  The Hubbard model on a cube was revisited and extended by both
nearest-neighbor (nn) Coulomb correlation and {nearest-neighbor} Heisenberg
exchange. The complete eigensystem was computed exactly for all electron
occupancies and all model parameters ranging from minus infinity to plus
infinity. For two electrons on the cluster the eigensystem is given in
analytical form. For six electrons and infinite on-site correlation U we
determinded the groundstate and the groundstate energy of the pure Hubbard
model analytically. For fixed electron numbers we found a multitude of ground
state level crossings in dependence on the various model parameters.
Furthermore the groundstates of the pure Hubbard model in dependence on a
magnetic field h coupled to the spins are shown for the complete U-h plane. For
the cubic cluster gas we calculated the density $n(\mu,T,h)$ and the
thermodynamical density of states from the grand potential. The ground states
and the various spin-spin correlation functions are studied for both attractive
and repulsive values of the three interaction constants. The anomalous
degeneration lines are calculated. For the cases where these lines end in
triple points we give the related parameter values. The influence of the
nn-exchange and the nn-Coulomb correlation onto the anomalous degeneration is
discussed.
