spectroscopy - Rotational degrees of freedom (3N-5 and 3N-6)

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).

Answer

Rotations around bonds are typically termed "internal rotations", and represent one of the most common problematic cases for the rigid-rotor-harmonic-oscillator (RRHO) model of internal molecular motion. This is because RRHO assumes that any vibrational amplitudes are "small," and internal rotations are most definitely not small! Such rotations still involve the internal degrees of freedom of the molecule, though, and thus (as Jan notes) are considered as part of the 'vibration' of the molecule, not its 'rotation.'

Internal rotations typically have some energy cost involved (as in your pentane example), and so cannot be treated as "free" rotations. They are usually termed "hindered rotations," and there is a great deal of literature studying them. Some citations I know offhand:

• Classic Pitzer (J Chem Phys 5 469, 1937), and Pitzer & Gwinn (J Chem Phys 10 428, 1942)


• Independent-rotations approximation for treating hindered internal rotation: Pfaendtner (Theor Chem Acc 118 881, 2007)


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  Other / more-advanced treatments of internal rotation:

  
  
  

One species of particular interest to me on this topic has been nitromethane, which has an extraordinarily small barrier to rotation around the $\ce{C}-\ce{N}$ bond (see Strekalov, above), such that at ambient temperature it can be considered essentially a free rotation.