Avoided Crossings and Resummation of Nearly Resonant Molecular Vibrations: Reconstruction of an Effective Secular Equation

摘要

Nondegenerate Rayleigh-Schrodinger perturbation theory diverges when applied to nearly resonant molecular vibrations. Traditionally, this divergence has been avoided by using quasidegenerate perturbation methods, such as partitioning, or Van Vleck's transformation approach. In this paper, we present a method of resummation, called reconstruction, that is appropriate for nearly degenerate systems displaying a strong mixing between a finite number of zeroth order states. In order to implement the approach, a set of strongly interacting states must be identified. The method then produces a pseudoconvergent power series expansion for an effective secular equation from the divergent perturbation expansion of the energies of the strongly interacting states. The effective secular equation has a low polynomial degree and may readily be solved numerically to yield resummed values of the energies. We compare reconstruction to diagonal Pade' approximation, and find that Pade' approximants resume energies in the vicinity of avoided crossings poorly, yielding levels that cross where the actual eigenvalues repel. Reconstruction, however, correctly reproduces the avoided crossing behavior. Reprints. (AW)