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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 theymove on the surface of a sphere), and hookium (in which they move in aquadratic potential). We find that the $\Ec$ values are strikingly similar,depending strongly on $D$ but only weakly on the external potential. Weconjecture that, for large $D$, the limiting correlation energy $\Ec \sim-\delta^2/8$ in any confining external potential, where $\delta = 1/(D-1)$.

Correlation energy of two electrons in the high-density limit