Variationally deorthogonalized localized molecular orbitals

Abstract

The Edmiston–Ruedenberg localized orbitals are shown to be solutions to the Hartree product of orthogonal orbital equations on the space of the occupied Hartree–Fock orbitals. The nonorthogonal Hartree orbitals on this space are thus a possible deorthogonalization of the localized orbitals. From the relationships of the Pauli principle to the orthogonality of the orbitals in the Hartree product wave function and to the full antisymmetry of the Hartree–Fock wave function, we propose an energy functional which imposes the Pauli principle on the Hartree product wave function without forcing the orbitals to be orthogonal. Minimization of this functional leads to the Pauli-constrained Hartree equations. An algorithm is developed for their solution on the space of the Hartree–Fock orbitals. The results are called the variationally deorthogonalized Hartree product orbitals. This algorithm is then extended to apply to other localization functionals. Variationally deorthogonalized Edmiston–Ruedenberg orbitals are obtained by maximizing a function of the sum of the orbital self-repulsions, and variationally deorthogonalized Boys localized orbitals are obtained by maximizing a function of the sum of the squares of the distances between orbital centroids. All three types of variationally deorthogonalized orbitals are obtained for H2O, cis- and trans-N2H2, C2H2, C2H4, eclipsed and staggered C2H6, BH3, and B2H6. Comparisons are made in terms of the population distributions and the magnitudes and phases of the overlap integrals.