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

In spectroscopy we described the electric energy with the approximative separability of internal motions as:

$$E=E_e+E_v+E_r+E_{ns}$$

(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).

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)
• 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.

trans-buta-1,3-diene is not a linear molecule. To be linear all atoms must be "in-line", e.g. $\ce{HCCH}$ is linear but $\ce{H2CCH2}$ is not.

The rotation around a bond axis is not free and is therefore described as a vibration. See for example this blog post. So, for example, for ethane the lowest of the $3N-6$ vibrational modes corresponds to a torsional motion about the $\ce{C-C}$ single bond. The mental picture here is not 2 methyl groups that rotate freely relative to one another, but rather a rocking back and forth motion in one minimum and then an occasional rotation in to the next minimum. The vibrationa motion is on the femto second time scale and every few nano-seconds the switch from one minimum to the next will occur. So the "rotation" will happen a thousand times per millisecond and look like free rotation.

• I now see that trans-buta-1,3-diene belongs to point group $\ce{C_{2h}}$ and should therefore be an asymmetric top ($\ce{I_{A}<I_{B}<I_{C}}$) and not be considered as linear molecule ($\ce{I_{A}<I_{B}=I_{C}}$) of point groups C_{∞v} or D_{∞h}. – laminin May 8 '15 at 20:15
• If I'm right the energy difference betweeen anti and staggerend around a C-C single bond is about 5 kcal/mol (converted: 1750 $cm^{-1}$). If there is not as much energy available, I see the "rotation" is like a vibration. I thought a vibrational motion is typically between 300 and 3000 $cm^{-1}$. What is meant with "not free"? Are translational, rotational and vibrational motion considered as "free" motions? – laminin Jun 13 '15 at 18:38
• The energy of the motion that describes rotational motion about a CC single bond is included in the vibrational energy ($E_v$). The frequency of this motion is typically < 500 cm$^-1$, so there is usually several bound levels. So the expression for $E_v$ for C-C bond rotation looks just that for, say, a C-C stretch. – Jan Jensen Jun 14 '15 at 8:43
• I'm a little bit surprized that my value of 1750 $\ce{cm^{-1}}$ (eclipsed activation barrier for butane around C(2)-C(3) bond) is different from your value of <500 $\ce{cm^{-1}}$. I don't know if the total "stretching vibration"-like rotation is defined by an average value of the different conformation activation energies or by the maximal value (5kcal/mol). Secondly, is the anharmonic oscillar $\omega_e(v+1/2)$-$\omega_e$ $x_e(v+1/2)^2$+.... still a good approximation to describe this complete (360°) rotation around the single bond C(2)-C(3) of pentane (as described in my initial post) ? – laminin Jun 15 '15 at 19:54
• I'll have a look at them tomorrow, gottta run – Jan Jensen Jun 19 '15 at 13:55