Fast modes

In the derivation we used the exact expansion for X t), but an approximate expression for the last two integrals, in which we approximate the potential derivative by a constant at Xq- The optimization of the action S with respect to all the Fourier coefficients, shows that the action is optimal when all the d are zero. These coefficients correspond to frequencies larger than if/At. Therefore, the optimal solution does not contain contributions from these modes. Elimination of the fast modes from a trajectory, which are thought to be less relevant to the long time scale behavior of a dynamical system, has been the goal of numerous previous studies. 

For example, the SHAKE algorithm [17] freezes out particular motions, such as bond stretching, using holonomic constraints. One of the differences between SHAKE and the present approach is that in SHAKE we have to know in advance the identity of the fast modes. No such restriction is imposed in the present investigation. Another related algorithm is the Backward Euler approach [18], in which a Langevin equation is solved and the slow modes are constantly cooled down. However, the Backward Euler scheme employs an initial value solver of the differential equation and therefore the increase in step size is limited. 

In order for the averaged momentum p to point along the slow manifold (i.e., for the components of momentum in the fast manifold to average to zero), one has to choose the averaging time r so as to be several times longer than the period of the fast modes but shorter than those of the slow modes. 

Figure 1. Main anharmonicities of the fast mode. F, fast mode S, slow mode (intrinsic anharmonic Morse potential) B, bending mode.

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 simplest hydrogen bond X-H Y model may be viewed as composed of two oscillators. The first one corresponds to the stretching X-H-Y of the valence bond X-H. We will refer to this mode as the fast mode of the... 

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]... 

It is then reasonable to introduce in the same way the equilibrium position qe of the fast mode [52,53] ... 

The potential energy U of the fast mode should be assumed to be harmonic, in a first approximation ... 

The general potential U (8) has not been used before 1999 [52] because its numerical matrix representation requires huge basis sets, incompatible with the common computers. In order to avoid this situation, an approximation has been undertaken in previous studies the adiabatic approximation [54,55], Following an idea of Stepanov [56], Marechal and Witkowski assumed that the fast mode follows adiabatically the slow intermonomer motions, just as the electrons are assumed to follow adiabatically the motions of the nuclei in a molecule. It has been shown [57] that the adiabatic approximation is only suitable for very weak hydrogen bonds, as discussed in the next section. 

The supplementary term exp(—yf) is added in order to account for the direct damping mechanism of the fast mode, in the spirit of the Rosch and Ratner results [58]. Indeed, the Hamiltonians used in this section do not account for relaxational mechanisms, which will be discussed later in Section VI. 

We restrict the study of the adiabatic approximation to the case where the modulation parameters (30,/0, and gD, defined in Eqs. (6) and (7), are neglected. The effective angular frequency of the fast mode reduces to ... 

Passing to the Boson operators by aid of Table II, and after neglecting the zero-point-energy of the fast mode, we obtain a quantum representation we shall name I, in which the effective Hamiltonians of the slow mode corresponding respectively to the ground and first excited states of the fast mode are... 

If we want to remove the driven term in the potential of the slow mode, when k = 0 (ground state of the fast mode), it is suitable to perform the following phase transformation ... 

Now, it may be of interest to look at the connection between the autocorrelation functions appearing in the standard and the adiabatic approaches. Clearly, it is the representation I of the adiabatic approach which is the most narrowing to that of the standard one [see Eqs. (43) and (17)] because both are involving the diagonalization of the matricial representation of Hamiltonians, within the product base built up from the bases of the quantum harmonic oscillators corresponding to the separate slow and fast modes. However, among the... 

Let us look at the standard Hamiltonian (13). Its representation restricted to the ground state and the first excited state of the fast mode may be written according to the wave operator procedure [62] by aid of the four equations... 

Bratos and Hadzi have developed another origin of the anharmonicity of the fast mode X-H -Y, the Fermi resonance, which is supported by several experimental studies [1,3,63-70], Widely admitted for strong hydrogen bonds [67], the important perturbation brought to the infrared lineshape by Fermi resonances has also been pointed out in the case of weaker hydrogen bonds [53,71-73]. 

It is of importance to note that we shall consider, in the present section, that the fast and bending modes are subject to the same quantitative damping. Indeed, the damping parameter of the fast mode yG and that of the bending mode y will be supposed to be equal, so that we shall use in the following a single parameter, namely y (= yG = y5). This drastic restriction cannot be avoided when going beyond the adiabatic approximation. 

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

Focusing on the 0 ) > 1 ) infrared transition of the fast mode, the dipole... 

We shall name standard the nonadiabatic treatment of Fermi resonances for which the modulations of the fast mode frequency and equilibrium position by... 

The effective Hamiltonian holds for a single excitation of the fast mode and... 

Now, comparing 7Sf and /Sf ex, we observe also some splitting modifications. We may attribute them to some additional changes in the frequency gap. Indeed, as shown in Section V.B, the full treatment of the Fermi coupling mechanism leads to a displacement of the potential of the fast mode (both in energy and position) that does not appear within the exchange approximation. The fast mode then involves an effective frequency that differs from oo0, which leads to an effective gap which differs from (oo0 — 2oog). 

The damped effective Hamiltonian l holds for a single excitation of the fast mode and involves, according to some unitary transformations, a driven term that describes the intermonomer motion. Within the sub-base (89b), it may be given by... 

The account for the relaxations in (100) and (101) was made through the damping parameters yG of the fast mode and that ys of the bending mode. The... 

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 ... 

We have seen that the exchange approximation consists mainly in ignoring the driven part of the fast mode motion, given by... 

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