Imperial College London

Henry Rzepa

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](http://en.wikipedia.org/wiki/Adolf_von_Baeyer){target="_blank"},
believed that cyclohexane too was flat, and what he said went. So when a
young upstart named [Hermann
Sachse](http://www.chem.yale.edu/~chem125/125/history/Baeyer/Sachse.html){target="_blank"}
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](http://dx.doi.org/10.1039/QR9561000044){target="_blank"}).
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](http://www-jmg.ch.cam.ac.uk/software/QRC/example.html){target="_blank"}
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](http://hdl.handle.net/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.

<table data-border="1">
<colgroup>
<col style="width: 33%" />
<col style="width: 33%" />
<col style="width: 33%" />
</colgroup>
<tbody>
<tr class="header">
<th>348<em>i</em> cm<sup>-1</sup> <span
style="vertical-align: baseline;">(B</span><sub>2g</sub><span
style="vertical-align: baseline;">)</span></th>
<th>244<em>i</em> (E<sub>2u</sub>)</th>
<th>244i (E<sub>2u</sub>)</th>
</tr>
&#10;<tr class="odd">
<td><div id="attachment_1588" class="wp-caption aligncenter"
style="width: 160px">
<img
src="http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2010/01/cx-c2h.jpg"
title="cx-c2h" class="size-full wp-image-1588" loading="lazy"
decoding="async" aria-describedby="caption-attachment-1588"
onclick="jmolInitialize(&#39;../Jmol/&#39;);jmolSetAppletColor(&#39;black&#39;);jmolApplet([450,450],&#39;load wp-content/uploads/2010/01/cyclohexane-d6h.log;frame 13;zoom 5;moveto 4 0 2 0 90 90;vectors on;vectors 4;vectors scale 5.0; color vectors green; vibration 6;animation mode loop;&#39;);"
width="150" height="129" alt="D6h to C2h for cyclohexane" />
<p>D6h to C2h for cyclohexane. Click for animation.</p>
</div></td>
<td><div id="attachment_1589" class="wp-caption aligncenter"
style="width: 160px">
<img
src="http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2010/01/cx-D2.jpg"
title="cx-D2" class="size-full wp-image-1589" loading="lazy"
decoding="async" aria-describedby="caption-attachment-1589"
onclick="jmolInitialize(&#39;../Jmol/&#39;);jmolSetAppletColor(&#39;black&#39;);jmolApplet([450,450],&#39;load wp-content/uploads/2010/01/cyclohexane-d6h.log;frame 14;zoom 5;moveto 4 0 2 0 90 90;vectors on;vectors 4;vectors scale 5.0; color vectors green; vibration 6;animation mode loop;&#39;);"
width="150" height="142" alt="D6h to D2" />
<p>D6h to D2. Click for animation.</p>
</div></td>
<td><div id="attachment_1590" class="wp-caption aligncenter"
style="width: 160px">
<img
src="http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2010/01/cx-c2v.jpg"
title="cx-c2v" class="size-full wp-image-1590" loading="lazy"
decoding="async" aria-describedby="caption-attachment-1590"
onclick="jmolInitialize(&#39;../Jmol/&#39;);jmolSetAppletColor(&#39;black&#39;);jmolApplet([450,450],&#39;load wp-content/uploads/2010/01/cyclohexane-d6h.log;frame 15;zoom 5;moveto 4 0 2 0 90 90;vectors on;vectors 4;vectors scale 5.0; color vectors green; vibration 6;animation mode loop;&#39;);"
width="150" height="140" alt="D6h to C2v" />
<p>D6h to C2v. Click for animation.</p>
</div></td>
</tr>
</tbody>
</table>

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 D~2~. The result is what we now call the
**twist-boat**. It is interesting, because the D~2~ 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~, D~2~ 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 D~2~ 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](http://www.ch.imperial.ac.uk/rzepa/blog/?p=7389){target="_blank"}
(DOI:
[10.1021/ja016622h](http://dx.doi.org/10.1021/ja016622h){target="_blank"})
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](http://www-jmg.ch.cam.ac.uk/software/QRC/example.html){target="_blank"}
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](http://www.ch.ic.ac.uk/rzepa/blog/?p=1603){target="_blank"}.

**Postscript.**  See [this more recent
post](http://www.ch.imperial.ac.uk/rzepa/blog/?p=7926){target="_blank"}.

::: {.wp_orcid_field}
[https://orcid.org/0000-0002-8635-8390](https://orcid.org/0000-0002-8635-8390){target="_blank"
rel="author"}
:::


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.

Messaggi di Rogue Scholar

The conformation of cyclohexane