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al. [94], who showed that in the Langevin regime the rate coefficients become independent of the vibrational level of the molecule. The third region is intermediate be- tween the Wigner and Langevin regimes, which typically corresponds to a quantum calculation with two or three partial waves. As shown in the lower panel of Fig. 17 the height of each J -resolved effective barrier depicted by vertical lines for J = 1 − 5 corresponds approximately to the maximum of each J -resolved quenching rate co- efficient. This indicates that the quenching process be- comes significant when the barrier height is energetically overcome. Whenever the elastic and quenching rate co- efficients have comparable values, the Langevin regime is reached, as shown in Fig. 17. Since the quenching prob- ability is almost equal to one and the elastic probability is almost equal to zero, elastic and quenching transition matrix elements are both equal to one and the cross sec- tions and rate coefficients for the two processes become similar. 

The above analysis shows that at energies above the onset of the Wigner regime the Langevin model can cor- rectly describe the dynamics for barrierless atom - dimer alkali-metal collisions because of the strong inelastic cou- plings. However, the classical model is not suitable for tunneling dominated reactions. Accurate quantum dy- namics calculations are computationally demanding for heavier systems such as Cs + Cs$_{2}$, Rb + Rb$_{2}$, Rb + RbCs or Cs + RbCs and the classical model may be used to qualitatively describe the dynamics of these sys- tems at energies above the s-wave regime. The tempera- ture dependence of the rate coefficients predicted by the Langevin model is given by the simple formula in atomic units: 

<img src='content_image/407322.jpg'>

where µ is the reduced mass of the atom - diatom colli- sional system and C$_{6}$ the dominant atom - diatom long range coefficient. In Fig. 18 we show the rate coefficients predicted by the Langevin model as functions of the tem- perature for different atom - diatom and diatom - di- atom collisions involving Rb or Cs atoms. For Rb − RbCs, Cs − RbCs, and RbCs − RbCs collisions we used the corre- sponding C$_{6}$ coefficients calculated by Hudson et al. [111]. In the absence of similar data for Cs + Cs$_{2}$ and Rb + Rb$_{2}$ collisions, we approximated the atom - dimer C$_{6}$ coeffi- cient by multiplying the corresponding atom - atom value by a factor of two. For Cs + Cs$_{2}$ collisions we used the C$_{6}$ coefficient for Cs − Cs interaction calculated by Amiot et al. [112] and for Rb + Rb$_{2}$ collisions we adopted the C$_{6}$ coefficient for Rb − Rb interaction reported by Derevianko et al. [113]. As shown in Fig. 18 the Langevin model pre- dicts rate coefficients within an order of magnitude of the experimental values for the different systems. The predicted results for the Cs + Cs$_{2}$, Rb + RbCs, and Cs + RbCs collisions agree with the corresponding experi- mental data of Refs. [106, 107, 111] within the reported error bars. For Rb + Rb$_{2}$ collisions the Langevin model 

predicts results in very close agreement with the experi- mental result of Wynar et al. [108]. Thus, the Langevin model appears to be valid for describing collisional prop- erties of these systems at µ K temperatures. The model also confirms that cold and ultracold collisions of alkali metal atoms and dimers are essentially characterized by the leading term in the long-range part of the interaction p otential. 

<img src='content_image/407317.jpg'>

## 2. Role of PES in determining ultracold reactions 

Potential energy surfaces are the key ingredients that enter in quantum dynamics calculations. While the two- body terms are generally well known and accurate, the three-body terms are more difficult to compute with high precision since they are non-additive and involve correla- tions between the three atoms. As a consequence, quan- tum dynamics calculations may suffer from the quality and degree of accuracy of the three-body terms. The sensitivity of the ultracold collision cross sections to the details of the PES has been investigated for atom - dimer alkali-metal systems by Qu´em´ener et al. [99] for Na + Na$_{2}$( v = 1 − 3 , j = 0) and by Cvitaˇs et al. [97] for Li + Li$_{2}$( v = 0 − 3 , j = 0). In these studies, a linear scal- ing factor λ has been included to tune the three-body interaction term in the PES and cross sections were cal- culated as a function of λ . Other studies have com- pared the dynamics with and without the three-body terms for Na + Na$_{2}$( v = 1 , j = 0) [98] and for Li + Li$_{2}$( v = 0 − 10 , j = 0) [96] collisions. 

The dependence of the total quenching cross sections on the three body term for Na + Na$_{2}$( v = 1 − 3 , j = 0) scattering [99] at a collision energy of 10 $_{−}$9 K is shown in Fig. 19. The contribution of each final vibrational level is also plotted. For the v = 1 vibrational level, the cross sections are very sensitive to the details of the three-body 

