10 -10 10 -9 10 -8 10 -10 10 -9 10 -8 Quenching cross section (cm 2 ) total v$_{f}$=1 v$_{f}$=0 0.98 0.99 1 1.01 1.02 10 -10 10 -9 10 -8 total v$_{f}$=0 v$_{f}$=1 v$_{f}$=2 Na + Na$_{2}$(v=1, j=0) Na + Na$_{2}$(v=2, j=0) Na + Na$_{2}$(v=3, j=0) 0 2e-11 4e-11 6e-11 8e-11 1e-10 0.98 0.99 1.00 1.01 1.02 0 4 8 12 16 cr os s  se cti on  (c m 2 ) ! j f a) v f  = 2 0 2e-11 4e-11 6e-11 8e-11 1e-10 0.98 0.99 1.00 1.01 1.02 0 4 8 12 16 20 24 cr os s  se cti on  (c m 2 ) ! j f b) v f  = 1 0 2e-11 4e-11 6e-11 8e-11 1e-10 0.98 0.99 1.00 1.01 1.02 0 4 8 12 16 20 24 28 cr os s  se cti on  (c m 2 ) ! j f c) v f  = 0 FIG. 20: Variation of state-to-state cross sections at a colli- sion energy of 10 − 9 K for Na + Na 2 ( v = 3 , j = 0) → Na + Na 2 ( v$_{f}$ = 2 , 1 , 0 , j$_{f}$ ) as a function of the parameter λ . See text for details. Reproduced with permission from Qu´ em´ ener et al. [99]. when inelastic couplings are strong, as in alkali-metal trimer systems. A generalization of this effect has been discussed by Hutson [114]. Larger modifications of the three-body term affect the cross sections significantly. This can also be seen in Fig. 21 for Li + Li$_{2}$ collisions [96] in which the three-body term is excluded from the dy- namics calculation. The state-to-state cross sections ex- hibit a stronger dependence on the three body term. Fig. 20 shows the dependence of the state-to-state cross sections on the three body term for the reaction Na + Na$_{2}$( v = 3 , j = 0) → Na + Na$_{2}$( v$_{f}$ = 2 , 1 , 0 , j$_{f}$ ) at a col- lision energy of 10 $_{−}$9 K. The oscillations in the cross sec- tions, when λ is modified, are due to Feshbach resonances which arise when a triatomic quasibound state (or a vir- tual state) crosses the energy threshold as the strength of the interaction potential is decreased (increased). When such a Feshbach resonance occurs, Cvitaˇs et al. [97] have argued, strong resonant peaks appear in the cross sec- tions if inelastic couplings are weak. In contrast, weak oscillations of one order of magnitude at best appear potential. A change of 1 % in λ leads to a significant change of 75 % in the cross sections. At the minimum of the Na$_{3}$ potential, a change of 1 % corresponds ap- proximately to 10 K. At present, ab initio calculations of PESs cannot be done with such accuracy. Thus it ap- pears that it is difficult to get an accuracy better than two orders of magnitude in the cross sections for Na + Na$_{2}$( v = 1 , j = 0) collisions. However, for v = 2 and 3, the total cross sections show a weaker dependence on the three-body term. The state-resolved cross sections do not show strong dependence on the three-body term, compared to collisions of molecules in v = 1. FIG. 19: Dependence of the quenching cross sections on the three-body term at a collision energy of 10 − 9 K for Na + Na 2 ( v = 1 − 3 , j = 0). Reproduced with permission from Qu´ em´ ener et al. [99]. 17