Multiconfiguration self-consistent-field theory for localized orbitals. II. Overlap constraints, Lagrangian multipliers, and the screened interaction field

Abstract

The general, standard, and localized forms of the multiconfiguration self‐consistent‐field orbital equations, with and without overlap constraints, are derived and compared. The equations are very similar in form to the corresponding single configuration (Hartree‐Fock) equations. The constrained equations may be formulated in terms of a single Hermitian orbital operator with a Lagrangian multiplier matrix which is always Hermitian. The equations for orbitals localized on an embedded fragment (an atomic shell, bond, atom, molecular fragment, or other subsystem that is part of a larger system) differ from the orbital equations for an isolated fragment only by the addition of a screened interaction field. Dropping the orbital overlap constraints introduces an additional condition on the Lagrangian multipliers that must be satisfied in order for the energy to be stationary with respect to variations in the overlap matrix.