The Hartree-Fock self-consistent-field approximation has provided an invaluable conceptual framework and a standard computational procedure for atomic and molecular quantum theory. Its shortcomings are significant however, and require remediation. Moller-Plesset perturbation theory offers a popular correction strategy: it formally expands eigenfunctions and eigenvalues as power series in a coupling parameter lambda that switches the Hamiltonian continuously between the Hartree-Fock form (lambda=0) and the electron-correlating "physical" Hamiltonian (lambda=1). Recent high-order Moller-Plesset numerical expansions indicate that the series can either converge or diverge at lambda=1 depending on the chemical system under study. The present paper suggests at least for atoms that series convergence is controlled by the position of a singularity on the negative real lambda axis that arises from a collective all-electron dissociation phenomenon. Nonlinear variational calculations for the two-electron-atom ground state illustrate this proposition, and show that series convergence depends strongly on oxidation state (least favorable for anions, better for neutrals, better yet for cations).