10 - 4 10 - 3 10 - 2 10 - 1 10 0 10 1 10 2 10 3 10 - 8 10 - 6 10 - 4 10 - 2 10 0 10 2 ! ( units  of  10 - 16 c m 2 ) E  ( units  of c m - $^{1}$) (a) Is o t o pe 3 Is o t o pe 4 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -6 10 -4 10 -2 10 0 10 2 !  (units of 10 -16 cm 2 ) E (cm$^{-1}$) (b) Isotope 3 Isotope 4 the spin-rotation coupling, while rotational transitions are sensitive to the fine-structure interactions. Table II provides a compilation of zero-temperature quenching rate coefficients for vibrational and rotational relaxation in several ion atom - molecule systems. III. CHEMICAL REACTIONS AT ULTRACOLD TEMPERATURES The last three decades have seen impressive progress in the experimental and theoretical descriptions of chem- ical reactions between atoms and small molecular sys- tems. While fully state-resolved experiments have been performed for a large number of collision systems, accu- rate quantum dynamics calculations have been restricted to systems involving light atoms such as H + H$_{2}$, F + H$_{2}$, Cl + H$_{2}$, C + H$_{2}$, N + H$_{2}$, O + H$_{2}$ and some of their system initial ( v, j ) k$_{T}$ =0 (cm $^{3}$s − $^{1}$) Ref. Ne + 2 + Ne ( v = 0 , j = 2) 3.3 × 10 − 10 [64] ( v = 0 , j = 4) 7.3 × 10 − 10 [64] ( v = 0 , j = 6) 7.0 × 10 − 10 [64] He + 2 + He ( v = 0 , j = 2) 6.7 × 10 − 10 [64] ( v = 0 , j = 4) 8.4 × 10 − 10 [64] ( v = 0 , j = 6) 1.2 × 10 − 9 [64] N + 2 + $^{3}$He ( v = 1 , j = 0) 4 × 10 − 14 [65] N + 2 + $^{4}$He ( v = 1 , j = 0) 3 × 10 − 15 [65] isotopic analogues [67, 68]. Most experimental studies of small molecular systems focussed on reactions at thermal or elevated collision energies though some recent mea- surements have been extended to temperatures as low as 10 K for some astrophysically relevant systems [69]. Due to the possibility of achieving coherent chemistry there has been substantial interest in understanding the behav- ior of chemical reactions at cold and ultracold tempera- tures. At these temperatures perturbations introduced by external electric and magnetic fields are significant compared to the collision energies involved and external fields can be employed to control and manipulate the reaction outcome. Over the last seven years a number of studies of ultracold atom - diatom chemical reactions have been reported both for reactions with and without an energy barrier. Chemical reactions at low temper- atures often behave quite differently from reactions at elevated temperatures. In particular, the weak van der Waals interaction potential which does not play any sig- nificant role at high temperatures may have a dramatic effect on the outcome of reactions at low temperatures. In the course of the last 10 − 15 years there has been much interest in understanding the role of res- onances in chemical reactions that involve an energy barrier for which the reactivity is primarily driven by tunneling at low temperatures. Recent studies on F + H$_{2}$/HD/D$_{2}$ [11, 70–80], Cl + HD [81, 82], H + HCl/DCl [83], Li + HF / LiF + H [84, 85], O + H$_{2}$ [86, 87], and F + HCl/DCl [88] reactions have demon- strated that decay of quasibound states of the van der Waals interaction potential in the entrance and exit chan- nels may give rise to narrow Feshbach resonances in the cross sections. The reactions of F with H$_{2}$ and HD have been the sub- ject of numerous quantum scattering calculations over the last two decades. The two reactions have emerged as benchmark systems for experimental and theoreti- cal studies of resonances in chemical reactions. A de- 9 TABLE II: Zero-temperature inelastic rate coefficients for dif- ferent ion atom - molecule systems. FIG. 8: Vibrational quenching cross sections for N + 2 ( v = 1 , j = 0) + $^{3}$He and N + 2 ( v = 1 , j = 0) + $^{4}$He (upper panel) and N 2 ( v = 1 , j = 0) + $^{3}$He and N 2 ( v = 1 , j = 0) + $^{4}$He (lower panel) systems. Reproduced with permission from Stoecklin et al. [65]. A. Tunneling dominated reactions