We have developed a method for adding the effect of the surface barrier to the Van HoveTong suite of low-energy electron-diffraction programs enabling us to analyze very-low-energy electron-diffraction (LEED) spectra from complex reconstructed systems. The method models the effect of the surface-potential barrier on the intensities and accurately replicates the fine-structure features found at low energies on many surfaces. It is free from many of the simplifying assumptions made by others attempting to model similar systems. We find that the fine-structure peak positions depend on the inelastic scattering in the barrier region as well as the shape of the real part of the potential barrier. This limits the accuracy of measurement of the shape of the barrier, but useful shape information can be derived from these analyses. We have studied the effect of oxygen overlayers on Cu(001) as well as the clean Cu(001) surface. These surfaces are well described by the model for a range of incident-beam directions. On the oxygen-exposed surface, it is necessary to assume that a c(2×2) structure exists as a precursor to the (2 ×2 2) R45°in order to explain all the data. Further, it is found that the surface-potential barrier moves away from the surface with establishment of the precursor c(2×2) structure, but then moves back towards the surface as the final (2 ×2 2) R45°forms with additional exposure to oxygen. This accounts for the changes in work function with exposure to oxygen found on this surface.