Analytic structure of ground-state energies and wave functions for the inhomogeneous electron liquid in non-relativistic He-like atomic ions with nuclear charge Ze

Abstract

The ground state of the He atom for fixed nucleus remains intractable so far as regards exact analytic solutions. However, some important results already exist pertaining to its ground-state wave function Ψ and corresponding electron density n(r). Here, we extend the existing studies by focussing attention on the non-relativistic series of He-like atomic ions with nuclear charge Z. We then find it instructive to start from the energy E(Z) of such a two-electron spin-compensated problem. This is known to have non-analytic behaviour at a critical Z, say Z c , equal to 0.911028. A form of Darboux transformation going back at very least to Brändas and Goscinski [E. Brändas and O. Goscinski, Int. J. Quantum Chem. 6, 59 (1972)] is refined somewhat here, and compared with a more intuitive approach of Callan [E. Callan, Int. J. Quantum Chem. 6, 431 (1972)]. The important 1/Z expansion of E(Z) is also invoked. The electron density n(r) and the ground-state wave function Ψ are then treated in turn, in a related manner; especially their asymptotic behaviour far from the nucleus. Finally, two exact wave functions for analytically solvable two-Fermion models are shown to sum the infinite series proposed by Fock [V. Fock, Izv. Akad. Nauk SSSR, Ser. Fiz. 18, 161 (1954)].