Aggregates of oxygen vacancies (F centers) represent a particular form of point defects in ionic crystals. In this study we have considered the combination of two oxygen vacancies, the M center, in the bulk and on the surface of MgO by means of cluster model calculations. Both neutral and charged forms of the defect M and M+ have been taken into account. The ground state of the M center is characterized by the presence of two doubly occupied impurity levels in the gap of the material; in M+ centers the highest level is singly occupied. For the ground-state properties we used a gradient corrected density functional theory approach. The dipole-allowed singlet-to-singlet and doublet-to-doublet electronic transitions have been determined by means of explicitly correlated multireference second-order perturbation theory calculations. These have been compared with optical transitions determined with the time-dependent density functional theory formalism. The results show that bulk M and M+ centers give rise to intense absorptions at about 4.4 and 4.0 eV, respectively. Another less intense transition at 1.3 eV has also been found for the M+ center. On the surface the transitions occur at 1.6 eV (M+) and 2 eV (M). The results are compared with recently reported electron energy loss spectroscopy spectra on MgO thin films.