  The ground state nature of the Falicov-Kimball model with unconstrained
hopping of electrons is investigated. We solve the eigenvalue problem in a
pedagogical manner and give a complete account of the ground state energy both
as a function of the number of electrons and nuclei and as a function of the
total number of particles for any value of interaction U. We also study the
energy gap and show the existence of a phase transition characterized by the
absence of gap at the half--filled band for U<0. The model in consideration was
proposed and solved by Farkasovsky for finite lattices and repulsive on-site
interaction U>0. Contrary to his proposal we conveniently scale the hopping
matrix to guarantee the existence of the thermodynamic limit. We also solve
this model with bipartite unconstrained hopping matrices in order to compare
with the Kennedy--Lieb variational analysis.
