Is there a definition of rotatable bond?

Question

Some intramolecular bonds are rotatable in the sense that the torsion angle around this bond is flexible (for example, $\psi$ and $\phi$ angles in a protein backbone). Others are essentially fixed (for example, the $\omega$ angle in the peptide bond in a protein backbone).

Is there an operational definition, or a published study where a general definition is given, which would allow me to recognize in a generic molecule (or at least an organic molecule) whether a given bond is rotatable or not?

If there is no general theory for this, what are the most common indicators of whether a bond can or cannot be rotated? Are there some rules of thumb?

Answer 1

Is there an operational definition, or a published study where a general definition is given, which would allow me to recognize in a generic molecule (or at least an organic molecule) whether a given bond is rotatable or not?

It depends on your purpose. Why do you want to identify some bonds and call the rotatable? For instance, PubChem, one of the biggest databases of molecules, defines rotatable bond as follows:

Rotatable bond is defined as any single non-ring bond, bounded to nonterminal heavy (i.e., non-hydrogen) atom.

To some extent all this boils down to the following IUPAC definition of free rotation:

free rotation (hindered rotation, restricted rotation)

In a stereochemical context the rotation about a bond is called 'free' when the rotational barrier is so low that different conformations are not perceptible as different chemical species on the time scale of the experiment. The inhibition of rotation of groups about a bond due to the presence of a sufficiently large rotational barrier to make the phenomenon observable on the time scale of the experiment is termed hindered rotation or restricted rotation.

At usual lab conditions a rotation around a single non-ring bond, bounded to nonterminal heavy atom will be indeed almost free, thus, PubChem definition can be used as a very simple rule-of-thumb to identify such bonds. But in general it depends on conditions. And purposes.

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With respect to protein backbones being rotatable or not: I'm not a speacialist in this area, but Wikipedia article on peptide bond claims that (emphasis mine)

Significant delocalisation of the lone pair of electrons on the nitrogen atom gives the group a partial double bond character. The partial double bond renders the amide group planar, occurring in either the cis or trans isomers.

So, this partial double bond character might contribute in protein backbones being not quite rotatable. Besides, the Wikipedia article also claims that (emphasis again mine):

In the unfolded state of proteins, the peptide groups are free to isomerize and adopt both isomers; however, in the folded state, only a single isomer is adopted at each position (with rare exceptions).

So, at least in the unfolded state protein backbones are rotatable. The fact that only two configurations (cis and trans) are possible is irrelevant. The notion of rotatable bond does not imply having few specific possible configurations.

Answer 2

The IUPAC Gold Book states at least somthing to free rotation:

free rotation (hindered rotation, restricted rotation)

In a stereochemical context the rotation about a bond is called 'free' when the rotational barrier is so low that different conformations are not perceptible as different chemical species on the time scale of the experiment. The inhibition of rotation of groups about a bond due to the presence of a sufficiently large rotational barrier to make the phenomenon observable on the time scale of the experiment is termed hindered rotation or restricted rotation.

What they don't tell is what causes a sufficiently large rotational barrier. Those are, e.g., ring strains or resonance stabilized bonds.

Answer 3

For every bond, there is an energy barrier required to rotate around the bond. Would the thermal energy (at room temperature) be enough to allow rotation, it can be considered rotatable. For the reasonable rate constant I would consider $s ^{-1}$ or faster. This means that single $\ce{C^{sp3}-C^{sp3}}$ bonds are rotatable, double bonds rarely, in other cases use common sense or quantum chemistry.

Medicinal chemists use more straightforward definition, http://pubs.acs.org/doi/abs/10.1021/jm020017n.