Dimensional expansions for two-electron atoms

Abstract

Approximate expansions in inverse powers of the dimensionality of space D are obtained for the ground-state energies of two-electron atoms. The method involves fitting polynomials in δ=1/D to accurate eigenvalues of the generalized D-dimensional Schrödinger equation. To the maximum order obtainable from the data, about δ7, the power series for nuclear charges Z=2, 3, and 6 all diverge at D=3. Asymptotic summation yields an energy for the Z=2 atom 1% in excess of the true value at D=3. However, expansions with a shifted origin, i.e., expansions in (δ−δ0), show improved convergence. Of particular interest is the case δ0=1, because the expansion coefficients can in principle be calculated by perturbation theory applied to the one-dimensional atom. Series in powers of (δ−1) appear to converge rapidly. Also the series in (δ−1) can be evaluated even for the hydride ion, with Z=1. For helium, this series is quite comparable to the more familiar expansion in powers of λ=1/Z, with errors in the partial sums decreasing by roughly an order of magnitude per term. Thus, for Z=2 the first four terms of the expansion in (δ−1) yield an energy within 0.02% of the true value at D=3. Similar results are found in an analogous treatment of accurate eigenvalues for the Hartree–Fock approximation. This provides a rapidly convergent dimensional expansion for the correlation energy.