## 2 Results and Discussion
### The Second Vinyl Bond of the bis-Styrene Molecule (DVB and BVPE)
\begin{table}
\begin{tabular}{c c c c c c} \hline  & \multicolumn{3}{c}{**WBI**} & \multicolumn{3}{c}{**Q\({}_{\rm NBO}\) (\(\epsilon\))**} \\ \cline{2-6}  & **B (Zr-H\({}_{1}\))** & **B (C\({}_{2}\)-C\({}_{3}\))** & **B (Zr-C\({}_{2}\))** & **B (C\({}_{3}\)-H\({}_{1}\))** & **Zr** & **C\({}_{2}\)** \\ \hline DVB[BVPE]-PP\({}_{1}\)+ [Zr]-H & 0.877 & 1.897 & & & 1.311 & \(-\)0.421 \\  & [0.877] & [1.898] & & & [1.311] & [\(-\)0.419] \\ IM\({}_{3,1,2}\) & [0.889] & [1.595] & 0.345 & 0.006 & 1.095 & \(-\)0.643 \\ TS\({}_{3,1,2}\) & [0.889] & [1.598] & [0.342] & [0.005] & [1.095] & [\(-\)0.639] \\ TS\({}_{3,1,2}\) & 0.604 & 1.392 & 0.529 & 0.283 & 1.148 & \(-\)0.703 \\  & [0.604] & [1.145] & [0.530] & [0.291] & [1.149] & [\(-\)0.704] \\ PR\({}_{1,2}\) & 0.144 & 1.024 & 0.788 & 0.782 & 1.319 & \(-\)0.806 \\ \hline  & \multicolumn{3}{c}{B (Zr-H\({}_{1}\))} & \multicolumn{1}{c}{B (C\({}_{2}\)-C\({}_{3}\))} & \multicolumn{1}{c}{B (Zr-C\({}_{3}\))} & \multicolumn{1}{c}{B (C\({}_{3}\)-H\({}_{1}\))} & \multicolumn{1}{c}{Zr} & \multicolumn{1}{c}{C\({}_{3}\)} \\ \hline IM\({}_{3,1}\) & 0.817 & 1.693 & 0.235 & 0.067 & 0.940 & \(-\)0.302 \\  & [0.819] & [1.697] & [0.232] & [0.065] & [0.948] & [\(-\)0.301] \\ \hline TS\({}_{3,1}\) & 0.636 & 1.497 & 0.383 & 0.261 & 0.890 & \(-\)0.352 \\  & [0.634] & [1.495] & [0.385] & [0.263] & [0.888] & [\(-\)0.353] \\ PR\({}_{2,1}\) & 0.181 & 1.081 & 0.650 & 0.747 & 1.043 & \(-\)0.469 \\  & [0.182] & [1.083] & [0.647] & [0.745] & [1.037] & [\(-\)0.465] \\ \hline \end{tabular}
\end{table} 
Table 2: The Wiberg bond indices (WBI) and natural charges (Q\({}_{\rm NBO}\)) for some key bonds and atoms for the reaction pathway of the second vinyl bond insertion; the values of BVPE are given in brackets. Figure 7: Optimized structures of key TSs and intermediates (IMs) in the BVPE reaction pathway shown in Figure 5. The key length is in angstroms. value of 3.311 and 3.264, respectively. The global reactivity index values of [Zr]-DVB-PP\({}_{1}\), [Zr]-[BVPE]-PP\({}_{1}\), and [Zr]-H reveal that they are electrophilic with global electrophilicity (\(\omega\)) values of 5.088, 5.128, and 5.260, respectively. The results also show that the \(\omega\) value of [Zr]-H is slightly higher than that of [Zr]-DVB-PP\({}_{1}\) and [Zr]-[BVPE]-PP\({}_{1}\), so that the electrophilicity of [Zr]-H is higher. Similar conclusions can be drawn from the local reactivity index values. The local electrophilicity (\(\omega_{local}\)) value of Zr in [Zr]-H (1.298) is higher than those of [Zr]-DVB-PP\({}_{1}\) (0.842) and [Zr]-[BVPE]-PP\({}_{1}\) (0.854), indicating that Zr in [Zr]-H has greater reactivity. The local nucleophilicity (\(N_{\text{Nu-local}}\)) index values of C\({}_{2}\) in DVB-PP\({}_{1}\) (0.473) and [BVPE]-PP\({}_{1}\) (0.308) are higher than those of C\({}_{3}\) (0.221 and 0.140), indicating that C\({}_{2}\) has greater reactivity; thus, it more easily reacts with Zr in [Zr]-H to form a Zr-C\({}_{2}\) bond to complete the 1,2-insertion reactions. The local nucleophilicity (\(N_{\text{Nu-local}}\)) index values of C\({}_{2}\) and C\({}_{3}\) in DVB-PP\({}_{1}\) (0.473 and 0.221) are higher those in [BVPE]-PP\({}_{1}\) (0.308 and 0.140), indicating that DVB-PP\({}_{1}\) has more reactivity; thus, it more easily reacts with [Zr]-H, consistent with the potential barrier of the DVB-PP\({}_{1}\) 1,2-insertion reaction process being lower than that of BVPE-PP\({}_{1}\) by 2.8 kcal/mol. The Fukui function is often used to predict reactions. The site with a larger Fukui function value has a higher reactivity. The Fukui function information for some molecules is shown in Table 3 and Figure 8. The results show that the Fukui function value (\(f^{*}\)) of Zr in [Zr]-H is higher than those of [Zr]-DVB-PP\({}_{1}\) and [Zr]-[BVPE]-PP\({}_{1}\), so the Zr in [Zr]-H is more reactive.