Partitioning Method and Van Vleck's Perturbation Theory

Abstract

1641 Energy expressions to fourth order are-given for a degenerate zeroth order state by the use of a partitioning process in relation to Van Vleck's transformation. Double perturbation is discussed as a natural extention of this treatment. The mathematical essence of Dalgarno's interchange theorem is reinterpreted. By using this interpretation, it is demonstrated that there are more than one accessible correlation expressions of the interchange theorem for degenerate zeroth order states. It is also discussed whether or not the difficulty in obtaining the interchange theorem for a degenerate case remains. § 1. Introduction In parallel with the Schrodinger and Heisenberg representations in quantum mechanics, there are two different approaches to perturbation theory. One is the direct Rayleigh-Schrodinger approach/' the other is the use of canonical transformation, 2 '' 3 ' The latter often called the method of contact tarnsformation by spectroscopists, has been conveniently utilized for treating vibration and rotation Hamiltonians.'> This method has been reviewed by Primas 8 > in operator form and extensively used by Robinson. 5 > Unfortunately, it is often difficult to find a proper. ·transformation for the entire spectru~ of a given Hamiltonian H. In connection with perturbation theory, however, the Van Vleck transformation 2 " 8 ' allows us to concentrate on a particular zeroth order energy and enables us to avoid unnecessary complexity. The first object of this article is to provide a resolvent form of the operator treatment of Van Vleck's perturbation theory, which gives a simple and compact form of the entire approach in a highly transparent way. 3 '' 7 ' The simple and lucid feature of this operator approach is particulary outstanding in the degenerate perturbation theory. 8 > The crux of this approach to Van Vleck's perturbation theory lies in the discovery of the recursion formula (3 ·10) which allows us to generate wave functions and energies corresponding to any required order of perturbation. Lowdin has given the relations, between various approaches to perturbation and his partitioning technique in a series of papers on perturbation theory. 7 '' 8 ' His treatment of perturbation has been primarily for nondegenerate cases. The author has treated degeneracies in the Schrodinger perturbation