On the Pairing Theorem and Its Extension

Abstract

AbstractThe mirror theorem in matrix theory is briefly reviewed. It says that if R and S are matrices of the order m × n and nxm, then the two products RS and SR have the same nonvanishing eigenvalues with the same multiplicities and canonical structures. The mirror theorem is used to show that if a = {ak} and b = {bl} are two sets of orbitals of order m and n, respectively, then there exist two unitary transformations U and V such that the orbitals a′ = aU and b′ = bV have the special property <ak′ | bl′> = λkδkl, i.e., they are all orthogonal to each other unless they belong to the same pair having k = l. This pairing theorem is of fundamental importance in treating the Coulomb correlation in molecular calculations by means of the so‐called Alternant Molecular Orbital (AMO) method, which uses for the occupied orbitals different orbitals for different spins. A simple extension of this approach to the virtual orbitals is also shown.