Spherically symmetric states of Hookium in a cavity

When a two-electron atom or ion is enclosed in an impenetrable spherical cavity, level crossings and avoided crossings are observed as the cavity radius changes. The locations of those crossings depend on the cavity radius and the nuclear charge of the system. The question arises as to whether this crossing behavior is unique to the one-electron Coulomb potential or whether it can be observed in other confined single-particle electron potentials. In this work we examined some low-lying singlet and triplet states of two-electron systems with isotropic harmonic singleparticle 3D potentials. The spherically symmetric S states are analyzed using variational energies calculated with Hylleraas-type function. The energy dependence of low-lying states is considered as a function of the cavity radius and the harmonic force constant. For positive force constants, there exist cavity radii where the 2(1)S and 1(3)S states are degenerate. Analogous points do not exist for the two-electron quantum dot where the one-electron potential corresponds to an infinite rectangular box. The structure of the energy spectrum for negative force constants is also studied. The similarities and differences of the two-electron S states for the Coulomb and harmonic potentials are considered.