Anharmonicity coupling

Figure 5. Reaction probabilities for a given instance of the noise as a function of the total integration time Tint for different values of the anharmonic coupling constant k. The solid lines represent the forward and backward reaction probabilities calculated using the moving dividing surface and the dashed lines correspond to the results obtained from the standard fixed dividing surface. In the top panel the dotted lines display the analytic estimates provided by Eq. (52). The results were obtained from 15,000 barrier ensemble trajectories subject to the same noise sequence evolved on the reactive potential (48) with barrier frequency to, = 0.75, transverse frequency co-y = 1.5, a damping constant y = 0.2, and temperature k%T = 1. (From Ref. 39.)...

On the other hand, one has to take into account the influence of the surrounding which must induce an irreversible evolution of the H-bond system when its fast mode is excited the fast mode may be directly damped by the medium that is the direct relaxation mechanism. It may be also damped through the slow mode to which it is anharmonically coupled, that is the indirect relaxation mechanism. A schematical illustration of these two damping mechanism is given in Fig. 2. Of course, the role played by damping must be more important for H bonds in condensed phase. 

The cornerstone of the strong anharmonic coupling theory relies on the assumption of a modulation of the fast mode frequency by the intermonomer distance. This behavior is correlated by many experimental observations, and it is undoubtly one of the main mechanisms that take place in a hydrogen bond. Because the intermonomer distance is, in the quantum model, represented by the dimensionless position coordinate Q of the slow mode, the effective angular frequency of the fast mode may be written [52,53]... 

Note that founder theoretical treatment of the strong anharmonic coupling has been done by Marechal and Witkowski [7] in the simplest case, obtained by neglecting the terms in p0,/0, and ga. 

Now, recall that for weak hydrogen bonds the high-frequency mode is much faster than the slow mode because 0 m 20 00. As a consequence, the quantum adiabatic approximation may be assumed to be verified when the anharmonic coupling parameter aG is not too strong. Thus, neglecting the diabatic part of the Hamiltonian (22) and using Eqs. (18) to (20), one obtains... 

In Fig. 4 we compare the adiabatic (dotted line) and the stabilized standard spectral densities (continuous line) for three values of the anharmonic coupling parameter and for the same damping parameter. Comparison shows that for a0 1, the adiabatic lineshapes are almost the same as those obtained by the exact approach. For aG = 1.5, this lineshape escapes from the exact one. That shows that for ac > 1, the adiabatic corrections becomes sensitive. However, it may be observed by inspection of the bottom spectra of Fig. 4, that if one takes for the adiabatic approach co0o = 165cm 1 and aG = 1.4, the adiabatic lineshape simulates sensitively the standard one obtained with go,, = 150 cm-1 and ac = 1.5. 

In this section we shall give the connections between the nonadiabatic and damped treatments of Fermi resonances [53,73] within the strong anharmonic coupling framework and the former theory of Witkowski and Wojcik [74] which is adiabatic and undamped, involving implicitly the exchange approximation (approximation later defined in Section IV.C). 

In the strong anharmonic coupling framework, the fast mode potential IJ is, according to Eq. (8),... 

First let us restrict the comparison to very weak H bonds because in such a case we can expect that the adiabatic approximation is fulfilled. We made some sample calculations shown in Figs. 8(c) and 8(d), with a small dimensionless anharmonic coupling parameter, ac = 0.6. The figures displays /sf ex [dashed line (d)] and 7Sf [dashed line (c)] at 300 K. Each one is compared with the adiabatic spectra 7f (superimposed full lines). Note that the adiabatic spectra (c)... 

When neglecting the strong anharmonic coupling—that is, in the situation of a pure Fermi coupling (no hydrogen bond)... 

Several studies of Fermi resonances in the absence of H bond have been made [76-80]. We shall account for this situation by simply ignoring the anharmonic coupling between the fast and slow modes (a = 0). The theory then describes the coupling between the fast mode and a bending mode through the potential Htf, with both of these modes being damped in the same way. Because aG = 0, the slow mode does not play any role, so that the total Hamiltonian does not refer to it ... 

Dealing with the restrictive situation of equal dampings, where yQ = y8 = y, the spectral density (107) may be written as the limit of /sf (oj) ex [Eq. (88)] when neglecting the anharmonic coupling—that is, when aG = 0 ... 

Then, neglecting the anharmonic coupling ac, the spectral density 7sf [Eq. (81)] reduces to... 

There are two kinds of damping that are considered within the strong anharmonic coupling theory the direct and the indirect. In the direct mechanism the excited state of the high-frequency mode relaxes directly toward the medium, whereas in the indirect mechanism it relaxes toward the slow mode to which it is anharmonically coupled, which relaxes in turn toward the medium. 

It may be shown [8] that both semiclassical [83,84], and full quantum mechanical approaches [7,32,33,58,87] of anharmonic coupling have in common the assumption that the angular frequency of the fast mode depends linearly on the slow mode coordinate and thus may be written... 

Now, return to Fig. 14. The right and left bottom damped lineshapes (dealing respectively with quantum direct damping and semiclassical indirect relaxation) are looking similar. That shows that for some reasonable anharmonic coupling parameters and at room temperature, an increase in the damping produces approximately the same broadened features in the RY semiclassical model of indirect relaxation and in the RR quantum model of direct relaxation. Thus, one may ask if the RR quantum model of direct relaxation could lead to the same kind of prediction as the RY semiclassical model of indirect relaxation. 

Figure 15 gives the superposition of RR (full line) and RY (dotted plot) spectral densities at 300 K. For the RR spectral density, the anharmonic coupling parameter and the direct damping parameter were taken as unity (a0 = 1, y0 = ffioo), in order to get a broadened lineshape involving reasonable half-width (a = 1 was used systematically, for instance, in Ref. 72). For the RY spectral density, the corresponding parameters were chosen aD = 1.29, y00 = 0.85angular frequency shift (the RY model fails to obtain the low-frequency shift predicted by the RR model) and a suitable adjustment in the intensities that are irrelevant in the RR and RY models. 

The anharmonic coupling parameter is able to reproduce a relatively large half-width of a weak H bond. 

The pure quantum approach of the strong anharmonic coupling theory performed by Marechal and Witkowski [7] gives the most satisfactorily zeroth-order physical description of weak H-bond IR lineshapes. 

The Rosch and Ratner [58] lineshapes, which are obtained within the adiabatic approximation, are not modified when working beyond the adiabatic approximation, if the anharmonic coupling between the slow and fast modes has a low magnitude. 

In the study of Watkins et al. (1990), the anharmonic coupling was identified with the cubic term of an expansion of the potential energy,... 

Thus, the required expression for the GF of the high-frequency mode taking into account the anharmonic coupling of the latter with the exchange deformation mode (characterized by a well-defined value of the reorientation barrier AU) takes the form (4.2.13) with restricted summation over the quantum numbers = <7 = 0, 1,. .. N of the subbarrier states. It is then expedient to rewrite Eq. (4.2.14) in the following form ... 

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