## 2 Results and Discussion
### The Second Vinyl Bond of the bis-Styrene Molecule (DVB and BVPE)
Relative to the DVB-PP\({}_{1}\) and [Zr]-H catalytic site, the potential barrier of the 1,2 insertion is 2.0 kcal/mol, while that of the 2,1 insertion is 7.4 kcal/mol. The total discharge energy for the 1,2 insertion is 12.2 kcal/mol, while that of the 2,1 insertion is 5.6 kcal/mol. Table 2 shows the Wiberg bond indices and natural charges for some key bonds and atoms for the reaction pathway of the second vinyl bond insertion. The WBI values of the Zr-C\({}_{2}\) bond (0.529 and 0.788) and C\({}_{3}\)-H\({}_{1}\) bond (0.283 and 0.782) for TS3\({}_{1,2}\) and PR\({}_{1,2}\) are larger than the corresponding values for the Zr-C\({}_{3}\) bond (0.383 and 0.650) and the C\({}_{3}\)-H\({}_{1}\) bond (0.261 and 0.747) for TS3\({}_{2,1}\) and PR\({}_{2,1}\). On the contrary, the WBI values of the Zr-H\({}_{1}\) bond (0.604 and 0.144) and the C\({}_{2}\)-C\({}_{3}\) bond (1.392 and 1.024) for TS3\({}_{1,2}\) and PR\({}_{1,2}\) are smaller than the corresponding values of the Zr-H\({}_{1}\) bond (0.636 and 0.181) and the C\({}_{2}\)-C\({}_{3}\) bond (1.497 and 1.081) for TS3\({}_{2,1}\) and PR\({}_{2,1}\). These all indicate an interaction between [Zr]-H and DVB-PP\({}_{1}\) in TS3\({}_{1,2}\), and PR\({}_{1,2}\) is stronger than that in TS3\({}_{2,1}\) and PR\({}_{2,1}\). A similar conclusion can be drawn from the natural charges. The Q\({}_{\rm{NBO}}\) values of the Zr atom in TS3\({}_{1,2}\) and PR\({}_{1,2}\) are 1.148 e and 1.319 e, which are much larger than the Q\({}_{\rm{NBO}}\) values of the Zr atom (0.890 e and 1.043 e) in TS3\({}_{2,1}\) and PR\({}_{2,1}\), and the Q\({}_{\rm{NBO}}\) values of the C\({}_{2}\) atom in TS3\({}_{1,2}\) and PR\({}_{1,2}\) are \(-\)0.703 e and \(-\)0.806 e, which are much smaller than the corresponding Q\({}_{\rm{NBO}}\) values of the C\({}_{3}\) atom (\(-\)0.352 e and \(-\)0.469 e) in TS3\({}_{2,1}\) and PR\({}_{2,1}\). The situation for BVPE is similar to that for DVB. The 1,2-mode insertion is thermodynamically preferred to the 2,1-mode insertion. The energy in the brackets in Figure 5 represents the energy in the BVPE reaction process, and Figure 7 lists the main TSs and the configurations of the intermediates. From an energy perspective, and for the relative stability of the activated zirconocene containing BVPE-PP\({}_{1}\) and [Zr]-H, the more feasible reaction pathway is the 1,2 insertion. The insertion of the vinyl bond of BVPE-PP\({}_{1}\) into [Zr]-H leads the transition state [TS3\({}_{1,2}\)] via [IM3\({}_{1,2}\)] before forming the product [PR\({}_{1,2}\)]. The potential barrier of the whole process is 4.8 kcal/mol (slightly higher than 2.0 kcal/mol for DVB-PP\({}_{1}\)), and the exothermic energy is 12.6 kcal/mol (similar to 12.2 kcal/mol for DVB-PP\({}_{1}\)). In the second method, BVPE-PP\({}_{1}\) needs to rotate 180\({}^{\circ}\) along the Zr-C\({}_{1}\) bond, BVPE-PP\({}_{1}\), and insert into [Zr]-H to form [IM3\({}_{2,1}\)], which transits to [TS3\({}_{2,1}\)], leading to the product [PR\({}_{2,1}\)]. The potential barrier of the whole process is 7.3 kcal/mol (similar to 7.4 kcal/mol for DVB-PP\({}_{1}\)), and the exothermic energy is 5.5 kcal/mol (similar to 5.6 kcal/mol for DVB-PP\({}_{1}\)). In comparison, the 1,2-insertion reaction path is more exothermic and has a lower energy barrier. Therefore, for the insertion of the second vinyl bond of the _bis_-styrene molecule [BVPE], the 1,2-insertion mode is the optimal insertion mode. To further investigate the reactions, the global reactivity index, local reactivity index, and Fukui functions values for some molecules were calculated (Table 3). The complete local reactivity index and Fukui function values of these molecules are given in the Supplementary Materials (Table S1). The global reactivity index values of DVB-PP\({}_{1}\) and [BVPE]-PP\({}_{1}\) clearly reveal the nucleophilicities of the molecules with a global nucleophilicity (\(N_{\rm Nu}\))