Critical Nuclear Charge for Two-electron Atoms

There has been a recent revival of interest in the critical nuclear charge $Z_c$ that is just sufficient to bind a nucleus of charge $Z$ and two electrons in the $1s^2\;^1S$ ground state [1--3]. It is conjectured that the inverse of critical charge is related to the radius of convergence $1/Z^*$ for a $1/Z$ expansion of the energy of the form $E(Z) = Z^2(E_0 + E_1/Z + E_2/Z^2 + \cdots)$. We have performed high precision variational calculations in Hylleraas coordinates, using the double basis set method [4], for values of $Z$ very close to $Z_c$, with basis sets containing up to 2809 terms ($\Omega = 24$). Our preliminary result is $Z_c = 0.911\,028\,224\,077\,260(15)$, corresponding to $1/Z_c = 1.097\,660\,833\,738\,555(18)$. Well-defined eigenvalues continue to appear for $Z < Z_c$, possibly corresponding to quasibound states in the scattering continuum due to a shape resonance induced by the polarization potential of the core.\\[4pt] [1] J.D. Baker et al., Phys.\ Rev.\ A {\bf 41}, 1247 (1990).\newline [2] N.L. Guevara and A.V. Turbiner, Phys. Rev. A {\bf 84}, 064501 (2011).\newline [3] J. Katriel et al. Phys.\ Rev.\ A {\bf 86}, 042508 (2012).\newline [4] G.W.F. Drake and Z.-C. Yan, Phys.\ Rev.\ A {\bf 46}, 2378 (1992).