Linked-Diagram and Coupled-Cluster Expansions for Multi-Configurational, Complete and Incomplete Model Spaces

The coupled-cluster or exp S formalism, where the wave operator is expressed in exponential form, is treated for a general, multi-Configurational model space. It is shown that the cluster operator, S, is rigorously connected when the model space is complete or "quasi-complete" in the sense that it contains all configurations that can be formed by distributing the valence electrons within certain groups of valence orbitals with given occupation number in each group. For this class of model spaces also the linked-diagram theorem is valid in the sense that the diagrams of the wave operator and the effective Hamiltonian do not contain any separate, closed part. The diagrams of the effective Hamiltonian are connected, while those of the wave operator may contain disconnected, open parts.

For a more general, multi-configurational (incomplete) model space a formal expansion of coupled-cluster type is still possible, but it is found that the cluster operator is no longer necessarily connected, which leads to unlinked diagrams in the expansion of the wave operator and the effective Hamiltonian. A general procedure for generating the cluster operator in this case is described and applied particularly in the pair approximation.