Force constant models for

The thirty-two silent modes of Coo have been studied by various techniques [7], the most fruitful being higher-order Raman and infra-red spectroscopy. Because of the molecular nature of solid Cqq, the higher-order spectra are relatively sharp. Thus overtone and combination modes can be resolved, and with the help of a force constant model for the vibrational modes, various observed molecular frequencies can be identified with specific vibrational modes. Using this strategy, the 32 silent intramolecular modes of Ceo have been determined [101, 102]. 

We can in fact fit the speed of sound and the zone-boundary mode to a simple nearest-neighbor force-constant model for the Weber system, just as we did in Table 9-2 for the tetrahedral solids. We obtain force constants... 

The Raman and infrared spectra for C70 are much more complicated than for Cfio because of the lower symmetry and the large number of Raman-active modes (53) and infrared active modes (31) out of a total of 122 possible vibrational mode frequencies. Nevertheless, well-resolved infrared spectra [88, 103] and Raman spectra have been observed [95, 103, 104]. Using polarization studies and a force constant model calculation [103, 105], an attempt has been made to assign mode symmetries to all the intramolecular modes. Making use of a force constant model based on Ceo and a small perturbation to account for the weakening of the force constants for the belt atoms around the equator, reasonable consistency between the model calculation and the experimentally determined lattice modes [103, 105] has been achieved. 

Table I summarizes some of the results of the dynamical calculations for adsorbed butane. The calculated surface vibratory mode frequencies are in reasonable agreement with the observed spectrum, lying in the range 50-125 cm"1 with the rocking mode about the chain axis having the highest frequency followed by the closely spaced bouncing and orthogonal rocking modes. Although there is some variation depending on the force-constant model used, the calculated frequencies are within 30 cm of the experimental values.

This is undertaken by two procedures first, empirical methods, in which variable parameters are adjusted, generally via a least squares fitting procedure to observed crystal properties. The latter must include the crystal structure (and the procedure of fitting to the structure has normally been achieved by minimizing the calculated forces acting on the atoms at their observed positions in the unit cell). Elastic constants should, where available, be included and dielectric properties are required to parameterize the shell model constants. Phonon dispersion curves provide valuable information on interatomic forces and force constant models (in which the variable parameters are first and second derivatives of the potential) are commonly fitted to lattice dynamical data. This has been less common in the fitting of parameters in potential models, which are onr present concern as they are required for subsequent use in simulations. However, empirically derived potential models should always be tested against phonon dispersion curves when the latter are available. 

From the simulations, we conclude that two hydrogen bonding force constants are a basic requirement for reproducing the measured spectrum. If a water-water potential generates sufficiently large force constant differences for the different proton configurations (or the different relative dipole-dipole orientations in water or ice), it should produce the same effect as seen in the LR model. The anisotropic properties of the classic potentials are a result of charge interaction and this anisotropy should increase in the polarisable potentials and hence they produce a broad optic peak. This broad peak indicates that the orientational variation of the potential function has been increased considerably but it may still be less than the critical value of 1.5 as we indicated in the section 6.1. One would, therefore, expect that a better polarisable potential would, eventually, be able to reproduce the split optic peaks in the measured INS spectrum. 

The vibration frequencies were calculated ab initio for a model structure comprising a bare metal cation interacting with the fliran ring. For both the experimental and calculated spectra the interaction of furan with the zeolite caused the out-of-plane vibrational modes to move to higher wave numbers, see Fig. 7.28. This result can be explained if one assumes an interaction, oriented perpendicular to the molecular plane of furan, between the zeolite cation and the 7t electron system of furan, which lead to an increase of the CH out-of-plane bending force constant. However, for the experimental spectra the wavenumbers of the out-of-plane modes then decrease in the series Li > Na > > Cs whereas the computed... 

In a critical evaluation of the results of anharmonic force constant calculations it is necessary to consider all the factors discussed above concerning the completeness, internal consistency, and accuracy of the data, the possible idiosyncrasies of the model potential used, the method of reducing the data, as well as the goodness of fit the force constants yield for the data points. It may be in order to mention that... 

Fig. 18. (a) PPP-SD-CI calculations of n-bond orders in the So (open circles) and T (closed circles) states for a model of all-/ra/is-P-carotene(docosaundecaene). The Sg-state (open circles) and the Tpstate (closed circles) stretching force constants (k) determined for (b) all-/wns-spheroidene in n-hexane solution and (c) 15-cis Spheroidene bound to the RC of a carotenoidless mutant Fb. sphaeroides R26 are also shown. Double circles indicate those force constants assumed for both the Sq and the T states. 

Figure 6. Dispersion relation for an isobaric crystal, where Mq = M. When /j = /2 for this force constant model, the diatomic crystal is equivalent to a monatomic crystal with lattice spacing a 2 or Brillouin zone boundary at 2ir/a. Hence the optical branch (solid line) appears as the portion of the acoustic branch from via to 2ir/a (dashed line) folded back to the zone center from the real Brillouin zone boundary (dotted vertical line).

Figure 31. Surface phonon dispersion for Cu(lll). The open circles are from HAS experiments, and the open triangles are from EELS experiments. The surface modes shown as solid lines and bulk band boundaries are based on a simple force constant model. The X and Y designations indicate the polarizations of the corresponding modes as identified in the reduced zone diagram in the inset. (Reproduced from Fig. 3 in Ref. 99, with permission.)...

Figure 32. Surface phonon dispersion for Nb(OOl). The data are the solid points which were taken at 900 K. Panels a and b correspond to slab dynamics calculations with two different force constant models the calculation in panel b uses the force constants from the bulk phonon fits. The solid lines represent the surface phonons and resonances polarized mainly longitudinally (or parallel), the lines with long dashes represent phonons polarized mainly perpendicularly, and those with short dashes are shear horizontal. (Reproduced from Fig. 6 of Ref. 107, with permission.)...

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