For the reaction M+(C2H4)(n-1) + C2H4 -> M+(C2H4)(n), where M = Ag, Au, the binding energies are predicted at the second order perturbation (MP2) and coupled cluster (CCSD(T)) levels of theory. As the basis set is systematically improved, the predicted M = Ag binding energies steadily improve, as compared to the experimental values. In fact, the complete basis set limit (CBS) predicted CCSD(T) binding energy for Ag(C2H4)(+) is within experimental error. For MP2, as the basis set is improved, the agreement with experiment worsens. Gold ions are predicted to bind more strongly than silver ions to ethene ligands. Mulliken population analyses of the silver and gold systems exhibit delocalization of the positive charges of the metal ions onto the ethene ligands. Reduced variational space analysis indicates that electrostatic interactions are the principal contributor to the bonding in these systems. Multiconfigurational self-consistent field calculations do not support the Dewar-Chatt-Duncanson model of transition metal-alkene bonding in Au(C2H4)(+).