We report a comprehensive density-functional theory ͑DFT͒ calculation of the lateral adiabatic potentialenergy surface ͑PES͒ of individual alkali atoms adsorbed on the Cu͑001͒ surface. The corrugation of the PES monotonically decreases by one order of magnitude from Li to Cs, the latter being mobile on the surface even at low temperatures. Substrate relaxations in metastable configurations are consistent with a harmonic response to the adsorbate and account for a significant fraction of the PES corrugation. A simple nonseparable cosine expression provides a good description of values computed between high-symmetry configurations. Results are in good agreement with published He atom scattering measures of surface diffusion, especially at the saddle point, and with structural characterization, pointing out the accuracy of DFT calculations for this class of systems.