  We show that recent high-precision measurements of relative on-site
interaction energies $\Delta U$ in a Mott insulator require a theoretical
description beyond the standard Hubbard-model interpretation, when combined
with an accurate coupled-channels calculation. In contrast to more
sophisticated lattice models, which can be elaborate especially for parameter
optimization searches, we introduce an easy to use effective description of $U$
valid over a wide range of interaction strengths modeling atomic pairs confined
to single lattice sites. This concise model allows for a straightforward
combination with a coupled-channels analysis. With this model we perform such a
coupled-channels analysis of high-precision $^7$Li spectroscopic data on the
on-site interaction energy $U$, which spans over four Feshbach resonances and
provide an accurate and consistent determination of the associated resonance
positions. Earlier experiments on three of the Feshbach resonances are
consistent with this new analysis. Moreover, we verify our model with a more
rigorous numerical treatment of the two atom system in an optical lattice.
