energies, then averaging over velocity distributions— it is possible for QCT calculations to directly resort to a canonical formalism, i.e. to select the initial velocities of the Monte-Carlo ensemble according to the relevant Maxwell-Boltzmann distribution and find the rates as: $$\label{eq:rate}
k_{J_1J'_1} = \left(\frac{8kT}{\pi\mu}\right)^{1/2}\,\pi
b_{max}^2\,\frac{N}{N_{\rm tot}}$$ where $b_{max}$ is the maximum impact parameter used (with the impact parameter $b$ distributed with the relevant $b\,\textmd{d}b$ probability density) and $N$ is the number of trajectories with the right final $J'_1$ value among all $N_{\rm tot}$ trajectories. The Monte-Carlo standard deviation is: $$\label{eq:error}
    \frac{\delta k_{J_1J_1'}}{k_{J_1J_1'}} = \left(\frac{N_{\rm
    tot}-N}{N_{\rm tot}N}\right)^{1/2} \quad ,$$ showing that the accuracy of the method improves for larger rates. The $b_{max}$ parameter was determined by sending small batches of $500$ to $1,000$ trajectories for fixed $b$ values; values of $20 \leq
b_{max} \leq 26$ Bohr were found. We then sent batches of $10,000$ trajectories for each temperature in the range $5-100$K, with a step of 5K. Trajectories are integrated by means of a Bürlich-Stoer algorithm [@numrec92], with a code similar to that of @faure05a. Precision is checked by conservation of total energy and total angular momentum.

Some illustrative results are shown in tables \[tab:rates\] and  \[tab:rates2\] and are illustrated in 