The Normal Modes of Vibration

Determine the number of vibrational degrees of freedom for the following molecules. 

The solutions above hint that symmetry has a great deal to do with the number of truly unique vibrational degrees of freedom. Consider benzene, which is planar and has Dgj, symmetry. Because of its symmetry, certain vibrations of benzene are identical to each other and have the same vibrational frequency. This means that there will be fewer than 30 unique vibrational frequencies in this molecule. (There are in fact only 20 unique frequencies.) Symmetry will have similar ramifications for vibrations in other molecules, too. 

The vibrations of all molecules can be described in terms of independent motions such that for each motion the frequency of vibration for all atoms is the same. These are the normal modes of vibration. Why are the normal modes so important There are several reasons, but perhaps the most important one is this To a good approximation, the frequencies of light that are absorbed due to vibrational motions of atoms in molecules are those that have the same frequencies as the normal modes of vibration. 

A single vibration agrees with the 3N — 5 expression for the number of vibrations in a molecule For HCl, 3N — 5 is 3(2) — 5, which equals 1. The two atoms in HCl can be thought of as a classical harmonic oscillator composed of two masses (the two atoms) connected by a spring (the chemical bond). This harmonic 

FIGURE 14.23 The single vibration of HCl has the hydrogen and chlorine atoms moving alternately back and forth. The hydrogens movement is much larger than the chlorine s, because of H s much lower mass. 

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Until 1962 the infrared and Raman spectra of thiazole in the liquid state were described by some authors (173, pp. 194-200) with only fragmentary assignments. At that date Chouteau et al. (201) published the first tentative interpretation of the whole infrared spectrum between 4000 and 650 cm for thiazole and some alkyl and haloderivatlves. They proposed a complete assignment of the normal modes of vibration of the molecule. 

The study of the infrared spectrum of thiazole under various physical states (solid, liquid, vapor, in solution) by Sbrana et al. (202) and a similar study, extended to isotopically labeled molecules, by Davidovics et al. (203, 204), gave the symmetry properties of the main vibrations of the thiazole molecule. More recently, the calculation of the normal modes of vibration of the molecule defined a force field for it and confirmed quantitatively the preceeding assignments (205, 206). 

In this paper, we review progress in the experimental detection and theoretical modeling of the normal modes of vibration of carbon nanotubes. Insofar as the theoretical calculations are concerned, a carbon nanotube is assumed to be an infinitely long cylinder with a mono-layer of hexagonally ordered carbon atoms in the tube wall. A carbon nanotube is, therefore, a one-dimensional system in which the cyclic boundary condition around the tube wall, as well as the periodic structure along the tube axis, determine the degeneracies and symmetry classes of the one-dimensional vibrational branches [1-3] and the electronic energy bands[4-12]. 

In order to find the normal modes of vibration, I am going to write the above equations in matrix form, and then find the eigenvalues and eigenvectors of a certain matrix. In matrix form, we write... 

This is clearly a matrix eigenvalue problem the eigenvalues determine tJie vibrational frequencies and the eigenvectors are the normal modes of vibration. Typical output is shown in Figure 14.10, with the mass-weighted normal coordinates expressed as Unear combinations of mass-weighted Cartesian displacements making up the bottom six Unes. 

In many of the normal modes of vibration of a molecule the main participants in the vibration will be two atoms held together by a chemical bond. These vibrations have frequencies which depend primarily on the masses of the two vibrating atoms and on the force constant of the bond between them. The frequencies are also slightly affected by other atoms attached to the two atoms concerned. These vibrational modes are characteristic of the groups in the molecule and are useful in the identification of a compound, particularly in establishing the structure of an unknown substance. 

In general, the first excited state (i.e. the final state for a fundamental transition) is described by a wavefunction pt which has the same symmetry as the normal coordinate (Appendix). The normal coordinate is a mathematical description of the normal mode of vibration. 

Assuming that a reasonable force field is known, the solution of the above equations to obtain the vibrational frequencies of water is not difficult However, in more complicated molecules it becomes very rapidly a formidable one. If there are N atoms in the molecule, there are 3N total degrees of freedom and 3N-6 for the vibrational frequencies. The molecular symmetry can often aid in simplifying the calculations, although in large molecules there may be no true symmetry. In some cases the notion of local symmetry can be introduced to simplify the calculation of vibrational frequencies and the corresponding forms of the normal modes of vibration. 

It is apparent from Fig. 4 that the normal modes of vibration of the water molecule, as calculated from the eigenvectors, can be described approximately as a symmetrical stretching vibration (Mj) and a symmetrical bending vibration... 

This condition on the so-called secular determinant is the basis of the vibrational problem. The roots of Eq. (59), X, are the eigenvalues of the matrix product GF, while the columns of L, the eigenvectors, determine the forms of the normal modes of vibration. These relatively abstract relations become more evident with the consideration of an example. 

The approach of the carbon atom to ethylene, and the conversion of 30 to 31, thus correspond to one of the normal modes of vibration of the cyclopropane ring, viz ... 

By the symmetry of a normal mode of vibration, we mean tbe symmetry of the nuclear framework under the distortion introduced by the vibration. Pictorially, the symmetry of the normal mode is equal to the symmetry of the pattern of arrows drawn to indicate the directions of the nuclear displacements. The normal modes of vibration of water are the symmetric and antisymmetric stretches, and the angle bend, shown in Figure 6-1. 

Since any general displacement is a superposition of translational, rotational and vibrational displacements it is possible to redescribe the motion of the molecule in terms of overall translations and rotations, and the normal modes of vibration. Using the character tables, it is possible to decompose the symmetry of the general displacement into the symmetries of the different types of motion. 

Subtracting the symmetries of the translational and rotational motions leaves the symmetries of the normal modes of vibration. For C2v, the symmetries of... 

Problem 7-13. Determine the symmetry species of the normal modes of vibration of the cyclopropilium cation, CsH. This molecule is planar and has Dsh symmetry. 

The normal mode calculation was used to elucidate the rotational isomerization equilibrium of the [C4CiIm]X liquids. In the wave number region near 800-500 cm, where ring deformation bands are expected, two Raman bands appeared at —730 cm and —625 cm in the [C4C4lm]Cl Crystal (1). In the [C4CiIm]Br these bands were not found. Here instead, another couple of bands appeared at —701 and —603 cm T To assist the interpretation of the spectra, the normal modes of vibrations calculated by Hamaguchi et al. [50] are shown in Figure 12.8. 

Figure 2.8 The geometty of pyrrole together with the displacement vectors of the normal modes of vibration appearing in the spectra of Figure 7. Reproduced with permission from Ref. [87]. Copyright (2011) AIP Publishing LLC.

The normal modes of vibration u are obtained as solutions of the Hessian eigenvalue problem,... 

Let us now look at the normal modes of vibration of a molecule which is as simple as possible and yet exemplifies all general features ordinarily encountered. The planar ion will serve for this purpose. As a nonlinear four-atomic species, it must have 3(4) -6 = 6 normal modes. In Figure 10.1 we have depicted these vibrations. In each drawing the length of an arrow relative to the length of another arrow in the same drawing shows how much the atom to which it is attached is displaced at any instant relative to the simultaneous displacement of the atom to which the other arrow is attached. 

The two characteristic features of normal modes of vibration that have been stated and discussed above lead directly to a simple and straightforward method of determining how many of the normal modes of vibration of any molecule will belong to each of the irreducible representations of the point group of the molecule. This information may be obtained entirely from knowledge of the molecular symmetry and does not require any knowledge, or by itself provide any information, concerning the frequencies or detailed forms of the normal modes. 

Figure 25 (a) Copper(II) stereochemical structural pathways and (b) the normal modes of vibration of an elongated tetragonal octahedral complex of copper(II), e g. [Cu(NH3)4(SCN)2]... 

Shimanouchi et al.105) also derive a set of force constants from the Raman LA mode data by analysing the normal modes of vibration of the alkane chains. These constants are then used to predict the chain modulus using a simple model for the planar zigzag chain. 

Example 9.1-1 Classify the normal modes of vibration of the carbonate ion C03 according to the IRs for which the normal coordinates form bases. 

Table 9.2. Derivation of the symmetry of the normal modes of vibration for the MLg molecule or complex ion with Oh symmetry. 

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