  The Hubbard model extended by either nearest-neighbour Coulomb correlation
and/or nearest neighbour Heisenberg exchange is solved analytically for a
triangle and tetrahedron. All eigenvalues and eigenvectors are given as
functions of the model parameters in a closed form. The groundstate crossings
and degeneracies are discussed both for the canonical and grand-canonical
energy levels. The grand canonical potential and the electron occupation of the
related cluster gases were calculated for arbitrary values (attractive and
repulsive) of the three interaction constants. In the pure Hubbard model we
found various steps in the electron occupation higher than one. It is shown
that the various degeneracies of the grand-canonical energy levels are
partially lifted by an antiferromagnetic exchange interaction, whereas a
moderate ferromagnetic exchange modifies only slightly the results of the pure
Hubbard model. A repulsive nn Coulomb correlation lifts these degeneracies
completely. The relation of the cluster gas results to extended systems is
discussed.
