Valence universal exponential ansatz and the cluster structure of multireference configuration interaction wave function

A rigorous algebraic formulation of open-shell coupled-cluster theory is presented. This formulation yields explicit formulas exhibiting the relationship between open-shell cluster amplitudes and linear coefficients of multireference CI wave functions. When the valence-universal exponential ansatz is used, the CI coefficients of states with n valence electrons contribute to the n-body and higher-order cluster operators. The implications of cluster conditions, requiring that the four-body cluster amplitudes be small, are investigated. It is shown that for valence-universal theories these conditions lead to approximate relations involving CI coefficients for states of systems differing in the number of electrons. For Lindgren’s ansatz these relations are linear in the CI coefficients corresponding to states with the largest electron number. For the valence-nonuniversal exponential ansatz of Jeziorski and Monkhorst, the cluster conditions do not mix wave functions for systems which differ in the number of electrons and are formally identical to those of the single-reference coupled-cluster theory. A detailed relationship between the cluster amplitudes of the valence-universal and valence-nonuniversal theories is also derived and discussed.