Wave function methods for the electronic Schrödinger equation

Transcription

1 Wave function methods for the electronic Schrödinger equation Zürich 2008

2 DFG Reseach Center Matheon: Mathematics in Key Technologies A7: Numerical Discretization Methods in Quantum Chemistry DFG Priority Program: Modern and Universal First-Principle-Methods for Many-Electron Systems in Chemistry and Physics DFG Priority Program: Extraction of Essential Information from Complex Systems

3 Basic model - electronic Schrödinger equation Electronic Schrödinger equation N nonrelativistic electrons + Born Oppenheimer approximation HΨ = EΨ The Hamilton operator H = 1 2 i i N i K ν=1 Z ν x i a ν N i,j 1 x i x j acts on anti-symmetric wave functions Ψ H 1 (R 3 {± 1 2 })N, Ψ(x 1, s 1,..., x N, s N ) R, x i = (x i, s i ) R 3 {± 1 2 }.

4 Goals To go beyond the accuracy of Density Functional Models one tries to approximate the eigenfuntion Ψ, of HΨ = EΨ. high precision: Correlation energy. E corr := E HF E 0 dynamic correlation: (closed shell R) HF is a good approximation, but the resolution of the electron- electron cusp hempers good convergence static correlation: a single Slater deteminant cannot provide a sufficiently good reference approximation, due e.g. open shells, requires multi configurational models excited states degenerate ground states

5 CI Configuration Interaction Method Approximation space for (spin) orbitals (x j, s j ) ϕ(x j, s j ) X h := span {ϕ i : i = 1,..., N }, ϕ i, ϕ j = δ i,j E.g. Canonical orbitals ϕ i, i = 1,..., N are eigenfunctions of the discretized Fock operator F := F h = N k=1 F k : V FCI V FCI Fϕ i λ i ϕ i, φ h = 0 φ h X h Full CI (for benchmark computations N = 18) is a Galerkin method w.r.t. the subspace V FCI = N X h = span{ψ SL = Ψ[ν 1,..ν N ] = 1 N! det(ϕ ν i (x j, s j )) N i,j=1, } i=1

6 CI Configuration Interaction Method N (discrete) eigenfunction ϕ i, Fϕ i λ i ϕ i, φ h = 0 φ h X h The first N eigenfunctions ϕ i are called occupied orbitals the others are called unoccupied orbitals (traditionally) ϕ 1,..., ϕ N, ϕ N+1,..., ϕ N Galerkin ansatz: Ψ = c 0 Ψ 0 + ν J c νψ ν H = ( Ψ ν, HΨ ν ), Hc = Ec, dim V h = N N Theorem (Brillouins theorem) Let Ψ 0 = Ψ HF and Ψ a i = Ψ S then Ψ a i, HΨ 0 = 0

7 Configuration Interaction Method Assumption: For any ϕ H 1 there exist ϕ h X h such that ϕ h ϕ H 1 0, if h 0 (roughly: lim h 0 X h = H 1 (R 3 {± 1 2 })) Theorem Let E 0 be a single eigenvalue and HΨ = E 0 Ψ and E 0,h, Ψ h V h V FCI be the Galerkin solution, then for h < h 0 hold Ψ Ψ h V c inf φ h V h Ψ φ h V E 0,h E 0 C inf φ h V h Ψ φ h 2 V. Since dim V h = O(N N ), (curse of dimension), the full CI method is infeasible for large N or N!!!!

8 Multi-Configuration Self Consistent Field Method - MCSCF Method Optimization of the orbital basis functions ϕ i, N M << N Y h := span {ϕ i : i = 1,..., M} X h := span {ϕ i : i = 1,..., N }, N V MCSCF = Y h = span{ψ SL = Ψ[ν 1,..ν N ] : ϕ ν Y h } i=1 For Ψ = c 0 Ψ 0 + ν J c νψ ν and E = Ψ, HΨ = J MCSCF (c, Φ) E 0 min{j MCSCF (c, Φ ) : c = 1, Φ S M } unknowns c, Φ := (ϕ i ) i=1,...,m S V M on Stiefel manifold S Exisitence results: Friesecke, Lewin(03)

9 Conclusions MCSCF can be solved with 2nd order optimization methods, e.g. trust region Newton methods Instead of FCI one can use a Complete Active Space MCSCF is for multi-configurational problems, ( where RHF is rather bad) static correlation Due to the e-e cusp, M ɛ 1/2 is large! FCI and MCSCF methods are scaling exponentially with N O( M N ) = O(e N )!!! restriction to single-double excitations etc. are not size consistent (see below)

10 Rayleigh-Schrödinger perturbation theory H = H 0 + U, Solution for H 0 is known H 0 ψ 0 = E 0 ψ 0. Adiabatic perturbation H(λ) = H 0 + λu. Assumption: E 0 is a simple eigenvalue of H 0 and λ E(λ), λ ψ(λ) are analytic in {λ C : λ 1 + ɛ} for some ɛ > 0. E(λ) = E k λ k, ψ(λ) = ψ k λ k k=0 k=0

11 Rayleigh-Schrödinger perturbation theory Intermediate normalization ψ = ψ(1) ψ, ψ 0 = 1 ψ l, ψ 0 = 0, l > 0. The projection P 0 onto the orthogonal complement of ψ 0 is defined by P 0 u = I ψ 0, u ψ 0. Inserting the ansatz into the Schrödinger equation k (H 0 E 0 )ψ k λ k = λuψ k λ k + ( E l ψ k l )λ k. k=0 k=0 k=1 l=1 for all λ 1 + ɛ, + sorting w.r.t. to powers of λ gives k (H 0 E 0 )ψ k = Uψ k 1 + E l ψ k l, k = 1,.... l=1

12 Rayleigh-Schrödinger perturbation theory Testing against ψ 0 gives 0 = ψ 0, (H 0 E 0 )ψ k = ψ 0, ( Uψ k 1 + ψ 0, Uψ k 1 + E k. k E l ψ k l ) = l=1 E k = ψ 0, Uψ k 1 k ψ k = P 0 (H 0 E 0 ) 1 P 0 ( Uψ k 1 + E l ψ k l ). For the computation of the energy contribution E k we need all ψ l up to l = k 1. l=1

13 Møller-Plesset Perturbation theory In the sequel we restrict ourselves finite dimensional subspace to V h = V FCI! N H = F + U = F i + ( 1 r i r j N (J i K i )) i=1 i<j i=1 where F is the Fock operator and U is called the fluctuation potential. Proposition Let Ψ = Ψ SL = Ψ[ν 1,..., ν N ] be a Slater determinant of canonical orbitals, then N N FΨ = εψ, ε = λ νi, E 0 = λ i. i=1 i=1

14 Let Ψ 0 be the HF wave function, the first order energy contribution is E 1 = Ψ 0, UΨ 0 = 1 N ij ij 2 i,j=1 Hartree-Fock energy E MP1 = E 0 + E 1 = E HF. Ψ 1 = P 0 (F E 0 ) 1 P 0 UΨ 0 and the second order contribution to the energy MP2: E 2 = Ψ 0, UΨ 1 = Ψ 0, UP 0 (F E 0 ) 1 P 0 UΨ 0, E MP2 = E 0 + E 1 + E 2 = E HF + E 2. Higher order contributions can be computed, e.g. MP3, MP4 etc. too. The convergence of the expansion is not guaranteed.

15 Second quantization Second quantization: annihilation operators: a j Ψ[j, 1,..., N] = Ψ[1,..., N] and = 0 if j not apparent in Ψ[...]. sign-normalization: j appears in the first place in Ψ[j, 1,..., N]. The adjoint of a b is a creation operator a b a b Ψ[1,..., N] = Ψ[b, 1,..., N] = ( 1)N Ψ[1,..., N, b] Lemma a k a l = a l a k, a k a l = a l a k, a k a l + a l a k = δ k.l

16 Excitation operators Single excitation operator e.g. X k 1 ( 1) p Ψ k 1 = Ψ[k, 2,..., N] = X k 1 Ψ 0 = X k j Ψ[1,...,..., N] = a k a 1Ψ 0 higher excitation operator X µ := X b 1,...,b k l 1,...,l k = k i=1 X b i l i, 1 l i < l i+1 N, N < b i < b i+1. A CI solution Ψ = c 0 Ψ 0 + µ J c µψ µ can be written by Ψ = c 0 + c µ X µ Ψ 0 = (I + T )Ψ 0 T = T 1 + T 2 + T µ J

17 Coupled Cluster Method - Exponential-ansatz Theorem (S. 06) Let Ψ 0 be a reference Slater determinant, e.g. Ψ 0 = Ψ HF and Ψ V FCI, or V, satisfying Ψ, Ψ 0 = 1 intermediate normalization. Then there exists an excitation operator (T 1 - single-, T 2 - double-,... excitation operators) N T = T i = t µ X µ i=1 µ J such that Ψ = e T Ψ 0

18 Baker-Campell-Hausdorff expansion We recall the Baker-Campell-Hausdorff formula e T Ae T = A + [A, T ] + 1 2! [[A, T ], T ] + 1 [[[A, T ], T ], T ] +... = 3! A + k=1 1 k! [A, T ] k. For Ψ V h the above series terminates, exercise** e T He T = H+[H, T ]+ 1 2! [[H, T ], T ]+ 1 3! [[[H, T ], T ], T ]+ 1 4! [H, T ] 4 e.g. for a single particle operator e.g. F there holds e T Fe T = F + [F, T ] + [[F, T ], T ]

19 Coupled Cluster energy Ψ V or V FCI = V FCI = span{ψ ν : ν J } Φ, (H E 0 )Ψ = 0 Φ V, V FCI due to Slater Condon rules and normalization Ψ, Ψ 0 = 1 E = Ψ 0, HΨ = E Ψ 0, H(I + T T )Ψ 0 = Ψ 0, H(I + T 1 +T 2 + T T )Ψ 0 Proposition E = Ψ 0, HΨ = Ψ 0, H(I + T 1 +T T 1 2 )Ψ 0

20 Projected Coupled Cluster Method amplitude equations Ψ µ, e T He T Ψ 0 = Ψ µ, e T HΨ = E Ψ µ, e T Ψ = 0 µ J The Projected Coupled Cluster Method consists in the ansatz l T = T k = t µ X µ µ J h k=1, 0 µ J h J, i.e. Ψ µ V h V FCI e.g. CCSD T = T 1 + T 2 = T (t) satisfying 0 = Ψ µ, e T He T Ψ 0 =: f µ (t), t = (t ν ) ν Jh, µ, ν J h These are L = J h << dimv FCI nonlinear equations for L unknown excitation amplitudes t µ.

21 Iteration method to solve CC amplitude equations Quasi-Newton method to solve CC amplitude equations We decompose the (discretized) Hamiltonian H = F + U, F - Fock operator, U - fluctuation potential. There holds [F, X µ ] = [F, X a 1,...,a k l 1,...,l k ] = ( k (λ aj λ lj ))X µ =: ɛ µ X µ. j=1 and [[F, X µ ], X µ ] = 0 together with (Bach-Lieb-Solojev) ɛ µ λ N+1 λ N > 0

22 Iteration method to solve CC amplitude equations The amplitude function t f(t) = (f µ (t)) µ Jh must be zero f µ (t) = Ψ µ, e T He T Ψ 0 = Ψ µ, [F, X µ ]Ψ 0 + Ψ µ, [U, T ]Ψ 0 = 0. The nonlinear amplitude equation f(t) = 0 is solved by Algorithm (quasi Newton-scheme) 1 Choose t 0, e.g. t 0 = 0. 2 Compute t n+1 = t n A 1 f(t n ), where A = diag (ɛ µ ) µ J > 0. The Coupled Cluster Method is size consistent!: H AB = H A +H B, e (T A+T B ) (H A +H B )e Ta+T B = e T A H A e T A +e T B H B e T B

23 Analysis of the Coupled Cluster Method We consider the projected CC as an approximation of the full CI solution! If h 0, then M and max ɛ µ! We need estimates uniformly w.r.t. h, N Definition Let M := dim V FCI dimensional parameter space V = R M equipped with the norm t 2 V := ɛ µ t µ Ψ µ 2 L 2 ((R 3 {± 1 2 })N ) = ɛ µ t µ 2 µ J µ J

24 Analysis of the Coupled Cluster Method Lemma (S.06) There holds t V T Ψ 0 H 1 ((R 3 {± 1 2 })N ) T Ψ 0 V. Lemma (S.06) For t l 2 (J ), the operator T := ν J t νx ν maps N T Ψ L2 t l2 Ψ L 2 Ψ V FCI L 2 (R 3 {± 1 2 }) i=1

25 Analysis of the Coupled Cluster Method Lemma (S.06) For t V, the operator T := ν J t νx ν maps T Ψ H 1 t V Ψ H 1 Ψ V FCI Corollary (S06) The function f : V V is differentiable at t V with the Frechet derivative f [t] : V V given by (f [t]) ν,µ = Ψ ν, e T [H, X µ ]e T Ψ 0 = ɛ ν δ ν,µ + Ψ ν, e T [U, X µ ]e T Ψ 0 All Frechet derivatives t f (k) [t] : V V, are Lipschitz continuous. In particular f (5) 0.

26 Convergence of the Coupled Cluster Method A function f : is called strictly monotone at t if f(t) f(t ), (t t γ t t 2 V for some γ > 0 and all t t V < δ. Theorem (S.06 (a priori estimate)) If f(t) = 0 and f is strictly monotone at t, then t h, resp. Ψ h = e T hψ 0 satisfy t t h V inf t v h V. v R J h P µ J v Ψ Ψ h H 1 inf Ψ e h µx µ Ψ 0 H 1. v R L

27 Convergence of the CC Method - Duality estimates Energy functional J(t) := E(t) := Ψ 0, H(1 + T T 1 2 )Ψ 0 where t solves the amplitude equations (f(t)) ν = Ψ ν, e T He T Ψ 0 = 0, ν J. Let us further consider the Lagrange functional L(t, a) := J(t) f(t), a, t V, a V. and its stationary points L t [t, a](r, b) := J [t]r f [t]r, a = 0, for all a V. and L a [t, a](r, b) = f(t), b = 0 for all b V.

28 Duality estimates for the Coupled Cluster Method The dual solution a satisfies f [t] a = (J [t]) V. Its Galerkin approximation is given by a h = (a µ ) µ Jh V h, f [t h ] a h, v h = (J [t h ]), v h, v h V h. The discrete primal solution t h = (t ν ) ν Jh V h solves f[t h ], b h = 0 for all b h J h. We define the corresponding residual r, r V (f µ ), µ J h (r(t h )) µ = 0, µ J h together Reinhold withschneider, the dual MATHEON residual TU Berlin

29 Duality estimates for the Coupled Cluster Method The derivatives J [t] and f [t] can be explicitly computed Ψ 0, HX µ e T Ψ 0 µ J 1 single (J [t]) = Ψ 0, UΨ µ, µ J 2 double 0, otherwise where T = t ν X ν, and (f [t]) µ,ν = ɛ µ δ µ,ν + Ψ µ, e T [U, X ν ]e T Ψ 0, µ, ν J. Lemma (dual weighted residual, Rannacher) Let x := (t, a) V V, x h := (t h, a h ) V h V h and e h = x x h. L (x) = 0, L (x h )y h = 0 y h V h V h Then L(x) L(x h ) = L [x h ](x y h ) + R 3, y h V h V h

30 Convergence of the Coupled Cluster Energies Theorem (S. 06 a priori estimate) The error in the energy J(t) J(t h ) can be estimated by E E h t t h V a a h V + ( t t h V ) 2 inf t u h V inf a b h V + u h V h b h V +( inf u h V h t u h V ) 2. E E h t t h V a a h V inf t u h V inf a b h V u h V h b h V All constants involved above are uniform w.r.t. N.

31 Conclusions Improvement by adding the e-e-cusp singularity explicitely, r 1,2, f 1,2 methods (Kutzelnigg-Klopper... ) CCSD and CCSD(T) are standard CCSDT; CCSDTQ etc. only for extremely accurate computations CC is the most powerful tool for computing dynamical correlation not good for multi-configurational problems, ( where RHF is rather bad) How to do Multi Reference Coupled Cluster???