Direct Minimization in Density Functional Theory

Transcription

1 Direct Minimization in Density Functional Theory FOCM Hongkong 2008

2 Partners joint paper with: J. Blauert, T. Rohwedder (TU Berlin), A. Neelov (U Basel) joint EU NEST project BigDFT together with Dr. Thierry Deutsch, Co-ordinator, Atomistic Simulation group at CEA-Grenoble, Prof. Stefan Goedecker, Computational Physics Group at the Basel University, Switzerland; Prof. Xavier Gonze, Uni. de Phy.-Ch.et de Phys. des Mat., Univ. Catholique de Louvain; Joint project with H.J. Flad (TUB) and W. Hackbusch (MPI Leipzig) DFG Priority Program: Modern and Universal First-Principle-Methods for Many-Electron Systems in Chemistry and Physics

3 Basic model - electronic Schrödinger equation Electronic Schrödinger equation N nonrelativistic electrons + Born Oppenheimer approximation HΨ = EΨ The Hamilton operator H = 2 i i i K ν= Z ν x i a ν + 2 x i x j i,j acts on anti-symmetric wave functions Ψ H (R 3 {± 2 })N, Ψ(x, s,..., x N, s N ), x i = (x i, s i ) R 3 {± 2 }.

4 Facts to know Output: ground-state energy E 0 = min Ψ,Ψ = HΨ, Ψ, Ψ = argmin Ψ,Ψ = HΨ, Ψ Kato,... ( ), energy space: Ψ H ((R 3 {± 2 })N ). HVZ-Theorem (..60..), existence: < E 0 < inf σ ess (H) Agmon ( ), decay: Ψ(x) = O(e a x ) if x. Kato (957), Thomas - von Ostenhoff et al. (98), cusp-singulariti linear Problem, but extremely high-dimensional + anti-symmetry constraints + singularities. traditional approximation methods (FEM, Fourier series, polynomials, MRA etc.): approximation error in R : n s, s- regularity, R 3N : n s 3N

5 Facts to know Output: ground-state energy E 0 = min Ψ,Ψ = HΨ, Ψ, Ψ = argmin Ψ,Ψ = HΨ, Ψ Kato,... ( ), energy space: Ψ H ((R 3 {± 2 })N ). HVZ-Theorem (..60..), existence: < E 0 < inf σ ess (H) Agmon ( ), decay: Ψ(x) = O(e a x ) if x. Kato (957), Thomas - von Ostenhoff et al. (98), cusp-singulariti linear Problem, but extremely high-dimensional + anti-symmetry constraints + singularities. traditional approximation methods (FEM, Fourier series, polynomials, MRA etc.): approximation error in R : n s, s- regularity, R 3N : n s 3N

6 Tensor product approximation Approximation by sums anti-symmetric tensor products (separation of variables): Ψ = k= c kψ k Ψ k (x, s ;... ; x N, s N ) = ϕ,k... ϕ N,k = N! det(ϕ i,k (x j, s j )), ϕ i,k, with ϕ i, ϕ j = s=± 2 R 3 ϕ i (x, s)ϕ j (x, s)dx =: δ i,j. Slater determinant: Ψ k is an (anti-symmetric) product of N orthonormal functions ϕ i, called spin orbitals ϕ i : R 3 {± 2 } C, i =,..., N, For most application it is sufficient to consider real valued functions

7 Slater Condon rules Abbreviations: a) Single particle operators A = h = N i= h i = N i= ( 2 i + M j= Z j x i R j ) = N i= ( 2 i + V core (x i )), h i = h j, i h j := ϕ i, h j ϕ j a)two particle operators G = N i N j>i x i x j, H = h + G a, b i, j := s,s =± 2 ϕ a (x, s)ϕ b (x, s )ϕ i (x, s)ϕ j (x, s ) x x dxdx

8 Slater Condon rules Abbreviations: a) Single particle operators A = h = N i= h i = N i= ( 2 i + M j= Z j x i R j ) = N i= ( 2 i + V core (x i )), h i = h j, i h j := ϕ i, h j ϕ j a)two particle operators G = N i N j>i x i x j, H = h + G a, b i, j := s,s =± 2 ϕ a (x, s)ϕ b (x, s )ϕ i (x, s)ϕ j (x, s ) x x dxdx

9 Slater-Condon Rules Single particle operators: ) Ψ = Ψ 2 Ψ = Ψ[..., ν i, ν j,...] Ψ 2 = Ψ[..., ν i, ν j,...] 2) Ψ 2 = T a j Ψ Ψ = Ψ[..., ν i, ν j,...] Ψ 2 = Ψ[..., ν i, ν a,...] 3) Ψ = T a,b i,j Ψ 2 Ψ = Ψ[..., ν i, ν j,...] Ψ, hψ = N l= Ψ 2, hψ = a or higher excitations Ψ 2 = Ψ[..., ν a, ν b,...] Ψ 2, hψ = 0 Proof: Leibniz formula + 2 Reinhold Schneider, MATHEON TU Berlin

10 Slater-Condon Rules Single particle operators: ) Ψ = Ψ 2 Ψ = Ψ[..., ν i, ν j,...] Ψ 2 = Ψ[..., ν i, ν j,...] 2) Ψ 2 = T a j Ψ Ψ = Ψ[..., ν i, ν j,...] Ψ 2 = Ψ[..., ν i, ν a,...] 3) Ψ = T a,b i,j Ψ 2 Ψ = Ψ[..., ν i, ν j,...] Ψ, hψ = N l= Ψ 2, hψ = a or higher excitations Ψ 2 = Ψ[..., ν a, ν b,...] Ψ 2, hψ = 0 Proof: Leibniz formula + 2 Reinhold Schneider, MATHEON TU Berlin

11 Slater-Condon Rules Single particle operators: ) Ψ = Ψ 2 Ψ = Ψ[..., ν i, ν j,...] Ψ 2 = Ψ[..., ν i, ν j,...] 2) Ψ 2 = T a j Ψ Ψ = Ψ[..., ν i, ν j,...] Ψ 2 = Ψ[..., ν i, ν a,...] 3) Ψ = T a,b i,j Ψ 2 Ψ = Ψ[..., ν i, ν j,...] Ψ, hψ = N l= Ψ 2, hψ = a or higher excitations Ψ 2 = Ψ[..., ν a, ν b,...] Ψ 2, hψ = 0 Proof: Leibniz formula + 2 Reinhold Schneider, MATHEON TU Berlin

12 Hartree-Fock approximation Simplest approximation by one Slater-determinant Ψ Ψ SL = Ψ 0 Hartree-Fock Model Minimize E FH = Ψ 0, HΨ 0 = J HF (Φ) = N i= 2 ϕ i, ϕ i + V core (x)ϕ i, ϕ i + 2 w.r.t. the constraint conditions ϕ i, ϕ j = δ i,j, i, j =,..., N. Density matrix: ρ(x, s, y, s ) := N i= ϕ i(x, s)ϕ i (y, s ), n(x) := s= pm ρ(x,s,x,s) is the electron density 2 With the Hartree potential V H and the exchange energy term W V H (x) = 4πn(x, x), Wu(x) = s ρ(x, s, y, s ) R 3 x y u(y, s )dy

13 Hartree-Fock approximation Simplest approximation by one Slater-determinant Ψ Ψ SL = Ψ 0 Hartree-Fock Model Minimize E FH = Ψ 0, HΨ 0 = J HF (Φ) = N i= 2 ϕ i, ϕ i + V core (x)ϕ i, ϕ i + 2 w.r.t. the constraint conditions ϕ i, ϕ j = δ i,j, i, j =,..., N. Density matrix: ρ(x, s, y, s ) := N i= ϕ i(x, s)ϕ i (y, s ), n(x) := s= pm ρ(x,s,x,s) is the electron density 2 With the Hartree potential V H and the exchange energy term W V H (x) = 4πn(x, x), Wu(x) = s ρ(x, s, y, s ) R 3 x y u(y, s )dy

14 Hartree-Fock approximation Simplest approximation by one Slater-determinant Ψ Ψ SL = Ψ 0 Hartree-Fock Model Minimize E FH = Ψ 0, HΨ 0 = J HF (Φ) = N i= 2 ϕ i, ϕ i + V core (x)ϕ i, ϕ i + 2 w.r.t. the constraint conditions ϕ i, ϕ j = δ i,j, i, j =,..., N. Density matrix: ρ(x, s, y, s ) := N i= ϕ i(x, s)ϕ i (y, s ), n(x) := s= pm ρ(x,s,x,s) is the electron density 2 With the Hartree potential V H and the exchange energy term W V H (x) = 4πn(x, x), Wu(x) = s ρ(x, s, y, s ) R 3 x y u(y, s )dy

15 Hartree-Fock approximation Simplest approximation by one Slater-determinant Ψ Ψ SL = Ψ 0 Hartree-Fock Model Minimize E FH = Ψ 0, HΨ 0 = J HF (Φ) = N i= 2 ϕ i, ϕ i + V core (x)ϕ i, ϕ i + 2 w.r.t. the constraint conditions ϕ i, ϕ j = δ i,j, i, j =,..., N. Density matrix: ρ(x, s, y, s ) := N i= ϕ i(x, s)ϕ i (y, s ), n(x) := s= pm ρ(x,s,x,s) is the electron density 2 With the Hartree potential V H and the exchange energy term W V H (x) = 4πn(x, x), Wu(x) = s ρ(x, s, y, s ) R 3 x y u(y, s )dy

16 Hartree-Fock- (HF) Approximation Ground state energy E 0 = min{ Hψ, ψ : ψ, ψ = } approximation of ψ by a single Slater determinant Ψ SL Φ (x, s,..., x N, s N ) := N! det(φ i(x j, s j )) Closed Shell Restricted HF (RHF): N := N 2 electron pairs minimization of the functional J HF (Φ) D E Φ J HF (Φ) := HΨ SL Φ, ΨSL Φ = + NX Z j= R 3 " φj (y) 2 φ i (x) 2 x y w.r.t. orthogonality constraints NX Z ` φi (x) 2 + 2V core(x) φ i (x) 2 + i= # φ i (x)φ i (y) φ j (x)φ j (y) dy dx 2 x y Φ = (φ i ) N i= H (R 3 ) N and φ i, φ j = δi,j

17 Hartree-Fock approximation Simplest approximation by one Slater-determinant Ψ Ψ SL = Ψ 0 Hartree-Fock Model Minimize E FH = Ψ 0, HΨ 0 = J HF (Φ) = N i= 2 ϕ i, ϕ i + V core (x)ϕ i, ϕ i + 2 w.r.t. the constraint conditions ϕ i, ϕ j = δ i,j, i, j =,..., N. Density matrix: ρ(x, s, y, s ) := N i= ϕ i(x, s)ϕ i (y, s ), n(x) := s=± ρ(x, s, x, s) is the electron density 2 With the Hartree potential V H and the exchange energy term W V H (x) = 4πn(x, x), Wu(x) = ρ(x, s, y, s ) u(y, s )dy s R 3 x y

18 Hartree-Fock approximation Simplest approximation by one Slater-determinant Ψ Ψ SL = Ψ 0 Hartree-Fock Model Minimize E FH = Ψ 0, HΨ 0 = J HF (Φ) = N i= 2 ϕ i, ϕ i + V core (x)ϕ i, ϕ i + 2 w.r.t. the constraint conditions ϕ i, ϕ j = δ i,j, i, j =,..., N. Density matrix: ρ(x, s, y, s ) := N i= ϕ i(x, s)ϕ i (y, s ), n(x) := s=± ρ(x, s, x, s) is the electron density 2 With the Hartree potential V H and the exchange energy term W V H (x) = 4πn(x, x), Wu(x) = ρ(x, s, y, s ) u(y, s )dy s R 3 x y

19 Hartree-Fock approximation Simplest approximation by one Slater-determinant Ψ Ψ SL = Ψ 0 Hartree-Fock Model Minimize E FH = Ψ 0, HΨ 0 = J HF (Φ) = N i= 2 ϕ i, ϕ i + V core (x)ϕ i, ϕ i + 2 w.r.t. the constraint conditions ϕ i, ϕ j = δ i,j, i, j =,..., N. Density matrix: ρ(x, s, y, s ) := N i= ϕ i(x, s)ϕ i (y, s ), n(x) := s=± ρ(x, s, x, s) is the electron density 2 With the Hartree potential V H and the exchange energy term W V H (x) = 4πn(x, x), Wu(x) = ρ(x, s, y, s ) u(y, s )dy s R 3 x y

20 Hartree-Fock approximation Simplest approximation by one Slater-determinant Ψ Ψ SL = Ψ 0 Hartree-Fock Model Minimize E FH = Ψ 0, HΨ 0 = J HF (Φ) = N i= 2 ϕ i, ϕ i + V core (x)ϕ i, ϕ i + 2 w.r.t. the constraint conditions ϕ i, ϕ j = δ i,j, i, j =,..., N. Density matrix: ρ(x, s, y, s ) := N i= ϕ i(x, s)ϕ i (y, s ), n(x) := s=± ρ(x, s, x, s) is the electron density 2 With the Hartree potential V H and the exchange energy term W V H (x) = 4πn(x, x), Wu(x) = ρ(x, s, y, s ) u(y, s )dy s R 3 x y

21 F F Φ = 2 + Hartree-Fock approximation Theorem (Lieb-Simon 77, P.L. Lions 89), e.g. for closed shell RHF, we write N for the number of electron pairs (N := N 2 ) Let Z N, then there exist a minimizer Φ = (ϕ,..., ϕ N ) [H (R 3 )] N, E There exists an unitary matrix U such that the functions ( ϕ i ) i=,.. with the Hamilton-Fock operator F The N lowest eigenvalues of F Φ satisfy λ λ 2... λ N < 0.

22 Kohn-Sham model Theorem (Kohn-Hohenberg) The ground state energy E 0 is a functional of the electron density n. 2 E 0 E KS = inf{j KS (Φ) : φ i, φ j = δ ij } minimization of the Kohn Sham energy functional J KS (Φ) J KS (Φ) = ( Z 2 NX Z φ i 2 + nv core + Z Z 2 i= n(x)n(y) x y dx dy E xc(n) φ i H (R 3 ), electron density n(x) := N i= φ i(x) 2 E xc (n) exchange-correlation-energy (not known explicitly) e.g. LDA (local density approximation) n E xc (n) : R R )

23 Effective single particle model The Hartree-Fock model is a prototype for improved approximations e.g. Kohn-Sham equation etc. Kohn -Hohenberg theorem: Ground state energy depends only on the electron density n(x) J KS (Φ) = N i= 2 ϕ i, ϕ i + V core ϕ i, ϕ i + 2 V Hϕ i, ϕ i + α V XC ϕ i, ϕ i Kohn Sham equations, e.g. closed shell: each orbital is for a pair of electron with opposite spin H Φ ϕ i = 2 ϕ i + V core ϕ i + V H ϕ i + αv XC ϕ i β 2 Wϕ i = λ i ϕ i The exchange correlation energy term in a simple LDA Model P(n(x))

24 Effective single particle model The Hartree-Fock model is a prototype for improved approximations e.g. Kohn-Sham equation etc. Kohn -Hohenberg theorem: Ground state energy depends only on the electron density n(x) J KS (Φ) = N i= 2 ϕ i, ϕ i + V core ϕ i, ϕ i + 2 V Hϕ i, ϕ i + α V XC ϕ i, ϕ i Kohn Sham equations, e.g. closed shell: each orbital is for a pair of electron with opposite spin H Φ ϕ i = 2 ϕ i + V core ϕ i + V H ϕ i + αv XC ϕ i β 2 Wϕ i = λ i ϕ i The exchange correlation energy term in a simple LDA Model P(n(x))

25 Effective single particle model The Hartree-Fock model is a prototype for improved approximations e.g. Kohn-Sham equation etc. Kohn -Hohenberg theorem: Ground state energy depends only on the electron density n(x) J KS (Φ) = N i= 2 ϕ i, ϕ i + V core ϕ i, ϕ i + 2 V Hϕ i, ϕ i + α V XC ϕ i, ϕ i Kohn Sham equations, e.g. closed shell: each orbital is for a pair of electron with opposite spin H Φ ϕ i = 2 ϕ i + V core ϕ i + V H ϕ i + αv XC ϕ i β 2 Wϕ i = λ i ϕ i The exchange correlation energy term in a simple LDA Model P(n(x))

26 Post-Hartree-Fock Methods Configuration Interaction (CI): Galerkin methods in V SD,... V CI. Møller Plesset Perturbation theory: H λ = F + λ(h F), λ =, F denotes the Fock operator Multi Configuration Self Consistent Field Method (MCSFC): Optimizing the energy within an approximation by K Slater-determinants Coupled Cluster Method (CC): Ansatz Ψ = e T Ψ 0. (CCSD: T = T + T 2 ) Reduced Density Matrix Minimization Semidefinite programing