  Strongly interacting fermionic atoms on optical lattices are studied through
a Hubbard-like model Hamiltonian, in which tunneling rates of atoms and
molecules between neighboring sites are assumed to be different. In the limit
of large onsite repulsion U, the model is shown to reproduce the t-J
Hamiltonian, in which the J coefficient of the Heisenberg term depends on the
particle-assisted tunneling rate g: explicitly, $J=4 g^2/U$. At half-filling, g
drives a crossover from a Brinkman-Rice paramagnetic insulator of fully
localized atoms (g=0) to the antiferromagnetic Mott insulator of the standard
Hubbard case (g=t). This is observed already at the intermediate coupling
regime in the number of doubly occupied sites, thus providing a criterion to
extract from measurements the effective value of g.
