Small amplitude harmonic motion - normal modes

Small Amplitude Harmonic Motion-Normal Modes (A)... 

Small Amplitude Harmonic Motion—Normal Modes... 

Small amplitude harmonic motion - normal modes... 

There will probably be some similarities, but also some fundamental differences. We have mainly considered small molecules with relatively rigid structures, in which the vibrational motions, although much different from the low-energy, near-harmonic normal modes, are nonetheless of relatively small amplitude and close to an equilibrium structure. (An important exception is the isomerization spectroscopy considered earlier, to which we shall return shortly.)... 

For polyatomic molecules, the problem is more complicated. For small-amplitude motion, certainly one can decompose the motion into independent harmonic (normal mode) motion and treat each of these as was done for a single oscillator. If the system has significant anharmonicity, then the good action-angle variables must first be found. Such techniques are available in the literature (8,21-25). 

In the harmonic approximation the problem of small amplitude vibrations (Chapters 6 and 7) reduces to the 3N — 6 normal modes N is the number of atoms in the molecule). Each of the normal modes may be treated as an independent harmonic oscillator. A normal mode moves all the atoms with a certain frequency about their equilibrium positions in a concerted motion (the same phase). The relative deviations (i.e. the ratios of the amplitudes) of the vibrating atoms from equilibrium are characteristic for the mode, while the deviation itself is obtained from them by multiplication by the corresponding normal mode coordinate Q (—oo, oo). The value Q = 0 corresponds to the equilibrium positions of all the atoms, Q and —Q correspond to two opposite deviations of any atom from its equilibrium position. 

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