Derivation of the Coupled Cluster Equations

``It is the need to remove the `unlinked clusters' and the introduction of Feynman diagrams which make MBPT [and CC theory] appear unfamiliar to quantum chemists.''[115] -- K. F. Freed

In this section we construct working equations for the coupled cluster singles and doubles (CCSD) method. Beginning from the approximation $\hat{T} \equiv \hat{T}_1 + \hat{T}_2$, we use algebraic and diagrammatic techniques to obtain programmable equations for the cluster amplitudes, t_i^a and t_ij^ab, in terms of the one- and two-electron integrals of the electronic Hamiltonian. As a first step we must introduce a few important tools of second quantization such as normal ordering and Wick's theorem to make the mathematical analysis much less complicated. The approach described here may easily be extended to higher-order cluster approximations (e.g., CCSDT and CCSDTQ, where the latter includes quadruple excitations), as well as many-body perturbation theory expressions.

As indicated in Karl Freed's quote above, the general quantum chemistry community has been slow to accept diagrammatic analyses of many-body perturbation theory and coupled cluster methods, and, until recently, these techniques have been used by relatively few researchers in the field. One of the goals of this review is to explain in straightforward terms one diagrammatic approach commonly used for the construction of coupled cluster equations. While attempting to be somewhat rigorous in the algebraic derivation of the coupled cluster equations, we present the corresponding diagrams with only minimal justification. For readers with a strong mathematical background who are interested in more detail, an extensive analysis of a similar diagrammatic technique may be found in the recent text by Harris, Monkhorst, and Freeman.[80]