Breakdown of the Normal Mode Approximation

Our development of the normal mode description of polyatomic vibrations in Sections 6.1-6.4 rested on the assumption that the potential energy function (6.4) is harmonic in the nuclear coordinates. As in diatomics, this assumption 

If one treats the anharmonicities and to second and first order respectively in a harmonic oscillator basis. Here q is the C-H bond displacement coordinate, and the expansion coefficients in the potential are related to the overtone spectrum parameters by 

In the local mode treatment [11] of the C-H stretching vibrations in benzene, the six bonds oscillate independently with energies (cf. Eq. 6.105) 

Another manifestation of vibrational anharmonicity occurs in Fermi resonance [8]. When two vibrational states of the same overall symmetry are accidentally degenerate, they can become strongly mixed by the anharmonic coupling terms between them. Their energies may be repelled considerably (in the language of degenerate perturbation theory), and the intensities of the spectroscopic transitions to these levels may be redistributed by the mixing. 

Marion, Classical Dynamics of Particles and Systems, Academic, New York, 1965. 

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This chapter begins with a classical treatment of vibrational motion, because most of the important concepts that are specific to vibrations in polyatomics carry over naturally from the classical to the quantum mechanical description. In molecules with harmonic potential energy functions, vibrational motion occurs in normal modes that are mutually uncoupled. Coupling between vibrational modes inevitably occurs in the presence of anharmonic potentials (potentials exhibiting cubic and/or higher order terms in the nuclear coordinates). In molecules with sufficient symmetry, the use of group theory simplifies the procedure of obtaining the normal mode frequencies and coordinates. We obtain El selection rules for vibrational transitions in polyatomics, and consider the rotational fine structure of vibrational bands. We finally treat breakdown of the normal mode approximation in real molecules, and discuss the local mode formulation of vibrational motion in polyatomics. 

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