  We numerically study the problem of two fermions in a three dimensional
optical lattice interacting via a zero-range Feshbach resonance, and display
the dispersions of the bound states as a two-particle band structure with
unique features compared to typical single-particle band structures. We show
that the exact two-particle solutions of a projected Hamiltonian may be used to
define an effective two-channel, few-band model for the low energy, low density
physics of many fermions at arbitrary s-wave scattering length. Our method
applies to resonances of any width, and can be adapted to multichannel
situations or higher-$\ell$ pairing. In strong contrast to usual Hubbard
physics, we find that pair hopping is significantly altered by strong
interactions and the presence of the lattice, and the lattice induces multiple
molecular bound states.
