In contrast, atoms at the second layer tend to relax outward. And both the contraction for the first interlayer and the expansion for the second interlayer exhibit oscillations of bilayer period, reflecting the effect of the quantum well states $^{25}$. We then check the reliability of our adiabatic calculations. The main reason for the wrong conclusions lead by adiabatic first-principles calculations on the adsorption of O$_{2}$ on sp -metals lies in that the lowest unoccupied molecular orbital (LUMO) state of O$_{2}$ is always aligned with the Fermi level at any distance between the molecule and metal surfaces such as Al(111), allowing a partial filling of the empty molecular orbital $^{39}$. Due to this unphysi- cal (metal-molecule or intramolecule) charge transfer, the adsorption of O$_{2}$ on the Al(111) surface is calculated to be dissociative without any energy barriers, which contradicts with the experimental observations of low initial sticking probability at low temperatures $^{8,9}$. In our previous work $^{19}$, we have shown that the unphysical partial filling of the LUMO does not happen during the adsorption of O$_{2}$ on the Be(0001) surface, and thus we obtained sizeable energy barriers for the dissociation of O$_{2}$. Here for the O$_{2}$/Pb(111) system, molec- ular adsorption precursor states exist, so we can not judge the reliability of our calculations by whether a sizable energy barrier exists or not. We calculate and show in Fig. 1(a) the electronic density of states (DOS) of an O$_{2}$ molecule in close to a 5 ML Pb(111) film. In comparison with the DOS of O$_{2}$ in close to the Mg(0001) and Al(111) surfaces depicted in Figs. 1(b) and (c), we find that the triplet ground state of the O$_{2}$ molecule is not influenced at all in close to the Pb(111) film, but totally disturbed by the Mg(0001) and Al(111) sur- faces due to the unphysical charge transfers. Therefore, we are sure that the adiabatic DFT calculations will not lead to unphysical conclusions for the O$_{2}$/Pb(111) system, as well as for the O$_{2}$/Be(0001) system. To facilitate our discussions, we give some notations at first. Formally, there are four high-symmetry sites on the Pb(111) surface, respectively the top (T), bridge (B), hcp (HH), and fcc (FH) hollow sites. Our previous study has already shown that the high-symmetry sites play crucial roles in the adsorption O$_{2}$ molecules on Pb(111) films $^{29}$. Therefore we do not consider other surface sites any more. At each surface site, an O$_{2}$ molecule has three different high-symmetry orientations, respectively along the x (i.e., [0 ¯ 11]), y (i.e., [ ¯ 211]), and z (i.e., [111]) directions. Therefore, we will use T- x, y, z , B- x, y, z , HH- x, y, z , and FH- x, y, z to respectively represent the 12 high symmetry adsorption channels for O$_{2}$. After geometry optimizations by initially setting the O$_{2}$ molecule at 4.0 ˚ A along each 5