  We prove that, in the large-dimension limit, the high-density correlation
energy $\Ec$ of two opposite-spin electrons confined in a $D$-dimensional space
and interacting {\em via} a Coulomb potential is given by $\Ec \sim -1/(8D^2)$
for any radial confining potential $V(r)$. This result explains the observed
similarity of $\Ec$ in a variety of two-electron systems in three-dimensional
space.
