## 4 Discussion  
Our analytical model in Section 3.1 indicates that degradation of the diffraction signal beyond \(q\) values of about 10-15 A\({}^{-1}\) is significant, with at least 50% of the signal lost by \(q=15\) A\({}^{-1}\). This is a conservative estimate, since it is based on width factors fitted to ground state harmonic vibrational wavefunctions. These widths reflect the shape of the initially excited wavepacket in the Franck-Condon region above the ground state equilibrium geometry, which is why the Wigner distribution is used for sampling initial conditions in semi-classical propagation schemes [57]. However, subsequent propagation of the excited wavepacket leads to further dispersion. This is a general phenomenon, as emphasized by our simulations in three different molecules: D\({}_{2}\), 1,3-cyclohexadiene, and ethylene. In all three examples, the scattering signal degrades further with time as the wavepacket evolves. In D\({}_{2}\) (Section 3.2.1), a diatomic molecule, we compare to a classically oscillating molecule and find that the diffraction signal is better defined and persists for all \(q\) when the nuclei are perfectly localized. Furthermore, we examine the effect of nuclear mass on the wavepacket, and find that the greater density of states associated with heavier atoms (or conversely a pump pulse with greater bandwidth) leads to greater initial localization. However, this sharper initial localization, which leads to better defined scattering, also leads to faster and more complete dispersion as the wavepacket evolves, degrading the scattering at longer times. In the ethylene molecule (Section 3.2.2), we examine the sensitivity of the scattering signal to the number of trajectories used in the simulations. We find that the number of trajectories correlates with the dispersion recorded in the scattering signal, such that a larger number of trajectories lead to an increased degradation of the signal. Strikingly, a small number of trajectories can qualitatively reproduce the scattering signal, indicating that a comparatively small number of trajectories can capture the essence of the reaction path and thus the diffraction signal, albeit while underestimating the dispersion at longer times. This finding is congruent with the analysis of recent ultrafast X-ray scattering experiments in 1,3-cyclohexadiene [6]. However, using not fully converged quantum molecular dynamics simulations, or equivalently just a small number of trajectories, introduces a sensitivity in the predicted diffraction patterns on the width parameters associated with each trajectory. In our simulations of 1,3-cyclohexadiene (Section 3.2.3), we examine this effect for the example of a larger, polyatomic molecule. Already with the standard and rather conservative width parameters, the scattering degrades notably at large values of \(q\), and an additional  
Figure 7: Contour plots of modified (elastic) scattering intensity, \(M(q,t)\), for the ring-opening reaction of 1,3-cyclohexadiene (CHD) [6]. Time in fs and momentum transfer \(q\) in Å\({}^{-1}\), with \(M(q,t)\) defined in Equation (16) (we use C-atom form factors in the denominator). (**a**) Perfectly localized trajectories with \(\gamma_{a}\rightarrow\infty\); (**b**) Standard values of the width parameters \(\gamma_{a}\); (**c**) Increased width, i.e., more delocalized wavepackets, obtained via half-value 0.5\(\gamma_{a}\) width parameters. 50% decrease in the width factors is sufficient to effectively quench the molecular diffraction signal for \(q>14\) A\({}^{-1}\). This suggests that width parameters must be considered whenever ultrafast diffraction data for the time-evolution of molecular geometry is analysed. Clearly, the concept of molecular geometry, in a classical sense, must be used with caution in the context of ultrafast processes and ultrafast diffraction in particular.