A molecule composed of N atoms has by definition 3N degrees of freedom comprising translation, rotation, and vibration. Of these, three degrees of freedom define the mean translational position of the molecule. In addition, linear molecules have two degrees of rotational freedom and all other molecules have three degrees of rotational freedom. Hence, a linear molecule has 3N — 5 vibrational degrees of freedom, while a nonlinear molecule has 3N — 6 vibrational degrees of freedom. Substituting for N it can be seen that a diatomic molecule has a single vibrational degree of freedom, a molecule with three atoms (such as CO2, or H2O) has either three or four vibrational modes depending on its linearity, a molecule with four atoms has six or seven vibrational modes, and so on. Clearly the number of possible vibrational modes increases rapidly with the number of atoms, although not all will contribute to the observable vibration-rotation spectra since not all will be able to interact with electric dipole radiation, as we shall now see.
Consider a simple tri-atomic molecule such as CO2. Since this is a linear molecule, there are 3 x 3 — 5 = 4 vibrational modes, which are shown in Figure 6.4. The first mode (^1) is called the symmetric stretch mode and the spacing of the vibrational energy levels is equivalent to a photon with a wavenumber of approximately 1,100 cm—1. However, the motion of the oxygen atoms associated with this mode do not change the center of charge of the molecule and thus this mode may not interact with electric dipole radiation. The next two vibrational modes (v2) are the bending modes, which have identical frequencies, and clearly vary the center of charge of the molecule giving rise to the fundamental vibration-rotation band centered at 667 cm—1 (shown in Figure 6.3). The last mode (v3) is the asymmetric stretch mode and may also interact with electric dipole radiation giving rise to the fundamental vibration-rotation band at 2,350 cm—1.
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