Hartree–Fock approximation for D-dimensional two-electron atoms

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Abstract
The Hartree–Fock method for two-electron atoms is generalized to spaces of arbitrary dimensionality. The problem is exactly soluble in two limiting cases, D→1 and D→∞, for any value of the nuclear charge Z. Numerical calculations of the ground-state energy are reported for a wide range of D and for Z=1 to 6, with an accuracy typically better than one part in 1010. Together with previous variational calculations employing the Pekeris method, these results permit the correlation energy to be evaluated with an accuracy typically better than one part in 106. The correlation energy is found to be largely independent of Z for any D. For a given Z, the correlation energy decreases smoothly as D increases; to a good approximation it is simply linear in 1/D. However, the correlation energy remains appreciable even in the limit D→∞; as a fraction of the total energy, for Z=2 the correlation energy varies from 2.28% at D=1 to 1.45% at D=3 to 0.99% as D→∞.