The Symmetry of Normal Vibrations

The complex, random, and seemingly aperiodic internal motions of a vibrating molecule are the result of the superposition of a number of relatively simple vibratory motions known as the normal vibrations or normal modes of vibration of the molecule. Each of these has its own fixed frequency. Naturally, then, when many of them are superposed, the resulting motion must also be periodic, but it may have a period so long as to be difficult to discern. 

The first question to be considered regarding the normal modes is that of their number in any given molecule. This, fortunately, is a very easy question, and doubtless many readers know the answer already. An atom has three degrees of motional freedom. It may move from an initial position in the x direction independent of any displacement that it may or may not undergo in the y or z direction, in the y direction independent of whether or not it moves in the x or z direction, and so on. In the molecule consisting of n atoms there will thus be 3n degrees of freedom. 

Let us now suppose that all n atoms move simultaneously by the same amount in the x direction. This will displace the center of mass of the entire molecule in the x direction without causing any alteration of the internal dimensions of the molecule. Thesame may of course be said of similar motions in the y and z directions. Thus, of the 3n degrees of freedom of the molecule, three are not genuine vibrations but only translations. Similarly, concerted motions of all atoms in circular paths about the jt, y, and z axes do not constitute vibrations either but are instead, molecular rotations. Thus, of the 3/i degrees of motional freedom, only 3n — 6 remain to be combined into genuine vibratory motions. 

We make note here of the special case of a linear molecule. In that instance rotation of the molecule may occur about each of two axes perpendicular to the molecular axis, but rotation of nuclei about the molecular axis itself cannot occur since all nuclei lie on the axis. Thus an //-atomic linear molecule has 3n - 5 normal modes. 

The lengths of the arrows relative to interatomic distances in the drawings, however, are exaggerated. 

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As stated in Secs. 1-7 and 1-8, it is possible, by using group theory, to classify the normal vibration into various symmetry species. Experimentally, measurements of the infrared dichroism and polarization properties of Raman lines of an orientated ciystal provide valuable information about the symmetry of normal vibrations (Sec. 1-23). Here we consider the polarization properties of Raman lines in liquids and solutions in which molecules or ions take completely random orientations. ... 

In Table B. 1 in Appendix B are given formulae, analogous to those derived for the C2 point group, for determining the number of normal vibrations belonging to the various symmetry species in all non-degenerate point groups. 

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 ... 

The determination of molecular orbitals in terms of symmetry-adapted linear combinations of atomic orbitals is analogous to the determination of normal vibrational modes by forming symmetry-adapted linear combinations of displacements. Both calculations are in reality the reduction of a representa-... 

The infrared active v (CH2), v (CH2), 8 (CH2), and yr (CH2) fundamentals can be readily assigned as a result of the extensive spectroscopic studies on hydrocarbons which have been undertaken [Sheppard and Simpson (795)]. In addition, because of the polarized radiation studies on single crystals of normal paraffins [Krimm (95)], it is possible to assign uniquely the components of the doublets found in the spectrum for these bands to symmetry species. Similarly, the Raman active va(CH. ), vs(CH2), (CHg), v+ (0), and v+ (n) fundamentals can be unambiguously assigned, the latter two on the basis of normal vibration calculations... 

Since two resonance lines at 39.0 and 47.7 ppm that correspond to those observed in the ttgg form and a resonance line at 49.0 ppm that corresponds to that in the tttt form are recognized in the gel spectrum, a coexistence of these two forms in the gel might be supposed. In an attempt to determine the possibility of the coexistence of the two forms in the gel, we measured the IR spectrum that is sensitive to the molecular conformation. The number of normal vibrational modes depends sensitively on the molecular conformation based on the selection rule of the symmetry species. Kobayashi et al. confirmed the vibrational modes assignable to the ttgg conformation in the IR spectrum for the gel from a sPP/carbon disulfide system [117]. However, since we used o-dichlorobenzene as solvent, we examined whether the gel structure depends on the solvent. 

In C70, because of its lower DSh symmetry, there are five kinds of non-equivalent atomic sites and eight kinds of non-equivalent bonds. This means that the number of normal vibrations increases for C70 in comparison to C60. Although there are now 204 vibrational degrees of freedom for the 70-atom molecule, the symmetry of C70 gives rise to a number of degenerate modes so that only 122 modes are expected. Of these 31 are infrared-active and 53 are Raman-active. 

Figure 6.6. Normal vibrations of tropolone molecule relevant for the tunneling tautomerization. The symmetry of each vibration is given in parentheses. The equilibrium bond length corresponds to the tropolone crystal. (From Redington et al. [1988].)...

In the present paper we assume that the molecule has the icosahedral symmetry. If one wants to consider a distortion of C 0+ or Cb0. the energy levels and their eigenvectors obtained here can be used as a starting point for the description of the Jahn-Teller effect in these systems. Indeed, the electron-phonon (or vibronic) coupling occurs if [.Tei]2 contains Fvib [19]. (Here Fd is the symmetry of an electronic molecular term, while J b is the symmetry of a vibrational normal mode.) Calculations using the terms in scheme of Ref. [4] have been performed in Ref. [20]. 

A molecule composed of A atoms has in general 3N degrees of freedom, which include three each for translational and rotational motions, and (3N — 6) for the normal vibrations. During a normal vibration, all atoms execute simple harmonic motion at a characteristic frequency about their equilibrium positions. For a linear molecule, there are only two rotational degrees of freedom, and hence (3N — 5) vibrations. Note that normal vibrations that have the same symmetry and frequency constitute the equivalent components of a degenerate normal mode hence the number of normal modes is always equal to or less than the number of normal vibrations. In the following discussion, we shall demonstrate how to determine the symmetries and activities of the normal modes of a molecule, using NH3 as an example. 

As another example, consider the NH3 molecule of C3V symmetry. Using the table given in Appendix 2, we find that m0 = 1 (nitrogen atom), m = 0 and mv = 1 (hydrogen atom). Thus, the number of normal vibrations in the 41, A2 and E species are 2, 0, and 2, respectively. 

General Formulas for Calculating the Number of Normal Vibrations in Each Symmetry Species... 

Thus, the force constants of the bonds, the masses of the atoms, and the molecular geometry determine the frequencies and the relative motions of the atoms. Fig. 2.1-3 shows the three normal vibrations of the water molecule, the symmetric and the antisymmetric stretching vibration of the OH bonds, and Va, and the deformation vibration 6. The normal frequencies and normal coordinates, even of crystals and macromolecules, may be calculated as described in Sec. 5.2. In a symmetric molecule, the motion of symmetrically equivalent atoms is either symmetric or antisymmetric with respect to the symmetry operations (see Section 2.7). Since in the case of normal vibrations the center of gravity and the orientation of the molecular axes remain stationary, equivalent atoms move with the same amplitude. 

Optimizing the structure and thermodynamic calculations were carried out for a separate molecule, which was considered in the main singlet state, with assistance of Gaussian 98. To increase the calculation accuracy, Tight option was added to Opt command. The calculations of normal vibration and thermodynamic parameters for all systems were done by Freq command. To exclude the symmetry error, which resulted in the qrpearance of false frequencies, the command NoSymm was used during the calculations of molecules of reagents and reaction products. 

In the Bom-Oppenheimer approximation the vibronic waveftmction is a product of an electronic waveftmction and a vibrational waveftmction, and its symmetry is the direct product of the symmetries of the two components. We have just discussed the symmetries of the electronic states. We now consider the symmetry of a vibrational state. In the harmonic approximation vibrations are described as independent motions along normal modes Q- and the total vibrational waveftmction is a product of functions, one waveftmction for each normal mode ... 

A more general method of finding the number of normal vibrations in each species can be developed by using group theory. The principle of the method is that all the representations are irreducible if normal coordinates are used as the basis for the representations. For example, the representations for the symmetry operations based on three normal coordinates, < , < 2, and Qy, which correspond to the p, and Vy vibrations in the HzO molecule of.Bg. 1-7, are as follows ... 

Carbon Dioxide, CO2. The molecule is linear and belongs to the point group. The number of atoms is 3, so the number of normal vibrations is (3 X 3) - 5 = 4. The set of Cartesian displacement vectors as basis for a representation is shown in Figure 5-9. The symmetry operations of the point group are also shown. The character table is given in Table 5-3. Recall (Chapter 4) that the matrix of rotation by an angle is... 

The fundamental vibrational spectra of crystals are properly interpreted on the basis of normal vibrations of the crystal as a whole. They are usually described as one- or multi-phonon transitions. These are governed by the laws of conservation of energy and wavevector in the photon-phonon system, e.g., for one phonon transition this latter rule allows only transitions creating phonons with the wavevector q = 0. In addition to these fundamental rules, there are selection rules based on symmetry considerations. For example, in elemental crystals which contain only 2 atoms per unit cell (such as Ge) the one optical phonon transition is forbidden (see Zallen (1968)). 

Table 1.5 lists the frequencies of normal vibrations of vinylidene and the transition state L of its rearrangement into acetylene La which has, in acordance with the symmetry requirements (Sect. 1.3.3.2), planar structure (E " = 6.4 kcal/mol). The calculation of the rearrangement rate using the data of Table 1.5 leads to the conclusion in favor of extremely fast tunnelling across the barrier even when the energy of XLIX does not exceed the energy of its zero vibrations, i.e., in the absence of any vibrational excitation whatsoever. The calculated life-time (reciprocal value of the rate constant) of vinylidene is a mere... 

To predict which bands wiU be Raman-active depend on the symmetry of the molecule. The symmetry elements of a molecule (i.e. centre, axes or planes) are associated with the symmetry operations which define all vibrational motions. For molecules with symmetry elements in its structure, the pattern of their normal vibrational modes will also have certain symmetry. Symmetry of normal vibrational modes influences a Raman tensor and consequently Raman activity of such vibrations. If we know the point group of the molecule and the symmetry labels for... 

Both F3 and O3 matrices are factored into symmetry blocks the dimensions of the blocks are equal to the number of normal vibrations in each symmetry species. The secular equations 5 or 7 take the form... 

Classification of the basic (normal) vibrations, according to their mode of symmetry— The operations of symmetry change specifically the mode of normal vibrations in a molecule. In this respect, a certain classification characterizes these vibrations according to their type of symmetry with respect to the particular operation and/or element of S5mnmetry that can be presented. 

Consider trans-C2H2Cl2. The vibrational normal modes of this molecule are shown below. What is the symmetry of the molecule Eabel each of the modes with the appropriate irreducible representation. 

Table 6.5 Number of normal vibrations of each symmetry species (Spec.) in the C2 point group...

One way to do so is to look at the normal mode corresponding to the imaginary frequency and determine whether the displacements that compose it tend to lead in the directions of the structures that you think the transition structure connects. The symmetry of the normal mode is also relevant in some cases (see the following example). Animating the vibrations with a chemical visualization package is often very useful. Another, more accurate way to determine what reactants and products the transition structure coimects is to perform an IRC calculation to follow the reaction path and thereby determine the reactants and products explicity this technique is discussed in Chapter 8. 

The 180° trans structure is only about 2.5 kcal/mol higher in energy than the 0° conformation, a barrier which is quite a bit less than one would expect for rotation about the double bond. We note that this structure is a member of the point group. Its normal modes of vibration, therefore, will be of two types the symmetrical A and the non-symmetrical A" (point-group symmetry is maintained in the course of symmetrical vibrations). 

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