## 3 Results and Discussion
### Frontier MOs and Electron Mobility  
The electron affinity energy and ionization potential of molecules are the important parameters to estimate the ability of oxidation and reduction. The calculated values of the adiabatic EA(\(a\)) and IP(\(a\)) at the wB97 XD/6-311G(\(d\),\(p\)) level and the corresponding experimental data [30] are presented in Table 3, as well as the calculated HOMO-LUMO energy gap (\(E_{\text{g}}\)). From Tables 3 and 4, it can be seen the EA(\(a\)) value of the polyethylene material grafted with the electron-donating group or the electron-withdrawing group is greater than that of the cross-linked polyethylene itself, because the cross-linked polyethylene grafted with other groups is equivalent to introducing a deep trap in the cross-linked polyethylene, which improves its electrical resistance. At the same time, we also found that the introduction of electron-withdrawing groups will lead to a sharp increase in electron affinity and increase the electrical tolerance. However, compared to the grafted electron-donating groups, it can also lead to a larger HOMO-LUMO energy gap, making it more difficult for electrons to be excited, resulting in a decrease in electron mobility. However, structures with lower energy gaps are more likely to be ionized by high-energy electrons, thereby playing a role in electron buffering. Therefore, a structure with both electron-withdrawing and electron-donating groups is our preferred choice, which is consistent with the results of [PERSON] and co-worker's [31] study on the enhanced electrical resistance of 3-Aminobenzoic acid (Aba) blended with polyethylene, supporting our calculation results. It is not difficult to see from Tables 3 and 4 that introducing functional groups with a benzene ring structure will reduce the energy gap between HOMO-LUMO, increase electron affinity, and reduce electron mobility,  
Figure 2: Optimized geometric structures of the studied molecules at the wB97 XD/6-311G(\(d\),\(p\)) level. Orange represents C atoms, white represents H atoms, red represents O atoms, blue represents N atoms, and green represents Cl atoms. such as \(E_{\text{gDap}}\) (10.94 eV) \(>E_{\text{gEbdg}}\) (8.92 eV), EA(\(a\))\({}_{\text{Dap}}\) (\(-\)2.91 eV) \(<\) EA(\(a\))\({}_{\text{Ebdg}}\) (\(-\)1.47 eV), \(\mu_{\text{Dap}}\) (1.31 \(\times\) 10\({}^{-11}\) m\({}^{2}\) V\({}^{-1}\) s\({}^{-1}\)) \(>\mu_{\text{Ebdg}}\) (1.96 \(\times\) 10\({}^{-18}\) m\({}^{2}\) V\({}^{-1}\) s\({}^{-1}\)); \(E_{\text{gAmp}}\) (11.56 eV) \(>E_{\text{gEba}}\) (9.22 eV), EA(\(a\))\({}_{\text{Amp}}\) (\(-\)2.82 eV) \(<\) EA(\(a\))\({}_{\text{Eba}}\) (\(-\)1.55 eV), \(\mu_{\text{Amp}}\) (1.20 \(\times\) 10\({}^{-12}\) m\({}^{2}\) V\({}^{-1}\) s\({}^{-1}\)) \(>\mu_{\text{Eba}}\) (2.23 \(\times\) 10\({}^{-19}\) m\({}^{2}\) V\({}^{-1}\) s\({}^{-1}\)); \(E_{\text{gNmp}}\) (10.57 eV) \(>E_{\text{gEbbb}}\) (8.95 eV), EA(\(a\))\({}_{\text{Nmp}}\) (\(-\)0.02 eV) \(<\) EA(\(a\))\({}_{\text{Ebbb}}\) (0.72 eV), \(\mu_{\text{Nmp}}\) (2.99 \(\times\) 10\({}^{-19}\) m\({}^{2}\) V\({}^{-1}\) s\({}^{-1}\)) \(>\mu_{\text{Ebbb}}\) (2.01 \(\times\) 10\({}^{-47}\) m\({}^{2}\) V\({}^{-1}\) s\({}^{-1}\)). This is because the benzene ring is both an electron-withdrawing and electron-donating group, resulting in molecules having both electron-donating and electron-withdrawing functional groups. It forms a large electron delocalization structure with the benzene ring, which can attract electrons to oscillate around the atomic nucleus, leading to energy dissipation, reducing electron mobility, and reducing the impact on polymer molecular chains. Therefore, when the stabilizer is uniformly dispersed in the polymer, buffering high-energy electrons can effectively suppress the initiation and growth process of the electrical tree in XLPE, which is consistent with the results of [PERSON] and co-worker [32; 33].