Critical Nuclear Charge of the Quantum Mechanical Three-Body Problem

The critical nuclear charge ($Z_c$) for a three-body quantum mechanical system consisting of positive and negative charges is the minimum nuclear charge that can keep the system in a bound state. Here we present a study of the critical nuclear charge for two-electron (heliumlike) systems with infinite nuclear mass, and also a range of reduced mass ratio ($\mu/m$) up to 0.5. The results help to resolve a discrepancy in the literature for the infinite mass case, and they are the first to study the dependence on reduced mass ratio. It was found that $Z_c$ has a local maximum with $\mu/m=0.352\:5$. The critical charge for the infinite mass case is found to be $Z_c = 0.911\:028\:224\:076\:8(1\:0)$. This value is more accurate than any previous value in the literature [1, 2, 3, 4], and agrees with the upper bound $Z_c=0.911\:03$ reported by Baker et al.\ [1]. The critical nuclear charge outside this range [0.5 -- 1.0] still needs to be investigated in future works.\\[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] F. H. Stillinger Jr.\ J. Chem.\ Phys.\ {\bf 45}, 3623 (1966).\newline [4] G. A. Arteca et al.\ J. Chem.\ Phys.\ {\bf 84}, 1624-1628 (1986).