Frequency internal mode

It should be possible to estimate r, the lifetime of the low-frequency Rh-C stretch using a2 and Am in Equation (5). The values for a2 = 1.2 THz and Am = 20 cm-1 yield a value for r of 0.75 ps. For a low-frequency mode that has a number of lower-frequency internal modes and the continuum of solvent modes to relax into, 0.75 ps is not an unreasonable value for the lifetime. A measurement of the Rh-C stretching mode lifetime would provide the necessary information to determine if the proposed dephasing mechanism is valid. In principle, the same mechanism will produce a temperature-dependent absorption line shift. However, other factors, particularly the change in the solvent density with temperature strongly influence the line position. Therefore, temperature-dependent line shifts cannot be used to test the proposed model. 

As follows from the recently performed analyses [3] the shape of the v (AH) IR absorption bands results usually from the indirect relaxation of the excited vibrational levels, i.e. through the coupling with the low frequency internal modes which next interact with phonons of the condensed phase. The broad bands are characteristic not only for condensed phases. They are also observed in the gas phase in the case of bulky expanded molecules, which can be treated as a bath. In this case a crucial is the coupling of v (A-H) vibrations with low frequency bridge vibrations. 

All P.M.R. spectra were measured with a Varian HA 100 spectrometer operating in the frequency-sweep mode with tetramethylsilane as the reference for the internal lock. The double and triple resonance experiments were performed using a Hewlett Packard 200 CD audio-oscillator and a modified Hewlett Packard 200 AB audio-oscillator (vide infra). Spectra were measured using whichever sweep width was required to ensure adequate resolution of the multiplets under investigation, generally 250 or 100 Hz, and sweep rates were selected as necessary. Extensive use was made of the Difference 1 and Difference 2 calibration modes of the instrument, both for the decoupling experiments and for the calibration of normal spectra. 

The presence of isotopic impurities causes clear effects in the vibrational spectra. Almost all modes studied so far show frequency shifts on S/ S substitution [81, 107]. The average shift of the internal modes is ca. 0.6 cm , and of the external modes it is 0.1-0.3 cm (Tables 3, 4 and 5). Furthermore, the isotopomers which are statistically distributed in crystals of natural composition can act as additional scattering centers for the phonon propagation. Therefore, in such crystals the lifetime of the phonons is shortened in comparison with isotopically pure crystals and, as a conse-... 

In order to simplify the expression for G, one has to employ a sufficiently simple model for the vibrational modes of the system. In the present case, the solvent contribution to the rate constant is expressed by a single parameter E, the solvent reorganization energy. In addition, frequency changes between the initial and final states are neglected and it is assumed that only a single internal mode with frequency co and with the displacement Ar is contributing to G. Thus the expression for G reduces to [124] ... 

Obviously, in the case of PS these discrepancies are more and more reduced if the probed dimensions, characterized by 2ti/Q, are enlarged from microscopic to macroscopic scales. Using extremely high molecular masses the internal modes can also be studied by photon correlation spectroscopy [111,112], Corresponding measurements show that - at two orders of magnitude smaller Q-values than those tested with NSE - the line shape of the spectra is also well described by the dynamic structure factor of the Zimm model (see Table 1). The characteristic frequencies QZ(Q) also vary with Q3. Flowever, their absolute values are only 10-15% below the prediction. 

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

It is often useful to have an approximate relation for VPIE s, especially when complete vibrational analysis is impossible. The AB approximation serves that purpose, and sometimes gives more physical insight than do detailed, but very complicated calculations using Equation 5.24. It is based on the observation that ordinarily condensed phase vibrations fall in two groups the first containing the high frequencies, m > 1 (most often the internal modes, uj = hcvj/kT), where the zero point (low temperature) approximation is appropriate, and... 

The A defined in Equation 5.30 is not to be confused with the Helmholtz free energy. Should the A frequencies be limited to the external hindered translations and rotations, vi g = vi g = 0, and this is an additional simplification. In some molecules, however, there are isotope sensitive low lying internal modes (often internal rotations or skeletal bends). In that cases both terms in Equation 5.30 contribute. 

As the oscillators of the OPP model vibrate independently of each other, the frequencies are dispersionless, that is, independent of a wavevector q. For the internal modes of a molecular crystal, this tends to be a very good approximation. For the external modes, the dispersion can be pronounced, as shown in Figs. 2.1 and 2.2. In order to obtain the mean-square vibrational amplitudes for the latter, a summation over all phonon branches in the Brillouin zone must be performed. 

A considerable simplification is achieved when molecules can be treated as rigid bodies, as was done for naphthalene and anthracene (Fig. 2.2), the frequency spectra of which were derived using atom-atom potential functions. The mean-square displacements due to the internal modes can be calculated from the experimental infrared and Raman force constants, and added to the values obtained with Eq. (2.58). The rigid-body model for thermal vibrations is further discussed in section 2.3.3. 

In molecular crystals, the separation between internal and external modes is of importance. Except for torsional oscillations in some types of molecules, the internal modes have much higher frequencies than the external modes. According to expressions such as Eqs. (2.51) and (2.58), the latter are then the dominant... 

Biirgi generalized the model, assuming a temperature independent high frequency term accounting for displacements due to the internal modes. By means of multi-... 

The matrices g and A transform the normal modes with frequencies Vj and eigenvectors V into atomic displacements (T), ADPs are the 3 x 3 diagonal blocks of and is a temperature-independent term accounting for the high-frequency vibrations (internal modes). 

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