  Several extensions to the Standard Model of particle physics, including light
dark matter candidates and unification theories, predict deviations from
Newton's law of gravitation. For macroscopic distances, the inverse-square law
of gravitation is well confirmed by astrophysical observations and laboratory
experiments. At micrometer and shorter length scales, however, even the
state-of-the-art constraints on deviations from gravitational interaction,
whether provided by neutron scattering or precise measurements of forces
between macroscopic bodies, are currently many orders of magnitude larger than
gravity itself. Here we show that precision spectroscopy of weakly bound
molecules can be used to constrain non-Newtonian interactions between atoms. A
proof-of-principle demonstration using recent data from photoassociation
spectroscopy of weakly bound Yb$_2$ molecules yields constraints on these new
interactions that are already close to state-of-the-art neutron scattering
experiments. At the same time, with the development of the recently proposed
optical molecular clocks, the neutron scattering constraints could be surpassed
by at least two orders of magnitude.
