We have performed first-principles total-energy calculations of low-dimensional sections of the electronically adiabatic potential energy surface (PES) that are relevant for the Eley-Rideal (ER) reaction of H atoms on a rigid Cu(111) surface. These calculations were performed within density-functional theory using a plane-wave and pseudopotential method and the generalized gradient approximation for the exchange-correlation energy. The calculated energy points for various configurations of one and two atoms on the Cu(111) surface were used to construct a model PES that can be used in ER reaction dynamics calculations.