## 2 Results
### Effective Potentials  
With the basic formalism at hand, we now proceed to the analysis of the particular case of the molecular ion scattering with a neutral atom. The first step needed to derive the potential matrix (3) is to calculate the two-body rovibrational wavefunctions of the molecular ion alone and their binding energies. State of the art ab initio methods can be used to calculate the interaction potential of Rb\({}_{2}^{+}\) in the electronic ground state with a finite precision of the order of 1% [32]. While this kind of accuracy is sufficient for spectroscopic data such as the vibrational spacing of the lower levels, it is unfortunately not enough to predict the binding energies of the weakly bound Rb\({}_{2}^{+}\) states. This is in part due to the large reduced mass of the system. For this reason we are allowed to choose a simple model potential to work with and obtain the knowledge about the general features of the ion-molecule systems. We have tested several model potentials such as the Lennard-Jones type interaction of the form \(V(x)=-\frac{C_{4}}{x^{4}}(1-(x_{0}/x)^{4})\) with the cutoff \(x_{0}\), the flat-bottom regularization \(V(x)=-\frac{C_{4}}{(x^{2}+x_{0}^{2})^{2}}\) for various potential depths, as well as used the analytical solutions of the polarization potential [33], and found no fundamental difference in the behavior of the resulting effective potentials. In what follows, we will use \(R^{\star}\) and \(E^{\star}\) of the atom-ion system as the units of length and energy, so that asymptotically the two-body ion-atom potentials approach \(-1/x^{4}\). One might expect that the effective atom-molecule potentials given by Equation (4) will follow the \(R^{-4}\) power law, but with a possibly varying coupling coefficient. This can be understood assuming the deeply bound states are pointlike compared to the range of the interparticle potential, and the contribution to the integral (4) comes only from \(r{\sim}0\). The next term which can be extracted perturbatively and takes into account the spatial structure of the molecule can be obtained by expanding the angle-averaged atom-molecule potential \(V_{\text{at-mol}}=V(\mathbf{r}_{13})+V(\mathbf{r}_{23})\) at large distances \(R\) in power series, which gives  
\[V_{\text{at-mol}}\overset{R\rightarrow\infty}{\longrightarrow}-\frac{1}{R^{4 }}+\frac{r^{2}/2+R_{6}^{6}}{R^{6}}+\ldots\,, \tag{10}\]  
with \(R_{6}\) denoting the characteristic distance corresponding to the van-der-Waals interaction \(R_{6}=(2\mu C_{6}/\hbar)^{1/4}\), which is typically an order of magnitude smaller than \(R^{\star}\)[34]. Thus, there is a contribution to the coupling matrix element proportional to \(\int dr\,\phi_{vj}(r)\phi_{v^{\prime}j^{\prime}}(r)\frac{r^{2}}{R^{6}}\). This term should be relevant for weakly bound states whose spatial extent cannot be neglected, both for the diagonal and nondiagonal matrix elements. This intuitively explains why the potential curves are largely independent of the exact form of the two-body potentials, as the short-range corrections only set in at distances smaller than \(R^{\star}\). Two of the diagonal matrix elements of Equation (4) are presented in the first panel of Figure 2. As expected, they decay according to the \(R^{-4}\) power law at long range with the fitted \(C_{4}\) coefficient equal to the original one. For the most weakly bound state, which in our case extends to around \(10R^{\star}\) and has the binding energy of only \({\approx}0.16E^{\star}\), the van-der-Waals term is also clearly visible. However, as this only provides a short-range correction we can conclude that the elastic scattering is just governed by the polarization potential with some unknown scattering length which depends on the short-range details. Two exemplary cases of the coupling potential curves are shown in the middle and the right panel of Figure 2. As expected, the couplings between deeply bound levels (middle panel) always follow the effective power-law potential with \(1/R^{4}\) dependence.