[en] Highlights: • The self-diffusion of H₂ molecules in MOF-508a was calculated. • The effect of temperature and concentration on the self-diffusion were investigated. • The obtained results in both Cartesian and crystallography frames were compared. • The quantum effects were considered at 77 K. • The reversible open-closed structure of MOF-508a can be useful in the storage process of gases. The mean square displacement and the self-diffusion coefficients of hydrogen molecules along both crystallographic and Cartesian coordinates inside the MOF-508a were calculated in the temperature range of 77–900 K. The self-diffusion is increased with temperature and also with hydrogen molecules loading up to a certain point after which it is decreased. Activation energy and binding energy of hydrogen molecules to the MOF sites were calculated for each loading. Analysis of the radial distribution functions shows that hydrogen molecules prefer to bind to the nitrogen and carbon atoms of the MOF. Furthermore, correlation between the hydrogen molecules increases with temperature.