  We study the low-lying energy clustering patterns of quantum antiferromagnets
with p sublattices (in particular p=4). We treat each sublattice as a large
spin, and using second-order degenerate perturbation theory, we derive the
effective (biquadratic) Hamiltonian coupling the p large spins. In order to
compare with exact diagonalizations, the Hamiltonian is explicitly written for
a finite-size lattice, and it contains information on energies of excited
states as well as the ground state. The result is applied to the
face-centered-cubic Type I antiferromagnet of spin 1/2, including
second-neighbor interactions. A 32-site system is exactly diagonalized, and the
energy spectrum of the low-lying singlets follows the analytically predicted
clustering pattern.
