Quantum indirect damping

Boulil et al. [8] have obtained the following expression for the quantum indirect damped autocorrelation function in representation II ... 

Another Approach of Quantum Indirect Damping Within Representation /// ... 

Taking into Account Quantum Indirect Damping by Aid of C2 symmetry [76]... 

An Approximation for Quantum Indirect Damping [80] when Davydov Coupling Occurring... 

Both these aspects will be the characteristics of the quantum indirect damping. 

Now, let us look at the incorporation of the quantum indirect damping in the quantum representation // of the H-bond bridge. It is necessary to introduce in the model of the weak H-bond working within the strong anharmonic coupling theory, an hypothesis on the nature and on the irreversible action of... 

Besides, Appendix I proves that in the presence of quantum indirect damping, according to Eqs. (1.6) and (1.12), the translation operator involved in Eq. (120) conserves its structure, but in such a way as its argument is modified according to... 

Main Lines for Obtaining the ACF (124) Involving Quantum Indirect Damping... 

Recall that in the presence of quantum direct damping, this ACF must be multiplied by an exponential decay e 7° as it appears by comparison of Eqs. (124) and (125). Also recall that the ACF (123) that is at the origin of Eq. (135), is an explicit expression of the formal one given by Eq. (114). We need to know what is the classical limit of this ACF (135) involving quantum indirect damping. 

As a consequence, the semiclassical limit of the ACF (142) involving quantum indirect damping is... 

Note that if quantum indirect damping has been taken into account in Eq. (144), on the other hand, the direct one has been ignored. Thus, in order to incorporate this last one into the semiclassical ACF, it is necessary, as above, to multiply Eq. (144) by an exponential decay ... 

Thus, there is the following expression for the semiclassical limit of quantum indirect damping ... 

The different situations illustrated in Fig. 6 correspond more and more to approximate models when passing up to down. The top of the figure given the case described by Eq. (126) and corresponds to the reference quantum indirect damping. Below, is depicted the situation described by Eq. (146), where the Dyson time ordering operator is ignored in the quantum model. Further below is given the behavior that corresponds to both Eqs. (152) and (174), that is, to weak approximations on the classical limit of the quantum model [Eq. (152)] and to the semiclassical model [Eq. (174)]. At last, at the bottom we visualize the semiclassical model of Robertson and Yarwood [described by Eq. (185)]. 

Thus, owing to the suitability of quantum theory of indirect damping and to its difficulty in being generalized to complex situations, it appears to be of interest to find a suitable approximation to take into account simply the quantum indirect damping. This is the aim of this section [67]. 

Previously, it was shown that the ACF of the H-bond with quantum indirect damping, but without direct damping, is given by Eq. (124), that is, by... 

Now, return to the situation where the quantum indirect damping is taken into account. 

This last expression may be viewed as the basic approximation in order to treat more easily the quantum indirect damping. 

Now, we show that the approximation that has been performed to treat the quantum indirect damping allows us to find other approximations for handling the quantum indirect relaxation in which the damping of the H-bond bridge is taken into account by aid of non-Hermitean effective Hamiltonians. 

Now, let us apply the approached model of quantum indirect damping to a situation where a Fermi resonance is possible to occur when the first excited state of the fast mode and the first overtones of some bending modes are close. The basic physical terms are defined in Table VIII. 

Now, consider the other simple method for the quantum indirect damping, where the Hamiltonians are Hermitean, but the eigenvalues become complex by addition of imaginary parts. 

Now, owing to the well-behaved character of the approached NHDH and CEL methods, it appears reasonable to apply them to more complex situations where, in the absence of Fermi resonance, the quantum indirect damping, and not only the direct damping [72], has to be treated beyond the adiabatic and harmonic approximations [71]. 

Besides, in a similar way, the second contribution, y which is a reflection of the quantum indirect damping, is assumed to be proportional to the indirect damping parameter y, times the excitation degree of the slow mode via ... 

In Section IE, a theoretical approach of the quantum indirect damping of the H-bond bridge was exposed within the strong anharmonic coupling theory, with the aid of the adiabatic approximation. In Section III, this theory was shown to reduce to the Marechal and Witkowski and Rosch and Ratner quantum approaches. In Section IV, this quantum theory of indirect damping was shown to admit as an approximate semiclassical limit the approach of Robertson and Yarwood. 

This section now deals with H-bonded species, where together with the strong anharmonic coupling and the quantum indirect damping, Davydov coupling and Fermi resonances may occur, that is, centrosymmetric H-bonded cyclic dimers the theory of which, for situations without damping,was first performed by Marechal and Witkowski [18]. 

As it appears, this ACF has the same structure as (108). This equation was found for a single H-bond bridge involving quantum indirect damping, except for the fact that b = b/y/2 in the Hamiltonian [IHl)/ given by Eq. (270) in place of b, which appears in the effective Hamiltonian L 0 given by Eq. (106). 

Here, [G))cnnl (l)] is the ACF of the g part of the system involving Fermi resonances and quantum indirect damping, whereas [G -1 (f)]M are the ACFs corresponding to the u parts involving neither Fermi resonance nor indirect damping that are given by Eq. (276). 

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