Dimensional interpolation for two-electron atoms

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Abstract
The ground-state electronic energy of helium-like atoms is calculated by interpolating between exact solutions for the limiting cases of one-dimensional and infinite-dimensional systems. In suitably scaled coordinates, the D=1 limit corresponds to replacing the Coulombic potentials by delta functions. With another choice of scaling, the D=∞ limit reduces to a classical electrostatic problem; the electrons take fixed positions that correspond to replacing wave functions with delta functions. The ground-state energy for arbitrary dimension D and nuclear charge Z can be represented as εD=ε∞[1+c1/D+c2/D2+F(D,Z)/D3]. The first three terms are obtained from a perturbation expansion about D=∞; these correspond, respectively, to the limiting rigid electronic structure (envisioned by G. N. Lewis!) and to harmonic and anharmonic vibrations about that structure. The interpolation function F(D,Z) is approximated as a geometric series, determined by fitting the accurately known energies for the D=1 limit and for D=5. This procedure yields an accuracy of 2 parts in 105 or better for D=3 and Z≥2. Study of the Z dependence reveals that at D=1 the perturbation expansion in powers of 1/Z has a self-similar structure. Also, for each D there exists a critical nuclear charge Z*, with magnitude near 12, for which F(D,Z*)=0; the first three terms of the expansion about D=∞ then give the exact energy.