We use classical and quantum mechanical methods to calculate the site-to-site hopping rate of hydrogen impurities in crystalline silicon over a wide range of temperatures. The calculations employ a parameterized version of a potential surface calculated via density functional methods, expanded through quadratic terms about a Cartesian reaction path with a flexible reference. The hopping rate is obtained from the time integral of a flux correlation function which is evaluated using classical molecular dynamics and real-time path integral techniques. The latter are based on the quasiadiabatic propagator discretization and utilize a combination of discrete variable representations and Monte Carlo sampling for the evaluation of the resulting multidimensional integrals. Our results indicate that quantum mechanical tunneling plays a significant role in the diffusion process even above room temperature. In addition, the calculated diffusion rate exhibits a reverse isotope effect in the domain between activated and tunneling dynamics which arises from the zero point energy of the hydrogen atom in the direction perpendicular to the line connecting two stable minima.