The conformation of cyclohexane

Like benzene, its fully saturated version cyclohexane represents an icon of organic chemistry. By 1890, the structure of planar benzene was pretty much understood, but organic chemistry was still struggling somewhat to fully embrace three rather than two dimensions. A grand-old-man of organic chemistry at the time, Adolf von Baeyer, believed that cyclohexane too was flat, and what he said went. So when a young upstart named Hermann Sachse suggested it was not flat, and furthermore could exist in two forms, which we now call chair and boat, no-one believed him. His was a trigonometric proof, deriving from the tetrahedral angle of 109.47 at carbon, and producing what he termed strainless rings.

Whilst the chair form of cyclohexane now delights all generations of chemistry students, the boat is rather more mysterious. Perhaps due to Sachse, it is still often referred to as a higher energy form of the chair (Barton, in the 1956 review that effectively won him the Nobel prize, clearly states that the boat is one of only two conformations free of angle strain, DOI: 10.1039/QR9561000044). Over the last 30 years or so, and especially with the advent of molecular modelling programs, the complexity of the conformations of cyclohexane has become realised. A nice recent illustration of that complexity is by Jonathan Goodman using commercial software. Here a slightly different take on that is presented.

The starting point is the flat Baeyer model for cyclohexane. Like benzene, it has D_6h symmetry. When subjected to a full force constant analysis using a modern program (in this instance Gaussian 09), this geometry is revealed (DOI: 10042/to-3708) to have three negative force constants, which in simple terms means it has three distortions which will reduce its energy. The eigenvectors of these force constants are shown below, and each set of vectors acts to reduce the symmetry of the species. Such symmetry-reduction is a well known aspect of group theory, and its analysis in the Lie symmetry groups is used in many areas of physics and mathematics, but it is a less used in chemistry.

The first of these symmetry-reducing vibrations (the B_2g mode) converts the geometry immediately to the chair conformation of cyclohexane. So in some ways, this use of symmetry is a modern equivalent of the trigonometry used by Sachse to prove his point.

The next two modes are degenerate in energy, and the first of these reduces the symmetry to D₂. The result is what we now call the twist-boat. It is interesting, because the D₂ group is one of the (relatively few) chiral groups, and the twist-boat exhibits disymmetric symmetry. In other words, following the vibrational eigenvectors in one direction leads to one enantiomer of the twist boat, and in the other direction to the other enantiomer. So (in theory only), one might actually be able to produce chiral cyclohexane (the experiment and resolution would have to be done at very low temperatures!). It is also interesting that theory nowadays could quite reliably calculate the optical rotation of this species (and its circular dichroism spectrum), so we certainly would know what to look out for.

The second component of the degenerate E imaginary mode leads directly to a species of C_2v symmetry, which we recognize as Sachse’s second possible form of cyclohexane. The symmetry-reductions of D_6h to C_2h, D₂ and C_2v all have paths on the grand diagram of the 32 crystallographic point groups and their sub groups, and is an interesting application of group theory to a mainstream topic in organic chemistry.

But the story is not quite complete yet. The C_2v boat is not the final outcome of the last distortion! It too is a transition state, connecting again the two D₂ forms. So the path from D_6h to C_2v is NOT a minimum energy reaction path, but a rather different type of path known as a valley-ridge inflection path. An example of such a surface can be seen for the dimerisation of cyclopentadiene (DOI: 10.1021/ja016622h) and effectively it connects one transition state to a second transition state, without involving any intermediates on the pathway. At some stage, the dynamics of the system takes over, and the symmetry breaks without the system ever actually reaching the second transition structure. This final aspect of the potential energy surface of cyclohexane was not discussed by Jonathan Goodman in his own article on the topic.

So symmetry-breaking is the topic of this blog, and its connection to physics and mathematics. And, I might add that the same approach can be taken for looking at the conformations of cyclobutane, pentane, heptane and octane. But that will be left for another post.

Postscript.  See this more recent post.