Calculation of the Vibrational Factor

Most spectroscopists are comfortable using RKR and Franck-Condon programs as black boxes. Since these programs are widely available, inexpensive to run, and generate more accurate vibrational factors than semiclassical or model potential approaches, the application-minded reader is advised to skip Sections 5.1.1 and 5.1.2. 

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The intensities of the infrared absorptions and of the inelastic scattered light (Raman) are determined by such electrical factors as dipole moments and polarizabilities. At the time of the pioneering studies on the infrared spectra of carbohydrates by the Birmingham school,7"11 calculations of the vibrational frequencies had been performed only for simple molecules of fewer than ten atoms.27,34,35 However, many tables of group frequencies, based on empirical or semi-empirical correlations between spectra and molecular structure, are available.32,34"37... 

The calculation of the structure factor in equation (8) assumes all atoms to be at rest in the positions corresponding to perfect crystal symmetry. In fact, atoms and molecules perform thermal vibrations and sometimes are statically disordered, that is, shifted from ideal positions, differently in different unit cells. Both effects disturb the long-range periodicity of a crystal and thus make it a poorer diffractor. Smearing of the electron distribution of an atom due to thermal vibration effectively reduces its scattering factor fj, so that it drops faster with sin 9IX than the corresponding curve for a stationary atom (/o), as shown in Figure 7. The decrease is described by an exponential temperature, or thermal, factor Q ... 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

C) The error in AE" /AEq is 0.1 kcal/mol. Corrections from vibrations, rotations and translation are clearly necessary. Explicit calculation of the partition functions for anharmonic vibrations and internal rotations may be considered. However, at this point other factors also become important for the activation energy. These include for example ... 

The isotope effects of reactions of HD + ions with He, Ne, Ar, and Kr over an energy range from 3 to 20 e.v. are discussed. The results are interpreted in terms of a stripping model for ion-molecule reactions. The technique of wave vector analysis, which has been successful in nuclear stripping reactions, is used. The method is primarily classical, but it incorporates the vibrational and rotational properties of molecule-ions which may be important. Preliminary calculations indicate that this model is relatively insensitive to the vibrational factors of the molecule-ion but depends strongly on rotational parameters. 

The first-principles calculation of NIS spectra has several important aspects. First of all, they greatly assist the assignment of NIS spectra. Secondly, the elucidation of the vibrational frequencies and normal mode compositions by means of quantum chemical calculations allows for the interpretation of the observed NIS patterns in terms of geometric and electronic structure and consequently provide a means of critically testing proposals for species of unknown structure. The first-principles calculation also provides an unambiguous way to perform consistent quantitative parameterization of experimental NIS data. Finally, there is another methodological aspect concerning the accuracy of the quantum chemically calculated force fields. Such calculations typically use only the experimental frequencies as reference values. However, apart from the frequencies, NIS probes the shapes of the normal modes for which the iron composition factors are a direct quantitative measure. Thus, by comparison with experimental data, one can assess the quality of the calculated normal mode compositions. 

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