## 2 Results
### Inelastic Scattering  
Let us now briefly recall the foundations of reactive scattering theory [26]. First let us introduce the Jacobi coordinates and denote the relative position vector between the center of mass of the molecule and the atom by \(\mathbf{R}\), and the internal coordinates of the molecular ion as \(\mathbf{r}\). The center of mass motion of the particles is not relevant. At large distances where the interaction can be neglected, the molecular ion can be described by its rovibrational states \(\phi_{ij}(r)\), where \(v\) and \(j\) label the vibrational and rotational quantum number of the molecule, respectively. It is now convenient to move to the body-fixed frame of the molecule [27; 28; 29], introducing the quantum numbers \((J,M)\) for the total angular momentum and its projection which are both conserved. The relative orbital angular momentum \(\ell\) fulfills the relation \(\boldsymbol{\ell}=\mathbf{J}-\mathbf{j}\). The rest of the basis states forming the basis for close-coupled equations can be labelled as \(\psi_{j\ell IM}(\omega_{m},\omega_{SF})\), where \(\omega_{m}\) and \(\omega_{SF}\) are the spherical angles describing the orientation of the molecule-atom pair and the molecule itself in the space-fixed frame. We note that the three-body potential landscape is only weakly anisotropic due to the lack of a dipole moment of the molecule and the short range of the van-der-Waals interaction. This validates the use of the coupled-states approximation [12; 30; 31], which states that the angular momentum projection \(\Omega\) on the molecular axis is conserved and that in general intermultiplet couplings are negligible so that one can decouple the molecular and orbital angular momenta. We can now calculate the potential matrix in the channel states basis simply by integrating the full three-body potential (2) over the molecular states. The rovibrational molecular state with quantum numbers \(v\), \(j\) has the corresponding binding energy denoted further as \(E_{vj}\). After decoupling the angular momenta, one obtains\[W^{JM}_{vj,\
u^{\prime}j^{\prime}}(R) = \left(\frac{2\mu^{\star}}{\hbar^{2}}E_{vj}+\frac{\ell(\ell+1)}{R^{2 }}\right)\delta_{vv^{\prime}}\delta_{jj^{\prime}}+\frac{2\mu^{\star}}{\hbar^{2} }V^{M}_{vj,\
u^{\prime}j^{\prime}}(R)\,, \tag{3}\]  
with the matrix element  
\[V^{M}_{vj,\
u^{\prime}j^{\prime}}(R) = 2\pi\int\phi_{v^{\prime}j^{\prime}}(r)Y_{j^{\prime}M}(\gamma,0) V(R,r,\gamma)\phi_{vj}(r)Y_{jM}(\gamma,0)\sin(\gamma)d\gamma dr \tag{4}\]  
containing the interaction between the free atom and the molecule as the interaction term within the molecule has been incorporated in the molecular energies. Here the integration over the polar angle gives a constant factor of \(2\pi\). Furthermore, \(\mu^{\star}=2m/3\) is the reduced mass of the atom-molecular ion pair and the orbital angular momentum (which is now also a good quantum number)  
\[\ell=\left(J(J+1)-j(j+1)-2M^{2}+1/4\right)^{1/2}-1/2\,. \tag{5}\]  
Furthermore, \(Y_{jk}\) denote spherical harmonics and \(\gamma\) is the Jacobi angle (see Figure 1). The nondiagonal terms of the potential matrix provide the coupling between the channels responsible for vibrationally inelastic collisions. Equation (3) provides the potential matrix for the multichannel scattering problem  
\[\frac{\partial^{2}\mathbf{F}}{\partial R^{2}}+\left(\frac{2\mu}{\hbar^{2}}E- \mathbf{W}(R)\right)\mathbf{F}=0 \tag{6}\]  
with the matrix \(\mathbf{F}\) containing the radial wavefunctions. Assuming the channels are open (as is the case if we start from the weakly bound state which has higher energy than all the others), at large distances the solution takes the canonical form  
\[\mathbf{F}(R)\overset{R\rightarrow\infty}{\longrightarrow}\left(\mathbf{J}(R )-\mathbf{N}(R)\mathbf{K}\right)\mathbf{A}\,, \tag{7}\]  
where matrices \(\mathbf{J}\), \(\mathbf{N}\) are diagonal and consist of spherical Bessel functions, e.g., \(J_{ij}=j_{\ell}(k_{i}r)\delta_{ij}\) with \(k_{i}=\sqrt{2\mu(E-E_{i})/\hbar^{2}}\) being the asymptotic channel wavenumber.