## 3 Results
### Simulations
#### 3.2.3 1,3-Cyclohexadiene (CHD)  
We consider the elastic scattering from the polyatomic molecule CHD during the electrocyclic ring-opening reaction triggered by an optical pump pulse [6; 15], as outlined in Section 2.2.3. A reduced-dimensionality representation of the dominant dynamics is shown in Figure 6, in terms of the length of the C-C bond that breaks during the ring-opening reaction. Here, we will focus on the effect of the wavepacket width parameters \(\gamma_{a}\) defined in Equation (6) on the diffraction signal. We show the scattering in terms of the modified molecular intensity, Equation (16), with \(f_{m}^{0}=f_{n}^{0}=f_{C}^{0}\). Note that the simulations are identical in all three scenarios discussed below, and only the \(\gamma_{a}\) parameters are varied. Note that fully converged simulations with an oversampled nuclear basis would not display a dependence on \(\gamma_{a}\) in the scattering.  
First, we examine the diffraction in the fully localized limit, i.e., for \(\gamma\to\infty\), shown in Figure 7a. This is also known as the bracket-averaged Taylor expansion (BAT) approximation [67], and is equivalent to perfectly localized trajectories not dissimilar to the trajectories obtained in surface hopping simulations. In this case, the damping of the molecular (interference) term is absent, and as expected, there is little deterioration of the diffraction signal even at large values of \(q\). In contrast, for the standard values of \(\gamma_{a}\) (taken from [PERSON] et al. [76]), there is a significant degradation of the diffraction signal at large values of \(q\) which persists at all times, as can be seen in Figure 7b. Given that the default width parameters are fitted to ground state molecules, it is not far-fetched that the most appropriate values will deviate from the standard \(\gamma_{a}\) values. We therefore consider, in the third scenario shown in Figure 7c, the diffraction when the width factors have been halved, i.e., \(\gamma_{a}/2\). This  
Figure 5: Contour plots of modified (elastic) scattering intensity, \(M(q,t)\), for the photoexcited dynamics of ethylene. Time in fs and momentum transfer \(q\) in Å\({}^{-1}\), with \(M(q,t)\) defined in Equation (16) (we use form factors for the C-atom in the denominator). (**a**) A small set of 20 trajectories; (**b**) A large set of 1000 trajectories, giving a better representation of dispersion of the wavepacket at long times.  
leads to an even more delocalized wavepacket and has a dramatic effect on the diffraction signal, with the signal becoming very weak for \(q>15\) A\({}^{-1}\) (intensity below the height of the lowest contour in Figure 7c). Again, considering that the standard values of \(\gamma_{\alpha}\) constitute a conservative estimate of the width of the wavepacket, this emphasizes the strong effect of the damping due to the delocalization of wavepackets.  
In addition to the effect of the width parameters discussed above, we also observe in Figure 7 that the dispersion plays an important role in degrading the signal over time, in particular for \(t>50\) fs, similarly to what was observed in the calculations for D\({}_{2}\) and ethylene. This 50 fs time-scale is consistent with Figure 6, which indicates that the dynamics is comparatively uniform at early times (the so-called 'ballistic' nature of this particular reaction [35]), but diverges at later times (\(t>50\) fs in Figure 6).  
Figure 6: Representative C\(-\)C bondlength distances for the bond that breaks during the 1,3-cyclohexadiene (CHD) ring-opening reaction, shown as a function of time. Reproduced from Ref. [6]. Trajectories in blue lead back to the ring-closed form, while trajectories in green lead to the ring-open form of the molecule.  
Figure 7: _Cont._