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vibrational levels involved in the collision, whether it in- volves ground state molecules, H$_{2}$( v = 0 , j = 0), vibra- tionally excited molecules, H$_{2}$( v = 1 , j = 0), rotationally excited molecules H$_{2}$( v = 0 , j = 2), or ro-vibrationally excited molecules, H$_{2}$( v = 1 , j = 2). H$_{2}$ molecules are weakly interacting and characterized by shallow van der Waals interaction at large separations and the com- puted rate coefficients are not representative of strongly interacting alkali-metal dimer systems. For example, Mukaiyama et al. [109] reported an inelastic rate coef- ficient of 5 . 1 × 10 $_{−}$11 cm 3 s $_{−}$1 for collisions between two weakly bound Na$_{2}$ molecules created by the Feshbach resonance method. In a similar study Syassen et al. [110] reported a rate coefficient of 3 × 10 $_{−}$10 cm 3 s $_{−}$1 for col- lisions between two Rb$_{2}$ molecules. Zahzam et al. [107] estimated a rate coefficient of about 10 $_{−}$11 cm 3 s $_{−}$1 for collisions between Cs$_{2}$ molecules. Ferlaino et al. [149] measured a rate coefficient of 9 × 10 $_{−}$11 cm 3 s $_{−}$1 for colli- sions between Cs$_{2}$ molecules in the non-halo regime. The latter authors also measured the variation of the inelastic rate coefficient as a function of the atom - atom scatter- ing length for collisions between tunable halo dimers of Cs$_{2}$. The large values of the inelastic rate coefficients for alkali-metal dimer systems are attributed to strong inelastic couplings and deeper potential energy wells. 

The state-to-state cross sections for three ro- vibrational combinations of the H$_{2}$ − H$_{2}$ system are presented in the lower panel of Fig. 25. The magnitude of the inelastic cross sections depends on the propensity of the diatom - diatom system to conserve the internal energy and the total rotational angular momentum of the colliding molecules [143]. The final state-to-state distribution in H$_{2}$( v = 1 , j = 0) + H$_{2}$( v = 0 , j = 0) collisions shows that there is no preferential popula- tion of rotational quantum number of either of the colliding molecules. The conservation of the total rotational angular momentum would entail a large change in the internal energy of the molecules. Thus, the purely vibrational transition (1000) → (0000) is not efficient because the energy gap is large. On the other hand, (near) conservation of the internal energy requires a large change in the total rotational angular momentum of the colliding molecules and the transition (1000) → (0800) is not dominant either. However, the state-to-state cross sections for H$_{2}$( v = 1 , j = 2) + H$_{2}$( v = 0 , j = 0) collisions indicate that the transition (1200) → (1000) is more efficient than all the other transitions combined. For this transition, the total rotational angular momentum change and the internal energy transfer are both minimized, leading to a more efficient energy transfer process. The state-to-state cross sections for H$_{2}$( v = 1 , j = 0) + H$_{2}$( v = 0 , j = 2) collisions present an interesting scenario in which the transition (1002) → (1200) is highly efficient and selective. In this case, the total rotational angular mo- mentum is conserved and the internal energy is almost unchanged (the energy gap between (1200) and (1002) is only 25.45 K). This creates a favorable situation 

leading to a near-resonant energy transfer process. This particular mechanism cannot occur in atom - diatom systems because simultaneous conservation of rotational angular momentum and internal energy of the molecule cannot occur. The mechanism is reminiscent of quasi-resonant energy transfer in collisions of rotation- ally excited diatomic molecules with atoms discussed previously [21, 53, 54, 60], but with a purely quantum origin. The near-resonant process may be an important mechanism for collisional energy transfer in ultracold molecules formed by photoassociation of ultracold atoms and for chemical reactions producing identical molecules. 

Table V provides a compilation of zero-temperature rate coefficients for rotational and vibrational quenching in molecule - molecule systems. 

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