On the application of Brillouin-Wigner perturbation theory to a relativistic and non-relativistic hydrogenic model problem

The application of Brillouin-Wigner perturbation theory to a hydrogenic model problem is described using both a relativistic and a nonrelativistic formalism. The nonrelativistic study is carried out by employing a basis set of Coulomb Sturmian functions whereas calculations within the relativistic formalism use basis sets of L-spinors. A two-state hydrogenic model problem provides a dramatic example of the dependence of the convergence behaviour of both the Brillouin-Wigner and the Rayleigh-Schrodinger perturbation expansions on the choice of basis set. Two types of Brillouin-Wigner perturbation expansion are compared in both the relativistic and the nonrelativistic formulations, the first using the exact energy in the denominator factors and the second employing total energies determined by a self-consistent procedure applied at finite order. Extrapolation procedures are also investigated. For the relativistic formulation the second-order energy is divided into two components, one arising from a sum over positive-energy states and the other corresponding to a sum over the negative-energy states. The application of correction terms, based on the known relation between the denominator factors occuring in Brillouin-Wigner perturbation theory and those in the Rayleigh-Schrodinger formulation, which restore linear scaling with particle number in many-body formulations, is investigated for the hydrogenic model problem.