Nonrigid dipole

Figure 2. Forms of the potential wells described in the chapter and the features pertinent to the corresponding molecular models, if and 2% denote, respectively, ensembles of librating and rotating dipoles p denotes a dipole moment VIB refers to the charges 8 of a nonrigid dipole vibrating along the H-bond p(t) denotes a given harmonically changing component of a dipole moment.

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. 

This expression is obtained by using only the Maxwell equations and the equations of classical mechanics and the condition (26) for q(t). Because of this, Eq. (27b) is valid irrespectively of the type of the projection = p g. This projection may correspond to reorientation of a rigid dipole or to vibration of bound charges of a nonrigid dipole. 

Section VI will present a first step for description (again in terms of the hat-curved model) of the collective (cooperative) effects in water due to the H bonds (i.e., following Walrafen [16]), resulting from the specific interactions. The dielectric spectra of ordinary and heavy water will be calculated in this section. For this purpose we shall apply (with some changes) recent investigation [6, 8] based on the concept of a nonrigid dipole. Other applications of the hat-curved model to water will be described in Sections VII, IX, and X. 

Figs. 32a-c illustrate the absorption spectra, calculated, respectively, for water H20 at 27°C, water H20 at 22.2°C, and water D20 at 22.2°C dotted lines show the contribution to the absorption coefficient due to vibrations of nonrigid dipoles. The latter contribution is found from the expression which follows from Eqs. (242) and (255). The experimental data [42, 51] are shown by squares. The dash-and-dotted line in Fig. 32b represents the result of calculations from the empirical formula by Liebe et al. [17] (given also in Section IV.G.2) for the complex permittivity of H20 at 27°C comprising double Debye-double Lorentz frequency dependences. 

Liquid T (K) A. Fitted Parameters Parameters Relevant to a Rigid Dipole Moment Parameters Relevant to a Nonrigid Dipole Moment ... 

It is interesting to compare the results obtained for ordinary and heavy water. To interpret the difference, we show in Fig. 33 by solid curves the total absorption attained in the R-band (i.e., near the frequency 200 cm-1). Dashed curves and dots show the components of this absorption determined, respectively, by a constant (in time) and by a time-varying parts of a dipole moment. In the case of D20, the R-absorption peak vR is stipulated mainly by nonrigidity of the H-bonded molecules, while in the case of H20 both contributions (due to vibration and reorientation) are commensurable. Therefore one may ignore, in a first approximation, the vibration processes in ordinary water as far as it concerns the wideband absorption frequency dependences (actually this assumption was accepted in Section V, as well is in many other publications (VIG), [7, 12b, 33, 34]. However, in the case of D20, where the mean free-rotation-frequency is substantially less than in the case of H20, neglecting of the vibrating mechanism due to nonrigid dipoles appears to be nonproductive. 

A principal drawback of the hat-curved model revealed here and also in Section V is that we cannot exactly describe the submillimeter (v) spectrum of water (cf. solid and dashed lines in Figs. 32d-f). It appears that a plausible reason for such a difference is rather fundamental, since in Sections V and VI a dipole is assumed to move in one (hat-curved) potential well, to which only one Debye relaxation process corresponds. We remark that the decaying oscillations of a nonrigid dipole are considered in this section in such a way that the law of these oscillations is taken a priori—that is, without consideration of any dynamical process. 

In Fig. 34c (for H20) and Fig. 34d (for D20), solid lines show the absorption coefficients a calculated in the R-band the estimated contributions to this a(v) due to reorientations and vibrations are shown by dashed and dashed-and-dotted curves, respectively. The resonance peak at vR 200 cm-1 found for both water isotopes is actually determined by a vibrating nonrigid dipole. The distinction between the curves calculated for ordinary and heavy water is substantial. In the case of H20, Fig. 34c, the contributions of reorientations and vibrations to the resulting a(v) curve are commensurable near the center of the R-absorption peak, while in the case of D20 (Fig. 34d) the main contribution to ot(v) near the R-band peak comes from vibration of the H-bonded molecules. 

The hat-curved-harmonic oscillator model, unlike other descriptions of the complex permittivity available now for us [17, 55, 56, 64], gives some insight into the mechanisms governing the experimental spectra. Namely, the estimated relaxation time of a nonrigid dipole (xovib 0.2 ps) is close to that determined in the course of very accurate experimental investigations and of their statistical treatment [17, 54-56]. The reduced parameters presented in Tables XIVA and XIVB and the form of the hat-curved potential well (determined by the parameters u, (3, f) do not show marked dependence on the temperature, while the spectra themselves vary with T in greater extent. We shall continue discussion of these results in Section X.A. 

Figure 44. Configuration of a nonrigid dipole formed by two charges oscillating along a straight line.

Using Eq. (359), we (1) introduce the factor g, which formally accounts for correlation of orientations of nonrigid dipoles (g will be found independently) and (2) express the polarization at any instant to of the last collision as a total dipole moment of a unit volume containing N bound charges ... 

The second (ionic) term in Eq. (387) is assumed to vanish in the limit co —> oo, just as does the term As (v) in Eq. (278) stipulated by oscillating charges of a nonrigid dipole. The first term in Eq. (387) will be calculated below in terms of the hybrid model, which was briefly described in Section IV.E. For the limit co —> oo we set this term to be equal to optical permittivity n2, the same as in pure water. 

In Section VILA a strongly idealized picture was described. The dielectric response of an oscillating nonrigid dipole was found in terms of collective vibrations of two charged particles. Now a more specific picture pertinent to an idealized water structure will be considered. Namely, we shall briefly consider thermal motions of a dipole as (i) pure rotations in Fig. 56b and (ii) pure translations in Fig. 58a. Item (i) presents the major interest for us, since we would like to roughly estimate on the basis of a molecular dynamics form of the absorption band stipulated by rotation of a dipole. Of course, even in terms of a simplified scheme, the internal rotations of a molecule should also be accompanied by its translations, so the Figs. 56a and 56b should somehow interfere. However, in Section IX.B.l we for simplicity will neglect this interference. This assumption approximately holds, since, as will be shown in Section IX.B.2, the mean frequencies of these two types of motion substantially differ. 

A new dielectric-relaxation mechanism, important in the THz region, is proposed, relevant to vibration of a nonrigid dipole in a direction perpendicular to that of the H-bond. 

The first mechanism (a) refers to dielectric relaxation pertinent to a permanent dipole influenced by a rather narrow hat intermolecular potential the next two (b, c) refer to the complex permittivity generated by two elastically vibrating hydrogen-bonded (HB) molecules. The last mechanism (d) refers to a nonrigid dipole vibrating in direction perpendicular to that of the undisturbed H-bond. 

The contribution of the mechanism a to the static permittivity is 0r,s = ss — Ags. The symbol A in A s refers to the contribution provided by the vibrating rigid and nonrigid dipoles pertinent to the VIB state. This symbol A is employed, since the optical constant n is included in the permittivity component or,s The value A s will be estimated in Section E. 

Figure 3 Contributions e" to the loss factor of water at 27°C (a, c, e) and of ice at —7°C (b, d, f) due to longitudinal harmonic vibration of a nonrigid dipole (a, b) harmonic reorientation of a permanent dipole (c, d) and nonharmonic transverse vibration of a nonrigid dipole (e, f). Symbols T and V refer, respectively, to the T- and V-bands.

It appears that the H-bond is polarized. In view of our estimate the mean moment iq of a nonrigid dipole is about 3 times larger than the moment ju0Y of a dipole librating in the hat well. 

Calculation gives the following approximate formula for the number m of the vibration cycles performed by a nonrigid dipole during the lifetime t rj /yj ... 

In Figs. 5 and 6, curves 1-4 refer, respectively, to mechanisms a-c—that is, to libration of rigid dipoles in the hat well (1), elastic reorientation of such dipoles (2), elastic translation of nonrigid dipoles (3), and their elastic transverse vibration (4). 

Thin lines in Figs. 5c-h and 6c-h refer to specific contributions due to nonharmonic reorientation of a permanent dipole in the hat potential (1), harmonic longitudinal vibration of HB nonrigid dipole (2), harmonic reorientation of a permanent HB dipole (3), and nonharmonic transverse vibration of a nonrigid HB dipole (4). 

The other time, Tq, pertinent to an elastic nonrigid dipole and comprising about 0.07 ps, decreases insignificantly with T in a wide temperature interval (see Table VI). This result may be interpreted as an evidence of its quantum nature . The fitted Tg(water) is less than the mean HB lifetime11 tHb mentioned by Teixeira [16]. We should remark that our lifetime xq actually means a period, during which longitudinal vibration remains coherent. 

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