  We study the properties of the Hooke's law correlation energy ($\Ec$),
defined as the correlation energy when two electrons interact {\em 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, which highlights the main differences of the $\Ec$ in
both the weakly and strongly correlated limits. Moreover, we show that the
Schr\"odinger equation of the spherium model is exactly solvable for two values
of the dimension ($D = 1 \text{and} 3$), and that the exact wave function is
based on Mathieu functions.
