Coordinates, Vibrational Wavefunctions, and Spectral Activities

Appendix Normal Coordinates, Vibrational Wavefunctions, and Spectral Activities (11, 13) 

Each normal mode of vibration can be described by a normal coordinate Qi which is a linear combination of nuclear displacement coordinates of the molecule. For the symmetric stretching vibration vi of C02, the normal coordinate is of the form 

The vibrational wavefunctions may be expressed as functions of the jth normal coordinate  

For nondegenerate vibrations all symmetry operations change Qj into 1 times itself. Hence Q/ is unchanged by all symmetry operations. In other words, Q and consequently y(O) behave as totally symmetric functions (i.e. the function is independent of symmetry). However, the wavefunction of the first excited state 3(1) has the same symmetry as Qj. For example, the wavefunction of a totally symmetric vibration (e.g. Qi of C02) is itself a totally symmetric function. 

The intensity of a spectral band is proportional to the probability that the associated transition could occur. The probability (and hence the intensity) of the fundamental transition  

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