TEST OF THE BARDEEN-COOPER-SCHRIEFFER AND THE RESONANT-VALENCE-BOND WAVE-FUNCTIONS AS SOLUTIONS OF THE HUBBARD-MODEL - A SMALL-CLUSTER CALCULATION

With the purpose of assessing the validity of commonly used approximat ions, best variational solutions are calculated for a four-site (tetra hedral-cluster) Hubbard model with periodic boundary conditions and co mpared with exact results. In particular, the optimum number-conservin g Bardeen-Cooper-Schrieffer-type variational solution is obtained for two, four, and six electrons, and the optimal resonating-valence-bond state for four electrons. For an attractive interaction U, the exact g round state of the system is obtained in all four cases. If the intera ction U is repulsive, the exact ground state is obtained only when the cluster contains two and six electrons. For four electrons the BCS ap proximation is only satisfactory for relatively small U. For any posit ive U the resonating-valence-bond state on the other hand, gives very good (albeit not exact) energies for the ground state.