energies $E_c < 1$ K, we propagate the elastic channel from $R=53 \ a_0$ to a sufficiently large distance, at which the interaction potential $U$ in Eq.  is less than $10^{-5} \times E_c$.

Vibrational relaxation in H$_2(v=1,j=0)$ + H$_2(v=0,j=0)$ collisions
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In the upper panel of Fig. \[XSEL-ULE-1000-FIG\] we show the $J$-resolved partial cross sections and their sum for the elastic channel for collision energies ranging from $10^{-3}-120$ K. Though only the $s$-wave contributes at energies below $10^{-2}$ K, the partial wave sum requires up to $J=10$ at $E_c=100$ K to yield converged cross sections. The state-to-state inelastic cross sections and their sum are shown in the lower panel of Fig. \[XSEL-ULE-1000-FIG\] for H$_2(v=1,j=0)$ + H$_2(v=0,j=0)$ collisions for $E_c=10^{-3}-120$ K. The cross sections of the inelastic processes are much smaller than that of elastic scattering. Due to the small anisotropy of the interaction potential with respect to the stretching of the molecule, the interaction potential matrix elements responsible for vibrational relaxation channels are quite small compared to the isotropic part responsible for elastic scattering. Thus, the couplings between different CMSs are rather weak and state-to-state transitions are less probable than elastic scattering. The diabatic potential curves in Fig. \[SPAG-FIG\] shows 