New Partitioning Perturbation Theory. I. General Formalism

By the use of partitioning techniques, a general formalism is developed for considering degenerate, almost-degenerate, and electron-exchange perturbation problems. In effect, we generalize the Van Vleck–Kirtman approach to arbitrary orders and arbitrary normalization and obtain three types of approximations: In the modified Kirtman treatment the functions through the Nth order are fully normalized and the energies are obtained as the roots of the secular equation. The DE–FOP–VIM approximation is the same except that the normalization of the functions is energy optimized. The Kirtman approximation uses the same functions as the modified Kirtman but the energies are obtained as the roots of a much simpler secular equation which results from a factorization of the original secular equation (except for terms of order 2N + 2). The Kirtman energies are not upper bounds. Löwdin's formalism is equivalent to the Modified Kirtman with the exception that Löwdin uses intermediate normalization. Electron exchange problems are considered more explicitly in a companion paper with the use of symmetry considerations.