Corresponding Orbitals and the Nonorthogonality Problem in Molecular Quantum Mechanics

Abstract

Given any two sets of spin orbitals ai and bj , there exist equivalent sets Iii and bj such that their overlap matrix is diagonal, i.e., (Iii I bi) =diiOij. This is the basis of the corresponding orbital transformation of Amos and Hall. Their transformation is shown to have widespread application to quantum chemistry. It leads to a simple generalization of the Slater-Condon rules for the expectation value of an operator between two determinantal wavefunctions when the spin orbitals of one function have no simple orthogonality relationship to those of the other function. In the case of single,determinantal wavefunctions, use of the corresponding orbital transformation and the integral Hellmann-Feynman formula leads to a very simple expression for the energy difference associated with two similar configurations of a molecular system. Extensions to limited configuration interaction expansions are discussed. Given single-determinantal wavefunctions for two related molecular systems, it is shown that the corresponding orbitals are those which are most nearly molecularly invariant in the sense of maximum overlap. A comparison of the Pitzer-Lipscomb wavefunctions for the staggered and eclipsed forms of ethane reveals that six of the nine corresponding orbitals have an overlap of no less than 0.999998 in the two configurations. Use of the corresponding orbital transformation overcomes various computational difficulties encountered with LOwdin's cofactor method for treating the nonorthogonality problem.

Block-Localized Density Functional Theory (BLDFT), Diabatic Coupling, and Their Use in Valence Bond Theory for Representing Reactive Potential Energy Surfaces

“…However, H uw is not a natural functional of the electron density, but it is a functional of both the electron densities ρ u ( r ) and ρ w ( r ) of states Ψ u and Ψ w . To evaluate the coupling matrix element using DFT, we make a comparison with the matrix element H uw from two nonorthogonal wave functions determined using the MOVB theory and in other calculations 3,13,14,21,55. First, we define the one-particle exchange (or transition) density matrix and the exchange (transition) electron density as follows $${\mathrm{boldD}}_{\text{italicuw}}={\mathrm{boldC}}^w{({\mathrm{boldS}}^{\text{italicuw}})}^{-1}({\mathrm{boldC}}^u{false)}^{\mathrm{normalT}}={\mathrm{boldC}}^w[({\mathrm{boldC}}^u{false)}^{\mathrm{normalT}}{\text{boldRC}}^w{false]}^{-1}({\mathrm{boldC}}^u{false)}^{\mathrm{normalT}}$$ $${\rho }_{\text{italicuw}}(\mathbf{r})={\displaystyle \sum _{\mu v}^m|{\chi }_{\mu }false(\mathrm{boldr}false)>false({\mathbf{D}}_{\mathit{\text{uw}}}{)}_{\mu v}<{\chi }_{v}false(\mathrm{boldr}false)|=\chi (\mathbf{r}){\mathrm{boldD}}_{\text{italicuw}}{\chi }^{\mathrm{normalT}}(\mathbf{r})}$$ …”

Calculation of transition matrix elements by nonsingular orbital transformations

“…This procedure is usually referred to as the corresponding orbital transformation [35,36]. By drawing a comparison between (51) and (53) we obtain…”

Superposition of nonorthogonal Slater determinants towards electron correlation problems

“…Corresponding orbitals have been employed for computing the matrix elements in nonorthogonal CI and transition moment calculations [22]. In the full CI (CAS) case, the model is invariant to unitary transformations of orbitals.…”