Helium- and Lithium-like ionic sequences: 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.

10 pages, LaTeX, typos corrected, Fig.1 added, a Note Added with calculated critical charge for $2^{1}S$ state for $(Z,e,e)$ system, $Z_{cr,2e}^{(2^{1}S)}=1.02$

References in corpus (4)

• Ground state wave function and energy of the lithium atom
• Analytic continuation of eigenvalues of a quartic oscillator
• Ground state of Li and Be${}^{+}$ using explicitly correlated functions
• An accurate few-parameter ground state wave function for the Lithium atom

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• Helium-like and Lithium-like ions: Ground state energy
• Ground state and polarization of an hydrogen-like atom near a Weyl semimetal
• Helium-like ions in $d$-dimensions: analyticity and generalized ground state Majorana solutions