  We consider the high-density-limit correlation energy $\Ec$ in $D \ge 2$
dimensions for the $^1S$ ground states of three two-electron systems: helium
(in which the electrons move in a Coulombic field), spherium (in which they
move on the surface of a sphere), and hookium (in which they move in a
quadratic potential). We find that the $\Ec$ values are strikingly similar,
depending strongly on $D$ but only weakly on the external potential. We
conjecture that, for large $D$, the limiting correlation energy $\Ec \sim
-\delta^2/8$ in any confining external potential, where $\delta = 1/(D-1)$.
