To understand the microscopic mechanism of H diffusion in tritium permeation barrier (TPB), we have explored the energetics and mobility of neutral hydrogen in I--Al2O3 with hexagonal structure as well as sequioxide Er2O3 with cubic bixbyite structure using first-principles density-functional calculations. The comparison of the most energetically favorable H interstitial positions between I--Al2O3 and Er2O3 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 I--Al2O3 and Er2O3 as D(T) = (2.37 A 10a7 m2/s) exp (a1.25 eV/kT) and D(T) = (1.72 A 10a7 m2/s) exp (a1.64 eV/kT), 1a3 orders of magnitude lower than the corresponding experimental data. The migration barrier for H diffusion between the planes defined by Er2O3 units along the a1 1 1a 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 Er2O3 is favorable along the a1 1 1a direction. Quantum effects on H diffusion through I--Al2O3 and Er2O3 are discussed.