Nonadiabatic transition region

Figure 1. Two basic elements of dynamics (l) propagation on a single adiabatic potential and (2) nonadiabatic transition. In the classically allowed case, the transition occurs at Ana- In the classically forbidden case, on the other hand, the transition region spans the interval (xi,Xf), where a , and Xf are the turning points. Taken from Ref. [9].

If the regions of nonadiabatic behavior are well localized in the configuration space M, an (F — l)-dimensional hypersurface can be defined at which the nonadiabatic transitions may take place this hypersurface is referred to as the crossing seam. The coupled relations, Eq. (14), describing the corresponding nonseparable electronic and nuclear motion, are to be solved at the seam. Elsewhere, the evolution of the polyatomic system can be then treated adiabatically (49,50). 

The expression of the Eq.(53) demonstrates the significant role of the diffraction type nonadiabatic transitions originating from the asymptotic region x —> oc. The contribution of this region to the amplitude of nonadiabatic transitions is equal to the inverse Massey parameter of Eq.(13) which has finite value at a —t oc because of the asymptotic degeneracy of potential energy curves. 

Another practical method is TSH (18), in which ordinary classical trajectories are run until they come close to the surface crossing region where the trajectories are jumped to the other surface with probability given by the Landau-Zener formula. This method is simple and convenient, but suffers from the following drawbacks all phases are completely neglected and only the probabilities (not the probability amplitudes) are handled the detailed balance is not necessarily satisfied and nonadiabatic transitions at energies... 

Although the Landau-Zener theory predicts all the experimental trends observed for E- V transfer, and, although more sophisticated techniques for handling the nonadiabatic transition are available, ° no calculations have been performed as yet for deactivation of I or Br by this mechanism. The major difficulty is that the theory is very sensitive to the details of the potential curves in the region of interaction. In general, these curves will be multidimensional surfaces. On the other hand, if our object is to learn about the potential from comparison of theory and experiment, it is just this sensitivity that we should demand from theory. It seems likely, therefore, that future theoretical work in this area will be very beneficial to our understanding of E- V transfer. 

Ao and Rammer [166] obtained the same result (and more) on the basis of a fully quantum mechanical treatment. Frauenfelder and Wolynes [78] derived it from simple physical arguments. Equation (9.98) predicts a quasiadiabatic result, = h k/ v 1 and the Golden Rule result, Pk = k/ v, in the opposite limit, which is qualitatively similar to the Landau-Zener behavior of the transition probability but the implications are different. Equation (9.98) is the result of multiple nonadiabatic crossings of the delta sink although it does not depend on details of the stochastic process Xj- t). This can be understood from the following consideration. For each moment of time, the fast coordinate has a Gaussian distribution, p Xf, t) = (xy — Xj, transition region, the fast coordinate crosses it very frequently and thus forms an effective sink for the slow coordinate. 

The situation is radically different in the inverted region, as well as in certain cases of nonequihbrium back transfer (see below), which are always nonadiabatic whatever the coupling strength is. For large V, the ET rate is no longer controlled by transport to the transition region but rather by nonadiabatic transitions between adiabatic states (see Fig. 9.1). Therefore, one should expect a decrease of the ET rate with increasing V to follow the solvent-controlled plateau. Usually, the Landau-Zener formula is used for the description of nonadiabatic transitions in the classical limit [162,163]. 

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