  Clathrate hydrates hold considerable promise as safe and economical materials
for hydrogen storage. Here we present a quantum mechanical study of H$_2$ and
D$_2$ diffusion through a hexagonal face shared by two large cages of clathrate
hydrates over a wide range of temperatures. Path integral molecular dynamics
simulations are used to compute the free-energy profiles for the diffusion of
H$_2$ and D$_2$ as a function of temperature. Ring polymer molecular dynamics
rate theory, incorporating both exact quantum statistics and approximate
quantum dynamical effects, is utilized in the calculations of the H$_2$ and
D$_2$ diffusion rates in a broad temperature interval. We find that the shape
of the quantum free-energy profiles and their height relative to the classical
free energy barriers at a given temperature, as well as the rate of diffusion,
are profoundly affected by competing quantum effects: above 25 K, zero-point
energy (ZPE) perpendicular to the reaction path for diffusion between cavities
decreases the quantum rate compared to the classical rate, whereas at lower
temperatures tunneling outcompetes the ZPE and as result the quantum rate is
greater than the classical rate.
