Collective vibrations

When contributions from the low frequency, collective vibrations of the solvent are included, eq. 11 becomes eq. 12. In eq. 

Contribution of dipolar dispersion laws to dephasing of high-frequency collective vibrations... 

Here we focus on the effect of dipolar dispersion laws for high-frequency collective vibrations on the shift and width of their spectral line, with surface molecules inclined at an arbitrary angle 6 to the surface-normal direction. For definiteness, we consider the case of a triangular lattice and the ferroelectric ordering of dipole moments inherent in this lattice type.56,109 Lateral interactions of dynamic dipole moments p = pe (e = (sin os, sin6fcin , cos )) corresponding to collective vibrations on a simple two-dimensional lattice of adsorbed molecules cause these vibrations to collectivize in accordance with the dispersion law 121... 

As a main point of this subsection,186 it is shown that due to sufficiently strong lateral interactions of adsorbed molecules on a two-dimensional triangular lattice, the spectral line for collective vibrations manifests some characteristic peculiar relationships between its dephasing-induced broadening and the resonance width 77 for the low-frequency mode The line width changes as Tj n(l/rf) for surface-normal and as rjm for surface-parallel molecular orientations, and takes nonzero values (independent of 77) for inclined molecules with the inclination angle ranging from 47° to 90°. 

The linear model can be extended to include more distant neighbours and to three dimensions. Let us consider an elastic lattice wave with wave vector q. The collective vibrational modes of the lattice are illustrated in Figure 8.6. The formation of small local deformations (strain) in the direction of the incoming wave gives rise to stresses in the same direction (upper part of Figure 8.6) but also perpendicular (lower part of Figure 8.6) to the incoming wave because of the elasticity of the material. The cohesive forces between the atoms then transport the deformation of the lattice to the... 

Figure 8.6 Schematic representation of transverse and longitudinal collective vibrational...

With the limitations above noted, let us return to the application of eqs 3-5 to IT transitions and to optical charge transfer (CT) in general. These equations, in fact, prove to be remarkably successful in providing a basis for understanding optical CT processes in a number of chemical systems. It was suggested above that the vibrational term, x> has both intramolecular and medium contributions. From dielectric loss and related measurements, the collective vibrations of the medium occur at low frequencies for most solvents, the energy spacings between levels are small, and equations based on the classical... 

In principle, solvent trapping is also included in the vibrational overlap intergral in equation (25). As noted above, the solvent dipole reorientational motions associated with solvent trapping can be treated as a series of collective vibrations of the medium using approaches devised for treating the collective vibrations of ions or atoms that occur in a crystalline lattice.32-33 However, the problem is mathematically intractable because of the many solvent molecules involved which lead to many normal modes and the existence of a near continuum of energy levels. In addition,... 

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. 

Diatomic molecules are the simplest condensed phase VER systems, for example, a dilute solution of a diatomic such as I2 or XeF in an atomic (e.g., Ar or Xe) liquid or crystal. Other simple systems include neat diatomic liquids or crystals, or a diatomic molecule bound to a surface. VER of a diatomic molecule can occur only by energy transfer to the collective vibrations of the bath, i.e., the phonons. Ordinarily VER is a high-order multiphonon process. Consequently there is an enormous variability in VER lifetimes, which may range from 56 s [liquid N2 (20)] to 1 ps [e.g., XeF in Ar (21)], and a high level of sensitivity to environment. Diatomic molecules have simple structures but complex VER mechanisms. 

Collective Vibrational Dynamics of Simple Liquids Michael Buchner and Thomas Dorfmiiller... 

Coherent Inelastic Scattering.—Inelastic neutron collisions with the solid can excite phonon modes (collective vibrations) and if the coherently scattered component can be detected variation with direction within the solid, i.e. the phonon dispersion curve, can be determined. This technique is well established for bulk solids and has been used recently to examine the properties of small particles (carbon black). 

Ex situ IR data are collected on dried, diluted powder films in a low vacuum enviromnent or one purged with a dry gas such as N2. Attenuated total reflectance (ATR)-IR spectroscopy provides surface-sensitive IR measurements and can be used for in situ studies of sorption phenomena. Raman spectroscopy is a related vibrational spectroscopy that provides complimentary information to IR. It can also be used to collect vibrational spectra of aqueous samples. Typical data reduction for vibrational spectra involves subtraction of a background spectmm collected under identical conditions from the raw, averaged sample spectrum. Data analysis usually consists of an examination of changes in peak position and shape and peak fitting (Smith, 1996). These and other spectral parameters are tracked as a function of maaoscopic variables such as pH, adsorption density, and ionic strength. 

The Hamiltonian of intramolecular collective vibrations of the quasi-one-dimensional system of coupled C=C bonds can be written as... 

In the case of the coherent vibronic state excitations, we shall take into account a change of the frequency of the collective vibrations in the polyene chain caused by the exciton excitation ... 

If we neglect the exciton-vibration interaction (i.e., the operator Hi), the wave functions of collective vibrations and of coherent excitonic states of the retinal molecule (they are specified by quasi-momentum Hq and are the proper functions of the Hamiltonian Hoq) form an orthonormal basis... 

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