Multiconfiguration Self-Consistent-Field Theory for Localized Orbitals. I. The Orbital Equations

Abstract

Variational equations are derived for the optimum orbitals for constructing a multiconfiguration expansion of the wave function for a many-electron system. Both the case where all Slater determinants are constructed from a common set of orbitals and the case where each determinant is constructed from an independent set of orbitals are considered. An equation with a single orbital operator is obtained in the former case; a separate operator for each determinant in the latter. The latter case corresponds to a physical picture in which the orbitals fluctuate as the system makes virtual transitions from one Slater determinant to another. The invariance properties of the wave function and of the variational equations with respect to linear transformations of the orbitals are analyzed, and various procedures are given for obtaining equations which select a unique set of orbitals. A prescription of Adams can be used in either case to localize the orbitals at a selected physical subsystem (an atomic shell, an atom, or a group of atoms). Another prescription can be used with a common set of orbitals to obtain an equation for natural spin orbitals.