Spectral density direct damping

Figure 16. Rosch and Ratner spectral density (direct damping) Rosch and Ratner lineshapes (lines) Lorentzian fit (circles) Gaussian fit (black dots).

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). 

The theories of the spectral density of direct and indirect damped H bond have been reviewed recently in an extensive way [8] so that we give here a... 

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. 

As seen, the spectral density involving direct damping is the double sum over m and n of Lorentzians centered on 0) = o>° (n m)il — 2ao2Q and having the same half-width y°, but different intensities, given by c x" Am (a°) 2. 

Moreover, the spectral density of the system involving both direct and indirect quantum damping, is the Fourier transform of Eq. (125), that is,... 

As a consequence, the classical spectral density taking into account both direct and indirect dampings become... 

This is performed in Fig. 6. For this purpose, two spectral densities are superposed in each case, one of reference involving only some lot of weak direct damping (grayed line shapes) and the other involving both direct and indirect damping (full lines). 

Figure 8. Spectral analysis involving direct and indirect dampings at T = 300 K. The direct damping parameter has been chosen greater (y° = 0.25 f ) when the indirect damping is missing, than (Y° — 0.025Si) when it is present (y — 0.1 SI) in order to distinguish clearly the spectral densities. Dirac delta peaks are corresponding to the situation without any damping. co° = 3000 cm-1,...

It appears that, as required, this ACF is twice that of the single H-bond bridge involving both direct and indirect damping which, after Fourier transform, leads to the spectral density (285). Note that this last SD reduces in turn to that of Boulil et al. [49] when the direct damping is missing, that is, when y° = 0 ... 

The spectral density (80), dealing with the bare H-bond without indirect damping was found with the aid of the set of simplified basic Eqs. (4-6). The aim of this appendix is to show that in this situation without direct and indirect dampings, the result (80) remains the same if the set of basic Eqs. (1-3) is used. 

Since the design time histories meet response spectrum enveloping requirement and power spectral density function requirement and the components of design time histories in each direction are statistically independent, the deisgn time histories are acceptable. The critical damping values are consistent with Reg. Guide 1.61 and ASME Code Case N-411-1. 

P. Blaise, O. Henri-Rousseau. Spectral density of medium strength H-bonds. Direct damping and intrinsic anharmonicity of the slow mode. Beyond adiabatic approximation. Chem Phys 256 85-106, 2000. 

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