Periodic orbits , nonadiabatic quantum dynamics

Second, the mapping approach to nonadiabatic quantum dynamics is reviewed in Sections VI-VII. Based on an exact quantum-mechanical formulation, this approach allows us in several aspects to go beyond the scope of standard mixed quantum-classical methods. In particular, we study the classical phase space of a nonadiabatic system (including the discussion of vibronic periodic orbits) and the semiclassical description of nonadiabatic quantum mechanics via initial-value representations of the semiclassical propagator. The semiclassical spin-coherent state method and its close relation to the mapping approach is discussed in Section IX. Section X summarizes our results and concludes with some general remarks. 

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

To perform a PO analysis of nonadiabatic quantum dynamics, we employ a quasi-classical approximation that expresses time-dependent quantities of a vibronically coupled system in terms of the vibronic POs of the system [123]. Considering the quasi-classical expression (16) for the time-dependent expectation value of an observable A, this approximation assumes that the integrable islands in phase space represent the most significant contributions to the dynamics of the observables considered [236]. As a consequence, the short-time dynamics of the system is determined by its shortest POs and can be approximated by a time average over these orbits. Denoting the A th PO with period 7 by qk t),Pk t) we obtain [123]... 

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