accurate quantum dynamics treatment of diatom - diatom collisions. Since H$_2$ is the most abundant molecular species in the universe, collisions between H$_2$ molecules leading to excitation and de-excitation of their rovibrational levels continue to be a topic of considerable interest in interstellar chemistry. However, an accurate calculation of rovibrational energy transfer rate coefficients in H$_2-$H$_2$ collisions for temperatures relevant to astrophysical environments is still a challenging problem. In recent years, due to the success in creating dense samples of ultracold molecules in magneto-optical traps, atomic and molecular collisions in cold and ultracold gases have attracted considerable experimental and theoretical attention [@Bahns00; @Masnou01; @Bethlem03; @Doyle04; @Hutson06; @Hutson07a; @roman-review; @Roman-pccp]. These unusual systems provide a fascinating opportunity to investigate atomic and molecular processes at temperatures close to absolute zero where the collisional outcomes are dramatically influenced by quantum effects. Recent studies of atom - diatom inelastic and reactive collisions at temperatures close to absolute zero have shown that such processes may occur with significant rate coefficients at ultracold temperatures [@bala-cpl-2001; @weck06; @Quemener08b; @bodo-review; @Soldan02; @Quemener04; @Quemener05; @Cvitas05a; @Cvitas05b; @Quemener07; @Quemener09]. The H$_2-$H$_2$ system serves as an ideal prototype to investigate molecule - molecule collisions at ultracold temperatures.

Several PESs of the 