J^\prime_1$, denote the initial and final angular momentum of the [$\mathrm{HC_3N}$]{}molecule, respectively, and $J_2$ denote the angular momentum of H$_2$. We also denote the largest value of [$J_1, J^\prime_1$]{} as $J_{1\rm up}$.

The most reliable approach to compute inelastic cross sections $\sigma_{J_1J^{\,\prime}_1}(E)$ is to perform quantum close coupling calculations. In the case of molecules with a small rotational constant, like [$\mathrm{HC_3N}$]{}[$B=4549.059\mbox{~MHz}$, see e.g. @thor00], quantum calculations become soon intractable, because of the large number of open channels involved. While observations at cm-mm wavelengths culminates with $J_{1\rm up}
\lesssim 24$ [@kahane94], sub-mm observations can probe transition as high as $J_{1\rm up} = 40$, at a frequency of 363.785 GHz and a rotational energy of $202.08 \:\rm
cm^{-1}$ [@pepe04; @charnley04; @cauxpc]. It is thus necessary to compute rates with transitions up to $J_1=50$ ($E=
386.8$ cm$^{-1}$), in order to properly converge radiative transfer models. Also, we aim at computing rates up to a temperature of 100 K for H$_2$. We resorted to two methods in order to perform this task. For $J_{1\rm up}\leq 15$, we performed quantum inelastic scattering calculations, as presented in next subsection \[par:molscat\]. For $J_{1\rm up} > 15$, we used the QCT method, as presented in subsection \[par:rates\].

For 