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=1,2,... do not play a distinguished role as would be associated with charge quantization. By definition, a critical charge Zcr 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, Zcr,2e=0.910850 and Zcr,3e=2.0090, respectively. Based on numerical analysis, the Puiseux expansion in fractional powers of (Z−Zcr) is constructed for both systems. Our results indicate the existence of a square-root branch point singularity at Zcr with exponent 3/2. A connection between the critical charge and the radius of convergence of 1/Z expansion is briefly discussed.