The vibrations of polyatomic molecules

We need to see how the concepts developed in previous sections can be used to interpret the information contained in the infrared and Raman spectra of biological macromolecules. 

How many modes of vibration, N, does a polyatomic molecule have We can answer this question by thinking about how each atom may change its location, and we show in the following Justification that 

A water molecule, H2O, is triatomic N = 3) and nonlinear and has three modes of vibration. Naphthalene, CioHg N = 18), has 48 distinct modes of vibration (some are degenerate in the sense of having the same frequency). Any diatomic molecule N = 2) has one vibrational mode carbon dioxide N = 3) has four vibrational modes. 

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The eigenvalue problem was introduced in Section 7.3, where its importance in quantum mechanics was stressed. It arises also in many classical applications involving coupled oscillators. The matrix treatment of the vibrations of polyatomic molecules provides the quantitative basis for the interpretation of their infrared and Raman spectra. This problem will be addressed tridre specifically in Chapter 9. 

Raman spectroscopy (Section 4.10) aids the study of the vibrations of polyatomic molecules. For a vibration to be Raman active, it must give a change in the molecular polarizability. For many molecules with some symmetry, one or more of the normal modes correspond to no change in... 

The vibrations of polyatomic molecules are more complicated, for the number of possible interactions rises sharply as the number of atoms increases. However, it is possible to handle the equations of motion governing even the most complicated vibrations by lineal combination (p. 48) of the equations of motions of rather simple vibrations. For example, all vibrations of the C02 molecule are said to be derived from superposition of the four modes of vibration indicated in Figure 25-4(a), whereas all vibrations of the SO2 molecule may be likewise broken down into combinations of one or more of the vibrations indicated in 25-4(b). 

The application of this result to the determination of bond stretching force constants in molecules encounters two difficulties. First, real molecules are not exactly harmonic oscillators. Secondly, although the only mode of vibration possible in diatomic molecules is a bond stretching motion, the vibrations of polyatomic molecules are much more complicated, and cannot be expressed as consisting only of a combination of bond stretching motions. We discuss these two problems in turn in the next two sections. 

Matsen and Franklin make use of an analogy between the formulation of monomolecular systems and normal mode analysis of the vibrations of polyatomic molecules. The too strict use of this analogy leads them to make an assumption that is wrong for monomolecular systems in general. Normal mode analysis of vibration spectra treats symmetric matrices except in very special cases, the rate constant matrix for the reactions of the various species A, is asymmetric when the amounts are expressed in the A system of coordinates. The heart of their formulation is expressed by Eqs. (2), (3), and (4) of their paper, which will be designated (MF2), (MF3), and (MF4) when expressed in our notation. Matsen and Franklin begin by assuming that a transformation matrix X exists such that... 

These properties of the normal coordinates enable us to treat the problem of the vibrations of polyatomic molecules by the methods of quantum mechanics. 

As pointed out in chapter 2, the Hamiltonian for the vibrations of polyatomic molecules (Eq. (2.47)) can be readily constructed as sums over momenta and positions if the vibrations are separable, a condition that leads to normal modes. 

The two bonds in this system are known as coupled oscillators . This type of system occurs often in the vibrations of polyatomic molecules in which two oscillators (which can be bonds or groups of bonds) couple to give symmetric and antisymmetric combinations. (In this case, where the system is exactly symmetrical, the symmetric combination will not be observed in the IR spectrum because the dipole moment of the molecule does not change.)... 

In the harmonic oscillator approximation the vibration of polyatomic molecules is that of normal modes, each acting like an independent harmonic oscillator. We number the normal modes with an index i, ranging from 1 to 3n — 5 for linear molecules and ranging from 1 to 3n — 6 for nonlinear molecules. The selection rules for vibrational tfansitions are ... 

The description of the vibrations of polyatomic molecules only becomes mathematically tractable by treating the system as a set of coupled harmonic oscillators. Thus a set of 3N - 6 (3N - 5 for linear molecules) normal modes of vibrations can be described in which aU the nuclei in the molecule move in phase in a simple harmonic motion with the same frequency, normal-mode frequencies are solved, the normal coordinates for the vibrations can be determined, and how the nuclei move in each of the normal modes of vibration can be shown. There are two important points that follow from this. First, each normal mode can be classified in terms of the irreducible representations of the point group describing the overall symmetry of the molecule [7, 8]. This symmetry classification of the... 

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