Hund and anti-Hund rules in circular molecules

We study the validity of Hund's first rule for the spin multiplicity in circular molecules---made of real or artificial atoms such as quantum dots---by considering a perturbative approach in the Coulomb interaction in the extended Hubbard model with both on-site and long-range interactions. In this approximation, we show that an anti-Hund rule always defines the ground state in a molecule with $4N$ atoms at half-filling. In all other cases (i.e., number of atoms not a multiple of four, or a $4N$ molecule away from half-filling) both the singlet and the triplet outcomes are possible, as determined primarily by the total number of electrons in the system. In some instances, the Hund rule is always obeyed and the triplet ground state is realized mathematically for any values of the on-site and long-range interactions, while for other filling situations the singlet is also possible but only if the long-range interactions exceed a certain threshold, relatively to the on-site interaction.