We report the computational results of hydrogen adsorption atop Mg and O atoms on the MgO(001) surface, followed by its absorption under the target atoms, using two approaches: tight-binding and ab initio methods. We present the energetic and electronic aspects of these interactions and discuss the qualities of the non-self-consistent field tight-binding results compared with the ab initio results. There is the qualitative reproduction of ab initio results in the hydrogen adsorption phase on both types of ions. The tight-binding results are found to be more accurate in the hydrogen absorption phase than in the adsorption phase. In the adsorption calculations the introduction of the surface dipole term in the tight-binding total energy would be required to compensate for the absence of electronic relaxation in the MgO(001) surface.