Hooke's law correlation in two-electron systems

We study the properties of the Hooke's law correlation energy (Ec), defined as the correlation energy when two electrons interact via a harmonic potential in a D-dimensional space. More precisely, we investigate the 1S ground-state properties of two model systems: the Moshinsky atom (in which the electrons move in a quadratic potential) and the spherium model (in which they move on the surface of a sphere). A comparison with their Coulombic counterparts is made that highlights the main differences of the Ec in both the weakly and strongly correlated limits. Moreover, we show that the Schrödinger equation of the spherium model is exactly solvable for two values of the dimension (D=1 and 3) and that the exact wave function is based on Mathieu functions.