Hund's multiplicity rule: from atoms to quantum dots

In 1965, Davidson has shown that the textbook explanation for the Hund's multiplicity rule in atoms, based on the Pauli principle, is wrong. The reason for the failure of the textbook proof, as has been given later by others and as appears today in modern textbooks, it is based on the need to introduce angular electronic correlation into the calculations. Here, we investigate an applicability of this argumentation for helium and for the case of two-electron spherically symmetric rectangular quantum dots (QDs). We show that, for helium and also for the QD, the differences between the singlet and triplet excited states can be explored by calculations within the framework of the mean-field approximation, and, surprisingly, without the need of introducing the angular electronic correlation. Moreover, our calculations have shown that the triplet state of the QD is lower in energy than the corresponding singlet state due to lower electronic repulsion contribution, exactly as being assumed in the oldest explanation of the Hund's rule based on the Pauli principle.