The quantum-to-classical correspondence is often quantified in dynamics by the out-of-time-order correlator (OTOC). In chaotic systems, the OTOC is expected to grow exponentially at early time, characteristic of a Lyapunov exponent; however, exponential growth can also occur for integrable systems. Here we investigate the OTOC for realistic diatomic molecular potentials in one degree of freedom, finding that the OTOC can grow exponentially near the dissociation energy of the molecule. Further, this dynamics is tied to the classical dynamics of the atoms at the outer classical turning point of the potential. These results should serve to guide and interpret dynamical chaos in more complex molecules.