Vibrations in molecules

A molecule with N atoms has 3N degrees of freedom, of which 3N-6 are vibrations. Some vibrations are skeletal, i.e. they involve almost the whole molecule. Other vibrations are more or less localized, confined to a small number of atoms of a functional group.

The other six degrees of freedom are three translations and three rotations of the whole molecule, which do not change the energy of the molecule as such.
If a molecule represents an energy minimum, the first derivatives of the energy w.r.t. the 3N cartesian coordinates are zero. Moreover, the second derivatives (a matrix of size 3N x 3N) will be positive. If this matrix is diagonalized we obtain the eigen vectors (normal coordinates) which describe the vibrations, and eigen values along the diagonal which are proportional to their frequencies. Six values will be zero.
Generally speaking vibrations that change the dipole moment will be visible in the infrared spectrum. The remaining ones may show up in a Raman spectrum.
A transition state is characterized by one non-positive (imaginary, printed as negative) eigen value, belonging to the reaction coordinate.

On the next pages we show a few examples.
Methyl amine, all 15 vibrations Chime Jmol
The C36 buckyball, a selection Chime Jmol

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