2 g \frac{h}{\mu } k^2 \overline{V}$$ $$\mathcal{K}^{\mathrm}{el}_{00} \to  g \frac{h}{2\mu}  (1+\nu^2) k\bar{a}^2, \,\,\,\,\, \mathcal{K}^{\mathrm}{el}_{1m} \to  g \frac{h}{2\mu}\frac{(1+\nu^2)^3}{(1-3\nu^2)^2} k^5 \overline{V}^2.$$ Due to the correction coming from the $\propto k^2$ term in the phase shift mentioned above, for $n=4$ the leading term in the $p$-wave elastic rate is proportional to $k^3$ instead.

High energy limits
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Reactive rate
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At high energies we first derive an approximate expression corresponding to the classical limit of the scattering. We assume $C^{-2}(E,\ell) =1$ and $\tan \lambda(E,\ell) =0$ for partial waves at which the collision takes place above the barrier, while for collisions below the barrier we take $C^{-2}(E,\ell) =0$. This neglects the effects of the quantum tunneling and of the quantum reflection. In this approximation we obtain $$\mathcal{K}^{\mathrm}{re} \stackrel{E\to\infty}{\longrightarrow}  \frac{h}{2 \mu} P^{\mathrm}{re} \ell_{\mathrm}{max}(E)\left[1+\ell_{\mathrm}{max}(E)\right]$$ where $\ell_{\mathrm}{max}(E)$ is the maximal angular momentum at which the top of the barrier is equal to the collision energy $E$. For a power-law potential $V(r) = - C_n/r^n$ this leads to $$\mathcal{K}^{\mathrm}{re} \stackrel{E\to\infty}{\longrightarrow}  \frac{h}{2 \mu k} P^{\mathrm}{re} \frac{n}{2} \left( \frac{E/E_n}{\frac{n}{2}-1}\right)^{(n-2)/n}.
\label{KLang}$$ In particular we notice that for van der Waals interaction the reactive rate constant behaves as $E^{1/6}$, while for polarization potential ($n=4$) it remains constant at high energy. This can 