# Hydrogen Deuteride for Cold Neutron Production: A Model for the Double Differential Cross Section
## Appendix B The Self Dynamic Structure Factor in the Gaussian Approximation  
The dynamical information conveyed by the VAF is a keypoint in the development of models for the self part of the DDCS of viscous dense fluids. In the present case of a molecular liquid, the considered velocity autocorrelation is that of the molecular centers of mass  
\[Z(t)=\langle\mathbf{v}_{\text{CM}}(0)\cdot\mathbf{v}_{\text{CM}}(t)\rangle,\] (A19)  
which is a complex-valued quantity satisfying \(Z(-t)=Z^{*}(t)\) and \(Z(-t)=Z(t+i\frac{\hbar}{k_{\text{B}}})\), with the latter property descending from the detailed balance principle. However, the output of an RPMD simulation is the canonical (or Kubo-transformed [47]) VAF:  
\[Z_{c}(t)=k_{\text{B}}T\int_{0}^{\frac{1}{k_{\text{B}}}}d\lambda\,\left\langle e ^{\lambda H}\mathbf{v}_{\text{CM}}(0)\,\cdot\,e^{-\lambda H}\mathbf{v}_{ \text{CM}}(t)\right\rangle\] (A20)  
where \(H\) is the Hamiltonian operator of the system. The canonical VAF is a real and even function of time. By Fourier transformation of \(Z(t)\) and \(Z_{c}(t)\), the respective frequency spectra \(p(\omega)\) and \(p_{c}(\omega)\) are obtained. Both functions are real and, while \(p_{c}(\omega)\) is even, \(p(\omega)\) can be decomposed into its symmetric and antisymmetric parts, \(p(\omega)=p_{\text{S}}(\omega)+p_{\text{A}}(\omega)\), which are the Fourier transforms of \(\text{Re}\,Z(t)\) and \(i\,\text{Im}\,Z(t)\), respectively. The three quantities just defined are all related to each other by the detailed balance requirement, \(p(-\omega)=\exp(-\frac{\hbar\omega}{k_{\mathrm{B}}T})p(\omega)\), and the following relationships hold [26]:  
\[p(\omega)=\frac{\hbar\omega}{k_{\mathrm{B}}T[1-\exp(-\frac{\hbar \omega}{k_{\mathrm{B}}T})]}\ p_{c}(\omega) \tag{10}\] \[p_{\mathrm{S}}(\omega)=\frac{\hbar\omega}{2k_{\mathrm{B}}T}\coth \left(\frac{\hbar\omega}{2k_{\mathrm{B}}T}\right)\ p_{c}(\omega)\] \[p_{\mathrm{A}}(\omega)=\frac{\hbar\omega}{2k_{\mathrm{B}}T}\ p_{ c}(\omega)\]  
The link between the above formalism about the VAF and the CM self intermediate scattering function \(F_{\mathrm{s}}(Q,t)\) is easily obtained by recalling that the latter is a Gaussian in wave vector \(Q\) in both the two limiting cases of hydrodynamic diffusion (\(Q\to 0\), \(t\to\infty\)) and the ideal gas behavior (\(Q\to\infty\), \(t\to 0\)) [25], with _exactly_ known functions of time alone entering the expression of \(F_{\mathrm{s}}(Q,t)\) for a classical system:  
\[F_{\mathrm{s}}(Q,t)=\exp\left(\ -Q^{2}Dt\right)\ \text{for}\ Q\to 0,t\to\infty \tag{11}\]  
\[F_{\mathrm{s}}(Q,t)=\exp\left(\ -Q^{2}\frac{k_{\mathrm{B}}T}{2M}t^{2}\right)\ \text{for}\ Q\to\infty,t\to 0 \tag{12}\]  
where \(D\) is the self diffusion coefficient. The Gaussian Approximation, introduced by [PERSON] in 1958 [33], simply extends the above functionality to any intermediate time and \(Q\) value, i.e., it assumes that  
\[F_{\mathrm{s}}(Q,t)\simeq\exp[-Q^{2}\ \gamma_{1}(t)]\ \text{for}\ \text{any}\ Q,\ \text{any}\ t. \tag{13}\]  
This assumption was more deeply formalized, even for quantum systems, by [PERSON] and co-workers [34], who demonstrated that the self intermediate scattering function can be expressed as an infinite series of the form  
\[F_{\mathrm{s}}(Q,t)=\exp\Bigg{[}\sum_{p=1}^{\infty}(iQ)^{2p}\gamma_{p}(t) \Bigg{]}, \tag{14}\]  
whose first term, coinciding with [PERSON]'s hypothesis, is related to the VAF spectrum. In particular, the final, and most significant expression of the GA in the [PERSON] \(etal\). quantum picture provides the function \(\gamma_{1}(t)\) of Equation (13) as:  
\[\gamma_{1}(t)=\frac{2}{3}\int_{0}^{+\infty}d\omega\frac{1}{\omega^{2}}\Bigg{\{} p_{\mathrm{S}}(\omega)[1-\cos(\omega t)]-ip_{\mathrm{A}}(\omega)\sin(\omega t )\Bigg{\}}. \tag{15}\]  
Such equations have been used in this work, starting from the RPMD simulation of \(Z_{c}(t)\), and developing the formalism previously described to implement the quantum GA in the DDCS algorithm.