Effect of phonon coupling on hydrogen tunneling rates at gas–surface interfaces

The centrifugal-dominant small-curvature semiclassical adiabatic tunneling approximation is used with variational transition state theory to calculate diffusion coefficients for hydrogen, deuterium, and tritium atoms on the (100) face of copper for temperatures in the range 80–1000 K. The system is modeled by the embedded cluster method, and the copper lattice is constructed with a lattice constant optimized for the assumed potential energy function. Calculations are presented with up to 171 nonfixed degrees of freedom. The results are well converged with respect to the number of lattice atoms whose motion is allowed to couple to the adatom. The difference between the diffusion coefficients calculated with rigid and nonrigid lattices at 300 K are 3.7, 3.3, and 3.1 for H, D, and T, respectively, increasing to factors of 24.1, 19.4, and 17.2 at 120 K. The effect levels off for lower temperatures, e.g., the ratio for H is 27.3 at 100 K and 24.4 at 80 K. The convergence with respect to the number of moving copper atoms is nonmonotonic; detailed examination of the intermediate results shows that such nonmonotonicities result from the nonsmooth cancellation of a large number of competing effects attributable to many surface phonon modes. We compare the present results to those predicted by path integral transition state theory and to those predicted by transition state theory with quantum effective potentials, and we find them to be in reasonably good agreement. This is very encouraging since tests of multidimensional semiclassical tunneling approximations have been limited to systems with only a few degrees of freedom in the past.