Coupled-cluster method types

An interesting point of the coupled cluster method concerns the treatment of quadruple excitations. If the CCD method is considered, in which only the T2 operator is retained in the exponent, the amplitudes for these excitations are given as products of amplitudes for double excitations according to the term Ij. In fact is a sum of the 18 products of type (with phase factors) which can be formed... 

Multireference coupled cluster methods, which started development more recently, are generally divided into two types. Hilbert space CC methods use multiple reference functions to obtain a description of a few states, including the reference state (for a review see (4)). Fock space methods (for a review see (5)), on the other hand, provide direct state-to-state energy differences, relative to some common reference state. The Fock space approach is particularly well-suited to the calculation of ionization potentials (IPs), electron affinities (EAs), and excitation energies (EEs). For principal IPs and EAs, FSCC is equivalent (6, 7) to the EOM-IP and EOM-EA CC methods (1, 2, 7, 8). In this paper, we will focus primarily on the IP problem. 

For atomic structure calculations, the four-component MCSCF approach was the method of choice for a long time. The implementation of methods for treating very large CSF spaces, particularly Davidson-type diagonalization techniques, produced a tool to compete with highly precise experimental measurements. The coupled-cluster method—or MBPT approach as it is usually called in the physics community—turned out to be a valuable, alternative, size-consistent method. 

P.G. Szalay, Towards state-specihc formulation of multireference coupled-cluster theory Coupled electron pair approximations (CEPA) leading to multireference configuration interaction (MR-CI) type equations, in R.J. Bartlett (Ed.), Modem ideas in coupled-cluster methods, World Scientific, Singapore, 1997, pp. 81-123. 

The ionization potentials and electron affinities of the atoms H, C, N, O and F have been computed by means of coupled-cluster methods using doubly augmented correlation-consistent one-electron basis sets in conjunction with explicitly correlated Slater-type geminals. Excitations up to the level of connected quintuples have been accounted for, and all orbitals in the core and valence shells have been correlated. Relativistic effects (spin-orbit as well as scalar) and diagonal Born-Oppenheimer corrections have been included. 

Couple cluster methods differ from perturbation theory in that they include specific corrections to the wavefunction for a particular type to an infinite order. Couple cluster theory therefore must be truncated. The exponential series of functions that operate on the wavefunction can be written in terms of single, double and triple excited states in the determinantl " . The lowest level of truncation is usually at double excitations since the single excitations do not extend the HF solution. The addition of singles along with doubles improves the solution (CCSD). Expansion out to the quadruple excitations has been performed but only for very small systems. Couple cluster theory can improve the accuracy for thermochemical calculations to within 1 kcal/mol. They scale, however, with increases in the number of basis functions (or electrons) as N . This makes calculations on anything over 10 atoms or transition-metal clusters prohibitive. 

Coupled cluster methods (CCSD (T) in particular) provide high-accuracy results (often within 0.1 kcal/mol) for many types of molecules (e.g., organic molecules), but have more difficulties with transition metal-containing species. The method scales as If and at present can conveniently be applied only to small molecules, where it is however quite valuable in producing benchmark results. 

The coupled cluster theory may be derived from the many-body perturbation theory which we have presented above. Each coupled cluster approximation can be obtained by summing certain well-defined types of diagrammatic terms through all orders of the perturbation expansion. We shall not present here the details of the relation between coupled cluster and many-body perturbation theories. For a detailed discussion, the reader is referred to the review by Paldus and Li [81], published in 1999, entitled A critical assessment of coupled cluster method in quantum chemistry and the chapter on coupled cluster theory by Paldus [82] in the Handbook of Molecular Physics and Quantum Chemistry. 

A number of types of calculations begin with a HF calculation and then correct for correlation. Some of these methods are Moller-Plesset perturbation theory (MPn, where n is the order of correction), the generalized valence bond (GVB) method, multi-conhgurational self-consistent held (MCSCF), conhgu-ration interaction (Cl), and coupled cluster theory (CC). As a group, these methods are referred to as correlated calculations. 

There are three main methods for calculating electron correlation Configuration Interaction (Cl), Many Body Perturbation Theory (MBPT) and Coupled Cluster (CC). A word of caution before we describe these methods in more details. The Slater determinants are composed of spin-MOs, but since the Hamilton operator is independent of spin, the spin dependence can be factored out. Furthermore, to facilitate notation, it is often assumed that the HF determinant is of the RHF type. Finally, many of the expressions below involve double summations over identical sets of functions. To ensure only the unique terms are included, one of the summation indices must be restricted. Alternatively, both indices can be allowed to run over all values, and the overcounting corrected by a factor of 1/2. Various combinations of these assumptions result in final expressions which differ by factors of 1 /2, 1/4 etc. from those given here. In the present book the MOs are always spin-MOs, and conversion of a restricted summation to an unrestricted is always noted explicitly. 

Perturbation methods add all types of corrections (S, D, T, Q etc.) to the reference wave function to a given order (2, 3, 4 etc.). The idea in Coupled Cluster (CC) methods is to include all corrections of a given type to infinite order. The (intermediate normalized) coupled cluster wave function is written as... 

---