Factorization of the Direct CI Coupling Coefficients into Internal and External Parts

Abstract

The direct CI coupling coefficients are factorized into a simple external part and a complicated internal part in the case of all single and double replacements from any set of reference configurations. The internal orbitals are the orbitals occupied in the reference configurations and are generally much fewer than the external orbitals. The external coupling coefficients are entirely determined in the external orbital space and are therefore independent of the choice and type of reference configurations. Explicit simple formulas for all possible external coupling coefficients are listed with the only assumption on the configurations that the external orbitals are spin-coupled first. These coefficients can be quickly evaluated as they appear in the diagonalization of the CI-matrix and therefore do not need to be stored, which solves the severe storage problem appearing in conventional CI-procedures. The internal coupling coefficients need to be stored but are always very few. The method is completely general in the selection of reference configurations. In a different CI-approach discussed in this paper a single reference state is used, which is composed of a linear combination of a large number of configurations, up to the order of thousands. Single and double replacements are constructed by applying the generators of the unitary group on the reference state. The simple factorization of the coupling coefficients appears also in this method with the same values for the external coupling coefficients. It is shown that the factorization can be used to greatly simplify the treatment of the matrix elements in this method by constructing intermediate quantities in two-step procedures.