  Using mean-field theory for the Bardeen-Cooper-Schriefer (BCS) to the
Bose-Einstein condensate (BEC) crossover we investigate the ground state
thermodynamic properties of an interacting homogeneous Fermi gas. The
interatomic interactions modeled through a finite range potential allows us to
explore the entire region from weak to strong interacting regimes with no
approximations. To exhibit the thermodynamic behavior as a function of the
potential parameters in the whole crossover region, we concentrate in studying
the contact variable, the thermodynamic conjugate of the inverse of the s-wave
scattering length. Our analysis allows us to validate the mean-field approach
across the whole crossover. It also leads to predict a quantum transition-like
in the case when the potential range becomes large. This finding is a direct
consequence of the k-dependent energy gap for finite interaction range
potentials.
