General Expression for the Interchange Theorem of Double Perturbation Theory

Abstract

The general expression for the interchange theorem in double perturbation theory is derived from the perturbation expansion. It gives a compact relation between two different formulations of the double perturbation energy ε(n,m). This relation will allow the exchange of integrals involving higher-order perturbed wavefunctions related to a complicated perturbation, by integrals involving higher-order perturbed wavefunctions related to the simpler perturbation. They can be applied, without qualifications, to the nondegenerate unperturbed state in any double perturbation problem. For cases where the unperturbed Hamiltonian and the more complicated perturbation are Hermitian, two more useful expressions of the interchange theorem are also obtained. These latter relations permit one to obtain the perturbation energy to (2n+1)th order with respect to the complicated perturbation from the knowledge of the wavefunction perturbed through the nth order. All the expressions of the interchange theorem we have obtained are valid to all orders with respect to both perturbations. They include Dalgarno's interchange theorem as the special lower-order cases.