results from our convergence studies. Figure \[XS-STTOST-FIG\] shows that the results are robust and converged with respect to all numerical parameters employed in the calculations. The upper panel shows cross sections for different values of $\lambda_1=\lambda_2$ in Eq. . It is seen that $\lambda_1=\lambda_2=8$ or higher is needed to accurately calculate cross sections for the dominant final CMS, (0800). The convergence of the results with respect to the cut-off energy, the size $\Delta R$ of the radial interval, the number of DVR points $N_{r_1}=N_{r_2} \equiv N_{r},N_{\theta_1}=N_{\theta_2} \equiv N_{\theta}$, and $N_{\phi}$ used for the vibrational and rotational wave functions are presented in the lower panel. These results are obtained with $\lambda_1=\lambda_2=8$ for the angular expansion of the interaction potential. For the results reported in the final production calculations for the BMKP PES we used $\lambda_1=\lambda_2=8$, $\Delta R = 0.05 \ a_0$, $N_{\theta}=14, ~ N_{\phi}=8$, and $N_{r}=24$.

![ Upper panel: Vibrational relaxation cross sections as functions of the collision energy for the H$_2(v=1,j=0)$ + H$_2(v=0,j=0)$ system. Lower panel: State-to-state cross sections at a collision energy of $E_c=0.01$ eV. \[XSIN-THERMAL-1000-FIG\]](XSELIN_H2-H2-v1j8-allJ-ecut10000-vj1000-tov0-0eVto1eV-ANG-BW.eps "fig:"){height="6cm"}\
![ Upper panel: Vibrational relaxation cross sections as functions of the collision energy for the H$_2(v=1,j=0)$ + H$_2(v=0,j=0)$ system. Lower panel: State-to-state 