Antisymmetric modes

There are at least two ways in which detailed information about electron-vibrational coupling strengths can be obtained for mixed-valence complexes. Both are based on the fact that such coupling will be reflected in modifications of the vibrational spectrum. Thus, for example, coupling to antisymmetric modes in a symmetric ion will modify intensities and frequencies of the modes involved. 

I have predicted that the very unusual low-frequency IR behavior for the Creutz-Taube ion calculated by Piepho, Schatz and Krausz [Piepho, S. B. Krausz, E. R. Schatz, P. N. J. Am. Chem. Soc. 1978, 100, 2996] on the assumption of only antisymmetric mode involvement in electron-vibrational interaction would not be found, and that it was an artifact of the method. The failure of experiments designed to locate such IR bands has subsequently been reported by Krausz, et al. 

Figure 10.14 Upper two pairs of CO groups centrically related both within each pair and between the pairs. denotes center of inversion. Lower symmetric and antisymmetric modes of vibration within each pair.

This way of expressing the overall modes for the pair of molecular units is only approximate, and it assumes that intramolecular coupling exceeds in-termolecular coupling. The frequency difference between the two antisymmetric modes arising in the pair of molecules jointly will depend on both the intra- and intermolecular interaction force constants. Obviously the algebraic details are a bit complicated, but the idea of intermolecular coupling subject to the symmetry restrictions based on the symmetry of the entire unit cell is a simple and powerful one. It is this symmetry-restricted intermolecular correlation of the molecular vibrational modes which causes the correlation field splittings. 

When definitely stabilized, the (EGDA)-Mo(VI) complex shows a doublet at 955-910 cm-1 which may be attributed to symmetric (Mo=0) the antisymmetric mode was not detected. The appearance of the symmetric Stretching-mode is consistent with a cis disposition of two terminal Mo=0 bonds (ref. 8). 

There are certain aspects of performance that make the Apm oscillators potentially attractive as chemical sensors. First of all, the fact that both surfaces contribute to the signal means that the sensitivity is higher than for the corresponding SAW device. The most important advantage follows from the fact that velocity of the lowest order of the antisymmetric mode is much slower than the compressional velocity of sound in most liquids (900-1,500 m s-1), which means that the energy... 

Suppose now that both types of vibrations are involved in transition. The symmetric modes shorten the effective tunneling distance to 2Q whereas the antisymmetric modes create the Franck-Condon factor in which the displacement 2Q0 now is to be replaced by the shorter tunneling distance 2Q, [Benderskii et al., 1991a] ... 

Contrary to the weak cage response, the guests properties experience considerable changes. In the encapsulated CH4 molecule, the C-H bonds are 0.014 A shorter than in the free molecule (Table 1). This entails a considerable increase in the C-H stretching frequencies a and t (by 140-150 cm-1) and a small increase in the deformation frequencies e and l (by 10-12 cm-1) (Table 2). For the stretching frequencies, similar shift values (138 cm-1 for symmetrical and 152 cm-1 for antisymmetrical modes) were calculated for the CH4 C60 cluster in Ref. [39] within the MP2 approximation. 

Figure 4.15 Illustration of the constructive and destructive interference pattern in a Mach-Zehnder waveguide modulator. Only if the outgoing waveguide is single mode, the resulting antisymmetric mode will be completely radiated into the substrate. A multimode waveguide cannot be used for an efficient Mach-Zehnder modulator...

The six normal modes for thiophosgene (Fig. 2.6) show the decomposition of the six normal modes into the three in-plane totally symmetric modes, 3A1 the one out-of-plane mode, Bi, and the two in-plane, antisymmetric modes, 2B2 ... 

Only the ten modes that are symmetric with respect to the symmetry plane of the toluene molecule can be CT-active. The five antisymmetric modes represent... 

Consider first the molecular adsorption structure. In this case, the essentially antisymmetric modes y = (2, 148) (notice the symmetry breaking due to the oxygen vacancy on the surface) do not participate in charge displacement between water and rutile (see Figs. 31 A and 32 A), so that the CT-reactivity information is basically limited to the four reactive IRM which originate from the two components on each reactant. However, in the transition-state struc-... 

Figure 3.39 (page 114) shows the phase velocities of the waves as a function of the product k4, where k, = 27t/A, A, is the wavelength of the bulk transverse (shear) wave in the medium of which the plate is made, and d is the plate thickness. The waves divide naturally into two sets symmetric waves (denoted by So, S],. ..) whose particle displacements are symmetric about the neutral plane of the plate, and antisymmetric waves (Aq, A, . ..), whose displacements have odd symmetry about the neutral plane. Figure 3.38 shows that for sufficiently thin plates (M < 1-6), only two waves exist — the lowest-order symmetric mode (Sq) and the lowest-order antisymmetric mode (Aq). These are the modes shown earlier in Figure 2.0d. The plate mode that we will emphasize here is the Ao mode, in which the elements of the plate undergo flexure as the wave propagates. The shape of a plate during propagation of this flexural mode has been likened to that of a flag waving in the wind.

Figure 3.40 Calculated phase velocity of flexural plate waves vs ratio of plate thickness, d, to wavelength. A, for silicon nitride. Material is assumed to have the elastic properties of Si3N4 and no residual tension. The mode shapes ate illustrated at the right with a greatly enlarged vertical scale for clarity. Ellipses at left show the retrograde elliptical particle motions of the lowest S3rmmetric and antisymmetric modes for d/A = 0.03. (Reprinted with pemtission. See Ref. (621.)...

In KHCOs, coupling of the two protons of a dimer gives rise to symmetric and antisymmetric modes (see below) that can be distinguished with INS [Fil-laux 1988], Owing to the limited resolution in energy, splittings are not visible in Fig. 6 but further experiments confirm that the effective mass of 1 amu holds for each normal mode, either symmetric or antisymmetric. 

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