4$. Since no such restriction is imposed in our calculations, the differences between the exact and CSA methods for CMSs with quantum numbers $j_{12} > 4$ and $|m_{j_{12}}| > 4$, such as (0404), (0600), (0602), (0604), (0800), are attributed to the restriction on $\Omega$ imposed in the CSA calculations. The good agreement between the full quantum and CSA methods for the BMKPE PES is attributed to the negligible contributions of the final states (0404), (0600), (0602), (0604) and (0800). Since the restriction on $\Omega$ primarily impacts these final states, it does not significantly affect the cross sections computed using the BMKPE PES.

![ (Color online) Rate coefficient for the vibrational relaxation of H$_2(v=1,j=0)$ in collisions with H$_2(v=0,j=0)$ as a function of the temperature. Results obtained using the BMKP and BMKPE PESs are compared with the experimental results of Audibert et al. [@Audibert75]. \[RATE-1000-FIG\]](RATE-BMKP-vj1000tov0.eps){height="6cm"}

The rate coefficients for the vibrational relaxation of H$_2(v=1,j=0)$ in collision with H$_2(v=0,j=0)$ as functions of the temperature are reported in Fig. \[RATE-1000-FIG\] for the BMKP PES (solid curve) and the BMKPE PES (dashed curve). The experimental results of Audibert et al. [@Audibert75] are also shown for comparison. For the reasons discussed above, the BMKP PES predicts 