We present a quantum mechanical study of the diffusion of CO molecules on the Cu(0 0 1) surface. We use the Strasbourg-Amsterdam-Postdam potential surface and a “non-tunnel”-variant hereof; to mimic an initial state that is localized in one adsorption well, a “local-potential-shift” concept is introduced; the Multi Configuration Time Dependent Hartree method to perform the calculations. Special emphasis is placed on the effect of the dimensionality of the models. Surprisingly, tunneling plays an important role typically 1 ps after the beginning of the dynamics; it dominates at around 1 ns and might, at least partly, explain the long diffusion rates measured experimentally for the system.