Three-body quantum Coulomb problem: analytic continuation

TL;DR

This paper analyzes the analytic structure of the ground state energy in the three-body quantum Coulomb system, predicting critical charges and excited states through analytic continuation and numerical methods.

Contribution

It provides precise calculations of the second critical charge and predicts an excited bound state of the negative hydrogen ion using analytic continuation.

Findings

Second critical charge Z_B^∞ = 0.904854 and Z_B^{m_p} = 0.905138.

Ground state energy has a square-root branch point at Z=Z_B.

Predicted excited state energy of H^- at -0.51554 a.u.

Abstract

The second (unphysical) critical charge in the 3-body quantum Coulomb system of a nucleus of positive charge $Z$ and mass $m_p$, and two electrons, predicted by F~Stillinger has been calculated to be equal to $Z_{B}^{\infty}\ =\ 0.904854$ and $Z_{B}^{m_p}\ =\ 0.905138$ for infinite and finite (proton) mass $m_p$, respectively. It is shown that in both cases, the ground state energy $E(Z)$ (analytically continued beyond the first critical charge $Z_c$, for which the ionization energy vanishes, to $Re Z < Z_c$) has a square-root branch point with exponent 3/2 at $Z=Z_B$ in the complex $Z$-plane. Based on analytic continuation, the second, excited, spin-singlet bound state of negative hydrogen ion H${}^-$ is predicted to be at -0.51554 a.u. (-0.51531 a.u. for the finite proton mass $m_p$). The first critical charge $Z_c$ is found accurately for a finite proton mass $m_p$ in the Lagrange mesh method, $Z^{m_p}_{c}\ =\ 0.911\, 069\, 724\, 655$.