Correlation energies for two interacting electrons in a harmonic quantum dot

The problem of correlated electrons in a quantum dot is interesting and exciting. Their confinement within a harmonic potential provides not only a needed environment but also a unique signature to the physics of the dynamic correlation. The energy level structure of the relative motion of two electrons in such a quantum dot is studied in detail. Simple but accurate analytic expressions for the correlation energy are found by a double-parabola approximation in a WKB treatment, whose results are shown to be in excellent agreement with those of exact numerical solutions. Based on the analytical study, however, a clear physical picture emerges. It is found that the internal energy levels of a given angular momentum m can be likened to a ladder of nearly equally spaced energy steps placed on a pedestal. The height of the pedestal is given by V-min(m,lambda), the minimum of the effective interaction potential associated with a given m and the Coulomb coupling strength lambda relative to the confinement. Superimposing ladders of all possible values of in then yields the entire level structure for the correlated relative motion. Owing to the unique nature of harmonic confinement, correlation thus manifests itself mostly through the lambda dependence of the classical-like V-min(m,lambda), in which the wave nature of the electrons plays no role except for the discrete values of m. [S0163-1829(98)03516-4].