where Ŝi and Ŝj are the spin angular momentum operators and J ij is the magnetic exchange coupling constant between the magnetic sites i and j. The symbol <i,j> indicates that the sum refers to the interactions between nearest neighbours only. When three spins are coupled, this results in one quartet (S = 3 2 ) state and two doublet states (S = 1 2 ) giving rise to two different quartet-doublet energy gaps. For the case of two spin-centers where only one coupling constant appears in the above Hamiltonian, it can be shown to be related to the energy gap between the resulting singlet and triplet states. This allows coupling constants to be extracted from electronic structure calculations of singlet and triplet states. For the case of symmetric couplings where J 12 = J 23 = J 13 = J, the two doublets will be degenerate allowing one to extract J from quartet-doublet energy gap as,