Microscopic derivation of Hubbard parameters for cold atomic gases

We study the exact solution for two atomic particles in an optical lattice interacting via a Feshbach resonance. The analysis includes the influence of all higher bands, as well as the proper renormalization of molecular energy in the closed channel. Using an expansion in Bloch waves, we show that the problem reduces to a simple matrix equation, which can be solved numerically very efficient. This exact solution allows for the precise determination of the parameters in the Hubbard model and the two-particle bound state energy. We identify the regime, where a single band Hubbard model fails to describe the scattering of the atoms as well as the bound states.