Hylleraas–Pekeris treatment of D-dimensional two-electron atoms

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Abstract
The algorithm of Pekeris for S states of two‐electron atoms is generalized to spaces of arbitrary dimensionality. Numerical calculations are reported for the ground state (1 1S) and first two excited states (2 3S and 2 1S) for a wide range of dimensions, 1<D<∞, and nuclear charge, 1≤Z≤6. The accuracy is typically better than one part in 108. The energy eigenvalues may be continued to arbitrary real values of the parameter δ=1/D. Real atoms, with D=3, connect smoothly with simple, exactly known limits at D→1 and D→∞. Analysis of the data permits several further terms in the 1/D expansion for the ground state energy to be determined, up to order D−12. This indicates that the expansion does not converge for D=3 but terms of third to sixth order do conform approximately to a geometric series form, as previously postulated in order to carry out dimensional interpolation. The excited state data exemplify near continuum motion at D→1 and quasivibrational asymmetric and symmetric stretching modes of electron motion as D→∞.