Abstract
The second (unphysical) critical charge in the three-body quantum Coulomb system of a nucleus of positive charge [Formula: see text] and mass [Formula: see text], and two electrons, predicted by Stillinger has been calculated to be equal to [Formula: see text] and [Formula: see text] for infinite and finite (proton) mass [Formula: see text], respectively. It is shown that in both cases, the ground state energy [Formula: see text] (analytically continued beyond the first critical charge [Formula: see text], for which the ionization energy vanishes, to [Formula: see text][Formula: see text][Formula: see text][Formula: see text][Formula: see text]) has a square-root branch point with exponent 3/2 at [Formula: see text] in the complex [Formula: see text]-plane. Based on analytic continuation, the second, excited, spin-singlet bound state of negative hydrogen ion H[Formula: see text] is predicted to be at −0.51554 a.u. (−0.51531 a.u. for the finite proton mass [Formula: see text]). The first critical charge [Formula: see text] is found accurately for a finite proton mass [Formula: see text] in the Lagrange mesh method, [Formula: see text].
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