Coupled Cluster Methods in Quantum Chemistry

Title: Coupled Cluster methods in quantum chemistry

Name: Thorsten Rohwedder, Reinhold Schneider

Al./Addr.: Technische UniversitatBerlin

Coupled Cluster methods in quantum chemistry

Short description

Coupled Cluster methods applied in quantum chemistry reformulate the electronic

Schrodinger equation as a nonlinear equation, enabling the computation of size- consistent high-precision approximations of the ground state solution for weakly correlated systems.

Introduction

The Coupled Cluster method (CC method) is one of the most successful and frequently used approaches for the computation of atomic and molecular electronic structure, i.e. for the solution of the stationary electronic Schrodinger equation, whenever high accuracy is required. In contrast to Hartree-Fock type methods (HF) or methods from density functional theory (see the respective entries in this work), high accuracy methods have to account in particular for the quantum-mechanical phenomenon of electronic correlation. If a preliminarily calculated reference solution { usually provided by a HF calculation { already is a good approximation to the sought ground state wave function, the problem is said to be weakly correlated. CC as a post-Hartree-Fock method

(also see the entry by M. Lewin) then enables an ecient, accurate and size-extensive description of solutions of the electronic Schrodingerequation. In this context, the 2 size-extensivity of the CC approach is a key aspect, re ecting the correct scaling of correlation energy with respect to the number of electrons.

CC methods were initially developed for the treatment of many body quantum systems in nuclear physics in the 1950s and were used for quantum chemical calculations since the 1966 initial work by Paldus and Czek,see (Czek,1991) for a historical overview.

For further information, cf. the excellent review (Bartlett/Musial, 2007) and the abun- dance of references therein. Recent extensions to linear response theory also allow the size-extensive computation of various physical and chemical properties like dipole moments, polarizabilities and hyperpolarizabilities, excitation energies etc., see (Ped- ersen/Koch/Hattig,1999). The CC method reformulates the electronic Schrodinger equation as a nonlinear equation by a parametrization via an exponential excitation operator { a proceeding explained in more detail in the next two sections.

Electronic Schrodingerequation and basis sets

Basic de nitions. We rst collect some basic facts required to de ne the Coupled

Cluster method. In chemistry, CC aims to solve the stationary electronic Schrodinger equation in its weak formulation, i.e. to compute a wave function © such that

h¨; H©i = E£h¨; ©i for all ¨ 2 H1: (1)

In this, © is obliged to be antisymmetric and to have a certain Sobolev regularity, so that 1 ^N 1 © 2 H1 := H1(R3 ¢ f¦ g; R)N \ L (R3 ¢ f¦ g; R) 2 2 2 i=1 with H1(X; R) denoting the set of real-valued one time Sobolev di erentiable functions on X, and where ^ descripts the antisymmetric tensor product of spaces; H : H1 ! H 1 is the weak Hamiltonian, xed by the numbers N;K of electrons and classical nuclei

3 of the system and by charge Z 2 N and xed position r 2 R of the latter. In atomic 3 units it is given by

XN XN XK XN XN 1 Z 1 1 H = ¡i + ; 2 jri aj 2 jri rjj i=1 i=1 =1 i=1 j=1 j6=i cf. the article by Yserentant in this work and in particular (Yserentant, 2010) for further information on the weak formulation.

Slater determinants. To discretize the above equation e.g. by Galerkin methods, a basis of H1 has to be constructed. As detailed in the contribution on Hartree-

Fock methods by I. Catto, this may be done by using a complete one-particle basis set

B := f' j p 2 Ng H1(R3 ¢ f¦ 1 g; R); to construct out of each N distinct indices p 2 p1 < : : : < pN 2 N a Slater determinant

1 ¡N ©[p1; : : : ; pN ] := p det 'p (xj) ; xi = (ri; i) N! i i;j=1

1 The set B := f©[p1; : : : ; pN ] j pi < pi+1 2 Ig then is a basis of the space H . In second quantization the creation operator is de ned by ab©[p1; : : : ; pN ] = ©[b; p1; : : : ; pN ]. Its

y adjoint is the corresponding annihilation operator ab©[b; p1; : : : ; pN ] = ©[p1; : : : ; pN ]

y and ab©[p1; : : : ; pN+1] = 0 if b 6= pi, i = 1;:::;N. With this notation at hand the

Hamiltonian can be expressed as

X X p y r:s y y H = hrapar + Vp;q apaqasar ; p;r p;q;r;s

p p;q with the one- and two-electron integrals hr, Vr;s . For more detailed information on second quantization formulation of electronic structure problems, confer e.g. (Helgaker et al., 2000).

In practice, the basis B (and thus B) is substituted by a nite basis set Bd, inducing a

1 Galerkin basis Bd for a trial space contained in H . A Galerkin method for the equation

(1) with Bd as basis for the ansatz space is termed Full-CI in quantum chemistry.

Because Bd usually contains far too many functions (their number scaling exponentially with the size of Bd), a subset BD of Bd is chosen for discretization. Unfortunately, 4 traditional restricted CI-methods, like the CISD method described in the entry by M.

Lewin on post-HF methods, thereby lose size-consistency, meaning that for a system

AB consisting of two independent subsystems A and B, the energy of AB as computed by the truncated CI model is no longer the sum of the energies of A and B. In practice, this leads to inaccurate computations with a relative error increasing with the size of the system. Therefore, size-consistency and the related property of size-extensivity are essential properties of quantum chemical methods (see e.g. (Helgaker et al., 2000)), which is why the linear parametrization of CI is replaced by an appropriate nonlinear ansatz, the CC ansatz.

Formulation of the CC ansatz

Excitation operators. The determinant ©0 := ©[1;:::;N], formed from the rst

N basis functions 'i (or occupied orbitals, in quantum chemist's language), is the so- called reference determinant of the ansatz. In practice, the above one-particle basis Bd is obtained from a preliminary Hartree-Fock computation, and ©0 then is the Hartree-

Fock approximation of the solution of Eq. (1); for the construction and analysis of the

CC method, it is only important that the reference is not orthogonal to © and that the occupied orbitals 'i (i < N) are L2-orthogonal to the virtual orbitals 'a, a > N,

£ £ so that the solution © can be then expressed as © = ©0 ¨ © , i.e. © is an orthogonal correction to ©0.

CC is formulated in terms of excitation operators

X := Xa1;:::;ar = Xa1 ¡¡¡ Xar = ay a ¡¡¡ ay a ; (2) i1;:::;ir i1 ir a1 i1 ar ir where r N, i1 < : : : < ir N, N + 1 a1 < : : : < ar. These X can also be charac- terized by their action on the basis functions ©[p1; : : : ; pN ] 2 B: If fp1; : : : ; pN g contains all indices i1; : : : ; ir, the operator replaces them (up to a sign factor ¦1) by the orbitals 5 a ; : : : ; a ; otherwise, Xa1;:::;ar ©[p ; : : : ; p ] = 0: Indexing the set of all excitation op- 1 r i1;:::;ir 1 N erators by a set M, we have in particular that B = f©0g [ f©j© = X©0; 2 Mg.

The convention that 'i?'a implies two essential properties, namely that excitation

2 operators commute (XX XX = 0), and are nilpotent, i.e. X = 0. Note that these only hold within the single-reference ansatz described here.

Exponential ansatz. The cluster operator of a coecient vector t 2 `2(M), t = P (t)2M is de ned as T (t) = 2M tX. The CC method replaces the linear P parametrization © = ©0 ¨ 2M tX©0 (of functions normalized by h©0; ©i = 1) by an exponential (or multiplicative) parametrization

P T (t) ( tX) © = e ©0 = e 2M ©0 = ¥2M(1 + tX)©0 :

1 Choosing a suitable coecient space V `2(M) re ecting the H -regularity of the solution, it can be shown that there is a one-to-one correspondence between the sets

£ £ 1 T (t) f©0 + © j ©0?© 2 H g; f©0 + T (t)©0 j t 2 Vg and fe ©0 j t 2 Vg:

Coupled Cluster equations. The latter exponential representation of all possible

£ solutions ©0 +© is used to reformulate Eq. (1) as the set of unlinked Full-CC equations for a coecient vector t 2 V,

T (t) h¨; (H E)e ©0i = 0; for all ¨ 2 B:

Inserting e T (t) yields the equivalent linked Full-CC equations

T T £ T T T h¨; e He ©0 i = 0 for all 2 M;E = h©0; He ©0i = h©0; e He ©0i:

For nite resp. in nite underlying one-particle basis B resp. Bd, both of these two sets of equations are equivalent to the Schrodingerequation (1) resp. the linear Full-

CI ansatz. For two subsystems A and B and corresponding excitation operators TA

T +T T T and TB, the exponential ansatz admits for the simple factorization e A B = e A e B .

Therefore, aside from other advantages, the CC ansatz maintains the property of size- consistency. 6

The restriction to a feasible basis set BD B corresponds in the linked formulation to a Galerkin procedure for the nonlinear function

¡ f : V ! V0; f(t) := h© ; e T (t)HeT (t)© i ; (3) 0 2M

£ T (t£) the roots t of which correspond to solutions e ©0 of the original Schrodinger equation. This gives the projected CC equations hf(tD); vDi = 0 for all vD 2 VD, where

VD = l2(MD) is the chosen coecient Galerkin space, indexed by a subset MD of M.

This is a nonlinear equation for the Galerkin discretisation f of the function f:

¡ T (tD) T (tD) f(tD) := h© ; e He ©0 i = 0: (4) 2MD

Usually, the Galerkin space VD is chosen based on the so-called excitation level r of the basis functions (i.e. the number r of one-electron functions in which © di ers from the reference ©0, see e.g. (Helgaker et al., 2000). For example, including at most twofold excitations (i.e. r 2 in (2)) gives the common CCSD (CC Singles/Doubles) method.

Numerical treatment of the CC equations

The numerical treatment of the CC ansatz consists mainly in the computation of a solution of the nonlinear equation f(tD) = 0. This is usually performed by quasi-

Newton methods,

(n+1) (n) 1 (n) (0) tD = tD F f(tD ); tD = 0 (5) with an approximate Jacobian F given by the Fock matrix, see below. On top of this method, it is standard to use the DIIS method (\direct inversion in the iterative subspace"), for acceleration of convergence. Convergence of these iteration techniques is backed by the theoretical results detailed later. We note that the widely used Mller-

Plesset second order perturbation computation (MP2), being the simplest post-Hartree-

Fock or wave function method, is obtained by terminating (5) after the rst iteration. 7 For application of the iteration (5), the discrete CC function (3) has to be evaluated.

Using the properties of the algebra of annihilation and creation operators, it can be shown that for the linked CC equation, the Baker-Campbell-Hausdor expansion ter- minates, i.e.

X1 1 X4 1 e T HeT = [H;T ] = [H;T ] n! (n) n! (n) n=0 n=0 with the n-fold commutators [A; T ](0) := A; [A; T ](1) := AT TA,[A; T ](n) :=

[[A; T ](n 1);T ]. It is common use to decompose the Hamiltonian into one- and two-body operators H = F + U, where F normally is the Fock operator from the preliminary self-consistent Hartree-Fock (or Kohn-Sham) calculation, and where the one-particle basis set 'p consists of the eigenfunctions of the discrete canonical Hartree-Fock (or

Kohn-Sham) equations with corresponding eigenvalues p. The CC equations (4) then read X4 1 F t h© ; [U; T ] © i = 0; for all 2 M ; (6) ; n! (n) 0 D n=0 P ¡ with the Fock matrix F = diag( ) = diag r ( ) . The commutators are then l=1 al il evaluated within the framework of second quantisation by using Wick's theorem and diagrammatic techniques, resulting in an explicit representation of f as a fourth order polynomial in the coecients t, see (Crawford/Schae er, 2000) for a comprehensible derivation.

Complexity. The most common variants of CC methods are the CCSD (see above) and for even higher accuracy the CCSD(T) method, see (Bartlett/Musial, 2007). In the latter, often termed the \golden standard of Quantum Chemistry", a CCSD method is converged at rst, scaling with the number N of electrons as N 6; then, the result is enhanced by treating triple excitations perturbatively by one step of N 7 cost in the

CCSDT basis set. While the computational cost for calculating small to medium sized molecules stays reasonable, it is thereby possible to obtain results that lie within the error bars of corresponding practical experiments. 8 Lagrange formulation and gradients

The CC method is not variational, which is a certain disadvantage of the method.

For instance, the computed CC energy is no longer an upper bound for the exact energy. The following duality concept is helpful in this context: Introducing a formal

Lagrangian

X T (t) T (t) T (t) L(t; ) := h©0; He ©0i + h©; e He ©0i; (7) 2M the CC ground state is E = inft2V sup2VL(t; ) : The corresponding stationary condition with respect to t reads X @L T (t) T (t) T (t) (t; ) = h©0;HXe ©0i + h©; e [H;X]e ©0i @t 2M = E0(t) + h; f 0(t)i = 0 (8)

for all 2 M, while the derivatives w.r.t. yield exactly the CC equations f(t) = 0

T (t) providing the exact CC wave function © = e ©0. Afterwards, the Lagrange multiplier

can be computed from equation (8). Introducing the functions

X X e e T £(t) T £(t) T (t) © := ©(t; ) = ©0 + e © = e (1 + X)©0 ; ©(t) = e ©0 ; where T £ is the adjoint of the operator T , there holds L(t; ) = h©e(t; ); H©(t)i together with the duality h©;e ©i = 1. As an important consequence, one can compute derivatives of energy with respect to certain parameters, e.g. forces, by the

Hellman-Feynman theorem. If the Hamiltonian depends on a parameter ! , H = H(!),

e then @!E = h©; (@!H)©i holds for the respective derivatives with respect to !. Ac- cording discrete equations are obtained by replacing V; M by their discrete counter- parts VD; MD: The above Lagrangian has been introduced in quantum chemistry by

Monkhorst; the formalism has been extended in (Pedersen/Koch/Hattig,1999) for a linear, size-consistent CC response theory. 9 Theoretical results: Convergence and error estimates

It has been shown recently in (Rohwedder/Schneider, 2011) that if the reference ©0 is suciently close to an exact wave function © belonging to a non-degenerate ground state and if VD is suciently large, the discrete CC equation (4) locally permits a unique solution tD. If the basis set size is increased, the solutions tD converge quasi- optimally in the Sobolev H1-norm towards a vector t 2 V parametrizing the exact

T (t) wave function © = e ©0. The involved constant (and therefore the quality of approximation) depends on the gap between lowest and second lowest eigenvalue and on k©0 ©kH1 . The above assumptions and restrictions mean that CC works well in the regime of dynamical or weak correlation, which is in agreement with practical experi- ence.

The error jE(t) E(tD)j of a discrete ground state energy E(tD) computed on Vd can be bounded using the dual weighted residual approach of Rannacher: Denoting by

(t; ) the stationary points of the Lagrangian (7) belonging to the full energy E£, and by tD the solution of the corresponding discretized equation f(tD) = 0; the error of the energy can be bounded by

¡2 jE(t) E(tD)j . d(t; Vd) + d(; Vd) and thus depends quadratically on the distance of the approximation subspace to the primal and dual solutions t; in V. Note that these estimates are a generalization of error bounds for variational methods, which allow for error bounds depending solely

2 on d(t; Vd) , and an improvement of the error estimates given in (Kutzelnigg, 1990).

Roughly speaking, this shows that CC shares the favourable convergence behaviour of the CI methods, while being superior due to the size-consistency of the CC approximation. 10 Outlook: Enhancements and simpli cations of the canonical CC method

To reduce the complexity or to remedy other weaknesses of the method, various variants of the above standard CC method have been proposed. We only give a short, incomplete overview.

Local CC methods. These techniques allow to accelerate the CCSD computations drastically by utilizing localized basis functions, for which the two-electron integrals RR p;q Vr;s = ('p(xi)'q(xj)'r(xi)'s(xj))=jri rjjdxidxj decay with the third power of the distance of the support of 'p'q and 'r's. Integrals over distant pairs can thus be neglected.

Explicitly correlated CC methods. Fast convergence of CC to the full basis set limit is hempered by the electron-electron-cusp, caused by discontinuous higher derivatives of the wave function where the coordinates of two particles coincide. Explicit incorporation of the electron-electron cusp by an r12 or f12 ansatz (Klopper et al., 2006) can improve convergence signi cantly. Recent density tting techniques (see

Sherrill, 2010) herein allow the ecient treatment of the arising three-body integrals.

Simpli ed approaches. The CCSD equation can be simpli ed by linearization and/or by leaving out certain terms in the CC equations. The random phase approximation

(RPA) or electron pair methods like CEPA methods may be derived this way, and these approaches may serve as starting point for developing ecient numerical methods providing almost CCSD accuracy within a much lower computational expense.

Multi-reference CC. Multi-reference methods (Bartlett, 2007) aim at situations where the reference determinant is not close to the true wave function, so that classical

CC methods fail. Unfortunately, multi-reference CC in its present stage is much more 11 complicated, less developed and computationally often prohibitively more expensive than usual Coupled Cluster.

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