value which gives good agreement with numerical calculations. For collisions with angular momenta larger than $\ell_{{\mathrm}{t}}$, we assume that tunneling is not important, thus we neglect the effects of the shape resonances. In this approximation we can set $y=0$ in the formula , obtaining $$K^{el,(1)} = \frac{2 h}{\mu k} \sum_{\ell > \ell_{{\mathrm}{t}}}(2 \ell +1)
\sin^2 \xi(E,\ell),$$ When the collision happens with the energy well below the top of the barrier, one can evaluate the phase shift $\xi(E,\ell)$, using an approximate expression derived in the semiclassical approximation [@Bransden1992] $$\label{semxi}
\xi(E,\ell) \approx - \frac{\mu}{\hbar^2} \int_{r_0}^{\infty} {\mathrm}{d}r \frac{V(r)}{\sqrt{k^2 -(\ell+\frac12)^2/r^2}}$$ This formula describes the contribution from the long-range part of the potential $V(r)$, where $r_0$ describes the classical turning point at large distances. In this way for the $1/r^n$ potential we obtain

$$K^{el,(1)}=\frac{\pi^2  h}{8 \mu k} \left(\frac{R_n k}{2\ell_{\mathrm}{t}+1}\right)^{2n-4}\frac{\Gamma(n-1)^2}{(n-2)\Gamma^4(n/2)}.$$

In the second regime relevant for collisions close to the top of the barrier or above the barrier we can assume the high-energy limit for MQDT functions, setting $C(E,l)\approx 1$ and $\tan \lambda(E,l)\approx 0$. In principle this approximation works well only for collisions with energies well above the centrifugal barrier. Nevertheless, we make only a small error making a similar approximation for a few partial waves 