Acetylene and its evolution products (CCH, C-2, and CCH2) on the Cu (001) surface are studied using the density functional method with cluster models. The binding energies, geometries, STM images, and vibrational frequencies for the adsorbates are obtained. The results agree well with the available experimental results. By comparing the calculated frequencies with the experimental ones, we identify the thermal evolution product of acetylene on Cu(001) in the HREELS experiment as CCH2, and establish its orientation relative to the surface. CCH2 is found to be more stable than C2H2 on Cu(001).