Amplitude of libration

In the mid 1950s Durwaxd W. J. Cruickshank " noted that atoms in, for example, a rotating molecule, are displaced towards the rotation axis. Rotational oscillations of molecules, such as found in crystalline benzene near its melting point, will cause an apparent displacement of atomic positions from their true positions because the best fit to the electron density should be curvilinear but, with the limitations of present-day techniques, is generally linear (Figure 13.15). If the root-mean-square amplitude of libration about an axis is o> (in radians), then the apparent (but not real) shortening of the bond, d, is ... 

If the mean-square amplitude of libration has been determined from a TLS analysis of the anisotropic displacement parameters, the force constant (for a harmonic oscillator) is... 

Emily Maverick and Jack Dunitz used the anisotropic displacement parameters from crystal structure analyses of metallocenes at various temperatures to estimate the barriers for the rotation of an individual C5H5 ring in the crystal about its five fold axis. The average mean-square amplitude of libration ( ) is is found to be 28° at 101 K, corresponding to an energy barrier of approximately 2 kcal moE (9.3 kJ... 

The IR spectra of ice and liquid water thus clearly show that the main difference between the H-bond networks of liquid water and ice is an appreciable increase, in liqnid water, of the amplitude of librations. Their especially important role has also been recently emphasized in analyses of femtosecond IR spectra (24). Let us note that this conclnsion does not contradict the usual simplified picture that H-bonds in liquid water are weaker than H-bonds in ice, as great rotational amplitudes of individual H2O molecules imply weakening of the H-bonds. It is more precise, as it points to the mechanism at the origin of this weakening, and will consequently allow a more precise description of the H-bond network of liquid water. It implies that the strengths of H-bonds in liquid water extend over a wide range and rapidly vary with time, at the frequency of librations. 

The thickness of the ice shell is still not clear. One method to estimate this value is to observe rotation variations (librations) of Europa. Calculations show, that the presence of an ocean increases the amplitude of libration by about 10%, depending mainly on the thickness of the icy shell are to be expected (van Hoolst et al., 2008 [345]). 

The r1L and 7js components of the spectra obtained are shown in Fig. 33. The vij of the T1L component is 114 kHz and is larger by 5 kHz than that of PBLG-fd2 at 19°C. This suggests that the amplitude of the libration in the stacked state is smaller than that in the free state. The Fq value of the 7js component is 109 kHz and is similar to that of PBLG-fd2. This is compatible with the assumption that the 7js component corresponds to the side chains in the free state. The fraction of the 7jL component is estimated to be about 60% from the integrated intensity of the spectrum at 19°C, which is comparable with the value estimated from T. ... 

The amplitude and correlation time tc of the libration in the stacked state were estimated by using the diffusion in a cone model. The semiangle of the cone 6c at 20° C were obtained to be 22° and 26° for the TiL component and PBG- d2 and PBLG- d2, respectively. The tc values were obtained to be about 10 12 s at 19°C in both states. The 6C of r1L component is 4° smaller than that of PBLG- d2, showing that the amplitude of the libration in the stacked state is smaller than that in the free state due to the steric hindrance between the adjacent side chains forming the stacking... 

The riL value of racemic PBG-ytL is several hundred ms and decreases gradually as temperature increases. These results show that the side chains participating in the stacking structure are not rigid. Tx at the y position below room temperature is governed by the rapid and small-amplitude libration as shown in Section 4.3. The libration is expected to exist in the stacked state as well as the free state. riL is considered to be governed by libration in the stacked state. In order to estimate the amplitude of the libration in the stacked state, we extracted the Tu component of the spectrum in the same way as that used for racemic PBG- d2. The spectra of the T1L and T, s components are shown in Fig. 37. The vq value of the r1L component is... 

Figure 7.8. Mean square amplitude of methyl group libration in CD3N02 as a function of temperature. The solid line corresponds to (2) for harmonic oscillator with frequency 42.7 cm-1, calculated from (2.82). (From Trevino et al. [1980].)...

In the exciton-photon interaction, the translational molecular motions have negligible effects owing to the small amplitude of the translation compared to the optical wavelength. In contrast, the molecular rotations may cause an important variation of the transition dipole the librations may be strongly coupled to the incident photon via its coupling to the exciton. If DX(R) is the transition dipole of an a molecule in a unit cell, the first-order expansion in the libration coordinate 8 around the u axis will give... 

We must remark that the amplitude of these processes is generally weak compared to the direct exciton-photon amplitude, owing to the small libration amplitudes (of the order of 1 °) at low temperatures. It is still smaller when the incident light polarization is parallel to the molecular transition dipole. For instance, in anthracene-crystal excitation, we expect the exciton-photon-phonon contribution to be more important for the a than for the b polarization. On the contrary, these processes become much more important in nonresonant excitations, in Raman scattering for instance (cf. Section II.D). 

The bottom H2Os librate in a very shallow energy well with an amplitude of over 100°. This is because the dipole-dipole interactions go from positive to slightly negative on rotation through 90°. At the same time, the attractive interactions become less negative. 

The anisotropic rigid-body displacements of a molecule can be described in terms of translation (T, vibration along a straight-line path), libration (L, vibration along an arc), and screw (S, a combination of vibration and translation that may be regarded as vibration along a helical path). The mean-square amplitude of translational vibration is usually referred to as a system of Cartesian coordinates and unit vectors. S is the mean correlation between libration about an axis and translation parallel to this axis. Each of these three components can be expressed as... 

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