The Coupled-Cluster Method

The one-particle excitation operator and the two-particle excitation operator T2 are defined by 

The effect of the operator in (16.15) is to express ipasa linear combination of Slater determinants that include d o and all possible excitations of electrons from occupied to virtual spin-orbitals. A full Cl calculation also expresses ip as a linear combination involving all possible excitations, and we know that a full Cl calculation with a complete basis set gives the exact ip. Hence, it is plausible that Eq. (16.15) is valid. 

The mixing into the wave function of Slater determinants with electrons excited from occupied to virtual spin-orbitals allows electrons to keep away from one another and thereby provides for electron correlation. 

In the CC method, one works with individual Slater determinants, rather than with CSFs, but each CSF is a linear combination of one or a few Slater determinants, and the CC and Cl methods can each be formulated either in terms of individual Slater determinants or in terms of CSFs. 

The aim of a CC calculation is to find the coefficients tf/, tijf. foralli,j,k,. and all a,b,c,. Once these coefficients (called amplitudes) are found, the wave function i/r in (16.15) is known. 

The coupled-cluster (CC) method for dealing with a system of interacting particles was introduced around 1958 by Coester and Kummel in the context of studying the atomic nucleus. CC methods for molecular electronic calculations were developed by Cfzek, Paldus, Sinanoglu, and Nesbet in the 1960s and by Pople and co-workers and Bartlett and co-workers in the 1970s. For reviews of the CC method, see R. J. Bartlett, J. Phys. Chem., 93,1697 (1989) R. J. Bartlett in Yarkony, Part II, Chapter 16 T. J. Lee and G E. Scuseria, in S. R. Langhoff (ed.). Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy, Kluwer, 1995, pp. 47-108. 

To apply the CC method, two approximations are made. First, instead of using a complete, and hence infinite, set of basis functions, one uses a finite basis set to express the spin-orbitals in the SCF wave function. One thus has available only a finite number of virtual orbitals to use in forming excited determinants. As usual, we have a basis-set truncation error. Second, instead of including all the operators fj, 72. r , one approximates the operator f by including only some of these operators. Theory shows (Wilson, p. 222) that the most important contribution to T is made by r2.The approximation T T 2 gives 

---

There is a variation on the coupled cluster method known as the symmetry adapted cluster (SAC) method. This is also a size consistent method. For excited states, a Cl out of this space, called a SAC-CI, is done. This improves the accuracy of electronic excited-state energies. 

Figures 11.9 and 11.10 compare the performance of the CCSD and CCSD(T) methods, based on either an RFIF or UHF reference wave function. Compared to the RMP plot (Figure 11.7), it is seen that the infinite nature of coupled cluster causes it to perform somewhat better as the reference wave function becomes increasingly poor. While the RMP4 energy curve follows the exact out to an elongation of 1.0A, the CCSD(T) has the same accuracy out to - 1.5 A. Eventually, however, the wrong dissociation limit of the RHF wave also makes the coupled cluster methods break down, and the energy starts to decrease.

The computational complexity of the coupled-cluster method truncated after a given excitation level m - for example, m = 2 for CCSD - may be discussed in terms of the number of amplitudes (Nam) in the coupled-cluster operator and the number of operations (Nop) required for optimization of the wavefunction. Considering K atoms, each with Nbas basis functions, we have the following scaling relations ... 

The coupled-cluster method is well-known by now, and only a brief account of aspects relevant to our applications is given here. 

This short sub-section on the coupled cluster method can clearly not cover everything of this important area, but a few more aspects should be pointed out here. First, a procedure is needed to obtain the cluster amplitudes For the CCSD wave-... 

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... 

The final topic we will discuss in this chapter is size-consistency, which has been mentioned several times already. A method is said to be size-consistent if the computed energy of the composite system A + B, with A and B at infinite distance from each other, yields the same energy as if the method is applied to A and B separately and the energies axe added, i.e. E(A+B)=E(A)+E(B). Some of the methods we have discussed are automatically size-consistent. This is true, for example, for the Hartree-Fock method and the complete Cl method, and it is also true for the methods discussed in the chapter on perturbation theory, such as the coupled cluster method. It is, however, not true for the SD-CI or the MR-CI method. We will in this section show that it is possible, by a slight modification of the formalism, to correct these Cl methods to be approximately size-consistent. The experience gathered over the past two decades on size-consistency corrections indicates that the calculated results axe much improved at the SD-CI level, whereas relative energies axe improved at the MR-CI level but the situation for geometries is less clear at this level. 

MBPT starts with the partition of the Hamiltonian into H = H0 + V. The basic idea is to use the known eigenstates of H0 as the starting point to find the eigenstates of H. The most advanced solutions to this problem, such as the coupled-cluster method, are iterative well-defined classes of contributions are iterated until convergence, meaning that the perturbation is treated to all orders. Iterative MBPT methods have many advantages. First, they are economical and still capable of high accuracy. Only a few selected states are treated and the size of a calculation scales thus modestly with the basis set used to carry out the perturbation expansion. Radial basis sets that are complete in some discretized space can be used [112, 120, 121], and the basis... 

The Soviet Union s military invasion of Czechoslovakia in August 1968 had a significant impact on the development of computational and theoretical chemistry in Canada. The troubles in Czechoslovakia led to Jiri Cizek and Joe Paldus accepting appointments at the University of Waterloo in 1968. Their many achievements include the first ab initio study of the coupled cluster method.104... 

The Configuration Interaction Approach to Electron Correlation - The Coupled Cluster Method... 

The second general approach to correlation theory, also based on perturbation theory, is the coupled-cluster method, which can be thought of as an infinite-order perturbation method. The coupled-cluster wave function T cc is expressed as a power series,... 

Monkhorst HJ (1977) Calculation of properties with the coupled-cluster method. Int J Quantum Chem Symp 11 421-432. 

Collins, C. L., Morihashi, K., Yamaguchi, Y, and Schaefer, H. R, Vibrational frequencies of the HF dimer from the coupled cluster method including all single and double excitations plus perturbative connected triple excitations, J. Chem. Phys. 103, 6051-6056 (1995). 

---