Analytic properties of the exact energy of the ground state of a two-electron atom as a function of 1/Z.

Abstract

Analytic properties of the ground-state energy of a two-electron atom as a function of \ensuremath{\lambda}=1/Z are studied. In addition to the previously known singular point of this function, ${\ensuremath{\lambda}}_{\mathit{s}}$\ensuremath{\approxeq}1.097 660 79, we find a new singular point ${\ensuremath{\lambda}}_{\mathrm{\ensuremath{\infty}}}$=\ensuremath{\infty} in the \ensuremath{\lambda} complex plane. We show that the function E(\ensuremath{\lambda}) has a branch-point singularity of the type ${\ensuremath{\lambda}}^{\ensuremath{\gamma}}$ at this point, where the exponent \ensuremath{\gamma}=2.06\ifmmode\pm\else\textpm\fi{}0.005. We find that the ansatz previously proposed to reproduce the asymptotic behavior of the large-order coefficients of the perturbation expansion [Baker et al., Phys. Rev. A 41, 1247 (1990)] can, with slight modification, reproduce the behavior of the exact ground-state energy of the two-electron atom around this singular point. We propose a representation of the exact ground-state energy of helium, possessing the required analytic properties.