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 cor- related 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 ac- curacy 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 meth- ods 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 func- tion, 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 meth- ods, 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 equa- tion. 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.