Method for direct determination of localized orbitals

Abstract

The orthogonalized Hartree method is put forward as a method for directly determining localized orbitals in atoms and molecules. To obtain the orbitals in this method one minimizes the energy functional for a Hartree product of orthonormal orbitals. If desired, the exchange energy for these orbitals is then subtracted. It is shown rigorously that this method gives orbitals more exchange-localized than can be obtained from the Hartree–Fock method, at a small cost in total energy. Specifically, if K represents the sum of the interorbital exchanges and E the total energy (including exchange), then 0 ? KorthHartree ? KHartree–Fock and 0 ? EorthHartree − EHartree–Fock ? KHartree–Fock − KorthHartree. The Edmiston–Ruedenberg method of localization minimizes KHartree–Fock, and so the orthogonalized Hartree method yields a K even smaller than the Edmiston–Ruedenberg K. From numerical evidence which is presented, previous work of others, and additional arguments, it is reasoned that the orthogonalized Hartree method, without exchange energy included, may provide good predictions of conformational energy changes in molecules—it is the orthogonality of the orbitals, not the exchange energy itself, which is more important for the prediction of energy changes. The near classical structure of the equations of the orthogonalized Hartree method is discussed.