 1 f 2... f i f N approximations: G[f] g(f 1,f 2,,f N ) G[f] g(c 1,c 2,,c N ) x 1 x 2... x i x N f(x) n c n φ n (x)

11 functional derivatives G[f 0 + ɛf 1 ]=G[f 0 ]+ɛ f 1 (x) δg δf(x) dx + O ( ɛ 2) f=f0 δg δf(x) f=f0 1 h g f i δg δf(x) = lim ɛ 0 G[f( )+ɛδ( x)] G[f( )] ɛ

12 the Hellmann-Feynman theorem Ĥ λ Ψ λ = E λ Ψ λ

13 the Hellmann-Feynman theorem Ĥ λ Ψ λ = E λ Ψ λ E λ = λ Ψ λ Ĥλ Ψ λ = Ψ λ Ĥ λ Ψ λ E λ = min Ψ Ĥλ Ψ {Ψ: Ψ Ψ =1} g(λ) =G[x(λ),λ] g(λ) =min x G[x, λ] G x g (λ) =x (λ) G x =0 x=x(λ) + G x=x(λ) λ

14 the Hellmann-Feynman theorem Ĥ λ Ψ λ = E λ Ψ λ E λ = λ Ψ λ Ĥλ Ψ λ = Ψ λ Ĥ λ Ψ λ E λ = min Ψ Ĥλ Ψ {Ψ: Ψ Ψ =1} g(λ) =G[x(λ),λ] g(λ) =min x G[x, λ] G x g (λ) =x (λ) G x =0 x=x(λ) + G x=x(λ) λ

15 conjugate variables & Legendre transforms E = E(V,x) P = E V V V Legendre transform: properties: H(P, x) =E + PV E convex V P ( ) H(P, x) = max E(V,x)+PV V H Hellmann-Feynman: x = E x H concave E(V,x) = min P ( H(P, x) PV )

16 Hohenberg-Kohn DFT H = 2 2 2m r 2 e i i 2 r i r j i i j V (r i ) E[V ] = min Ψ ˆK + Ŵ + ˆV Ψ Ψ = min [ Ψ ˆK + Ŵ Ψ + Ψ ] ρ(r)v (r)dr properties: consequences: E[V ] is convex (requires some work to demonstrate) ρ(r) = δe δv (r) (from Hellmann-Feynman) V (r) ρ(r) (1st HK theorem) F [ρ] =E V (r)ρ(r)dr is the Legendre transform of E [ ] E[V ] = min F [ρ]+ V (r)ρ(r)dr (2nd HK theorem) ρ

17 Hohenberg-Kohn DFT [ ] E[V ]=min ρ F [ρ]+ V (r)ρ(r)dr

18 exchange-correlation energy functionals local-density approximation (LDA, Kohn and Sham, 1965) ( ) E xc [ρ] = ɛ xc ρ(r) ρ(r)dr generalized-gradient approximation (GGA, Becke, Perdew, and Wang, 1986) E xc = ρ(r)ɛ GGA (ρ(r), ρ(r) ) dr DFT+U (Anisimov and others, early 90 s) meta-gga (Perdew and others 2003) E mgga = ρ(r) ( ɛ mgga ρ(r), ρ(r),τs (r) ) dr τ s (r) = 1 2 ψ i (r) 2 2 hybrid functionals (B3Lyp, PBE0, Becke 1993) i E hybr = αe x HF +(1 α)e x GGA + E c E DFT +U [ρ] =E DFT + Un(n 1)

19 KS equations from functional minimization E[{ψ}, R] = 2 2m v ψ v(r) 2 ψ v (r) r 2 dr + e 2 2 V (r, R)ρ(r)dr+ ρ(r)ρ(r ) r r drdr + E xc [ρ] ( ) E(R) =min E[{ψ}, R] {ψ} ψu(r)ψ v (r)dr = δ uv

20 KS equations from functional minimization E[{ψ}, R] = 2 2m v ψ v(r) 2 ψ v (r) r 2 dr + e 2 2 V (r, R)ρ(r)dr+ ρ(r)ρ(r ) r r drdr + E xc [ρ] ( ) E(R) =min E[{ψ}, R] {ψ} ψu(r)ψ v (r)dr = δ uv δe KS δψ v(r) = u Λ vu ψ u (r) ) ( 2 2m 2 + v KS [ρ](r) ψ v (r) =ɛ v ψ v (r)

21 solving the Kohn-Sham equations ψ v (r) = j c(j, v)ϕ j (r) ψ v (r) c(v, j) δe KS δψ v(r) = uv Λ vu ψ u (r) h KS [c](i, j)c(j, v) =ɛ v c(i, v) j ċ(i, v) = h KS [c](i, j)c(j, v)+ j Λ vu c(i, v) u

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