The density matrix in self-consistent field theory - III. Generalizations of the theory

Abstract
The self-consistent field theory of a previous paper (McWeeny 1956a) is generalized so as to apply to a system in which some orbitals are doubly occupied and others singly. Starting from an initial approximation, the density matrices for both the closed shell and the 'open' shell are simultaneously improved by an iterative method until self-consistency is achieved. A second generalization deals with the case in which several shells can profitably be distinguished, not by a difference of orbital occupation numbers but by virtue of their weak coupling (e.g. the atomic K, L, M shells of argon). In this case it is best to consider each shell individually (their number and type now being unrestricted), achieving approximate self-consistency in one shell at a time. The nature of the solutions is discussed. In cases of high symmetry, degeneracy occurs and a single determinant is not an acceptable state function; at the same time the self-consistent solution has awkward symmetry properties. This latter difficulty can be avoided with only slight loss of accuracy by applying suitable constraints. And the wider use of constraints, as a means of facilitating solution, is discussed. Finally, it is shown how, having obtained self-consistent density matrices, the orbitals themselves may be extracted in readiness for a configuration interaction calculation; and the degenerate case, in which interaction is essential, is briefly discussed.

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