  We present the theory of an atomic gas in an optical lattice near a Feshbach
resonance. We derive from first principles a generalized Hubbard model, that
incorporates all the relevant two-body physics exactly, except for the
background atom-atom scattering. For most atoms the background interactions are
negligible, but this is not true for lithium-6, which has an exceptionally
large background scattering length near the experimentally relevant Feshbach
resonance at 834 G. Therefore, we show how to include background atom-atom
scattering by solving the on-site two-body Feshbach problem exactly. We apply
the obtained solution to lithium-6 and find that the background interactions
indeed have a significant effect in this case.
