Collision dynamics time-dependent perturbation

The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic, vibrational, rotational, and nuclear spin). Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions. 

The Time Dependent Processes Section uses time-dependent perturbation theory, combined with the classical electric and magnetic fields that arise due to the interaction of photons with the nuclei and electrons of a molecule, to derive expressions for the rates of transitions among atomic or molecular electronic, vibrational, and rotational states induced by photon absorption or emission. Sources of line broadening and time correlation function treatments of absorption lineshapes are briefly introduced. Finally, transitions induced by collisions rather than by electromagnetic fields are briefly treated to provide an introduction to the subject of theoretical chemical dynamics. 

With increasing anisotropy, lOSA approximation breaks very early which indicates, of course, the importance of the dynamical contribution of rotation to the vibrational transition. One should expect that this contribution will show up in a substantial modification of the Landau-Teller exponent. Once it is realized that the Ehrenfest-Landau-Teller semiclassical exponential factor is proportional to the square of the Fourier component of the external time-dependent perturbation, and that the respective generalized Landau-Teller exponent can be recovered from the classical exponent, the strategy of finding the most efficient energy-transfer pathway becomes clear one should simply look for a mode which will provide the largest high-frequency Fourier components of the time-dependent perturbation that simulates a collision. 

The time evolution in Eq. (7.75) is described by the time-dependent Schrodinger equation, provided the molecule is isolated from the rest of the universe. In practice, there are always perturbations from the environment, say due to inelastic collisions. The coherent sum in Eq. (7.75) will then relax to the incoherent sum of Eq. (7.74), that is, the off-diagonal interference terms will vanish and cn 2 — pn corresponding to the Boltzmann distribution. As mentioned earlier, the relaxation time depends on the pressure. In order to take advantage of coherent dynamics it is, of course, crucial that relaxation is avoided within the duration of the relevant chemical dynamics. 

The coupling functions 1 and still depend on the molecular vibrational and rotational degrees of freedom as well as the relative molecule-perturber separation, R. Since the experiments imply that the physical origin of the collision-induced intersystem crossing resides in long-range attractive interactions, we may adopt a semiclassical approximation where the quantum-mechanical variables for the relative translation is replaced by a classical trajectory, R(l), for the relative molecule-perturber motion. The internal dynamics is then influenced by the time-dependent interactions f s[ (0] and Fj-j-fR(r)], which are still functions of molecular rotational and vibrational variables. For simplicity and for illustrative purposes we consider only the pair of coupled levels S and T and a pure triplet level T, which represents the molecular state after the collision. Note T may differ in rotational and/or vibrational quantum... 

The time-dependent Schrodinger equation for the effect of the collision dynamics on the probability amplitudes for being in S, T, and T is most conveniently represented in the mixed state basis (3.3), as this makes all coupling matrix elements vanish in the limit of infinite molecule-perturber separations. Within the mixed basis set, the matrix elements of V can be worked out to give... 

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