Atomic vs. quantum dot open shell spectra

Hund's rules apply to pairs of atomic states which, in the limit of weak interelectronic repulsion, belong to a common configuration. We examine the asymptotic behavior of the energy difference between states belonging to such pairs (ΔE) as a function of the nuclear charge (Z) at both Z→∞ (ignoring relativistic effects) and at Z→Z_c, where Z_c is the critical charge below which the outermost electron is not bound. We find that the ratio ΔEZ² is non-monotonic, having a maximum at some intermediate value of Z. This non-monotonicity is shown to account for the sign reversal in the difference between the corresponding interelectronic repulsion energies. For analogous open shell systems with confining potentials (such as Hooke's atom or the infinite spherical well), the corresponding function of the energy difference, in terms of which the interelectronic repulsion energy is given as a first derivative with respect to the parameter that characterizes the strength of the one-body potential (such as the harmonic force constant or the well radius, respectively), is monotonic in this parameter. This implies that the relative magnitude of the interelectronic repulsion energies that characterizes the strong one-body limit (usually, the ordering naively expected on the basis of Hund's rule) remains valid along the whole isoelectronic sequence.