Since its introduction into quantum chemistry in the late 1960s by Cízek and Paldus[1,2,3], coupled cluster theory has emerged as perhaps the most reliable, yet computationally affordable method for the approximate solution of the electronic Schrödinger equation and the prediction of molecular properties. The purpose of this chapter is to provide computational chemists who seek a deeper knowledge of coupled cluster theory with the background necessary to understand the extensive literature on this important ab initio technique.
In spite of the method's present utility and popularity, the quantum chemical community was slow to accept coupled cluster theory, perhaps because the earliest researchers in the field used elegant but unfamiliar mathematical tools such as Feynman-like diagrams and second-quantization to derive working equations. Nearly ten years after the essential contributions of Paldus and Cízek, Hurley presented a re-derivation of the coupled cluster doubles (CCD) equations[4] in terms which were more familiar to quantum chemists. Soon thereafter Monkhorst[5] developed a general coupled cluster response theory for calculating molecular properties. By the end of the 1970s, computer implementations of the theory for realistic systems began to appear as the groups of Pople[6] and Bartlett[7] each developed and tested spin-orbital CCD programs. A few years later, Purvis and Bartlett derived the coupled cluster singles and doubles (CCSD) equations and implemented them in a practical computer program.[8] Since that pioneering achievement, the popularity of coupled cluster methods has blossomed, and tremendous efforts have been made in the construction of highly efficient CCSD energy codes[9,8,10,11,,13,14], inclusion of higher excitations in the coupled cluster wavefunction,[15,16,17,18,19,20,,22,23,24,25,,27,28,29,30,31,,33,34] spin-adaptation of open-shell methods[35,36,37,38,39,40,,42], as well as development of analytic first[43,44,45,46,47,,49,50,51,52,,54] and second[55,56,,58,59] energy derivatives, and methods to treat excited states.[60,61,62,63,,65,66,67,,69,70,71,,73,74]
In the following section, we will use the cluster function approach developed by Sinanoglu[75] to justify the well-known exponential form of the coupled cluster wavefunction. This task requires use of the mathematical technique known as second-quantization (also called ``occupation-number'' formalism), and we introduce important concepts as they are needed. We then construct the operator equations of coupled cluster theory and address issues such as the Hausdorff expansion, variational approaches, and an eigenvalue perspective on the coupled cluster problem. In the next section, we develop a set of algebraic and diagrammatic tools needed to derive programmable equations for the CCSD method, and, using these tools, we discuss the property of the energy known as size extensivity. Next, we examine the relationship between the coupled cluster equations and those of finite-order many-body perturbation theory, leading to an explanation of the popular (T) correction implemented in many quantum chemical program packages. We then discuss some of the issues associated with an efficient computer implementation of coupled-cluster-like equations, such as matrix formulations, intermediate factorization, spin and spatial symmetry simplifications, and atomic-orbital-based algorithms. Finally, we describe some of the latest developments in the theory, including the implementation of open-shell Brueckner methods, an area of coupled cluster theory which in recent years has proven to be valuable for a number of difficult open-shell symmetry-breaking problems.
We would like to stress that this chapter is a review of coupled cluster theory. It is not primarily intended to provide an analysis of the numerical performance of the coupled cluster model, and we direct readers in search of such information to several recent publications[76,77,78,79]. Instead, we offer a detailed explanation of the most important aspects of coupled cluster theory at a level appropriate for the general computational chemistry community. Although many of the topics described here have been discussed by other authors[80,77,81,78], this chapter is unique in that it attempts to provide a concise, practical introduction to the mathematical techniques of coupled cluster theory (both algebraic and diagrammatic), as well as a discussion of the efficient implementation of the method on high-performance computers, in a manner accessible to newcomers to the field.