Nonadiabatic tunneling transition

The problem of nonadiabatic tunneling in the Landau-Zener approximation has been solved by Ovchinnikova [1965]. For further refinements of the theory beyond this approximation see Laing et al. [1977], Holstein [1978], Coveney et al. [1985], Nakamura [1987]. The nonadiabatic transition probability for a more general case of dissipative tunneling is derived in appendix B. We quote here only the result for the dissipationless case obtained in the Landau-Zener limit. When < F (Xe), the total transition probability is the product of the adiabatic tunneling rate, calculated in the previous sections, and the Landau-Zener-Stueckelberg-like factor... 

The method is composed of the following algorithms (1) transition position is detected along each classical trajectory, (2) direction of transition is determined there and the ID cut of the potential energy surfaces is made along that direction, (3) judgment is made whether the transition is LZ type or nonadiabatic tunneling type, and (4) the transition probability is calculated by the appropriate ZN formula. The transition position can be simply found by... 

Figure 45. Schematic picture representing the nonadiabatic tunneling-type transition.

The formulas derived in the time-independent framework can be easily transferred into the corresponding time-dependent solutions. The formulas in the time-independent linear potential model, for example, provide the formulas in the time-dependent quadratic potential model in which the two time-dependent diabatic quadratic potentials are coupled by a constant diabatic coupling [1, 13, 147]. The classically forbidden transitions in the time-independent framework correspond to the diabatically avoided crossing case in the time-dependent framework. One more thing to note is that the nonadiabatic tunneling (NT) type of transition does not show up and only the LZ type appears in the time-dependent problems, since time is unidirectional. 

Three factors are relevant for the formation of the outer-sphere transition state the reagents must approach each other, the electronic interaction must be large enough —or a nonadiabatic tunneling mechanism needs to be operative—and the restriction imposed by the lack of nuclear motion during electronic motion must be satisfied. The energy of the transition state is determined by these three factors. 

C. Zhu and H. Nakamura, Theory of nonadiabatic transition for general two-state curve crossing problems. I Nonadiabatic tunneling case, J. Chem. Phys. 101 10630 (1994). 

H. Nakamura, Semiclassical treatment of nonadiabatic transitions multilevel curve crossing and nonadiabatic tunneling problems, J. Chem. Phys. 87 4031 (1987). 

To deal with the ET rate in such a case, our strategy is to combine the generalized nonadiabatic transition state theory (NA-TST) and the Zhu-Nakamura (ZN) nonadiabatic transition probability.The generalized NA-TST is formulated based on Miller s reactive flux-flux correlation function approach. The ZN theory, on the other hand, is practically free from the drawbacks of the LZ theory mentioned above. Numerical tests have also confirmed that it is essential for accurate evaluation of the thermal rate constant to take into account the multi-dimensional topography of the seam surface and to treat the nonadiabatic electronic transition and nuclear tunneling effects properly. 

Since the nonadiabatic transition is not a main subject in this book, the expressions of the matrices I, 0 = transpose of I), and N are not given here. The reader interested in these should refer to Reference [48], and especially Appendix A of this book. The nonadiabatic tunneling problem is, however, discussed in Chapter 5. 

Multidimensional theory is not yet available, unformnately, not only for the nonadiabatic tunneling problem but also for general nonadiabatic transition problems. For practical applications, the Zhu-Nakamura formulas for transition amplitude including phases can be incorporated into classical or semiclassical propagation... 

Classical trajectories are the only feasible means to explicitly treat all atoms in a dynamical study of a unimolecular reaction. Trajectories have been used extensively to interpret A + BC bimole-cular reactions and a considerable amount of literature exists with respect to these studies. Excitation functions, scattering angles, product energy distributions, and other dynamical properties are usually quantitatively determined by the trajectory calculations. The semiclassical studies of Marcus and Miller have in general confirmed the accuracy of classical trajectories in calculating dynamical properties for bimolecular reactions. However, the trajectories do not describe quantum mechanical effects such as interferences, tunneling, and nonadiabatic electronic transitions. 

It is to be emphasized that, despite the formal similarity, the physical problems are different. Moreover, in general, diabatic coupling is not small, unlike the tunneling matrix element, and this circumstance does not allow one to apply the noninteracting blip approximation. So even having been formulated in the standard spin-boson form, the problem still remains rather sophisticated. In particular, it is difficult to explore the intermediate region between nonadiabatic and adiabatic transition. 

In the nonadiabatic limit ( < 1) B = nVa/Vi sF, and at 1 the adiabatic result k = k a holds. As shown in section 5.2, the instanton velocity decreases as t] increases, and the transition tends to be more adiabatic, as in the classical case. This conclusion is far from obvious, because one might expect that, when the particle loses energy, it should increase its upside-down barrier velocity. Instead, the energy losses are saturated to a finite //-independent value, and friction slows the tunneling motion down. 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

---