Origin of the first Hund rule and the structure of Fermi holes in two-dimensional He-like atoms and two-electron quantum dots

The lowest singlet-triplet pair of states of the two-electron two-dimensional quantum dots and the corresponding pair of states of the two-dimensional helium-like systems have been studied by the full configuration interaction method focusing on the origin of the first Hund rule. The one- and two-electron components of the singlet-triplet energy gap show distinct trends for the systems studied in the regime of small nuclear charge Zn or of small confinement strength ω. The (0σg)(1πu) singlet state in quantum dots is characterized by a larger electron repulsion than its counterpart triplet state for all values of ω, while this relationship gets inverted for the corresponding (1s)(2p) singlet-triplet pair of He-like systems for small values of Zn, such as Zn = 2 or 3. The internal part of the full configuration interaction wavefunctions has been extracted and visualized in the three-dimensional internal space (r1, r2, ϕ-) to rationalize the observed trends. The singlet probability density of He-like systems located originally near the Fermi holes is shown to migrate into regions where either r1 or r2 are large while the corresponding singlet probability of quantum dots stays close to the Fermi holes. Their differences and their observed trends are rationalized on the basis of the structure of the genuine and conjugate Fermi holes.