Simultaneous calculation of several interacting electronic states by generalized Van Vleck perturbation theory

Abstract

We present an accurate many‐body perturbation theory for the simultaneous calculation of an arbitrary set of interacting electronic states based on the generalized Van Vleck method. This treatment yields an effective Hamiltonian for the interacting states (or model space) which is Hermitian and energy‐independent. A special provision is made for external states that are energetically adjacent to the model space. The perturbation formulas for the effective Hamiltonian matrix contain ordinary Rayleigh–Schrödinger type energy denominators which include zeroth‐order splitting between the various interacting states. These energy denominators factor in all orders just as they do in the nondegenerate problem. A convenient diagrammatic representation of the perturbation expansion is developed. In carrying out the expansion there is some freedom associated with the orthonormality condition for states in the model space. We make the simplest possible choice with regard to evaluation of the perturbation formulas. This leads to a fully linked cluster expansion through fourth order. In fifth‐order unlinked clusters begin to appear off‐diagonal. They are few in number and very easily calculated. To illustrate the treatment we consider a model space containing only single substitution configurations. For a space of dimension d one has to compute on the order of d 2 off‐diagonal matrix elements of the effective Hamiltonian. However, the computational effort to obtain the entire set is roughly comparable to that required for a single diagonal element. Furthermore, the most extensive perturbation sums that occur in the diagonal elements are exactly the same for each state and need be computed only once. These sums also appear in an ordinary Rayleigh–Schrodinger ground state calculation.