Libration barriers

The Debye phenomenology is consistent with both gas-like and solidlike model representations of the reorientation mechanism. Reorientation may result either from free rotation paths or from jumps over libration barriers [86]. Primary importance is attached to the resulting angle of reorientation, which should be small in an elementary step. If it is... 

As one would expect, the rate of orientational relaxation in the jump model is activated, and the higher the libration barrier U0, the lower the rate. However, the Hubbard relation obtained as a result of Eq. (1.123) used in Eq. (2.96) does not involve this characteristic parameter of the solid-like model ... 

Fig. 7.2. The radial dependence of the anisotropic part of the intermolecular potential (a) variation of height of the librational barrier in any diametrical cross-section of the cage and its rectangular approximation (b) the corresponding rectangular approximation of F(r) separation between the region of libration and that of free rotation inside the cage.

In simpler substituted ethanes in symmetrical conformations, the librational minima will be nearer to each other and they coincide at the perfectly staggered position in ethane, which has no librational barrier. If the simpler ethane has an unsymmetrical conformation, e.g. gauche butane, one librational minimum may disappear or be merely a point of inflection, so that the sixfold potential becomes an unsymmetrical threefold one with more substituted carbon-carbon bonds with other than simple methyl substituents, the six possible minima should be considered as three librational pairs. Whether any librational pair is so unsymmetrical that it can be considered as a single minimum, and thus simplified, can be decided by calculation, or perhaps by intuition. The libration may not be suitable for experimental study, but awareness of the... 

The bulk of knowledge of libration in hydrocarbons is derived from calculations, which show how unusually high the librational barrier for tri-tert-butylmethane is. In a set of 1,2-dialkyltetramethylethanes, calculations show that all libration barriers are less than 2.2kcalmol" (Table 4), with different values for the gauche and anti conformations. 

The results for the two trigonal complexes are shown below. The fade out of the resonances indicates the onset of rapid reorientation and the surmounting of the librational barrier. This is particularly clear for the "equatorial" chlorine atoms in the Dianin complex, and is consistent with the egg-shaped form of the cavity in this compound. 

The libration barriers were estimated from the far infrared spectra using the Brot-Darmon model (see Sect. 4.1). Taking in Eq. (8) U((p) = Q, one gets... 

A remarkable relationship between Qu r and coh was obtained, namely, Qubr E bhfX which points to the essential contribution to libration barriers from intermolecular interactions. The contribution into Q, , from the barrier to internal rotation <2c-c is substantially less. This is apparently what accounts for an approximate coincidence between the peak position of librational absorption in the far infrared spectra of polymers and low-molecular-weight liquids close to the structure to the respective monomer units, e.g. of PS and CgHg or PCS and CgHjCl (Table 2). 

The envelope of the Stark structure of the rotator in a constant orienting field, calculated quantum-mechanically in [17], roughly reproduces the shape of the triplet (Fig. 0.5(c)). The appearance of the Q-branch in the linear rotator spectrum indicates that the axis is partially fixed, i.e. some molecules perform librations of small amplitude around the field. Only molecules with high enough rotational energy overcome the barrier created by the field. They rotate with the frequencies observed in the... 

Fig. 1.22. Angular-dependent potential U for one-dimensional libration over barriers of height U0. The arrows show the way of libration below barriers and random translations from one well to another due to high-energy fluctuations. The broken line presents the approximation of the parabolic well valid at the bottom.

Biphenyl, terphenyl, and quaterphenyl all have room-temperature structures that behave similarly, and also all have low-temperature phases in which only one conformer is present. The torsion angle about the central C-C bond in biphenyl is estimated to be about 10° in the low-temperature form, which is appreciably less than that in the gas phase. In the case of terphenyl, it has been established (41) that the room-temperature structure is disordered each molecule librates in a double-well potential, with the barrier height being about 0.6 kcal/ mol. In the low-temperature form the molecule is stabilized in one of the two minima of the well, and has its terminal rings rotated in the same sense, so that the molecule conserves its center of symmetry. This alternation of rotations between adjacent rings is found (42) also in the low-temperature form of quaterphenyl and results in the molecule being noncentric (even though the crystal structure has a center of symmetry). 

A simplified version of this model, termed the hybrid model (VIG, p. 305) [32-34, 39] (see also Section IV.E) was proposed for the case of a small cone angle p. In this model the rotators move freely over the barrier U0 as if they do not notice the conical surface the librators move in the diametric sections of a cone—that is, they librate. The hybrid model was widely used for investigation of dielectric relaxation in a number of nonassociated and associated liquids, including aqueous electrolyte solutions (VIG, p. 553) [53, 54]. The hat model was recently applied to a nonassociated liquid [3] and to water [7, 12c]. 

The FC or Born-Oppenheimer approximation is physically clear if the activation energy barrier is in the Dielectric Continuum. The reacting ion is activated by some collisional or vibrational-librational means from the classical Boltzmann thermal pool, so that the rate of activation is equal to the rate of arrival of energy, which is equal to a characteristic classical electrolyte frequency. The electron transfers when its energy exceeds that of the barrier due to the inertia of the solvent permanent dipoles. Marcus4149 consistently supposed that the medium may be regarded as a dense gas phase with a collision frequency, which in its... 

Simply to multiply correlations, such as have just been derived, by a suitable exp (—at) is obviously improper as it gives incorrect behaviour at small times. It may be empirically useful to allow such a correlation to run down nearly to its y(oo) and then switch in the exponential decay with an arbitrary smoothing at the transition. One way to do this while preserring the initial slope is to multiply by exp[- aifit + e — 1)], but this spoils the initial curvature. In any case, the switching in of the exponential ought to arise naturally from the analysis of the problem. Libration in a two-well potential. For simplidty we consider two equivalent equilibrium positions separated by tt radians. The similar potential wells are taken to be harmonic to a height much greater than kT, and separated by equal potential barriers on either side. 

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