Symmetry of vibrational modes

The symmetry of any property of a molecule, ineluding a mode of vibration, can be determined by seeing how it behaves when operated on by the various symmetry elements that make up the overall symmetry point group of the molecule (Section 2.3.1). This behavior can be described in terms of the characters, Xv( ) for a 

To obtain a simple description of atomic motions (so-called symmetry coordinates), it is essential to treat symmetry-related atoms together, and it is helpful to remember that the number of stretching modes is equal to the number of bonds. In SiClaHa, therefore, there are two Si-H and two Si-Cl stretches. For example, the stretching vibration of the two Si—H bonds of SiCl2H2 in phase is symmetric with respect to the identity ( ), C2 and both Oy operations, so its representation is 

Symmetry representations of (a) the stretching, (b) the scissors deformations and (c) the wag, rock and twist motions of SiCFHz. 

The character table shows that the z axis also has aj symmetry, so each of these aj vibrations will involve a dipole change along z, and will be IR-active (i.e. give rise to a fundamental band in the IR spectrum). The character table also shows that there are components a x, Uyy and of the polarizability tensor that have ai 

Finally, local symmetry can be used to analyze the vibrations of quite complex molecules. This is illustrated in the on-line supplement to Chapter 8. 

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Both show a greater number of vibrational modes than predicted for pure Tj symmetry, but show good correspondence with the expected distribution for a distorted D2d cube (24,43). This finding Is In agreement with the x-ray crystallographic results which Indicate that the 4Fe-4S clusters In these two proteins, as well as In a number of model compounds, have a compressed tetragonal structure with 4 short and 8 long Fe-S bonds (41,45). 

The study of dynamics of a real polymer chain of finite length and containing some conformational defects represents a very difficult task. Due to the lack of symmetry and selection mles, the number of vibrational modes is enormous. In this case, instead of calculating the frequency of each mode, it is more convenient to determine the density of vibrational modes, that is, the number of frequencies that occur in a given spectral interval. The density diagram matches, apart from an intensity factor, the experimental spectmm. Conformational defects can produce resonance frequencies when the proper frequency of the defect is resonating with those of the perfect lattice (the ideal chain), or quasi-localized frequencies when the vibrational mode of the defect cannot be transmitted by the lattice. The number and distribution of the defects may be such... 

In order to apply group-theoretical descriptions of symmetry, it is necessary to determine what restrictions the symmetry of an atom or molecule imposes on its physical properties. For example, how are the symmetries of normal modes of vibration of a molecule related to, and derivable from, the full molecular symmetry How are the shapes of electronic wave functions of atoms and molecules related to, and derivable from, the symmetry of the nuclear framework ... 

Problem 10-14. If a molecule has a center of symmetry, no vibrational mode can be active in both the infrared and Raman. Verify this for the groups D2h and >oo/d by considering all possible symmetry species of the normal modes. Can you see why the statement will always be true Can you prove it ... 

These group species symbols correspond to the symmetry of vibrational normal modes of the same sort. 

Fig. 3. Excitation of vibrational modes due to different reaction channels. Concerted double proton transfer leads to a symmetric stretching vibration and symmetry breaking single proton transfer to an antisymmetric bending motion. Damping of the vibrational motion by internal vibrational redistribution is indicated by IVR .

The number of vibrational modes of a molecule composed of N atoms is 3N — 6 (or 3N — 5 if linear). We may find which of these are infrared and Raman active by the application of a few simple symmetry arguments. First, infrared energy is absorbed for certain changes in the vibrational energy levels of a molecule. For a vibration to be infrared active, there must be a change in the dipole moment vector... 

In carrying out the procedure for a tetrahedral species, it is convenient to let four vectors on the central atom represent the hybrid orbitals we wish to construct (Fig. 3.26). Derivation of the reducible representation for these vectors involves performing on them, in turn, one symmetry operation from each class in the Td point group. As in the analysis of vibrational modes presented earlier, only those vectors that do not move will contribute to the representation. Thus we can determine the character for each symmetry operation we apply by simply counting the number of vectors that remain stationary. The result for AB4 is the reducible representation, I",. 

We have followed a phenomenological approach and used the cluster model [18]. In this model the eg-type distortion interacts more strongly with the electronic state of an octahedral coordinated Cr3+ ion than the distortions of t2g symmetry. According to Ham [19], we assume that the continuum of vibrational modes with eg character can be approximated by a single mode with an effective frequency o>, mass /r and coupling constant V. The collective coordinates of the eg mode are conventionally known as Qe x2 — y2) and Qs ( 3z2 — r2). The linear Jahn-Teller Hamiltonian in equation (1) for the X state is [18] ... 

In order to evaluate quantitatively the orientation of vibrational modes from the dichroic ratio in molecular films, we assume a uniaxial distribution of transition dipole moments in respect to the surface normal, (z-axis in Figure 1). This assumption is reasonable for a crystalline-like, regularly ordered monolayer assembly. An alternative, although more complex model is to assume uniaxial symmetry of transition dipole moments about the molecular axis, which itself is tilted (and uniaxially symmetric) with respect to the z-axis. As monolayers become more liquid-like, this may become a progressively more valid model (8,9). We define < > as the angle between the transition dipole moment M and the surface normal (note that 0° electric field of the evenescent wave (2,10), in the ATR experiment are given by equations 3-5 (8). 

Vibrational absorption spectroscopy occurs in the infrared spectrum. The location of such an absorption will be directly linked to the frequency of vibrational modes. The symmetry of the modes of vibration will also affect the absorption and this is evident upon inspection of equation (5.1). These symmetries are contained in the forms of the wave function themselves, which are symmetric or antisymmetric with respect to planes within a molecule. Upon inspection of the integrand of equation (5.1), it is apparent that certain products of the wave functions with the dipole function can lead to an asymmetric result, causing the integral to vanish. Such transitions are forbidden and will not appear in the infrared spectrum. 

Finally, it is useful to mention here a systematic way to derive the symmetries of the vibrational modes for a linear molecule. Even though formally carbon suboxide has T>ooh symmetry, its vibrational modes may be derived by using the l)2u character table. The procedure is illustrated below for the carbon suboxide molecule ... 

Monodentate attachment to a metal ion lowers the symmetry of perchlorate to C3v and bidentate attachment to C2v (15-17). Consequently the number of vibrational modes should increase (Table I). In addition, a metal-oxygen stretching frequency would also be expected in the far-IR region and has been located in the range 360-290 cm-1 (18). These effects resulting on coordination, particularly the increase in the number of vibrational modes, may be used for identifying coordination of perchlorate. 

Figure 6-40. The symmetry-reducing vibrational mode of eg symmetry for an octahedron.

The indole molecule is a planar asymmetric rotor with symmetry and 29 planar and 13 nonplanar fundamentals. Selected fundamental vibrations of indole in the gas and liquid phase have been re-examined < 1995SAA1291 >. The IR overtone/combination region from 1600 to 2000 cm was used to establish the wave numbers of nonplanar, hydrogenic-wagging modes for which the active IR and Raman fundamental is weak. A complete assignment of vibrational modes for indole by application of DFT and a hybrid Hartree-Fock/ DFT method has been provided <1996J(P2)2653>. 

Analysis of the symmetry of the chemical species under study (i.e. the point group for a free molecule, the space and factor groups for a crystal [44]), according to the site symmetry of every atom, allows the determination of the irreducible representation of the total modes and, after the subtraction of the translational and rotational modes (the acoustic modes for the crystals), the irreducible representation of the vibrational (or optical ) modes can be obtained. This means that the number of vibrational modes belonging to the symmetry species associated with the molecular or crystal symmetry can be counted. Consequently, the number of active modes can be counted, according to the symmetry selection rules of the different techniques (in particular IR and Raman). 

Examination of the columns on the far right in the character table shows that translation along thex, y, and z directions is A + B + B2 (translation is motion along the X, y, and z directions, so it transforms in the same way as the three axes) and that rotation in the three directions R, Ry, R ) is Aj + Bi + B2- Subtracting these from the total above leaves 2Aj -t- B, the three vibrational modes, as shown in Table 4-10. The number of vibrational modes equals 3N — 6, as described earlier. Two of the modes are totally symmetric (Ai) and do not change the symmetry of the molecule, but one is antisymmetric to C2 rotation and to reflection perpendicular to the plane of the molecule (fii). These modes are illustrated as symmetric stretch, symmetric bend, and antisymmetric stretch in Table 4-11. 

The usual JT effect in 3D point groups exemplifies the second case, Tyr C Tnm- These cases involve molecules that are more involved than the simplexes, which implies that the site which is stabilized by a maximal subgroup contains more nuclei than the one that is considered in the simplex. As a result all the possible JT interaction symmetries are represented at least by one normal mode, but in addition the space of vibrational modes also contains inactive modes. A case in point are centrosymmetric molecules where only gerade modes can be JT active, the odd modes are found in the remainder space Tnm — jt-... 

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