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both $^{3}$He and $^{4}$He. For both systems reasonably accu- rate intermolecular potentials have been reported. The initial calculations on the He − H$_{2}$ system employed the potential energy surface (PES) of Muchnick and Russek (MR) [26]. For He − H$_{2}$ vibrational excitation of the H$_{2}$ molecule has a dramatic effect on the zero-temperature quenching rate coefficients. As illustrated in Fig. 1, the vibrational quenching rate coefficients increase by about three orders of magnitude between v = 1 and v = 10 of the H$_{2}$ molecule [19]. 

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## FIG. 1: Rate coefficients for the quenching of H 2 ( v, j = 0) by collisions with He atoms as functions of the temperature for v = 1 − 10 of the H 2 molecule. Reproduced with permission from Balakrishnan et al. [19]. 

The quenching rate coefficients exhibit a minimum at around 10 K which roughly corresponds to the depth of the van der Waals interaction potential. This behavior appears to be a characteristic of vibrational quenching rate coefficients. For incident energies lower than the well depth the rate coefficient exhibits a minimum and with subsequent decrease in temperature the rate coefficient begins to increase before attaining the Wigner limit. For systems with deeper van der Waals wells the minimum is shifted to higher temperatures. Measurements of vibra- tional relaxation rate coefficients for the H$_{2}$ − CO system have confirmed this behavior [27]. 

Balakrishnan, Forrey, and Dalgarno [28] also investi- gated vibrational relaxation of H$_{2}$ in collisions with H atoms for vibrational quantum numbers v = 1 − 12 of the H$_{2}$ molecule. They adopted a non-reactive scattering formalism and neglected the rotational motion of the H$_{2}$ molecule. The calculations showed that vibrational relaxation rate coefficients are strongly dependent on the initial vibrational level of the H$_{2}$ molecule. The relaxation rate coefficients were found to increase by about seven orders of magnitude between vibrational levels v = 1 and v = 12. The dramatic variation in the rate coefficients with increase in vibrational excitation was explained in terms of the matrix elements of the 

interaction potential between the vibrational wavefunc- tions as functions of the atom - molecule center-of-mass separation. 

Vibrational relaxation rate coefficients in atom - molecule systems are often influenced by van der Waals complexes formed during the collision process. Decay of these complexes leads to resonances in the energy dependence of the relaxation cross sections (see Fig. 2 for Ar − D$_{2}$ collisions). Applying effective range theory to describe ultracold collisions of the He − H$_{2}$ system, Bal- akrishnan et al. [19] and Forrey et al. [20] demonstrated that vibrational pre-dissociation lifetimes of resonances that lie close to the energy threshold can be derived accurately from the value of the zero-temperature quenching rate coefficient. This formalism was extended to describe vibrational relaxation of trapped molecules and it was shown that the vibrational relaxation rate is controlled by the most weakly bound state of the van der Waals complex [22]. In a related work Dashevskaya et al. [29] have shown that vibrational quenching of H$_{2}$( v = 1 , j = 0) at low temperatures can be described using a two channel approximation within the quasi- classical method provided appropriate parameters are employed in the calculations. Subsequently, Cˆ ot´ e e t al. [30] generalized this method to predict vibrational relaxation lifetimes of atom - diatom van der Waals complexes with energies near the dissociation threshold. 

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Unlike in thermal energy collisions, the presence of a weakly bound state in the vicinity of a channel threshold can dramatically influence the cross sections in the ultracold regime. This is illustrated in Fig. 2 where the cross sections for vibrational relaxation of 

