Critical Nuclear Charge for Two-Electron Atoms

The critical nuclear charge Z(c) required to bind a nucleus plus two electrons in a heliumlike atom has recently been an area of active study, resulting in a disagreement with earlier calculations and with the value obtained from the radius of convergence 1/Z* of a 1/Z expansion of the energy. In order to resolve the disagreement, have performed high-precision variational calculations in Hylleraas coordinates. With the double basis set method, we have been able to obtain good convergence for Z very close to Z(c), which together with the Hellmann-Feynman theorem yields the value Z(c) = 0.911 028 224 077 255 73(4), corresponding to 1/Z(c) = 1.097 660 833 738 559 80(5). This value is in agreement with the value obtained by Baker et al. [Phys. Rev. A 41, 1247 (1990)]. A significant feature of the results is that the outer electron remains localized near the nucleus, even at Z = Z(c), and the bound state evidently changes smoothly into a shape resonance for Z < Z(c). A qualitative polarization potential is proposed to account for the resonance, and the radial distribution function for the electron density is calculated.