Hamiltonian modes starting equations

In the full quantum mechanical approach [8], one uses Eq. (22) and considers both the slow and fast mode obeying quantum mechanics. Then, one obtains within the adiabatic approximation the starting equations involving effective Hamiltonians characterizing the slow mode that are at the basis of all principal quantum approaches of the spectral density of weak H bonds [7,24,25,32,33,58, 61,87,91]. It has been shown recently [57] that, for weak H bonds and within direct damping, the theoretical lineshape avoiding the adiabatic approximation, obtained directly from Hamiltonian (22), is the same as that obtained from the RR spectral density (involving adiabatic approximation). 

Although in principle the microscopic Hamiltonian contains the infonnation necessary to describe the phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct a reduced description. Generally, a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a usefiil starting point. The equation of motion of the macrovariables can, in principle, be derived from the microscopic... 

We start from the well-known Hamiltonian of the linear E e JT effect, comprising the four e2g modes v15 — vK. For later purposes this is written immediately for both the X2Eig and B2E2g states of Bz+, and including the linear inter-state coupling term between them. From our earlier work (see Fig. 2 of Ref. [23] and equation (7) of Ref. [24]) this is adopted to be... 

In order to obtain a more compact formulation of the mixed quantum-classical equations we use a Hamilton-Jacobi-like formalism for the propagation of the quantum degree of freedom as in earlier studies [23], A similar approach has been introduced by Nettesheim, Schiitte and coworkers [54, 55, 56], TTie formalism presented here is based on recent investigations of the present authors [23], This formalism can be summarized as follows. Starting from the Hamiltonian Eqn. (2.2) and averaging over the x- and y-mode, respectively, gives... 

As the three effective mode model starts from the linear vibronic coupling Hamiltonian (LVC) [9] it may also have some relevance to generalize it and start from the quadratic vibronic coupling Hamiltonian (QVC) to obtain the appropriate quadratically extended (three)-effective mode equations. The motivation for this work has arisen that, in addition to the numerous applications of the LVC model, some other works in which the QVC model is used are also available [32,35], Our aim is to proceed along this direction. Following [21], we set up the QVC three-effective mode Hamiltonian and, using it for the pyrazine molecule we can calculate the autocorrelation function, the spectrum and the diabatic populations. The obtained results can be compared to those calculated by the LVC three-effective mode method. 

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