Using density-functional theory, we provide the first conclusive evidence of the existence of a molecularly chemisorbed state for oxygen on the Rh (111) surface. Four species are identified: a paramagnetic state above the bridge site with a binding energy of 1.95 eV, a more weakly bound paramagnetic state above the top site with a binding energy of 0.95 eV, and two nonmagnetic states above the face-centered-cubic (fcc) and hexagonal-close-packed (hcp) hollow sites each with a binding energy of 1.98 eV. We compare these results with our calculations of the binding energy for atomic oxygen on the fcc and hcp hollow sites and an upper bound on the dissociation barrier to understand major portions of the dissociation reaction coordinate. Combining our data with the experimental and theoretical results for oxygen dissociation on many other fcc (111) metal surfaces, we conclude that all these metal surfaces possess similar minima corresponding to physisorption, molecular chemisorption, and dissociative chemisorption. Despite these similarities, the differing binding energies and barrier heights account for the varying mechanisms of oxygen dissociation on different fcc (111) surfaces.