## 3 Elementary Steps: The Interconversion Reaction of Hydroxyacetone and 2-Hydroxypromal
The quantity \(P_{\text{tunn}}(E^{\prime})\) is the tunneling coefficient at the energy \(E^{\prime}\), and is given by the expression  
\[P_{\text{tunn}}(E^{\prime})=\frac{\sinh(a)\sinh(b)}{\sinh^{2}((a+b)/2)+\cosh^ {2}(c)} \tag{4}\]  
where, \(a\), \(b\), and \(c\) are parameters defined by:  
\[a=\frac{4\pi\sqrt{E^{\prime}+V_{0}}}{h\
u_{i}(V_{0}^{-\frac{1}{2}}+V_{1}^{- \frac{1}{2}})},\ b=\frac{4\pi\sqrt{E^{\prime}+V_{1}}}{h\
u_{i}(V_{0}^{-\frac{ 1}{2}}+V_{1}^{-\frac{1}{2}})},\ c=2\pi\sqrt{\frac{V_{0}V_{1}}{(h\
u_{i})^{2}}- \frac{1}{16}}. \tag{5}\]  
Here, \(V_{1}\) is the classical energy barrier for the reverse reaction, and \(\
u_{i}\) is the magnitude of the imaginary frequency of the saddle point (in Equation (5), \(h=1\) if the energies are expressed, as in this work, in cm\({}^{-1}\)). For illustrative purposes, we computed the microcanonical rate coefficient for the direct and inverse reaction of Figure 3, both with and without tunneling correction. To this purpose, the three molecular species were modeled by density-functional theory with the B2 PLYP-D3/jun-cc-pVTZ model chemistry. According to our calculations, the forward reaction is exoergic by 1719 cm\({}^{-1}\) with a barrier of 16448 cm\({}^{-1}\) (the barrier for the backward reaction is 14729 cm\({}^{-1}\)). The resulting microcanonical rate coefficients \(k(E)\) are plotted in Figure 4 (on a logarithmic scale) for the forward (blue line) and backward (red line) reaction as a function of the energy relative to the hydroxyacetone zero-point energy. In the same figure, the dashed curves are the tunneling-corrected ones. As apparent, the tunneling correction enhances the reaction rate both in the forward and backward direction, more visibly nearby the threshold region, thus lowering the actual value of the reaction threshold in both directions. The thermal rate coefficient can easily be computed from the microcanonical rate coefficients using the following equation:  
\[k(T)=\frac{1}{Q(T)}\int_{0}^{\infty}k(E)\rho(E)e^{-E/k_{\text{B}}T}\text{d}E\, \tag{6}\]  
with \(Q(T)\) being the partition function of the reactant species. The computed thermal rate coefficients for the forward reaction (isomerization reaction of hydroxyacetone to 2-hydroxyproman) in the temperature range 151-501 K are given as Arrhenius plot (\(\log_{10}k(T)\) versus \(1/T\)) in Figure 5.  
Results are reported both neglecting (blue triangles) and including (blue circles) tunneling. These data can be fitted to the popular Arrhenius equation:  
\[k(T)=Ae^{-\frac{E_{\text{a}}}{kT}} \tag{7}\]  
(with \(A\) being the pre-exponential factor, \(E_{\text{a}}\) the activation energy, and \(R\) the gas constant) or to the more refined Arrhenius-Kooij formula [33] (also known as modified Arrhenius equation [34]) allowing for a temperature dependence of the pre-exponential factor:  
Figure 2:0  
which essentially implies a linear variation of the activation energy with the temperature, \(E_{\text{a}}/R=\gamma+\beta T\). The Arrhenius best-fitting curve for both the tunneling-corrected and no-tunneling data are shown in Figure 5 as dashed black line and solid black line, respectively. The Arrhenius-Kooij best-fitting curve for the tunneling-corrected data is also reported as a red dashed line in the same  
Figure 2:1 Microcanonical rate coefficients for the backward (red color) and forward (blue color) isomerization reaction of hydroxyacetone and 2-hydroxypromanal (Figure 3) as a function of the energy relative to the reactant zero-point energy. Dashed lines are the tunneling-corrected versions of the rate coefficients. figure, while the best-fitting parameters together with the associated coefficient of determination \(\mathcal{R}^{2}\) are given in Figure 2:2  
As evident from Figure 5 and Table 1, the Arrhenius equation perfectly fits the thermal rate coefficients calculated by neglecting tunneling, yielding a \(\mathcal{R}^{2}=1.0000\) and an activation energy of 16428 cm\({}^{-1}\) that compares well with the computed reaction barrier.