A Generalized Self-Consistent Field Method

A self-consistent field method is set up, general enough to be used in case we are treating configuration interaction between any number of states each represented by a single determinantal wave function. Thus it is more general than the Hartree-Fock method, which is limited to a single determinantal wave function. We cannot use a variation method, for if a sufficiently extensive configuration interaction is carried out, an equally good final value of the energy and wave function will be obtained irrespective of the one-electron functions used. Instead, we return to the original postulate of Hartree, that each electron is assumed to move in the averaged charge distribution of all other electrons, and all nuclei. It is shown that when this condition is properly interpreted, it leads to a unique potential for the self-consistent field, all the oneelectron orbitals being solutions of the same Schrödinger equation, and hence orthogonal to each other. This field is somewhat different from that of Hartree, who did not literally follow this prescription for finding the potential. For the case where we are using a single determinantal function, the present method reduces to the simplification of the Hartree-Fock method recently proposed by the writer.