Two-electron atoms near the one-dimensional limit

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Abstract
If the Hamiltonian of a two-electron atom is generalized in a natural way to arbitrary spatial dimension D, an especially simple case is found in the D=1 limit. While the ground state energy is singular at this point, a scaling argument reduces the problem to a limiting Hamiltonian with only two degrees of freedom in which the Coulombic potentials all reduce to δ functions. Since the singularity at D=1 dominates the energy at nearby dimensions, this limit forms the basis for an expansion in (D−1)/D which is reasonably accurate at D=3. By combining results from this expansion with the 1/D expansion about the D→∞ limit, estimates of the energy at D=3 are obtained with accuracy orders of magnitude better than that of either series alone. The simplicity of the D=1 and large-D limits and the accuracy of this method allow some qualitative insight into the physical features contributing to correlation effects in small atoms. Analysis of other singularities suggests that the 1/D series has zero radius of convergence for two-electron atoms. We conclude with a discussion of excited states and larger atoms and make some appealing connections with the orbital picture.