  The properties of two-component Fermi gases become universal if the
interspecies s-wave scattering length $a_s$ and the average interparticle
spacing are much larger than the range of the underlying two-body potential.
Using an explicitly correlated Gaussian basis set expansion approach, we
determine the eigen energies of two-component Fermi gases in a cubic box with
periodic boundary conditions as functions of the interspecies s-wave scattering
length and the effective range of the two-body potential. The universal
properties of systems consisting of up to four particles are determined by
extrapolating the finite-range energies to the zero-range limit. We determine
the eigen energies of states with vanishing and finite momentum. In the
weakly-attractive BCS regime, we analyze the energy spectra and degeneracies
using first-order degenerate perturbation theory. Excellent agreement between
the perturbative energy shifts and the numerically determined energies is
obtained. For the infinitely large scattering length case, we compare our
results - where available - with those presented in the literature.
