content_image/390810.jpg
cross sections approach finite values in the Wigner regime for the bosonic system of OH molecules and the fermionic system of OD molecules, the inelastic cross sections ex- hibit a totally different behavior. At ultra-low energies when molecules interact with an electric field, s-wave scattering yields an inelastic cross section diverging as E $_{−}$1 / 2 c oll for bosonic systems, while p-wave scattering yields an inelastic cross section vanishing as E 1 / 2 c oll for fermionic systems. Thus, the inelastic processes are suppressed for the fermionic system. The differences are attributed to the bosonic/fermionic character of the molecules and to the applied electric field. In the absence of an applied electric field, the elastic cross section of the fermionic sys- tem would decrease as E 2 c $_{oll}$, faster than the inelastic cross section, which decreases as E 1 / 2 c $_{oll}$. Ticknor and Bohn [138] subsequently studied OH − OH collisions in the presence of a magnetic field. They showed that magnetic fields of several thousand gauss reduce inelastic collisions by about two orders of magnitude. Based on these results, they concluded that magnetic trapping may be favorable for OH molecules. 

The theoretical studies presented above for molecule - molecule collisions refer to rigid rotor molecules. Pogreb- nya and Clary [139, 140] have investigated vibrational re- laxation in collisions of hydrogen molecules using a full- dimensional quantum dynamics formalism with an an- gular momentum decoupling approximation in the body fixed frame. They studied H$_{2}$( v = 1 , j ) + H$_{2}$( v = 0 , j $^{′}$) collisions involving both para-para and ortho-ortho com- binations for collision energies of 1 meV (11.6 K) to 1 eV (11604 K). Time-dependent quantum mechanical calcu- lations based on the multiconfiguration time-dependent Hartree approach have been recently applied to study full-dimensional quantum dynamics of the H$_{2}$ − H$_{2}$ sys- tem [141, 142]. These methods are suitable at high collision energies and have not been applied to cold and ultracold collisions. Very recently, vibrational en- ergy transfer mechanism for ultracold collisions between para-hydrogen molecules has been explored by Qu´em´ener et al. [143] using a full-dimensional quantum theoret- ical formalism implemented in a new code written by Krems [144]. The quantum scattering calculations are based on the theory described by Arthurs and Dal- garno [145], Takayanagi [146], Green [147], and Alexan- der and DePristo [148]. The H$_{2}$ molecules were initially in different quantum states characterized by the vibra- tional quantum number v and the rotational angular mo- mentum j . A combination of two ro-vibrational states of H$_{2}$ was referred to as a combined molecular state (CMS). A CMS, denoted as ( vjv $^{′}$j $^{′}$), represents a unique quan- tum state of the diatom - diatom system before or after a collision. 

The cross sections for H$_{2}$( v = 1 , j = 0) + H$_{2}$( v = 

## B. Vibrationally inelastic transitions 

<img src='content_image/390810.jpg'>

0 , j = 0), H$_{2}$( v = 1 , j = 2) + H$_{2}$( v = 0 , j = 0) and H$_{2}$( v = 1 , j = 0) + H$_{2}$( v = 0 , j = 2) collisions are pre- sented in the upper panel of Fig. 25. Using the CMS notation, this corresponds, respectively, to initial CMSs (1000), (1200) and (1002). The elastic cross sections are almost independent of the different initial ro-vibrational states of the molecules, but the inelastic cross sections are strongly dependent on the initial rotational and vibra- tional levels of the H$_{2}$ molecules. At 10 $_{−}$6 K, the inelastic relaxation of H$_{2}$( v = 1 , j = 0) is almost six orders of mag- nitude more efficient in collisions of H$_{2}$( v = 0 , j = 2) than in collisions with H$_{2}$( v = 0 , j = 0) and two orders of mag- nitude more efficient than in collisions of H$_{2}$( v = 1 , j = 2) with H$_{2}$( v = 0 , j = 0). At 25.45 K, which corresponds to the energy difference between the CMSs (1002) and (1200), the inelastic cross sections become comparable for H$_{2}$( v = 1 , j = 2) + H$_{2}$( v = 0 , j = 0) and H$_{2}$( v = 1 , j = 0) + H$_{2}$( v = 0 , j = 2) collisions. The inelastic scatter- ing depends on the type and the combination of ro- 

