The summed quantity is none other than the quadratic Casimir operator for representation of the spin algebra on the three consecutive sites j − − 1 , j , j + 1 , which in turn can be decomposed into a direct sum of spin 1/2 and 3/2 representations. It has the eigenvalues 1 2 ( 1 2 + 1 ) = 3 / 4 for the spin 1/2 subspace and 3 2 ( 3 2 + 1 ) = 15 / 4 for the spin 3/2 subspace.