Energy conservation, nonadiabatic quantum

All approaches for the description of nonadiabatic dynamics discussed so far have used the simple quasi-classical approximation (16) to describe the dynamics of the nuclear degrees of freedom. As a consequence, these methods are in general not able to account for processes or observables for which quantum effects of the nuclear degrees of freedom are important. Such processes include nuclear tunneling, interference effects in wave-packet dynamics, and the conservation of zero-point energy. In contrast to quasi-classical approximations, semiclassical methods take into account the phase exp iSi/h) of a classical trajectory and are therefore capable—at least in principle—of describing quantum effects. 

The structure of C given above, consists of two distinct components (i) classical propagation on mean surfaces accompanied by quantum mechanical phase oscillations with frequency Lvaa> = (Ea — Ea>)/h, and (ii) nonadiabatic transitions accompanied by changes in the momentum of the environment in order to conserve energy. The classical Liouville operator... 

Let us first consider the population probability of the initially excited adiabatic state as shown in Fig. 6. Within the first 20 fs, the quantum-mechanical result is seen to decay almost completely to zero. The result of the QCL calculation matches the quantum data only for about 10 fs and is then found to oscillate around the quantum result. A closer analysis of the calculation shows that this flaw of the QCL method is mainly caused by large momentum shifts associated with the divergence of the nonadiabatic couplings = (4> d/df j 5 ). We therefore chose to resort to a simpler approximation and require that the energy of each individual trajectory is conserved. Employing this scheme, the corresponding QCL result for the adiabatic population matches at least qualitatively the quantum data. 

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