Coupled Cluster - Cluster Operator

Cluster Operator

The cluster operator is written in the form,

,

where is the operator of all single excitations, is the operator of all double excitations and so forth. In the formalism of second quantization these excitation operators are conveniently expressed as

and so forth.

In the above formulae and denote the creation and annihilation operators respectively and i, j stand for occupied and a, b for unoccupied orbitals. The creation and annihilation operators in the coupled cluster terms above are written in canonical form, where each term is in normal order. Being the one-particle excitation operator and the two-particle excitation operator, and convert the reference function into a linear combination of the singly and doubly excited Slater determinants, respectively. Solving for the unknown coefficients and is necessary for finding the approximate solution .

Taking into consideration the structure of, the exponential operator may be expanded into Taylor series:

This series is finite in practice because the number of occupied molecular orbitals is finite, as is the number of excitations. In order to simplify the task for finding the coefficients t, the expansion of into individual excitation operators is terminated at the second or slightly higher level of excitation (rarely exceeding four). This approach is warranted by the fact that even if the system admits more than four excitations, the contribution of, etc. to the operator is small. Furthermore, if the highest excitation level in the operator is n,

then Slater determinants excited more than n times may (and usually do) still contribute to the wave function because of the non-linear nature of the exponential ansatz. Therefore, coupled cluster terminated at usually recovers more correlation energy than configuration interaction with maximum n excitations.