A theoretical study of H-2 and D-2 desorbing from Cu(111) is reported. The study makes use of the LEPS PES of Dai and Zhang [J. Chem. Phys. 1995, 102, 6280]. The LEPS parameters have been modified in order to lower the barrier threshold in conformity with accurate ab initio electronic structure calculations. The topological study of the modified PES by the CHAIN method reveals unambiguously that the transition state (TS) is located at the top of a unique early barrier along the desorption path. The adsorbed H atoms are supposed to be in thermal equilibrium with the metal surface. Batches of classical trajectories (CT) are then carried out from the TS onto the products with their initial conditions canonically distributed and the effect of their possible recrossing of the TS taken into account according to Keck's method [Discuss. Faraday Soc. 1962, 33, 173]. Product state distributions are then calculated using the Gaussian weighting procedure [Chem. Phys. Lett. 2004, 397, 106] to account for the quantization of the vibration motion of the desorbed diatom. These distributions are in overall good agreement with experimental measurements. On average, the early barrier to desorption results in a significant vibrational excitation of the final diatom and a strong deexcitation of its rotational angular momentum J from the TS onto the products. Moreover, the orientation of the rotation plane is roughly random for low values of J (both cartwheel and helicopter motions are observed) while it is more likely parallel to the metal surface for large values of J (predominance for helicopter motion). These findings are analyzed in some details.