Small-amplitude vibrations, normal-mode coordinates

Small-amplitude vibrations, normal-mode coordinates 

In the following we show that a simple description of the (quantum or classical) dynamics can be obtained in a multidimensional system close to a stationary point. Thus, the system can be described by a set of uncoupled harmonic oscillators. The formalism is related to the generalization of the harmonic expansion in Eq. (1.7) to multidimensional systems. 

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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. 

The a tensor is the polarizability of the material. Considering its modulation by the normal atomic vibrations of the material, for small amplitude oscillations near the equilibrium, the polarizability dependence on the normal coordinate Q associated with a normal mode of vibration can be written as ... 

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