Adsorption and chemisorption of H-2 in mordenite is studied using ab initio density functional throry (DFT) calculations. The geometries of the adsorption complex, the adsorption energies, stretching frequencies, and the capacity to dissociate the adsorbed molecule are compared for different active sites. The active centers include a Bronsted acid site, a three-coordinated surface Al site, and Lewis sites formed by extraframework cations: Na+, Cu+, Ag+, Zn (2+), Cu2+, Ga3+, and Al3+. Adsorption properties of cations are compared for a location of the cation in the five-membered ring. This location differs from the location in the six-membered ring observed for hydrated cations. The five-membered ring, however, represents a stable location of the bare cation. In this position any cation exhibits higher reactivity compared with the location in the six-membered ring and is well accessible by molecules adsorbed in the main channel of the zeolite. Calculated adsorption energies range from 4 to 87 kJ/mol, depending on electronegativity and ionic radius of the cation and the stability of the cation-zeolite complex. The largest adsorption energy is observed for Cu+ and the lowest for Al3+ integrated into the interstitial site of the zeolite framework. A linear dependence is observed between the stretching frequency and the bond length of the adsorbed H-2 molecule. The capacity of the metal-exchanged zeolite to dissociate the H-2 molecule does not correlate with the adsorption energy. Dissociation is not possible on single Cu+ cation. The best performance is observed for the Ga3+, Zn2+, and Al2+ extraframework cations, in good agreement with experimental data.