Ab initio calculations on surface diffusion of Cu adatoms on Cu(001) are presented. The hopping mechanism with a calculated energy barrier of 0.69 eV is found to be favorable over the exchange mechanism with 0.97 eV. We find from the geometry relaxations that adatoms are significantly attracted to the surface and push away nearest-neighbor atoms in the surface. Lateral relaxations of atoms in the surface are larger than vertical ones.