Multiplication generalization, vibration-rotation

In general, if a bond has a dipole moment, its stretching frequency causes an absorption in the IR spectrum. If a bond is symmetrically substituted and has zero dipole moment, its stretching vibration is weak or absent in the spectrum. Bonds with zero dipole moments sometimes produce absorptions (usually weak) because molecular collisions, rotations, and vibrations make them unsymmetrical part of the time. Strongly polar bonds (C=0 groups, for example) may absorb so strongly that they also produce overtone peaks, which are relatively small peaks at a multiple (usually double) of the fundamental vibration frequency. 

Here, Vexm is a purely multiplicative operator called the extra potential term. The volume elements used to normalize the wavefunction reads dV = sin (3dad0d f dqi... dqsN-6- This ensures that the momentum operators Pi are Hermitian. The matrices G, C and F, which stand for internal vibrations, Coriolis couplings and overall rotation, respectively, are, in general, functions of the internal coordinates. 

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