Atomic energies from the large-dimension limit

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Abstract
Analytic approximations to nonrelativistic atomic ground state energies are obtained from the first two terms of the 1/D expansion for the N‐electron atom. These two terms describe the equilibrium structure (D→∞ limit) and normal mode oscillations (1/D term) of a completely symmetric N‐dimensional configuration of localized particles. The connection between these large‐D results and real atoms is established through the vibrational state, which is restricted by antisymmetry requirements at D=3. Convergence considerations lead us to consider three different approximations, depending on whether all, none, or part of the results obtained from the 1/D term are used (in addition to those obtained from the D→∞ limit); the maximum errors are respectively about 8%, 3%, and 1%. In all three approximations the dependence of neutral atom energies on the nuclear charge Z is roughly Z12/5 for physical Z (as observed for real atoms) and roughly Z7/3 for very large Z (in agreement with the known asymptotic result). The best approximation, which utilizes the 1/D term up to lowest nonvanishing order in 1/Z, is comparable in accuracy to single‐ζ Hartree–Fock calculations.

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