## 3 Results
### Simulations
#### 3.2.1 Deuterium (D\({}_{2}\))  
We begin by considering the vibrational wavepacket in the \(B\)-state of the diatomic molecule D\({}_{2}\). The probability distribution of the wavepacket is shown in a contour plot in Figure 3a as a function of time and internuclear distance \(R\). Upon excitation at the inner turning point by the 20 fs pump pulse, the wavepacket oscillates between the inner and outer turning points of the \(B\)-state potential with \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Atom** & **H** & **C** & **N** & **O** & **F** & **S** & **Cl** \\ \hline H & 0.22 (0.17) & 0.40 (0.33) & 0.39 (0.31) & 0.35 (0.27) & 0.31 (0.21) & 0.38 (0.28) & 0.29 (0.17) \\ C & & 0.73 (0.65) & 0.71 (0.61) & 0.64 (0.54) & 0.57 (0.40) & 0.69 (0.54) & 0.53 (0.33) \\ N & & & 0.69 (0.58) & 0.62 (0.50) & 0.55 (0.38) & 0.67 (0.51) & 0.52 (0.32) \\ O & & & & 0.56 (0.44) & 0.50 (0.33) & 0.61 (0.45) & 0.47 (0.28) \\ F & & & & & 0.44 (0.25) & 0.54 (0.34) & 0.41 (0.21) \\ S & & & & & & 0.66 (0.45) & 0.50 (0.28) \\ Cl & & & & & & & 0.39 (0.17) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The damping factor (see Equation (14)) at \(q=10\) Å\({}^{-1}\) for pairs of atoms, including: H, C, N, O, F, S, and Cl. A damping factor with value 1 corresponds to no damping (100% of the signal remains), and 0 to complete damping (the signal vanishes). In parenthesis, the value of the damping factor for \(\gamma_{\alpha}-\sigma_{\alpha}\), with \(\sigma_{\alpha}\) the standard deviation, is given to provide an approximate lower bound. The numerical values for \(\gamma_{\alpha}\) and \(\sigma_{\alpha}\) are taken from Ref. [76]. Figure 2: Damping factors (see Equation (14)) as a function of momentum transfer \(q\) (Å\({}^{-1}\)) calculated using frozen Gaussian widths \(\gamma_{\alpha}\) from [PERSON] et al. [76] for scattering from the following pairs of atoms: C\(-\)H, C\(-\)C, C\(-\)N, C\(-\)O, and C\(-\)S. The damping is given for \(\gamma_{\alpha}\pm\sigma_{\alpha}\), where \(\sigma_{\alpha}\) are the standard deviations in the fitted widths [76]. a period of 82 fs. Due to the anharmonic nature of the potential, the wavepacket gradually spreads across the entire \(B\)-state potential. It is worth emphasizing that this wavepacket has been observed in a recent experiment using a strong-field probe [27]. The elastic X-ray scattering corresponding to this wavepacket is shown in Figure 3b as a function of time and momentum transfer \(q\). In order to focus on the structural component of the diffraction, and to remove the atom-specific effects of the form factors, we plot the _modified molecular intensity_[74],  
\[M(q,t)=qI_{\mathrm{mol}}(q,t)/\left|f_{m}^{0}(q)\right|\ \left|f_{n}^{0}(q) \right|, \tag{16}\]  
with \(f_{m}^{0}=f_{n}^{0}=f_{\mathrm{H}}^{0}\) in the present case. The molecular intensity, \(I_{\mathrm{mol}}(q,t)\), is calculated as \(I_{\mathrm{mol}}(q,t)=\langle\Psi(t)|\sum_{i\
eq j}f_{i}^{0}(q)f_{j}^{0}(q) \sin qR_{ij}/qR_{ij}|\Psi(t)\rangle\), which thus combines the elastic scattering approximation in Equation (10) and the IAM in Equation (12). The modified molecular intensity makes it easier to visualize the elastic scattering intensity at larger values of \(q\), which otherwise decays rapidly due to the exponential decay of the atomic form factors \(f^{0}(q)\)[74]. We assume an instant X-ray probe pulse, \(\delta(t-t_{P})\), to avoid convolution over the temporal X-ray pulse envelope, which would otherwise act to further '_soften_' the diffraction patterns and thus obscure the effect of the wavepacket on the scattering. Since D\({}_{2}\) has a small reduced mass the wavepacket is strongly delocalized from the beginning, even when excited by a bandwidth limited pulse as in the present example. The resulting diffraction signal is thus strongly damped from the outset, as can be seen at high \(q\) values in Figure 3b. It is enlightening to compare the diffraction pattern in Figure 3b to that of a classical D\({}_{2}\) molecule, shown in Figure 3c, in which the nuclei are perfectly localized at all times.