Three-body quantum Coulomb problem: analytic continuation

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 $ReZ<Z_c$) 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, $Z_c^{m_p}=0.911069724655$.

12 pages, 1 figure, 3 tables: title changed and Figure modified, several explanatory sentences added, text improved for better understanding, some typos fixed, to be published at Mod Phys Lett A

References in corpus (2)

• Nuclear critical charge for two-electron ion in Lagrange mesh method
• On $1/Z$ expansion, the critical charge for two-electron system and the Kato theorem

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