The results of Hartree-Fock (with relativistic effective core potentials) followed by variational and second-order multireference perturbational configuration interaction (CIPSI) calculations are reported for the nine lowest-lying electronic states of the CuRu molecule and the eight low-lying states of the CuRuH2 molecule belonging to the C-2v symmetry group which evolve from the H-2 side-on interaction with the ruthenium atom. The effect of the copper atom on the ability of the ruthenium atom toward the capture and dissociation of the H-2 molecule is analyzed through comparison with the results previously reported for the simpler systems RuH2 and RuH2+. For all the reaction channels investigated here, the copper atom depresses the ability of ruthenium to dissociate the H-2 molecule.