molecule keeps its triplet electronic structure. As we have discussed, the LUMO of O$_{2}$ is above the Fermi energy, proving the reliability of our adiabatic DFT calculations. The highest occupied molecular orbital (HOMO) and LUMO are the spin-up and spin-down antibonding π ⋆ p molecular orbitals, respectively. Besides of HOMO and LUMO, p electrons of O$_{2}$ make up of two bonding orbitals, σ$_{p}$ and π$_{p}$ . For both spins, the σ$_{p}$ orbitals are lower in energy than the π$_{p}$ orbitals. When the O$_{2}$ molecule evolves into the transition state, electrons transfer from the σ$_{p}$ and π$_{p}$ molecular orbitals to Pb through hybridizations. As shown in Fig. 6(b), the σ$_{p}$ and π$_{p}$ peaks are diminished. We can also see from Fig. 6(b) that the HOMO and LUMO of O$_{2}$ are broadened through hybridizations with electronic states of Pb. During the electronic hybridizations, the spin splittings for all the O$_{2}$ molecular orbitals are all decreased. In the final dissociated adsorption state, the total spin of the adsorption system becomes zero. We can see from Fig. 6(c) that the molecular orbitals of O$_{2}$ no longer exist after electronic hybridizations. As discussed in our previous work $^{30}$, the interactions between O and Pb atoms display a mixed ionic/covalent character. 

The dissociation of O$_{2}$ from the molecular adsorption precursor states are more compli- cated than from O$_{2}$ gases because the formation and breaking of O-Pb bonds during the dissociation of O$_{2}$ also need to be considered. Detailed information on the atomic adsorp- tion of oxygen on the Pb(111) surface can help us to simplify the problem. In our previous study $^{30}$, we have found that the O atoms choose to adsorb at hollow sites on the Pb(111) surface. Thus we only need to consider the dissociation of an adsorbed O$_{2}$ molecule into different hollow sites. For an O$_{2}$ molecule in the HH1 or HH2 state, there are two different dissociation paths, respectively to the atomic adsorption of oxygen at two neighboring fcc and hcp hollow sites and to the atomic adsorption of oxygen at two fcc hollow sites. The calculated 2D PES cuts for the adsorbed O$_{2}$ molecules in HH1 and HH2 states to dissociate into two neighboring hollow sites are shown in Figs. 7(a) and (b), while the calculated 2D PES cuts for O$_{2}$ to dissociate into two fcc hollow sites are also shown in Figs. 7(c) and (d). The calculated energy barriers for these four paths are respectively 0.22, 0.26, 0.59 and 0.37 eV. These values indicate that the most energetically favorable dissociation path for an adsorbed O$_{2}$ molecule is from the HH1 adsorption state into the atomic adsorption of oxygen at two neighboring fcc and hcp hollow sites. The saddle point in Fig. 7(a) corresponds to the adsorption structure of two oxygen atoms near two neighboring bridge sites, with their horizontal distance to be 1.40 ˚ A. The smallest energy barriers for an O$_{2}$ molecule in the 

