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We present ground state calculations for low-density Fermi gases described bytwo model interactions, an attractive square-well potential and a Lennard-Jonespotential, of varying strength. We use the optimized Fermi-Hypernetted Chainintegral equation method which has been proved to provide, in the densityregimes of interest here, an accuracy better than one percent. We first examinethe low-density expansion of the energy and compare with the exact answer byHuang and Yang (H. Huang and C. N. Yang, {\em Phys. Rev.\/} {\bf 105}, 767(1957)). It is shown that a locally correlated wave function of theJastrow-Feenberg type does not recover the quadratic term in the expansion ofthe energy in powers of $\a0\KF$, where $\a0$ is the vacuum $s$-wave scatteringlength and $\KF$ the Fermi wave number. The problem is cured by addingsecond-order perturbation corrections in a correlated basis. Going to higherdensities and/or more strongly coupled systems, we encounter an instability ofthe normal state of the system which is characterized by a divergence of the{\em in-medium\/} scattering length. We interpret this divergence as aphonon-exchange driven dimerization of the system, similar to what one has atzero density when the vacuum scattering length $\a0$ diverges. We then study,in the stable regime, the superfluid gap and its dependence on the density andthe interaction strength. We identify two different corrections to low-densityexpansions: One is medium corrections to the pairing interaction, and the otherone finite-range corrections. We show that the most important finite-rangecorrections are a direct manifestation of the many-body nature of the system.

Correlations in the low-density Fermi gas: Fermi-Liquid state, Dimerization, and BCS Pairing