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. 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 Table V provides a compilation of zero-temperature rate coefficients for rotational and vibrational quenching in molecule - molecule systems. system initial ( v, j, v $^{′}$, j $^{′}$) k$_{T}$ =0 (cm $^{3}$s − $^{1}$) Ref. H 2 + H 2 ( v = 1 , j = 0 , v ′ = 0 , j ′ = 0) 8.9 × 10 − 18 [143] ( v = 2 , j = 0 , v ′ = 0 , j ′ = 0) 3.9 × 10 − 17 [143] ( v = 3 , j = 0 , v ′ = 0 , j ′ = 0) 1.2 × 10 − 16 [143] ( v = 4 , j = 0 , v ′ = 0 , j ′ = 0) 1.8 × 10 − 16 [143] ( v = 5 , j = 0 , v ′ = 0 , j ′ = 0) 6.9 × 10 − 16 [143] ( v = 6 , j = 0 , v ′ = 0 , j ′ = 0) 2.9 × 10 − 15 [143] ( v = 1 , j = 0 , v ′ = 1 , j ′ = 0) 2.7 × 10 − 16 [143] ( v = 2 , j = 0 , v ′ = 1 , j ′ = 0) 1.3 × 10 − 14 [143] ( v = 3 , j = 0 , v ′ = 1 , j ′ = 0) 1.6 × 10 − 14 [143] ( v = 4 , j = 0 , v ′ = 1 , j ′ = 0) 9.0 × 10 − 15 [143] ( v = 2 , j = 0 , v ′ = 2 , j ′ = 0) 6.1 × 10 − 16 [143] H 2 + H 2 ( v = 0 , j = 2 , v ′ = 0 , j ′ = 0) 3.9 × 10 − 14 [143] ( v = 1 , j = 2 , v ′ = 0 , j ′ = 0) 3.7 × 10 − 14 [143] ( v = 2 , j = 2 , v ′ = 0 , j ′ = 0) 8.7 × 10 − 14 [143] ( v = 3 , j = 2 , v ′ = 0 , j ′ = 0) 1.5 × 10 − 13 [143] ( v = 4 , j = 2 , v ′ = 0 , j ′ = 0) 2.5 × 10 − 13 [143] ( v = 1 , j = 2 , v ′ = 1 , j ′ = 0) 8.8 × 10 − 14 [143] ( v = 2 , j = 2 , v ′ = 1 , j ′ = 0) 6.1 × 10 − 14 [143] H 2 + H 2 ( v = 1 , j = 0 , v ′ = 0 , j ′ = 2) 5.6 × 10 − 12 [143] ( v = 2 , j = 0 , v ′ = 0 , j ′ = 2) 3.4 × 10 − 12 [143] ( v = 3 , j = 0 , v ′ = 0 , j ′ = 2) 2.7 × 10 − 12 [143] ( v = 4 , j = 0 , v ′ = 0 , j ′ = 2) 2.1 × 10 − 12 [143] ( v = 2 , j = 0 , v ′ = 1 , j ′ = 2) 5.2 × 10 − 12 [143] H 2 + CO ( v = 0 , j = 0 , v ′ = 0 , j ′ = 1) 2.0 × 10 − 12 [132] ( v = 0 , j = 0 , v ′ = 0 , j ′ = 2) 3.0 × 10 − 11 [132] ( v = 0 , j = 0 , v ′ = 0 , j ′ = 3) 1.2 × 10 − 10 [132] H 2 + CO ( v = 0 , j = 1 , v ′ = 0 , j ′ = 1) 1.2 × 10 − 11 [132] ( v = 0 , j = 1 , v ′ = 0 , j ′ = 2) 4.0 × 10 − 11 [132] ( v = 0 , j = 1 , v ′ = 0 , j ′ = 3) 8.5 × 10 − 11 [132] TABLE V: Zero-temperature inelastic rate coefficients for dif- ferent molecule - molecule systems. 23