## 3 Results  
### Simple Model  
We begin by examining the effect of the delocalization of the nuclear wavepacket on the diffraction signal via a simple analytical model. For this, we use frozen Gaussian width parameters, \(\gamma_{\alpha}\), fitted to calculated ground state harmonic vibrational wavefunctions in a large set of organic molecules by [PERSON] et al. [76]. Consider a Gaussian nuclear wavepacket, \(v(\mathbf{R})\), expressed as a product of three-dimensional Gaussian wavefunctions, one for each atom. The probability density for such a wavepacket is,  
\[\left|v(\mathbf{R}_{1},\ldots,\mathbf{R}_{N_{\text{at}}})\right|^{2}=\prod_{ \alpha=1}^{N_{\text{at}}}\left(\frac{\gamma_{\alpha}}{\pi}\right)^{\frac{3}{2 }}e^{-\gamma_{\alpha}(\mathbf{R}_{\alpha}-\mathbf{Q}_{\alpha})^{2}}, \tag{13}\]  
where \(\mathbf{R}_{\alpha}=(R_{ax},R_{ay},R_{az})\) are the Cartesian nuclear coordinates, \(\mathbf{Q}_{\alpha}\) is the coordinate that determines the center of the wavepacket, \(N_{\text{at}}\) is the number of atoms, and \(\gamma_{\alpha}\) the width parameter of the Gaussian associated with each atom. There is a large body of theoretical work that shows that such Gaussian wavepackets constitute a sensible basis for nuclear dynamics (see e.g., Refs. [40; 43; 44; 47; 77]). Time-dependent propagation of the wavepacket generally involves propagation (classical, semi-classical, or quantum) of the phase-space coordinates and sometimes the width parameter \(\gamma_{\alpha}\). The X-ray scattering signal will be proportional to \(\left\langle v(\mathbf{R})\right|\left|f_{\mathrm{IAM}}(q)\right|^{2}\left|v( \mathbf{R})\right\rangle_{\mathbf{R}}\), according to the elastic approximation in Equation (10). Solving this integral analytically, including rotational averaging and thus assuming no preferred orientation of the molecule, results in,  
\[I_{\mathrm{IAM}}^{\mathrm{wavepacket}}(q)\propto\sum_{\alpha=1}^{N_{\mathrm{ at}}}\left|f_{\alpha}^{0}(q)\right|^{2}+\sum_{\beta\
eq\alpha}^{N_{\mathrm{at}}}f_{ \alpha}^{0}(q)f_{\beta}^{0}(q)\frac{\sin\left(qQ_{\alpha\beta}\right)}{qQ_{ \alpha\beta}}e^{-q^{2}/2\gamma_{\alpha\beta}}, \tag{14}\]  
with \(Q_{\alpha\beta}=\left|\mathbf{Q}_{\alpha}-\mathbf{Q}_{\beta}\right|\) the distance between centers of the Gaussian wavepackets for each pair of atoms, and with the damping factor \(\exp\left(-q^{2}/2\gamma_{\alpha\beta}\right)\) proportional to the combined Gaussian width of the two atoms given by \(\gamma_{\alpha\beta}=2\gamma_{\alpha}\gamma_{\beta}/\left(\gamma_{\alpha}+ \gamma_{\beta}\right)\). The consequence of introducing the wavepacket in Equation (13) is thus an exponential damping of the molecular term in the scattering, which leaves the atomic term unaffected. It is instructive to contrast Equation (14) with the standard rotationally averaged IAM expression for scattering, originally derived by [PERSON] [78],  
\[I_{\mathrm{IAM}}(q)\propto\sum_{\alpha=1}^{N_{\mathrm{at}}}\left|f_{\alpha}^{0 }(q)\right|^{2}+\sum_{\beta\
eq\alpha}^{N_{\mathrm{at}}}f_{\alpha}^{0}(q)f_{ \beta}^{0}(q)\frac{\sin\left(qR_{\alpha\beta}\right)}{qR_{\alpha\beta}}, \tag{15}\]  
where \(R_{\alpha\beta}\) are the distances between atoms (i.e., \(\mathbf{R}_{\alpha}\) are the fixed nuclear positions). This expression is recovered from the wavepacket-damped expression in Equation (14) for strong localization (\(\gamma_{\alpha}\rightarrow\infty\)). Conversely, strong _d_elocalization (\(\gamma_{\alpha}\to 0\)) extinguishes the structural interference in the molecular term, which is responsible for structural information. It is interesting to note that the damping in Equation (14) takes the same form as the temperature damping derived in electron diffraction [2], but has nothing to do with temperature; it is a consequence of the inherently delocalized nature of the wavepacket [21]. Numerically, we can examine the damping factor \(\exp-q^{2}/2\gamma_{\alpha\beta}\) using the Gaussian widths determined by [PERSON] et al. [76].