We review the adsorption and desorption of molecular oxygen on a reduced TiO2 (110) surface. This system is known to play a fundamental role in heterogeneous photocatalysis. Periodic calculations are performed with the objective of characterizing the variety of stable species of O-2 that are known to exist on the TiO2 surface. The implications of our results for recent experiments are discussed. We also consider a direct optical excitation mechanism for the ultraviolet (UV) light-desorption process and model the most stable O-2/TiO2-x system as a cluster. Hi-h-level ab initio calculations of the excited states and interaction matrix elements are performed using different orbitals, separately optimized for the target states. The nonadiabatic and dipole-moment couplings are calculated directly from the correlated wave functions by a special transformation to bi-orthonormat (dual) orbital sets to preserve their structure. The method used for the electronic structure calculations is described in detail. Finally, the effect of the electronic coupling in the UV-photodesorption dynamics is analyzed in detail.