Semiclassical perturbation theory for two electrons in a D-dimensional quantum dot

Dimensional perturbation theory is applied to the two-electron D-dimensional quantum dot, obtaining accurate values for the ground- and excited-state energies. The expansion parameter is $1/\kappa ,$ where $\kappa =D+2\mathrm{}|l|,$ D is the effective spatial dimensionality of the quantum dot environment, and l is the relative-motion angular momentum quantum number. In this method, no approximations are made in the treatment of correlation. Analytic approximations for ground- and excited-state energies are obtained from the zeroth- and first-order terms of the perturbation expansion; thus, constituting a semiclassical approach to the quantum dot from a perturbation formalism. Using this analytic form of the energy, parametrized by D, the effects of the effective quantum dot dimensionality on the energy spectra may be investigated. Systematic corrections are made to the semiclassical approximation by adding higher-order perturbation terms. The method described may be extended to obtain analytic approximations to the ground-state energy of the many-electron D-dimensional quantum dot Hamiltonian by truncating the $1/D$ expansion to low order.