Nonadiabatic transition semiclassical theory

Surface Hopping Model (SHM) first proposed by Tully and Preston [444] is a practical method to cope with nonadiabatic transition. It is actually not a theory but an intuitive prescription to take account of quantum coherent jump by replacing with a classical hop from one potential energy surface to another with a transition probability that is borrowed from other theories of semiclassical (or full quantum mechanical) nonadiabatic transitions state theory such as Zhu-Nakamura method. The fewest switch surface hopping method [445] and the theory of natural decay of mixing [197, 452, 509, 515] are among the most advanced methodologies so far proposed to practically resolve the critical difficulty of SET and the primitive version of SHM. 

In this chapter the basic semiclassical theory of nonadiabatic transition is reviewed and explained. Semiclassical theory has a history of more than 60 years since the pioneering work of Landau (13), Zener (14), and Stiickelberg (15) in 1932. Not only a brief historical survey of the theory, but also the most recent theoretical achievements are... 

This chapter is organized as follows. Section II interprets the underlying concept of nonadiabatic transition and its interdisciplinarity. Properties of nonadiabatic couplings are summarized. The terms diabatic, adiabatic, and nonadiabatic are explained. Section III briefly surveys the history of semiclassical theories of nonadiabatic transition... 

Actually, the semiclassical formulas introduced in the previous section, although valid only asymptotically, are in the form which appears to be suitable for extensions in the complex q-plane. It would be interesting to further investigate this aspect, since analytic continuation plays a role in theories of nonadiabatic transition [27], a role which has not been firmly assessed until now because in actual problems the analytic structure of numerically generated eigenvalues is poorly understood [28. ... 

Another basic theory of nonadiabatic transitions is the semiclassical Ehrenfest theory (SET). Although it can cope with multidimensional nonadiabatic electronic-state mixing, it inevitably produces a nuclear path that runs on an averaged potential energy surface after having passed across the nonadiabatic region, which is totally unphysical. Unfortunately, since SET seems intuitively correct, a naive and conventional derivation of this theory obscures how this critical difficulty arises. 

The semiclassical Ehrenfest theory coupled with this representation was applied in an electron flux analysis in chemical reactions where large charge transfer occurs caused by significant nonadiabatic transition. The chemical systems treated in the summaries below are Na - - Cl and formic acid dimer (FAD). The time shift flux operator stated in the previous subsection was utilized in an analysis of the microscopic electron d3mamics in this chemically representative case. 

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... 

For example, the ZN theory, which overcomes all the defects of the Landau-Zener-Stueckelberg theory, can be incorporated into various simulation methods in order to clarify the mechanisms of dynamics in realistic molecular systems. Since the nonadiabatic coupling is a vector and thus we can always determine the relevant one-dimensional (ID) direction of the transition in multidimensional space, the 1D ZN theory can be usefully utilized. Furthermore, the comprehension of reaction mechanisms can be deepened, since the formulas are given in simple analytical expressions. Since it is not feasible to treat realistic large systems fully quantum mechanically, it would be appropriate to incorporate the ZN theory into some kind of semiclassical methods. The promising semiclassical methods are (1) the initial value... 

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