Helium- and Lithium-like ionic sequences: Critical charges

TL;DR

This study investigates Coulomb systems with a fixed nucleus and two or three electrons, identifying critical charges where bound states cease to exist, and analyzing the nature of singularities at these points.

Contribution

The paper determines the critical charges for helium- and lithium-like ions and analyzes the singularity structure of energy curves at these points.

Findings

Critical charge for two-electron system: 0.91085

Critical charge for three-electron system: 2.009

Energy curves are smooth with a square-root branch point at critical charges

Abstract

In non-relativistic quantum mechanics we study the Coulomb systems of infinitely massive center of charge Z and two-three electrons: $(Z,e,e)$ and $(Z,e,e,e)$. It is shown that in both cases the total energy curve in $Z$ is smooth, without any visible irregularities. Thus, for both systems the physical integer charges $Z=1,2,...$ do not play a distinguished role as would be associated with charge quantization. By definition, a critical charge $Z_{cr}$ is a charge which separates a domain of the existence of bound states from a domain of unbound ones (continuum). For both systems the critical charges are found, $Z_{cr,2e}=0.91085$ and $Z_{cr,3e}=2.009$, respectively. Based on numerical analysis, the Puiseux expansion in fractional powers of $(Z-Z_{cr})$ is constructed for both systems. Our results indicate the existence of a square-root branch point singularity at $Z_{cr}$ with exponent 3/2. A connection between the critical charge and the radius of convergence of 1/Z-expansion is briefly discussed.