Corresponding Orbitals and the Nonorthogonality Problem in Molecular Quantum Mechanics

Given any two sets of spin orbitals ai and bj, there exist equivalent sets âi and b̂j such that their overlap matrix is diagonal, i.e., 〈âi | b̂j〉=d̂iiδij. 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 Löwdin's cofactor method for treating the nonorthogonality problem.