We have used spin-polarized density functional theory (DFT) to study O(2) evolution on a clean partially reduced rutile TiO(2)(110) surface (i.e., a surface having oxygen vacancies) and its interaction with Au(1) or Au(2) cluster adsorbed on it. We assume that the total spin of the electronic wave function is related to the number of unpaired spins (N(s)) and calculate the binding and the activation energies involved in O(2) evolution for fixed values of N(s). In addition to keeping N(s) constant, we assume that reactions in which the N(s) of the reactants differs from that of the products are very slow. The potential energy surfaces obtained in this way depend strongly on N(s). For example, O(2) dissociation at the vacancy site on a clean partially reduced TiO(2)(110) surface is exothermic by 0.85 eV in the triplet state and the highest activation energy in the chain of reactions leading to the O(2) dissociation is 0.67 eV. In the singlet state, O(2) dissociation is endothermic by 0.11 eV and the activation energy leading to dissociation is 1.30 eV. These observations are in qualitative agreement with scanning tunneling microscopy experiment in which O(2) dissociation on a partially reduced rutile TiO(2)(110) surface is observed at temperature as low as 120 K. In contrast, O(2) dissociation is predicted to be endothermic and is prevented by an activation barrier larger than 1 eV in all the previous DFT calculations, in which the DFT program varies N(s) to get the lowest energy state. We find that on a partially reduced rutile TiO(2)(110) with Au(1) and Au(2) preadsorbed on its surface, O(2) dissociates at the vacancy site: One oxygen atom fills the oxygen vacancy and the other becomes available for oxidation chemistry. This means that Au(1) and Au(2) supported on a partially reduced TiO(2)(110) surface is not an oxidation catalyst since the presence of oxygen turns it into a stoichiometric Au(n)/TiO(2)(110) surface. Finally, we find that the evolution of oxygen on Au(1) and Au(2) in the gas phase is very different from the evolution on the same clusters supported on the partially reduced TiO(2)(110) surface. For example, the molecular adsorption of O(2) is favored in the gas phase (except on Au(1) (-) and Au(2) (-) in the quartet state), while the dissociative adsorption is favored by more than 1 eV when Au(1) and Au(2) are supported on the partially reduced TiO(2)(110). Furthermore, the activation energies associated with O(2) dissociation in the gas phase (DeltaE(act)>2.4 eV) are reduced by at least a factor of 2 when the clusters are supported on TiO(2)(110).