  We prove that the common Mie-Lennard-Jones (MLJ) molecular potentials,
appropriately normalized via an affine transformation, converge, in the limit
of hard-core repulsion, to the Toda exponential potential. Correspondingly, any
Fermi-Pasta-Ulam (FPU)-like Hamiltonian, with MLJ-type interparticle potential,
turns out to be $1/n$-close to the Toda integrable Hamiltonian, $n$ being the
exponent ruling repulsion in the MLJ potential. This means that the dynamics of
chains of particles interacting through typical molecular potentials, is close
to integrable in an unexpected sense. Theoretical results are accompanied by a
numerical illustration; numerics shows, in particular, that even the very
standard 12--6 MLJ potential is closer to integrability than the FPU potentials
which are more commonly used in the literature.
