Heliumlike and lithiumlike ionic sequences: Critical charges

Abstract

In nonrelativistic 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 $Z=1,\phantom{\rule{0.222222em}{0ex}}2,...$ do not play a distinguished role as would be associated with charge quantization. By definition, a critical charge ${Z}_{\text{cr}}$ 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, ${Z}_{\text{cr},2e}=0.910850$ and ${Z}_{\text{cr},3e}=2.0090$, respectively. Based on numerical analysis, the Puiseux expansion in fractional powers of $(Z\ensuremath{-}{Z}_{\text{cr}})$ is constructed for both systems. Our results indicate the existence of a square-root branch point singularity at ${Z}_{\text{cr}}$ with exponent 3/2. A connection between the critical charge and the radius of convergence of $1/Z$ expansion is briefly discussed.