on Pb(111) films, and discuss the QSE on the corresponding energy barriers. And finally in Sec. VI, we give our conclusions. 4 II. COMPUTATIONAL METHODS III. THE ADSORPTION PROPERTIES OF O$_{2}$ ON THE PB(111) SURFACE The geometries of ultrathin Pb(111) films are firstly relaxed before our study for the adsorption and dissociation of O$_{2}$ molecules. It is found that the surface relaxations display well-defined QSE $^{29}$. Generally speaking, atoms at the outmost layer tend to relax inward. The DFT total energy calculations are carried out using the Vienna ab initio simu- lation package $^{31}$. The generalized gradient approximation (GGA) of Perdew et al . 32 and the projector-augmented-wave (PAW) potentials 33 are employed to describe the exchange- correlation energy and the electron-ion interaction, respectively. The so-called “repeated slab” geometries 34 are employed to model the clean Pb(111) films ranging from 4 to 8 mono- layers (ML). A vacuum layer of 20 ˚ A is adopted to separate the Pb(111) slabs in two adjacent cells, which is found to be sufficiently convergent from our test calculations. The O$_{2}$ was placed on one side of the slab, namely on the top surface, whereas the bottom layer was fixed. All other Pb layers as well as the oxygen atoms are free to relax until the forces on them are less than 0.02 eV/ ˚ A. The plane-wave energy cutoff is set to 400 eV. A Fermi broadening 35 of 0.1 eV is chosen to smear the occupation of the bands around E$_{F}$ by a finite- T Fermi function and extrapolating to T =0 K. Integration over the Brillouin zone is done using the Monkhorst-Pack scheme 36 with 13 × 13 × 1 grid points. The calculated lattice constant of bulk Pb is 5.03 ˚ A, in good agreement with the experimental value of 4.95 ˚ A $^{37}$. The binding energy of spin-polarized O$_{2}$ is calculated to be D =5.78 eV per atom and the O-O bond length is about 1.235 ˚ A. These results are typical for well-converged DFT-GGA calculations. Compared to the experimental 38 values of 5.12 eV and 1.21 ˚ A for O binding energy and bonding length, the usual DFT-GGA result always introduces an overestimation, which reflects the theoretical deficiency for describing the local orbitals of the oxygen. We will consider this overbinding of O$_{2}$ when drawing any conclusion that may be affected by its explicit value.