Librational band calculations

Figure 47 Absorption frequency dependence in the libration band calculated for ice at the temperature -7°C for the spatial (a) and planar (b) hat models with the parameters u = 8.5, / = 0.15, y = 0.8 / = 23.5° (spatial model) and 21.2° (planar model)...

The latter is determined by the oscillation frequency, decaying coefficient, and vibration lifetime. This nonrigid dipole moment stipulates a Lorentz-like addition to the correlation function. As a result, the form of the calculated R-band substantially changes, if to compare it with this band described in terms of the pure hat-curved model. Application to ordinary and heavy water of the so-corrected hat-curved model is shown to improve description (given in terms of a simple analytical theory) of the far-infra red spectrum comprising superposition of the R- and librational bands. 

The calculated width of the librational band is wider than the recorded one, cf. solid and dashed lines in Figs. 15a and 15b. This disagreement is... 

Models 1-4 have a fundamental drawback The librational absorption band calculated for water appears to be too wide. This drawback at first glance could be overcome, if one employs the so-called field models, in which the static potential presents a smooth well (where a notion of a collision of a dipole with a wall actually has no physical sense). However, from the discussion given just below we shall see that this reasonable idea does not work properly with respect to calculating the wideband spectra in water. 

In our early work33 [50] the constant field model was applied to liquid water, where the harmonic law of particles motion, corresponding to a parabolic potential, was actually employed in the final calculations of the complex permittivity. In this work, qualitative description of only the libration band was obtained, while neither the R-band nor the low-frequency (Debye) relaxation band was described. Moreover, the fitted mean lifetime x of the dipoles, moving in the potential well, is unreasonably short ( ().02 ps)—that is, about an order of magnitude less than in more accurate calculations, which will be made here. 

Figure 24. Absorption coefficient (a, c) and wideband diecltric loss (b, d) calculated for liquid H20 water at 22.2°C (a, b) and 27°C (c, d) for the hat-curved model (solid lines). The experimental a(v) dependencies [17, 42, 56] are shown by dashed lines. The horizontal lines in Figs, (a) and (c) denote the maximum absorption recorded in the librational band.

In the librational band we have attained now a satisfactory agreement of the theoretical and experimental absorption frequency dependences. Comparing Figs. 32a and 32b with Figs. 26a and 26c calculated in Section V for a pure ... 

Figures 32d-f, placed on the right-hand side of Fig. 32, demonstrate a wideband dielectric-loss frequency dependence. This loss is calculated (solid lines) or measured [17, 42, 51, 54] (dashed lines) for water H20 and D20 at the same temperatures, as correspond to the absorption curves shown on the left-hand side of Fig. 32. Our theory gives a satisfactory agreement with the experimental data, obtained for the Debye region, R- and librational bands, to which three peaks (from left to right) correspond. However, in the submillimeter wavelength region (namely, from 10 to 100 cm ) the calculated loss is less than the recorded one. The fundamental reason for this difference will be discussed at the end of the next section.

Ab initio calculations have given vibrational wavenumbers for Cn.H2/D2 complexes,675 and for HCl(NH3)n clusters, where n = 1 - 4.676 High-resolution far-IR spectroscopy for the OC H35C1 heterodimer showed that the band origin of the HC1 libration band, vf, was 201.20464(27) cm-1.677 DFT calculations gave vibrational wavenumbers for the adduct H20.C10.678... 

Thus, in the case of heavy water the librational-band maximum is shifted to low frequencies due to increase of the moment of inertia 7or, while the translational-band maximum is located at approximately the same frequency vq 200 cm-1 though doubling of the moment of inertia 7vib (see the note in Table II). The calculated microwave/THz spectra of OW and HW are also rather close (cf. Figs. 4 and 5, Figs. 6 and 7). This result corresponds to the experiment by Zelsmann [21]. 

The MD simulation by Marchi [8] of the librational band (Fig. 18) is rather poor in this example. A better simulation is for the translational band (solid line in Fig. 19). The dashed curve represents here the energy loss function determined by the dielectric constant s and calculated in the cited work by Marchi. The shift between the solid and dashed curves represents a specific parameter of the fluid... 

For the frequency range 10-1000 cm-1 we present in Fig. 20a, b by solid lines the far-IR ice spectra of the absorption coefficient a(v) and of dielectric loss e"(v) calculated for the temperature — 7°C. The symbols V, T, L refer, respectively, to the V-, translational, and librational bands. The open circles mark the experimental data by Warren [49] these data are reproduced in Table IX. The fitted model and molecular parameters, used in this calculation, are given in Table X. 

Application of the relationship (210) for calculation of the libration band in water and in ice made evident the barest necessity to simplify the hat model, specified in Section VIII.A. 1. 

In Fig. 45a we show the dimensionless absorption calculated for the librational band for the parameters, typical for liquid water. Here distinction of the planar model (solid lines) from the spatial one (dashed line) turns out to be noticeable. For ice (Fig. 45b) the indicated distinction becomes less. 

In the new calculation scheme the hat model may be involved twice. First, this model may be applied for describing the relaxation frequency band, characterized by the Debye time td- Second, this model may be used for describing the libration band, characterized by the lifetime Tor. Therefore, the parameters of the two used hat models (or similar to them) should be quite different. We emphasize that in the present calculation scheme, applied to water, we employ the hat model only once—for describing both Debye and libration bands, for which the same set of the model parameters is used. 

It would be important to find analogous mechanism also for description of the main (librational) absorption band in water. After that it would be interesting to calculate for such molecular structures the spectral junction complex dielectric permittivity in terms of the ACF method. If this attempt will be successful, a new level of a nonheuristic molecular modeling of water and, generally, of aqueous media could be accomplished. We hope to convincingly demonstrate in the future that even a drastically simplified local-order structure of water could constitute a basis for a satisfactory description of the wideband spectra of water in terms of an analytical theory. 

Using this result, we may simplify calculation of the spectral function Liz) by neglecting the precessional contribution to L. We shall estimate also in this approximation the peak frequencies X ib and xrot of the absorption bands determined by the librational and the rotational subensembles. 

Nevertheless, normal coordinate model calculations of the H2O librations, as reported recentlyare very valuable for understanding the vibrational modes, the mean square amplitudes (see also Ref. 144), and the relative order of these water bands (see Sect. 4.4). The force constants obtained, however, must be taken with caution (see the discussion given above). 

In Ih phase, the selection rules for D6h are violated due to the proton disorder. So the observed band shape represents mostly the phonon density of states (DOS). As shown in Fig.2 (b), observed Raman spectra in the librational region in Ih phase surprisingly agrees with results of the neutron scattering (IINS) and MD calculation. In XI phase, Raman spectra show mostly the first order scattering around the T -point but qualitative agreement with neutron and MD studies is also seen in XI phases. 

Phonon wings were introduced in the context of their impact on the internal vibrational spectrum of molecules but the librational mode is an external mode, it is itself a phonon. However, this is only a question of classification and semantics. The phonon wing treatment is simply one approach to calculating the intensities of combination bands. It is the method of choice when detailed information on the external mode atomic displacements is absent. We now proceed to apply the phonon wing treatment of 2.6.3, from which we shall obtain the value of the mean square displacement of the ammonium ion due to the translational vibrations of the lattice, own (which is but one of the contributions to the full ext.)... 

The analysis so far has all been based on the isolated molecule approximation. For maleic anhydride this is valid as the agreement between the gas phase calculation and the solid state spectrum shows. However, it does not account for some of the most intense bands in the spectrum, those below 200 cm in the lattice mode region. It is possible to assign the bands, if it is assumed that the librational modes are more intense and occur at higher fi equency than the translational modes. It also requires the assumption that the factor group splitting is small (there... 

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