Three-mode model, nonadiabatic quantum

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]. 

Let us briefly discuss the characteristics of the nonadiabatic dynamics exhibited by this model. Assuming an initial preparation of the S2 state by an ideally short laser pulse. Fig. 1 displays in thick lines the first 500 fs of the quantum-mechanical time evolution of the system. The population probability of the diabatic state shown in panel (b) exhibits an initial decay on a timescale of w 20 fs, followed by quasi-periodic recurrences of the population, which are damped on a timescale of a few hundred femtoseconds. Beyond 500 fs (not shown) the S2 population probability becomes quasi-stationary, fluctuating statistically around its asymptotic value of 0.3. The time-dependent population of the adiabatic S2 state, displayed in panel (a), is seen to decay even faster than the diabatic population — essentially within a single vibrational period — and to attain an asymptotic value of 0.05. The finite asymptotic value of is a consequence of the restricted phase space of the three-mode model. The population Pf is expected to decay to zero for systems with many degrees of freedom. 

To obtain a first impression of the nonadiabatic wave-packet dynamics of the three-mode two-state model. Fig. 34 shows the quantum-mechanical probability density P (cp, f) = ( (f) / ) (p)(cp ( / (f)) of the system, plotted as a function of time t and the isomerization coordinate cp. To clearly show the... 

The goal of this review is to critically compare — from both a concep-tional and a practical point of view — various MQC strategies to describe non-Born-Oppenheimer dynamics. Owing to personal preferences, we will focus on the modeling of ultrafast bound-state processes following photoexcitation such as internal-conversion and nonadiabatic photoisomerization. To this end, Sec. 2 introduces three model problems Model I represents a three-mode description of the Si — S2 conical intersection in pyrazine. Model II accounts for the ultrafast C B X internal-conversion process in the benzene cation, and Model III represents a three-mode description of ultrafast photoisomerization triggered by a conical intersection. Allowing for exact quantum-mechanical reference calculations, all models have been used as benchmark problems to study approximate descriptions. 

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