In spectroscopy we described the electric energy with the approximative separability of internal motions as:
\begin{equation} E=E_e+E_v+E_r+E_{ns} \end{equation}
(energies: electronic, vibratory, rotatory; nuclear spin (neclected))
With the Born Oppenheimer approximation of nuclei and electrons you get a formula which describes the degrees of freedoms for vibration motions:
- 3N-5 for linear molecules
- 3N-6 for non-linear molecules
This formula can be understood by:
- the coordinates of 3N atoms.
- 3 degrees of motions for the three translation of the centre-of-mass motion
- 2 (for linear) respective 3 (for non-linear) degrees of freedoms reduced for the rotation of the whole molecule
I guess trans buta-1,3-diene is here described as a linear molecule. [added: in the rigid rotor approximation]
First I was thinking why we are so much interested on the number of freedoms for vibrations. We consider only the rotation of the whole molecule but not around bonds.
Why do we not consider rotation about a bond axis, e.g. for pentane a rotation of an ethyle around the propyle group? (rotation about the C(2)-C(3) axis)
May be because it's much another energy range as rotational energy of about 1-100 cm-1 (typical period of 0.1-10 ps).