Dimensional perturbation theory for excited states of two-electron atoms

Large-order dimensional perturbation theory, which yields high accuracy for ground-state energies, is applied here to excited states of the two-electron atom. Expansion coefficients are computed recursively using the moment method, which we formulate in terms of normal coordinates. We consider the first two excited S states of helium, corresponding, at the large-dimension limit, to one quantum in either the antisymmetric-stretch normal mode or the symmetric-stretch normal mode. Comparison with the hydrogenic limit has identified these states as 1s2s ${}_{}{}^3{}_{}{}^{}$S and 1s2s ${}_{}{}^1{}_{}{}^{}$S, respectively. We sum the 1/D expansions at D=3, using summation procedures that take into account the dimensional singularity structure of the eigenvalues, and find convergence at D=3 to the eigenvalues predicted by the hydrogenic assignments, despite apparent qualitative differences between the eigenfunctions at large D and those at D=3. In the D→∞ limit, the electrons are equidistant from the nucleus. Our results for 1s2s energies appear to imply that the shell structure is properly accounted for by terms in the expansion beyond the lowest order. This robustness of the 1/D expansion suggests that the method will be applicable to many-electron systems.