Invariance of the correlation energy at high density and large dimension in two-electron systems

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