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 {\it always} defines the ground state in a molecule with $4N$ atoms at half-filling. In all other cases (i.e. number of atoms {\it not} 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 {\it 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.