Binding energies of the lithium isoelectronic sequence approaching the critical charge

Abstract

The Simon-Zhislin-Hunziker theorem implies that ${Z}_{c}$, the critical charge below which the three electron atom is not bound, is at most 2. The vanishing electron affinity of He implies that ${Z}_{c}$ is not less than 2. Hence, ${Z}_{c}=2$. To elucidate the approach to the critical charge, we calculated nonrelativistic binding energies for the third electron in the ground state, $1{s}^{2}2s{\phantom{\rule{0.16em}{0ex}}}^{2}S$, and in the first and second excited states, $1{s}^{2}2p{\phantom{\rule{0.16em}{0ex}}}^{2}P$ and $1{s}^{2}3s{\phantom{\rule{0.16em}{0ex}}}^{2}S$, for nuclear charges approaching ${Z}_{c}$. At this limit the quantum defects for both ${}^{2}S$ states are found to approach unity. This implies that the orbital specifying the outer ($ns,\phantom{\rule{0.28em}{0ex}}n=2,3$) electron becomes a very diffuse $(n\ensuremath{-}1)s$-type orbital, except within the relatively tiny space occupied by the inner two-electron shell. For the ${}^{2}P$ state the quantum defect approaches zero both as $Z\ensuremath{\rightarrow}\ensuremath{\infty}$ and as $Z\ensuremath{\rightarrow}2$. An expression for the $s$-$p$ splitting at $Z\ensuremath{\rightarrow}2$ is suggested, that improves upon earlier results based on energies computed (or measured) at integer values of $Z$. Rigorous large $Z$ asymptotic expressions for the quantum defects in the $1{s}^{2}ns{\phantom{\rule{0.16em}{0ex}}}^{2}S$ states are presented, exhibiting the expected mild dependence on the principal quantum number.