Convergence properties of the Brillouin-Wigner type of perturbation expansion

Abstract

We study the Brillouin-Wigner perturbation expansion of the model-space effective Hamiltonian corresponding to the full Hamiltonian H( x) = H 0 + xH 1, H 0 and H 1 being respectively the unperturbed and the interaction Hamiltonian and x being a strength parameter. The radius of convergence for the perturbation expansion is related to the poles of the energy-dependent effective interaction, and the location of these poles in the complex x-plane is discussed. The situation with poles lying off the real x-axis is examined. In terms of the spectrum of the unperturbed Hamiltonian H 0, some necessary conditions for convergence are derived, and the effects of intruder states are discussed. It is shown that the BW expansion of the ground-state energy can always be made convergent by a shift of the unperturbed energy spectrum.