Nonlinear variational study of perturbation theory for atoms and ions

Abstract

Using the $\frac{1}{Z}(=\lambda )$ perturbation format, analytical properties of eigenstates in the complex $\lambda$ plane have been studied for two- and three-electron atoms, using simple nonlinear variational calculations. The ground states for both systems are included, as well as four excited states for two electrons. The results suggest a general partitioning of bound states into two categories: (i) If the singly charged anion is bound, analytical continuation in $\lambda$ along the positive real axis causes the energy to penetrate the continuum, and subsequently to terminate at a branch point. (ii) If the singly charged anion is unbound, the energy is tangent to the continuum edge, and analytic continuation in $\lambda$ to larger positive values creates a non-normalizable wave function.