To understand the microscopic mechanism of H diffusion in tritium permeation barrier (TPB), we have explored the energetics and mobility of neutral hydrogen in alpha -Al sub(2)O sub(3) with hexagonal structure as well as sequioxide Er sub(2)O sub(3) with cubic bixbyite structure using first-principles density-functional calculations. The comparison of the most energetically favorable H interstitial positions between alpha -Al sub(2)O sub(3) and Er sub(2)O sub(3) shows that crystal structure plays a critical role in determining migration barriers. Combining static and molecular-dynamics calculations with nudged elastic band method, we derive the temperature-dependent diffusivity of hydrogen or deuterium in alpha -Al sub(2)O sub(3) and Er sub(2)O sub(3) as D (T) = (2.37 x 10 super(-7) m super(2)/s) exp (-1.25 eV/kT) and D (T) = (1.72 x 10 super(-7) m super(2)/s) exp (-1.64 eV/kT), 1-3 orders of magnitude lower than the corresponding experimental data. The migration barrier for H diffusion between the planes defined by Er sub(2)Ch units along the (111) direction is found to be very small at 0.16 eV, while higher migration barriers of 0.41 eV and 1.64 eV are found for the diffusion across the planes. These results indicate that H diffusion in Er sub(2)O sub(3) is favorable along the (111) direction. Quantum effects on H diffusion through alpha -Al sub(2)O sub(3) and Er sub(2)O sub(3) are discussed.