I ) Direct minimization in Hartree Fock and DFT - II) Converence o

Transcription

1 I ) Direct minimization in Hartree Fock and DFT - II) Converence of Coupled Cluster Computational methods in many electron systems : MP2 Workshop Göteborg 2009

2 Ab initio computation Reliable computation of atomisitic molecular phenomena will play an important role in modern material science, chemistry, (molecular biology) Ab initio computation is based on first principles of quantum mechanics

3 Electronic Structure Calculation Electronic Schrödinger equation N stationary 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, s i ) R 3 {± 1 2 }.

4 Goals Output: ground-state energy E 0 = min Ψ,Ψ =1 HΨ, Ψ, Ψ = argmin Ψ,Ψ =1 HΨ, Ψ most quantities for molecules (chemistry) and crystals (solid state physics) can be derived from E 0, e.g. atomic forces, molecular geometry, bonding and ionization energies etc. these quantities are (small) differences E 0,a E 0,b excited states are required for opto-electronic effects accuracy is limited due to neglecting relativistic and non-born-openheimer effects.. is beyond the present presentation

5 Basic Problem - Curse of dimensions linear eigenvalue problem, but extremely high-dimensional + anti-symmetry constraints + lack of regularity. traditional approximation methods (FEM, Fourier series, polynomials, MRA etc.): approximation error in R 1 : n s, s- regularity, R 3N : n s 3N, (s < 5 2 ) with n DOFs For large systems N >> 1 ( N > 1) the electronic Schrödinger equation seems to be intractable! But 70 years of impressive progress has been awarded by the Nobel price 1998 in Chemistry: Kohn, Pople

6 For present applications it is sufficient to consider real valued Tensor product approx. - separation of variables Approximation by sums of anti-symmetric tensor products: Ψ = c k Ψ k k=1 Ψ k (x 1, s 1 ;... ; x N, s N ) = ϕ 1,k... ϕ N,k = 1 N! det(ϕ i,k(x j, s j )) with ϕ i,k {ϕ j : j = 1,...}, w.l.o. generality ϕ i, ϕ j = ϕ i (x, s)ϕ j (x, s)dx = δ i,j. R 3 s=± 1 2 A Slater determinant: Ψ k is an (anti-symmetric) product of N orthonormal functions ϕ i, called spin orbital functions ϕ i : R 3 {± 1 } R, i = 1,..., N, 2

7 Spin functions and spatial orbitals Example of Slater determinant (N = 2) Ψ[φ 1, φ 2 ]((x, s 1 ; y, s 2 ) = 1 2 (φ 1 (x, s 1 )φ 2 (y, s 2 ) φ 2 (x, s 1 )φ 1 (y, s 2 )). spin functions χ and spatial orbital functions φ: ϕ(x, s) = φ α (x)χ α (s) + φ β (x)χ β (s) with spin functions χ α (+ 1 2 ) = 1, χ α( 1 2 ) = 0, χ β(s) = 1 χ α (s) Closed shell RHF (Restricted Hartree Fock) : φ α,i = φ β,i = φ i, i = 1,..., N = N 2.

8 Hartree-Fock- (HF) Approximation Ground state energy E 0 = min{ Hψ, ψ : ψ, ψ = 1} approximation of ψ by a single Slater determinant Ψ SL Φ (x 1, s 1,..., x N, s N ) := 1 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=1 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=1 # 1 1 φ i (x)φ i (y) φ j (x)φ j (y) dy dxa 2 x y Φ = (φ i ) N i=1 ( H 1 (R 3 ) ) N and φi, φ j = δi,j

9 Kohn-Sham model Theorem (Kohn-Hohenberg) The ground state energy E 0 is a functional of the electron density n. 1 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 1 2 NX Z φ i 2 + nv core + 1 Z Z 2 i=1 n(x)n(y) x y dx dy E xc(n) φ i H 1 (R 3 ), electron density n(x) := N i=1 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 )

10 Φ := (φ 1,..., φ N ) (H 1 (R 3 )) N = V N = V Gelfand triple V := H 1 (R 3 ) L 2 (R 3 ) H 1 (R 3 ) = V Φ T Ψ := ( φ i, ψ j ) i,j R N N scalar product Φ, Ψ := tr Φ T Ψ = N i=1 φ i, ψ i R AΦ := (Aφ 1,..., Aφ N ), A : V V Simplified Problem: minimize J SCF (Φ) := N Aφ i, φ i = tr Φ T AΦ = Φ, AΦ i=1 w.r.t. to orthogonality constraints Φ T Φ = I. I.e. finding the invariant subspace for the first N eigenfunctions.

11 Geometry of the admissible set Definition (Stiefel and Grassman manifolds) Stiefel manifold V V,N := V := {Φ = (φ i ) N i=1 φ i V, φ i, φ j = δ i,j } Grassmann manifold is a quotient manifold G V,N := G := V V,N /, Φ Φ Φ = ΦU, U U(N) (identify ONB spanning the same subspace span Φ) Density matrix operator projects onto span Φ := span{φ i }, N D Φ := φ i, φ i i=1 There is a one-to-one correspondence between [Φ] G D Φ (density matrix operator)

12 Tangent space Theorem (Edelman, Arias, Smith (98); Blauert, Neelov, Rohwedder, S. (08)) tangent space T [Φ] G = {δψ V N (δψ) T Φ = 0 R N N } T [Φ] G = {(δψ i ) N i=1 : δψ i V, δψ i span {φ i : i = 1,..., N}} (I D Φ ) : V N T [Φ] G, is an orthogonal projection onto the tangent space T [Φ] G tangent space T Φ S = T [Φ] G + {ΦA : A T = A} = {Θ V N : Θ T Φ = Φ T Θ }

13 Definition (Gradient J KS (Ψ) = F KS Ψ Ψ ) Kohn-Sham Fock operator F KS n = F KS Φ : V V defined by FΦ KSϕ(x)= 1 2 ϕ(x) + Vcore(x)ϕ(x)+ R n(y) dyϕ(x) + vxc(n)(x)ϕ(x) x y Unitary invariance: J KS (Φ) = J KS (ΦU) Theorem (Necessary 12t order conditions) If [Ψ] = argmin {J (Φ) : [Φ] G} V N (V N h ) then F [Ψ] Ψ, δφ = 0 δφ T [Ψ] G V N (V N h ) F[Ψ] KS ψ(k) i N j=1 (I D Ψ )F [Ψ] Ψ, δφ = 0 δφ V N (V N h ) ψ (k) j λ ji = 0 i = 1,..., N SCF-iteration Cances -LeBris (01)

14 Projected Preconditioned Gradient Step Algorithm Algorithm. 1 Guess initial Φ (0) = (ϕ 1,..., ϕ N ) V 2 repeat 1 compute Λ (k) by λ (k) ij 2 ˆϕ (k+1) i = ϕ (k) i = ϕ i, F KS [Φ (k) ] ϕ(k) j B 1 k (F KS [Φ (k) ] ϕ(k) i N appropriate preconditioner B k j=1 ϕ(k) j λ (k) ji ) with 3 project ˆΦ (k+1) onto V glob : orthonormalize ˆΦ (k+1) to Φ (k+1) until convergence reached

15 Projection onto the Stiefel manifold Projection P : V N S,(resp. G), P Φ =: Φ V, s.t. span { ˆφ i : i = 1,..., N} = span {φ i : i = 1,..., N} Lemma Löwdin transformation Φ = L 1 Φ where LL T = Φ T Φ Diagonalization of Λ (n+1) = Φ T F [Φ] Φ = ( ϕi, F [Φ] ϕ i ) N i=1, yields the first N eigenvalues λ (n) 1... λ (n) N of F [Φ]. At the minimizer [Ψ], if F [Ψ] has a spectral gap λ N < Λ N+1, then F [Ψ] is V -elliptic on T [Ψ] φ, F [Ψ] φ φ 2 V φ span {ψ i; i = 1,..., N}.

16 Comment on direct minimization The algorithm computes an ON basis of the invariant subspace of F Ψ corresponding to the N lowest eigenvalues. everything is valid if V := V h is a finite dimensional subspace (Galerkin approximation) improvement by subspace acceleration: e.g. DIIS gradient directed convergence with Armijo line search B : V V, φ 2 B = φ, φ B := Bφ, φ φ 2 H 1 e.g.: B C, e.g. multigrid or convolution by FFT for a fixed operator A it is a block PINVIT iteration cf. e.g. (Bramble & Knyazev et al.) use for SCF Iteration

17 Convergence results Definition measure of the error between subspaces spanned by Φ = (ϕ 1,..., ϕ N ), Ψ (I D Ψ )Φ 2ˆB := N φ i D Ψ φ i 2ˆB i=1 Theorem ( Blauert, Neelov, Rohwedder, S. (08)) If Φ (0) U δ (Ψ), and (J (Ψ) Λ)Φ, Φ γ Φ 2 H 1N for all Φ T [Ψ] G, then there exists χ < 1 such that (I D Ψ )Φ (n+1) ˆB χ (I D Ψ )Φ (n) ˆB

18 Energy error Theorem ( Blauert, Neelov, Rohwedder, S. (08)) Let J C 2 (V N, R), then J (Φ (n) ) J (Ψ) + R 2 = 2 (Ψ Φ (n) ), (A [Φ (n) ] Φ(n) Φ (n) Λ (n) ) (I D Ψ )Φ (n) 2 V N where R 2 = O( (I D Ψ )Φ (n) 2 ). V N Corresponding results for a priori and a posterriori estimates for the Galerkin solution together with S. Schwinger and W.Hackbusch (08).

19 Energy error Constrained optimization problem (revisited): u = argmin{j(v) : G(v) = 0} Lagrangian L(x) := L(u, Λ) = J(u) ΛG(u) (x = (u, Λ) X) Theorem (Rannacher et al. ) If L (x)y = 0 y X and L (x n )x n = 0, L(x) L(x n ) = 1 2 L (x n )(x x n ) + O( x x n 3 X ) J(u) J(u n ) = 1 J 2[ (u n )(u u n ) Λ n G (u n )(u u n ) (Λ Λ n )G(u n ) ] Here L(Φ, Λ) = J (Φ) + trλ( Φ T, Φ I)

20 Orbital based functional L(Φ, Λ) = J HF (Φ) + trλ( Φ T, Φ I) error measure on G : [Φ] [Ψ] := inf{u U(N) : Φ ΨU V N } Theorem J (Ψ h ) J (Ψ) + R 3 (ΦU Φ h ) = 2 (ΨU Φ h ), A Ψh Ψ h Ψ h Λ Φ h V h But U = argmin U U(N) R 3 (ΦU Φ h ), R 3 = O( Ψ h ΨU 3 V N ) is unknown.

21 Interpolation estimates For local basis functions like Finite Elements or wavelets (Interpolation estimates) Let Ω k := suppψ k Ω k, V h := span{ψ k }, h k diamω k diam Ω k, then there ex. an operator P h : V V h reproducing polynomials such thtat u P h u L2 (Ω k ) h 2 k u H 2 ( Ω k ) e.g. P h (Clermont, Scott-Zhang quasi) interpolation operator Lemma (H 2 regularity) The minimizer Φ = (ϕ i ) of J HF, J SCF is in (H 2 (R 3 )) N. This is assumed to hold also for J KS

22 A posteriori error estimates Let R := (R i ) N i=1 := (F [Ψ h ] Λ h )Φ h V N be the residual. The local residuals r k, ρ k are defined by N i,k := R i 2 L 2 (Ω k ), ρ2 i,k r 2 i=1 := h 1/2 k N [ n ϕ h,i ] 2 L 2 (e k ) i=1 where [ n ϕ h,i ] ek denotes the jump of the normal derivatives across the edges in Ω k if ψ k C 1. The error estimator depends on the representation Φ V of [Φ] G. it allows an individual discretization of ϕ i

23 A posteriori error estimates Theorem (S. &Schwinger ) If Φ G, Φ V N, Φ i V N h J = J HF, J KS, J SCF, then be the minimizers of J (Φ h ) J (Φ) (ri,k 2 + ρ2 i,k )h4 k i,k 1/2 Φ (H 2 ) N it reveals the eigenvalue error estimator of Larsen The H 2 -regularity assumption simplifies the proof, but may not be required, see (Heuveline & Rannacher).

24 BigDFT Project T. Deutsch (CEA Grenoble) S. Goedecker (Uni Basel) X. Gonze (UC Louvain) R. Schneider (U Kiel TU Berlin) Aims: implementing an electronic structure calculation program with wavelet bases usable on massively parallel computer embed this code in the existing program package ABINIT ( linear scaling code with respect to the number of electrons

25 Scaling Function ϕ and Wavelet ψ p = p = Figure: Daubechies scaling functions ϕ with p = 2 and 5. 2 p = 2 p =

26 Adaptivity two computational regions: coarse and fine region (sufficient with pseudopotentials) fine region: scaling functions and wavelets (8 basis functions per grid point) coarse region: only scaling functions (1 basis function per grid point)

27 Adaptivity

28 Adaptivity

29 Adaptivity

30 Numerical results example: C 19 H 22 N 2 O (N = 55), Figure:

31 Numerical results Convergence history for the direct minimization scheme and together with DIIS acceleration 10 1 steepest descent 10 1 history size for DIIS = hgrid = 0.3 hgrid = 0.45 hgrid = hgrid = 0.3 hgrid = 0.45 hgrid = 0.7 absolute error absolute error iter, iter Figure: convergence

32 Accuracy Depending on h grid convergence rate: O(h 14 grid )

33 Wavelets vs. Plane Waves (Degrees of Freedom)

34 Wavelets vs. Plane Waves (Runtime) Runtimes for Cinchonidine Abinit CPMD BigDFT 3000 sec abs. prec. error

35 Computing Times

36 Summary Tensor product approximation is the key to treat the high dimensional problem Hartree Fock is a rank one anti-symmetric tensor product approximation In theory, Density Functional Theory (DFT) can provide the exact ground state energy and density Due to the unknown exchange correlation potential there remains an unavoidable modelling error The high dimensional (d = 3N) linear eigenvalue problem is reduced by HF and DFT to a (system of) low dimensional (d = 3) nonlinear eigenvalue type problems relatively large systems can be treated if one accepts the modeling error, complexity is O(N 2 dim V h ) O(N 3 ) O(N) for large systems. for more accurate computations one has to consider the original highdimensional problem Quantum Monte Carlo Methods or Wave Function - Post Hartree Fock Methods

37 Full 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 } H 1 (R 3 {± 1 2 }), ϕ i, ϕ j = δ i,j, Full CI (for benchmark computations N = 18) is a Galerkin method w.r.t. the subspace V FCI = N i=1 X h = span{ψ SL = Ψ[ν 1,..ν N ] = 1 N! det(ϕ νi (x j, s j )) N i,j=1 } Galerkin ansatz: Ψ = c 0 Ψ 0 + ν J c νψ ν H = ( Ψ ν, HΨ ν ), Hc = Ec, but dim V h = N N!

38 FCI Method - canonical orbitals Let Ψ 0 = Ψ[1,.., N] = 1 det(ϕ N! i (x j, s j )) N i,j=1, be a reference These first N orbital functions ϕ i are called occupied orbitals the others are called unoccupied orbitals ϕ 1,..., ϕ N, ϕ N+1,..., ϕ N they are eigen functions of F : X X,X := H 1 (R 3 {± 1 2 }) Fϕ i λ i ϕ i, φ h = 0 φ h X h, f p r := Fϕ r, ϕ p = δ r,p, r, p, N Lemma The Fock operator F := F h = N k=1 F k : V FCI V FCI admits ( N ) FΨ[ν 1,..., ν N ] = λ νi Ψ[ν 1,..., ν N ] i=1

39 Some remarks about choie of ϕ H = F + U, U is a two particle operator FCI exponentially scaling wr.t. to N, one has to confine to a small subspace Usually N 10N but one hat to resolve the e-e cusp F Fock operator for (HF or KS), precomputation H. Yserentant has shown the existence of F where the eigen-functions provide a sparse grid basis (to avoid curse of dimensions) adaptive schemes (Griebel et al., Flad, Rohwedder & S) ϕ k can be choosen optimally (MRSCF), not eigen fuctions of a single operator, existence proved by Friesecke, Levin but still exonential complexity of FCI Localized basis functions reduced complexity of matrix elements linear scaling (?!)

40 Second quantization Second quantization: annihilation operators: a j Ψ[j, 1,..., N] := Ψ[1,..., N] and := 0 if j not apparent in Ψ[...]. The adjoint of a b is a creation operator v a b Ψ[1,..., N] = Ψ[b, 1,..., N] = ( 1)N Ψ[1,..., N, b] Theorem (Slater-Condon Rules) H : V V resp. H : V FCI V FCI reads as (basis dependent) H = F + U = fr p a r a p + urs pq a r a s a qa p p,q p,q,r,s

41 Excitation operators Single excitation operator, Let Ψ 0 = Ψ[1,..., N] be a reference determinant then e.g. X k 1 Ψ 0 := a k a 1Ψ 0 ( 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 operators k X µ := X b 1,...,b k l 1,...,l k = X b i l i, 1 l i < l i+1 N, N < b i < b i+1. i=1 A CI solution Ψ h = c 0 Ψ 0 + µ J h c µ Ψ µ can be written by Ψ h = c 0 + c µ X µ Ψ 0, c 0, c µ R, J h J. µ Jh

42 Coupled Cluster Method - Exponential-Ansatz Theorem (S. 06, (Coester & Kümmel 1959)) Let Ψ 0 be a reference Slater determinant, e.g. Ψ 0 = Ψ HF and Ψ V FCI, V, satisfying Ψ, Ψ 0 = 1 intermediate normalization. Then there exists an unique 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

43 CC Energy and Projected Coupled Cluster Method Let Ψ V FCI satisfying HΨ := H h Ψ = E 0 Ψ, then, due to the Slater Condon rules and Ψ, Ψ 0 = 1 E = Ψ 0, HΨ = Ψ 0, He T Ψ 0 = Ψ 0, H(I + T 1 +T T 2 1 )Ψ 0 The Projected Coupled Cluster Method applies the ansatz l Ψ h = e T h Ψ 0, T h := T := T k = t µ X µ, 0 µ J h J µ J h k=1 Ψ µ subspace of V FCI, CCSD T = T 1 + T 2 = T (t) and (Galerkin) projection onto subspace.

44 Projected Coupled Cluster Method Let T = lk=1 T k = µ J h t µ X µ, 0 µ J h J using 0 = Ψ 0, (H E)Ψ = Ψ 0, (H E(t h )e T (t h) Ψ 0 The unlinked projected Coupled Cluster formulation 0 = Ψ µ, (H E(t h ))e T (t h) Ψ 0 =: g µ (t), t = (t ν ) ν Jh, µ, ν J h The linked projected Coupled Cluster formulation consists in 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 << N nonlinear equations for L unknown excitation amplitudes t µ. Theorem ( Kümmel 1959, S. 09) The (projected) CC Method is size consistent!: H AB = H A + H B EAB CC = E A CC + E B CC.

45 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, 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 ]

46 Iteration method to solve CC amplitude equations 0 = f µ (t) = Ψ µ, e T He T Ψ 0 = Ψ µ, [F, X µ ]Ψ k=0 1 k! Ψ µ, [U, T ] k Ψ 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. ɛ µ < λ N λ N+1 < 0 (Bach-Lieb-Sololev) the matrix A > 0

47 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

48 Analysis of the CC Methods - Lemmas 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

49 Analysis of CC Method - Lemmas 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.

50 Convergence of the Coupled Cluster Method Lemma Let Ψ h = e T hψ 0 where t t h V inf t v h V. v R J h then P µ J v Ψ Ψ h H 1 inf Ψ e h µx µ Ψ 0 H 1. v R L Definition A function g : is called strongly monotone at t if g(t) g(t ), (t t γ t t 2 V for some γ > 0 and all t t V < δ.

51 Local existence and quasi-optimal convergence Strict monotonicity of f is not known yet! Let T (t) := µ t µx µ we consider g : V V, g(t) ν := Ψ ν, (H E(t)e T (t) Ψ 0. Theorem (S. 2008) Let E be a simple EV. If Ψ Ψ 0 V < δ sufficiently small, and J h excitation complete, then 1 for E = E(t h ) := Ψ 0, He T (th) Ψ 0, there holds g(t h ), v = 0, v V h f(t h ), v = 0, v V h 2 g is strongly monontone at t t V δ 3 there ex. t h V h with g(t h ), v = f(t h ), v = 0, v V h, t t h V inf v V h t v h V.

52 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 u h V h t u h V a a h V +( inf u h V h t u h V ) 2. E E h inf u h V h t u h V a a h V +( inf u h V h t u h V ) 2 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.

53 Numerical examples CCSD versus CISD The relative difference in the correlation energy between CI and CC for several molecules in bonding configuration is plotted over the total number of electrons N and the number of valence electrons. All computations were performed with MOLPRO The lack of size consistency suggests a behavior N rel. correlation energy difference rel. correlation energy difference nuclear charge number of valence electrons

54 Conclusions - CC is a most powerful wave function method If the reference Ψ 0 is sufficiently close to the exact solution Ψ, granted by the results of Yserentant, then we have quasipotimal convergence w.r.t the wave function and super-optimal convergence w.r.t. the energy + size consistency CCSD and CCSD(T) are standard, CCSDT; CCSDTQ etc. only for extremely accurate computations not good for multi-configurational problems, e.g. (near-) degenerate ground state ( where RHF is rather bad)