Møller-Plesset perturbation theory expresses the energy as a function E(z) of a perturbation parameter, z. This function contains singular points in the complex z-plane that affect the convergence of the perturbation series. A review is given of what is known in advance about the singularity structure of E(z) from functional analysis of the Schrödinger equation, and of techniques for empirically analyzing the singularity structure using large-order perturbation series. The physical significance of the singularities is discussed. They fall into two classes, which behave differently in response to changes in basis set or molecular geometry. One class consists of complex-conjugate square-root branch points that connect the ground state to a low-lying excited state. The other class consists of a critical point on the negative real z-axis, corresponding to a dissociation phenomenon. These two kinds of singularities are characterized and contrasted using quadratic summation approximants. A new classification scheme for Møller-Plesset perturbation series is proposed, based on the relative positions in the z-plane of the two classes of singularities. Possible applications of this singularity analysis to practical problems in quantum chemistry are described.
ASJC Scopus subject areas
- Physical and Theoretical Chemistry