Dimension dependence of correlation energies in two-electron atoms

Abstract

Correlation energies (CEs) for two-electron atom ground states have been computed as a function of the dimensionality of space D. The classical limit D→∞ and hyperquantum limit D→1 are qualitatively different and especially easy to solve. In hydrogenic units, the CE for any two-electron atom is found to be roughly 35% smaller than the real-world value in the D→∞ limit, and about 70% larger in the D→1 limit. Between the limits the CE varies almost linearly in 1/D. Accurate approximations to real CEs may therefore be obtained by linear interpolation or extrapolation from the much more easily evaluated dimensional limits. We give two explicit procedures, each of which yields CEs accurate to about 1%; this is comparable to the best available configuration interaction calculations. Steps toward the generalization of these procedures to larger atoms are also discussed.