Considerations on Hund's first rule in a planar two-electron quantum dot

We give a detailed analysis of the applicability of Hund's first rule for harmonic planar two-electron quantum dots by means of entanglement witnesses. We find that for purely harmonic confinement there is only one pair of singlet and triplet states for which it can be applied. We also discuss the origin and validity for this case and extend the discussion to a quartic confining potential, a hard-wall potential, and a combination of harmonic confinement with quartic perturbation. A generalized rule, the alternating rule, is found to be applicable and valid for vanishing angular momentum states in all these cases. Furthermore, we are able to clarify the role of entanglement in general harmonic two-electron models for vanishing interaction strength. This behavior can be attributed to the special separability properties of these models.