Coupled Cluster Theory

The theoretical framework of Coupled Cluster (CC) theory was developed in the late 1960s [39,40], but it was not until the late 1970s that the practical implementation began to take place and until 1982 that the corner stone of modern implementation, CCSD [41] (CC including all single and double excitations), was presented.

CC solves the size consistency problem of CI by forming a wavefunction where the excitation operators are exponentiated,
 $$\begin{align} \Psi_{CC} = \exp(\hat{T})\Psi_{0} \end{align}$$
where
$$\begin{align}\hat{T}=\hat{T}_{1}+\hat{T}_{2}+\hat{T}_{3}+\dots \end{align}$$
and $\hat{T}_{n}$ is a linear combination of all n-type excitations, for example,
 $$\begin{align} \begin{split} \hat{T}_{1}\Psi_{0} = \sum_{i}\sum_{a}C_{i}^{a}\Psi_... ...}\Psi_{0} = \sum_{i>j}\sum_{a>b}C_{ij}^{ab}\Psi_{ij}^{ab} \end{split}\end{align}$$
where C_i^a and C_ij^ab are the coefficients to be determined. Substituting equation (1.62) into equation (1.60) yields the CCSD wavefunction
$$\begin{align}\begin{split} \Psi_{CCSD} = \Psi_{0}+ & \sum_{a}\sum_{i}C_{i}^{a}\P... ...\sum_{k>l}C_{ab}^{ij}C_{cd}^{kl}\Psi_{ijkl}^{abcd}+\ldots \end{split}\end{align}$$
This reveals the advantage of CC theory: higher excitations are partially included, but their coefficients are determined by the lower order excitations. The coefficients are determined by projecting Schrödinger's equation on the left with the configurations generated by the $\hat{T}$ operator. This replaces the eigenvalue problem by a non-linear simultaneous system, requiring iterative solution. Luckily, convergence is fast in most cases [42].

As mentioned above, the addition of pure triple excitations is required for some chemical problems. However, CCSDT scales as O(N⁸), which is impractical for all but the simplest of systems. A more practical alternative is CCSD(T) [37] where the effect of triples is estimated through perturbation theory with a non-iterative O(N⁷) cost.

With a large enough basis set CCSD typically recovers 95% of the correlation energy for a molecule at equilibrium geometry, while CCSD(T) sees a further five- to ten-fold reduction in error [43]. With such accuracy CC has become the method of choice for accurate small-molecule calculations, even though the method is not variational (property 6, above).

A method closely related to CCSD is Brueckner Doubles (BD) [44], which uses the Brueckner orbitals [45] rather than the HF orbitals for a CCSD treatment. The Brueckner orbitals are defined as the set of orbitals for which the single excitation coefficients are zero. Finding these orbitals makes the theory slightly more computationally intensive (BD and BD(T) still scale as O(N⁶) and O(N⁷) respectively). However, BD theory promises a slight increase in accuracy above CCSD.