We study the properties of the Hooke's law correlation energy (Er), 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 E_c in both the weakly and strongly correlated limits. Moreover, we show that the Schrodinger 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.
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