6 adsorption channel, we have got 6 molecular adsorption states for O$_{2}$ $^{29}$. By using numbers 1, 2, and 3 to represent the O$_{2}$ orientation along the x, y, and z directions, we name the six adsorption states respectively as HH1, HH2, HH3, FH1, FH2, and FH3. The atomic geometries for the HH1, HH2 and HH3 states are listed in Figs. 2(b), (c) and (d) respectively. The geometries for the FH1, FH2 and FH3 states are very similar to HH1, HH2 and HH3 states, with the adsorbed oxygen atoms at fcc hollow sites instead of hcp hollow sites. Among the six precursor states, HH1 and HH2 are the most stable and two nearly degenerate states. In our previous work, we have already discussed in detail the interactions between oxygen and Pb for the HH1 state $^{29}$. Here we further analyze the electronic interactions for the HH2 state to discuss why the two adsorption states with different O$_{2}$ orientations (i.e. HH1 and HH2 states) are almost degenerate. Figure 3 shows the calculated difference charge density Δ ρ ( r ) for the HH2 state on the 5 ML Pb(111) film, which is obtained by subtracting the electron densities of noninteracting component systems, ρ $^{Pb(111)}$( r ) + ρ $^{O$_{2}$}$( r ), from the density ρ ( r ) of the HH2 state, while retaining the atomic positions of the component systems at the same location as in HH2. A positive Δ ρ ( r ) here represents charge accumulation, whereas a negative Δ ρ ( r ) represents charge depletion. Figure 3(a) clearly shows a charge transfer from Pb to O$_{2}$, which is consistent with the electronegativity difference between O and Pb atoms, and same with what we found for the HH1 state $^{29}$. The contour plot for Δ ρ ( r ) in the plane containing the two O atoms and parallel to the Pb surface is shown Fig. 3(b), we can see that there is an electron depletion area in the middle of the two oxygen atoms, indicating the donation of electrons from bonding orbitals of O$_{2}$ to Pb. We can also see two electron accumulation areas around the two oxygen atoms indicating the back donation of electrons from Pb to the π ⋆ p antibonding orbital of O$_{2}$. This interaction mechanism of O$_{2}$-to-surface donation and surface-to-O$_{2}$ π ⋆ p back donation is just the same with that in the HH1 state $^{29}$. So it is no wonder that the two adsorption states are nearly degenerate and have similar adsorption energies. In comparison with the Be(0001), Mg(0001) and Al(111) surfaces, Pb(111) is unique in that a parallel O$_{2}$ molecularly adsorbs only on it. This is because of the different surface electronic structures. As shown in Fig. 4, the p electronic states around the Fermi energy strongly hybridize with s states for both the Mg and Al surfaces (also for the Be(0001) surface $^{19}$), but does not hybridize at all for the Pb(111) surface. As we have seen $^{29}$, the