the rigid rotor approximation. Though the agreement between these calculations were generally good, important differences were found for some transitions, especially for (0000) $\to$ (0202) and (0000) $\to$ (0404) transitions. In Fig. \[ST2STXS-THERMAL-0000-FIG\] we show rotational excitation cross sections in H$_2(v=0,j=0)$ + H$_2(v=0,j=0)$ collisions leading to final CMSs (0200), (0202), (0400), (0402), and (0404). To obtain converged cross sections, these calculations include contributions from $J=0-100$. Also included are TIQM rigid rotor results from the present work, and full-dimensional time-dependent calculations of Otto et al. [@Otto08]. Figure 4 of Ref. [@Otto08] presents similar comparison with the results of Lee et al. [@Lee06], Lin et al. [@Lin02], and Sultanov et al. [@Sultanov06a]. Figure \[ST2STXS-THERMAL-0000-FIG\] shows that our results are generally in good agreement with those of Otto et al. for the (0000) $\to$ (0200) and (0000) $\to$ (0400) transitions. The agreement is better for collision energies lower than 0.5 eV. The rigid rotor calculations from the present study underestimate the cross sections. The largest discrepancy between the present work and the time-dependent calculations occur for the (0000) $\to$ (0202) transition. Overall, for all of the transitions shown in Fig. \[ST2STXS-THERMAL-0000-FIG\] the agreement with the time-dependent results is better at low collision energies 