and HH2 (d) states to the atomic adsorption at two fcc hollow sites, as functions of the O$_{2}$ bond length d$_{O}$$_{−}$$_{O}$ and distance h (from the surface). 14 Fig.8 (color online) The projected density of states around the two oxygen atoms at the initial (a), transition (b) and final states (c) of the dissociation path from the HH1 state to the atomic adsorption of two oxygen atoms at neighboring hcp and fcc hollow sites. The Fermi energies are set to 0. Fig.9 (color online) Color-filled contour plots of the potential energy surface for O$_{2}$ dissociation on the 4 (a), 5 (b), 6 (c) and 7 ML (d) Pb(111) films from the HH1 state to the atomic adsorption of two oxygen atoms at neighboring hcp and fcc hollow sites, as functions of the O$_{2}$ bond length d$_{O}$$_{−}$$_{O}$ and distance h (from the surface).