Correlated electronic structure models suggested by the large-dimension limit

Abstract

We describe three simple models for electronic structure in many-electron atoms and ions. Each model is parametrized by the spatial dimensionality D, which is ordinarily set to 3. All procedures are motivated by the solution for D -+ 00, a classical limit which can be solved exactly either with or without electron correlation. Each model modifies the D -+ 00 limit treatment to reflect important finite-D effects, but retains the classical character of the limit; the modifications, in order of increasing realism, are designated post-scaling, pre-scaling, and pre-structuring. The three models yield pointlike electronic structures somewhat reminiscent of pre-quantal atomic models. However, all electronic positions have components outside the D-dimensional physical subspace, a feature which significantly enhances the ability of a localized structure to represent the true solution and to model it quantitatively. Specific calculations reported are total and correlation energies for atoms with Z :s;; 92 (computed by post-scaling and pre-scaling), correlation corrections to ionization potentials and electron affinities for Z :s;; 54 (post-scaling), and the asymptotic behavior of correlation energies for Z -+ 00 (post-scaling).

Bohr model and dimensional scaling analysis of atoms and molecules

“…It thus is surprising that such an unrealistic model nonetheless offers useful information for determining D ¼ 3 results, particularly about electron correlation. An instructive analysis has been given by Loeser and colleagues [131]. A crucial feature is that the pointwise large-D structure exists in the scaled space, not the 'real world'.…”

Continuation of the Wave Function for Higher Angular Momentum States toDDimensions

“…Despite the simplicity of this procedure, the large D limit incorporates a number of features of the D=3 problem, such as correlation and some aspects of electron geometry. However, some features of the D=3 solution are notably absent, in particular Pauli principle effects such as shell structure; although some shell structure can be incorporated in the energy by an aufbau process which folds in an appropriate dimensional scaling for each shell [48,73]. One systematic way to improve upon the large D limit is to use perturbation theory as we have outlined above.…”

“…One systematic way to improve upon the large D limit is to use perturbation theory as we have outlined above. Another approach is to improve the zeroth order solution by incorporating some finite D effects while still keeping the simplicity of a non-differential Hamiltonian [47,48,99]. These nondifferential Hamiltonians are termed sub-Hamiltonians and can incorporate increasing levels of structure found in the D=3 solution, i.e.…”

“…These nondifferential Hamiltonians are termed sub-Hamiltonians and can incorporate increasing levels of structure found in the D=3 solution, i.e. shells, subshells and then orbitals [48]. The minima of the sub-Hamiltonians correspond to the maxima of the electron probability density and the energy is given by a weighted sum of the energies of each minima.…”

Interdimensional degeneracies for a quantum three-body system in D dimensions

“…As is well known, dimensional scaling methods provide a powerful approach for studying atomic and molecular systems in D dimensions [5,[17][18][19][20][21][22], but much work has been confined to S-wave states and some P e states [7][8][9] which were obtained by exploiting a known interdimensional degeneracies between S e states in 5 dimensions…”