Imperial College London

Henry Rzepa

My last comment as [appended to the previous
post](http://www.ch.imperial.ac.uk/rzepa/blog/?p=9512&cpage=1#comment-35668){target="_blank"} promised
to analyse two so-called extended porphyrins for their topological
descriptors. I start with
the Cãlugãreanu/[Fuller](http://links.jstor.org/sici?sici=0027-8424%28197104%2968%3A4%3C815%3ATWNOAS%3E2.0.CO%3B2-2){target="_blank"}
theorem  which decomposes the topology of a space curve into two
components, its twist (Tw) and its writhe (Wr, this latter being the
extent to which coiling of the central curve has relieved local
twisting) and establishes a topological invariant called the [linking
number](http://en.wikipedia.org/wiki/Linking_number){target="_blank"}\[cite\]10.1021/ja710438j\[/cite\]

** Lk = Tw + Wr **

<table class="aligncenter" data-border="0" data-align="center">
<colgroup>
<col style="width: 50%" />
<col style="width: 50%" />
</colgroup>
<tbody>
<tr class="odd">
<td><div id="attachment_9558" class="wp-caption aligncenter"
style="width: 220px">
<img
src="http://www.ch.imperial.ac.uk/rzepa/blog/wp-content/uploads/2013/02/hiytal.jpg"
class="size-full wp-image-9558" decoding="async"
aria-describedby="caption-attachment-9558"
onclick="jmolInitialize(&#39;../Jmol/&#39;,&#39;JmolAppletSigned.jar&#39;);jmolSetAppletColor(&#39;white&#39;);jmolApplet([450,450],&#39;load wp-content/uploads/2013/02/HIYTAL.mol;&#39;);"
width="210" alt="Click for 3D." />
<p>HIYTAL. Click for 3D.</p>
</div></td>
<td><div id="attachment_9559" class="wp-caption aligncenter"
style="width: 220px">
<img
src="http://www.ch.imperial.ac.uk/rzepa/blog/wp-content/uploads/2013/02/selquw.jpg"
class="size-full wp-image-9559" decoding="async"
aria-describedby="caption-attachment-9559"
onclick="jmolInitialize(&#39;../Jmol/&#39;,&#39;JmolAppletSigned.jar&#39;);jmolSetAppletColor(&#39;white&#39;);jmolApplet([450,450],&#39;load wp-content/uploads/2013/02/SELQUW.mol;&#39;);"
width="210" alt="SELQUW. Click for 3D." />
<p>SELQUW. Click for 3D.</p>
</div></td>
</tr>
</tbody>
</table>

Visual inspection of the models above (I really do encourage you to
click on the images to load the 3D coordinates) reveals that
HIYTAL\[cite\]10.1002/chem.200701909\[/cite\] has a major coil that
forms one and a half helical turns in a clockwise direction, and a loop
connecting the ends of the coil which forms a half-helical turn in an
anti-clockwise direction.
SELQUW\[cite\]10.1002/chem.200600158\[/cite\] has a major coil
comprising one and half helical turns in an anti-clockwise direction and
a connecting loop which also coils anti-clockwise. So the former
sustains a total of **one** full (clockwise) helical turn and the latter
**two** full (anti-clockwise) helical turns.

The [nomenclature for helical
molecules](http://en.wikipedia.org/wiki/Axial_chirality){target="_blank"}
includes a chiral descriptor P (for a positive helical turn) and M (for
a negative helical turn). What such a descriptor does not do is quantify
the total number of helices describing the topology. So I suggest we use
instead the linking number Lk. Instead of P and M, we have positive and
negative integers (in units of 2π) providing this quantitative
information.

The linking number analysis for these two molecules comes out as
below.^‡^ I have multiplied the linking number unit from 2π to 1π for a
reason that I will explain shortly:

|        |             |     |        |       |             |
|--------|-------------|-----|--------|-------|-------------|
|        | π-electrons | Lk  | Tw     | Wr    | Δ~r~ (meso) |
| SELQUW | 56=4n       | -4  | -1.34  | -2.66 | 0.048       |
| HIYTAL | 62=4n+2     | +2  | +0.46  | +1.54 | 0.045       |

You can see that the linking numbers (and their signs) correspond
exactly to the visual analysis of the helical turns above. My reason for
including the factor of 2 is that it enables us to make a further link
to the Hückel aromaticity rule:

1.  [Cyclic conjugated systems are **aromatic** if they contain 4n+2
    π-electrons and have an even or zero linking number (in units of
    1π). ]{style="line-height: 13px;"}
2.  Cyclic conjugated systems are **aromatic** if they contain
    4n π-electrons and have an odd linking number (in units of 1π). 
3.  Cyclic conjugated systems are **anti-aromatic** if they contain 4n
    π-electrons and have an even or zero linking number (in units of
    1π). 
4.  Cyclic conjugated systems are **anti-aromatic** if they contain 4n+2
    π-electrons and have an odd linking number (in units of 1π). 

By these rules, SELQUW contains (by the shortest path) 56 π-electrons,
belongs to the 4n electron rule (n=14) and hence is formally
anti-aromatic (rule 3 above). HIYTAL has a path of 62-electrons, belongs
to the 4n+2 rule (n=15) and hence is formally aromatic (rule 1 above). 

For systems with so many (correlated) electrons, it is probably tenuous
to make a connection between the bond-length alternation at the meso
position and the aromaticity (or lack of it). I comment only that HIYTAL
converts more of the coiling into writhing of the central curve than
does SELQUW, and this destroys less π-π overlap by reducing the overall
degree of twisting. I might also speculate that nevertheless a modest
degree of twisting may impact upon the intrinsic distortivity
of π-electrons in cyclically conjugated systems (such as that in
benzene\[cite\]10.1021/cr990363l\[/cite\]), as noted in this [earlier
post](http://www.ch.imperial.ac.uk/rzepa/blog/?p=485){target="_blank"}.
Such effects may make the interpretation of bond-alternation in such
helical systems more difficult.

------------------------------------------------------------------------

^‡ A program for calculating these components can be found
[here](https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:link){target="_blank"}.
For a fun-packed journey through linking numbers and the association
with valentine cards, [go see this
post](http://www.ch.imperial.ac.uk/rzepa/blog/?p=3492){target="_blank"}
here!^

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


My last comment as appended to the previous post promised to analyse two so-called extended porphyrins for their topological descriptors. I start with the Cãlugãreanu/Fuller theorem

Messaggi di Rogue Scholar

Linking numbers, and twist and writhe components for two extended porphyrins.