The origin of the superlattice present in the commensurate hexagonal structure of self-assembled monolayers of n-alkanethiols on gold and the question of whether the thiols are adsorbed onto the surface as dimers (disulfides) or monomers (thiolates) have been under debate for many years. Looking for a better understanding of the structural properties of these systems, we have performed a theoretical study of the molecular and dissociative adsorption of dimethyl disulfide on Au(111) as a function of coverage (0.25 less than or equal to Theta less than or equal to 1), using gradient-corrected density functional (DFT) calculations with a slab geometry. For the dissociated state, our results indicate that the hcp hollow site is much less favorable than the fee site. For the latter site, we find that, because of surface gold atom relaxation, the adsorption energy depends strongly on Theta, changing from similar to 18 kcal/mol at Theta = 0.25 to similar to3 kcal/mol at Theta = 1. For the bridge site, instead, the adsorption energy is a weak function of Theta, and for all investigated coverages, this site is by far the most stable. According to our DFT approach, the adsorption of dimethyl disulfide is dissociative with a thermodynamic gain, at Theta = 1, of similar to 13 kcal/mol with respect to the adsorbed molecular state. We also find, however, that the energy of c(4 x 2) structures containing at least two inequivalent CH3S groups per unit cell (with a minimum S-S distance of similar to3.7 Angstrom) is, within the accuracy of our approach, indistinguishable from the pure (root3 x root3) hexagonal structure. Our results suggest that the full solution of this thorny problem will require, also for the shortest chains, an estimate of the energetic contribution of dispersion forces that are not included in the DFT calculations.