## 4 Multi-Step Reactions: The Dissociation of Vinyl Cyanide  
The dissociation of vinyl cyanide (VC, C\({}_{3}\)H\({}_{3}\)N), is particularly interesting because it involves multiple reaction channels and different sets of products (HCN, HNC, HCCH, and :CCH\({}_{2}\)) and hence \begin{table}
\begin{tabular}{c c c} \hline \multicolumn{3}{c}{**Arrhenius (Equation (7)), No Tunneling**} \\ \hline \(A\) (s\({}^{-1}\)) & \(E_{\text{a}}/R\) (K) & \(\mathcal{R}^{2}\) \\ \(1.01\times 10^{12}\) & 23636.0 & 1.0000 \\ \hline \multicolumn{3}{c}{**Arrhenius (Equation (7)), with Tunneling**} \\ \hline \(A\) (s\({}^{-1}\)) & \(E_{\text{a}}/R\) (K) & \(\mathcal{R}^{2}\) \\ \(9.12\times 10^{9}\) & 21752.9 & 0.9973 \\ \hline \multicolumn{3}{c}{**Arrhenius–Kooij (Equation (8)), with Tunneling**} \\ \hline \(\alpha\) (s\({}^{-1}\)) & \(\beta\) & \(\gamma\) (K) & \(\mathcal{R}^{2}\) \\ \(1.20\times 10^{1}\) & 21.4 & 16000.4 & 0.9992 \\ \hline \end{tabular}
\end{table} 
Table 1: Results of the fit of the Arrhenius and Arrhenius–Kooij equations to the computed thermal rate coefficients for the isomerization reaction of hydroxyacetone to 2-hydroxyponal in the temperature range 151–501 K.  
Figure 5: Arrhenius plot of the computed thermal rate coefficient (both neglecting and including tunneling) for the isomerization reaction of hydroxyacetone to 2-hydroxyponal (Figure 3) and of the Arrhenius and Arrhenius–Kooij best-fitting curves. it serves as a very good test case for master-equation based kinetic models. The potential-energy surface for this reaction has been investigated in a recent work by [PERSON] et al. [35] through ab initio CCSD and CCSD(T) calculations with the 6-311+G(2d,2p) and 6-311++G(3 df,3 pd) basis sets. In the same work, a reaction scheme involving ten unimolecular steps, three of which reversible, was proposed. The ten reaction steps are summarized in Table 2, while the associated reaction diagram is given in Figure 6.  
As shown in Figure 6 and Table 2, VC can directly dissociate to product sets :CCH\({}_{2}\) + HCN and HCCH + HCN through steps 1 and 2 (the only direct dissociation paths), or lead to formation of reaction intermediates Int1-III (the most stable one), Int1-IV, and Int1-V, further evolving to products. On the other hand, product HNC can only be formed via indirect dissociation paths involving the above-mentioned intermediates. A screenshot of VMS showing the structures of all the molecular species involved in this reaction is given in Figure 7.  
Within a master-equation approach (see for instance [36]), to determine the time evolution of the relative abundance of the involved species, initially a matrix, \(\mathbf{K}\), is set up by opportunely combining the microcanonical rate coefficients at a specified energy. In particular, the diagonal elements \(K_{ii}\) contain the loss rate of species \(i\), while the off-diagonal elements \(K_{ij}\) contain the rate of formation of species \(i\) from species \(j\). The rate of change in the concentration of each species is given by the vector differential equation:  
\[\frac{\mathrm{d}\mathbf{c}}{\mathrm{d}t}=\mathbf{Kc} \tag{9}\] \begin{table}
\begin{tabular}{c l l} \hline \hline  & **Reaction Step** & \(\mathbf{E_{a}/cm^{-1}}\) \\ \hline
1 & VC \(\xrightarrow{k_{1f}}\) :CCH\({}_{2}\) + HCN & 35185 \\
2 & VC \(\xrightarrow{k_{2f}}\) HCCH + HCN & 41271 \\
3 & VC \(\xleftarrow{k_{3f}}\) Int1-III & 20076 \\
4 & Int1-III \(\xrightarrow{k_{4f}}\) HCCH + HCN & 32947 \\
5 & VC \(\xleftarrow{k_{5f}}\) Int1-IV & 37494 \\
6 & Int1-IV \(\xrightarrow{k_{6f}}\) HCCH + HNC & 12521 \\
7 & VC \(\xleftarrow{k_{7f}}\) Int1-V & 36724 \\
8 & Int1-V \(\xleftarrow{k_{8f}}\) :CCH\({}_{2}\) + HNC & 19727 \\
9 & Int1-III \(\xrightarrow{k_{8f}}\) :CCH\({}_{2}\) + HNC & 31128 \\
10 & Int1-III \(\xrightarrow{k_{10f}}\) HCCH + HNC & 32912 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Reaction steps involved in the dissociation mechanism of vinyl cyanide considered in this work.