  Full 3D calculations of small two-component Fermi gases under
highly-elongated confinement, in which unlike fermions interact through
short-range potentials with variable atom-atom s-wave scattering length, are
performed using the correlated Gaussian approach. In addition, microscopic 1D
calculations are performed for effective "atomic" and "molecular" 1D model
Hamiltonian. Comparisons of the 3D and 1D energies and excitation frequencies
establish the validity regimes of the effective 1D Hamiltonian. Our numerical
results for three- and four-particle systems suggest that the effective 1D
atom-dimer and dimer-dimer interactions are to a good approximation determined
by simple analytical expressions. Implications for the description of quasi-1D
Fermi gases within strict 1D frameworks are discussed.
