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We calculate numerically the exact energy spectrum of the six dimensionalproblem of two interacting Bosons in a three-well optical lattice. Theparticles interact via a full Born-Oppenheimer potential which can be adaptedto model the behavior of the s-wave scattering length at Feshbach resonances.By adjusting the parameters of the corresponding Bose-Hubbard (BH) Hamiltonianthe deviation between the numerical energy spectrum and the BH spectrum isminimized. This defines the optimal BH parameter set which we compare to thestandard parameters of the BH model. The range of validity of the BH model withthese parameter sets is examined, and an improved analytical prediction of theinteraction parameter is introduced. Furthermore, an extended BH model andimplications due to the energy dependence of the scattering length andcouplings to higher Bloch bands at a Feshbach resonance are discussed.

Ab-initio determination of Bose-Hubbard parameters for two ultracold atoms in an optical lattice using a three-well potential