  We derive effective lattice models for ultracold bosonic or fermionic
nonreactive molecules (NRMs) in an optical lattice, analogous to the Hubbard
model that describes ultracold atoms in a lattice. In stark contrast to the
Hubbard model, which is commonly assumed to accurately describe NRMs, we find
that the single on-site interaction parameter $U$ is replaced by a
multi-channel interaction, whose properties we elucidate. The complex,
multi-channel collisional physics is unrelated to dipolar interactions, and so
occurs even in the absence of an electric field or for homonuclear molecules.
We find a crossover between coherent few-channel models and fully incoherent
single-channel models as the lattice depth is increased. We show that the
effective model parameters can be determined in lattice modulation experiments,
which consequently measure molecular collision dynamics with a vastly sharper
energy resolution than experiments in an ultracold gas.
