the case of collisions between ground state molecules, the elastic cross sections from rigid rotor calculations of Lee et al. [@Lee06] using the DJ PES and the present full-dimensional results on the BMKP surface are in good agreement with each other. However, the inelastic cross sections show significant differences. The inelastic rotational de-excitation cross section on the DJ PES is about an order of magnitude larger than that derived from the BMKP PES. This confirms conclusions derived from previous calculations [@Lee06; @Otto08] that the BMKP PES is not reliable in predicting accurate values of rotational excitation cross sections, especially for the $\Delta j=2$ rotational transitions. In the lower panel we show elastic and inelastic rotational de-excitation cross sections for H$_2(v=0,j=2)$ + H$_2(v=0,j=2)$ collisions. Interestingly, (0202) $\to$ (0000) inelastic cross sections are comparable in magnitude for the BMKP PES, the DJ PES, and also for the ZR PES. However, for the (0202) $\to$ (0200) transition, the BMKP potential yields smaller cross sections than DJ and ZR potentials for collision energies below 2 K. The agreement improves for higher collision energies, especially with that from the ZR potential.

The elastic cross section of the H$_2(v=0,j=0)$ + H$_2(v=0,j=0)$ collision is shown in Fig. 