The structures of molecularly and dissociatively adsorbed hydrogens on a ZrO2 surface are studied by abinitio SCF calculations for Zr3O8(H-8)-H-2 cluster embedded in an electrostatic potential representing the Madelung potential. The side-on and end-on molecular adsorption stares on the Zr and O atoms, respectively, are calculated to have adsorption energies of 7.5 and 3.1 kcal/mol, and the heterolytic dissociation state is calculated with a heat of adsorption of 22.7 kcal/mol, while the experimental value is 8 kcal/mol. The IR frequencies of the H-H, Zr-H, and O-H stretching vibrations are calculated and compared with the experimental values. The reaction path for the dissociation of H on the surface is calculated and the electronic mechanism of the dissociation is discussed. The calculated activation energy is 29.5 kcal/mol, while the experimental one is 10 kcal/mol. The mechanism for the chemisorption and dissociation of a hydrogen molecule on the ZrO2 surface is discussed, and analysis of the electron density is presented.