The expectation values of radial and angular quantities for two-electron atoms at the critical nuclear charge where the ground state ceases to exist are calculated employing the Hylleraas-configuration interaction basis functions. The radial quantities achieve better convergence than previous predictions and accurate angular quantities are reported for the first time. Based on these quantities, the geometric structure of the system is examined to support the criteria that the critical behavior of the system can be modeled by the inner and outer electrons separately. The work of King et al. [Phys. Rev. A 93 022509 (2016)] has shown that the inner-electron probability density distribution closely resembles that of a hydrogenic atom. Here we further show that the outer electron can be reasonably modeled by a shifted exponential polarization potential with soft truncation in the short range. The model potential proposed here reproduces very well the radial expectation value of the outer electron as well as the peak position, maximum magnitude, and long-range asymptotic behavior of the outer-electron radial density distribution.
Critical stability of the metastable bound 2p2 3Pe state of two-electron atoms near the critical nuclear charge Zc (smaller than which the bound state ceases to exist) is investigated by employing the explicitly correlated Hylleraas-configuration interaction basis function. Our numerical calculation shows that, when the nuclear charge decreases to Zc, the ionization energy of the system approaches zero linearly and this behavior is fully consistent with the Hellmann-Feynman theorem. Radial and angular physical quantities as well as the inner and outer electron radial density distributions are calculated to reveal the classical geometric structure of the two-electron atom. The inner electron can be well modeled by a hydrogenic model in the 2p state and its density distribution does not change much as the nuclear charge is changed. The outer electron, which is still localized in a finite region near the nucleus before the system transforms into a well-defined resonance, is shifted notably into a far-distance region. The critical stability of the system near the bound-continuum limit is therefore dominated by the outer electron.
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