  We analyze the BCS-BEC crossover transition of a balanced two component
mixture of fermions interacting via a finite range potential, within a mean
field approach. For the analysis we consider three finite range potentials
cases describing the interaction between different Fermi species: a square
well, an exponential and a Yukawa potential. The T = 0 thermodynamics analysis
along the BCS-BEC crossover allow us to recognize the proper variables, for
finite range interactions, that capture the transition from a thermodynamic
non-universal behavior at unitarity, to its universal restoration. On the other
side, the determination of the pair functions along the crossover suggests that
the smooth transition occurs always between the scattering resonance and the
change of sign of the chemical potential. This identification follows directly
from the pair wave functions behavior, which in the BCS and BEC sides become
exponentially localized and oscillatory slowly decaying respectively.
