Does Møller–Plesset perturbation theory converge? A look at two‐electron systems

We study the convergence or divergence of the Møller–Plesset Perturbation series for systems with two electrons and a single nucleus of charge Z > 0. This question is essentially to determine if the radius of convergence of a power series in the complex perturbation parameter λ is greater than 1. The power series is centered at λ = 0, so we try to find whether or not the singularity closest to λ = 0 is inside the closed unit disk in the complex plane. We give a description of possible causes for divergence in the general problem and then examine two Helium‐like models. The first model is a simple one‐dimensional model with delta functions in place of Coulomb potentials. The second is the realistic three‐dimensional model. For each model, we show rigorously that if the nuclear charge Z is sufficiently large, there are no singularities for real values of λ between −1 and 1. Using a finite difference scheme, we present numerical results for the delta function model. The numerics are consistent with proven results and also suggest that the closest singularity occurs where λ is real and negative. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009