Imperial College London

Henry Rzepa

Something important happened in chemistry for the first time about 100
years ago. A molecule was built (nowadays we would say synthesized)
specifically for the purpose of investigating a theory. It was
cyclo-octatetraene or (CH)~8~, and it was made by Willstätter and
Waser\[cite\]10.1002/cber.191104403216\[/cite\] to try to find out if
benzene, (CH)~6~, was an **aromatic** one-off or whether it might be a
member of a series, envisaged as (CH)~n~. Of course, a hell of a
surprise was in store for Willstätter and
Waser\[cite\]10.1002/cber.191104403216\[/cite\]! Prior to this
synthesis, (CH)~8~ had never existed; nature had not gotten there first.
In that sense, chemistry became much like mathematics had before it; it
was OK to make molecules because they *might be interesting*, and for
the purpose of investigating possible patterns in nature. So it is in
this spirit that I suggest an interesting molecule here. It is a
molecular trefoil, constructed by joining 15 pyrrole units together into
a ring with appropriate linkers and in effect tying a knot in that ring.
A trefoil knot to be specific.

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![](http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2010/05/metallatrefoil.jpg "metallatrefoil"){.size-full
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A molecular trefoil knot, shown with a Mg at the centre. Click to view
in 3D
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Why might such a molecule be interesting? These are **ten reasons**:

1.  It would make an interesting ligand for a metal
2.  It has lots of interesting groves and dimples for transition states
    to nest in
3.  It would be an extended porphyrin (a pentadecaphyrin to be precise).
    Nature likes to make molecules out of tetraphyrins (chlorophyll,
    haemoglobin, etc), and so we are pushing beyond nature's own
    boundaries. Both a penta and a hexadecaphyrin have already been
    made\[cite\]10.1002/chem.200701909\[/cite\],\[cite\]10.1002/chem.200701909\[/cite\],\[cite\]10.1021/ja005588o\[/cite\]
4.  The trefoil knot is a most interesting object in a branch of
    mathematics called **knot theory**, and it is also related to
    another fascinating object, the **Möbius band**.
5.  The pyrrole units in such a molecule are *conjugated* via the
    π-system, and the molecule above is potentially fully conjugated
    across its entire length. This could make it aromatic, and hence it
    is interesting for the same reason that
    Willstätter\[cite\]10.1002/cber.191104403216\[/cite\] found
    cyclo-octatetraene so.
6.  The system above, if carefully counted, would have 74 π-electrons in
    cyclic conjugation. This would make it a 4n+2 aromatic (n=18), just
    like benzene, but not at all like (CH)~8~ (which as Willstätter and
    Wases\[cite\]10.1002/cber.191104403216\[/cite\] found, is not
    aromatic).
7.  It seems highly twisted. Indeed the title of this post is
    super-twisted. But is it really? We learn from topology that twist
    is not the only property that cyclic bands or strips can have. They
    can also exhibit **writhe**. So is it writhed as well as twisted?
8.  Aromatic molecules have one rather mysterious behaviour. The ring
    bonds in in such systems resemble neither double nor single bonds,
    but aromatic bonds, and in this they have a length intermediate
    between the single and the double, and this applies to all of the
    bonds. The origins of this *delocalization* continue to provoke
    controversy (see [this
    post](http://www.ch.ic.ac.uk/rzepa/blog/?p=485){target="_blank"
    rel="noopener noreferrer"}). Thus it is thought that only (planar)
    carbon rings with around 26 or less π-electrons can exhibit such
    equal lengths (boron rings can apparently go much
    further.\[cite\]10.1039/B911817A\[/cite\]. More than that, and
    distortion sets in which makes the lengths alternate. The molecule
    above has 74 π-electrons. What will its bonds do, and is what they
    do related to the twist (or the writhe) of the system?
9.  The trefoil is **chiral**. It cannot be superimposed upon its mirror
    image. But how chiral (whatever that means)?
10. The system has many design handles, including the number of pyrrole
    (or thiophene) units, the number of N-H vs =N motifs, and the scope
    for templating using a metal cation (Mg in the example above).

So what might be the properties of our trefoil knot? I am going to list
only two here.

1.  A theorem emerged from mathematics in the 1970s known as the White,
    Cãlugãreanu, Fuller Theorem. It defines the topological properties
    of bands in terms of a quantity known as the linking number (Lk).
    The theorem states that: **Lk = Tw + Wr**, where Lk is an integer,
    being the sum of two properties **Tw** (the total twist of the band)
    and **Wr** (the total writhe of the band). This theory was recently
    extended to the analysis of twisted conjugated molecular
    rings\[cite\]10.1021/ja710438j\[/cite\], for which Lk adopts integer
    values (in units of π). Thus a conjugated Möbius π-cycle has a value
    Lk = 1π (specifically when describing the band formed by the
    π-electrons). Most of this value is composed of twist rather than
    writhe. What of our molecule? Well, it has Lk =6π, and this
    comprises Tw = \~-0.8π and Wr \~+6.8π (yes the two can be either
    positive or negative, and do not have to be the same sign). The
    surprise is that it is (overall) hardly twisted! The knot is
    composed almost entirely of writhe. So much for the title of this
    post!
2.  What about the bond lengths? The best way of analyzing
    these\[cite\]10.1021/ol703129z\[/cite\] is to compare pairs of
    so-called *meso*-bonds, being the coupler unit connecting any two
    pyrrole rings. Around the cycle, all the C-C meso-pairs are \~1.4Å
    and the C-N pairs are both \~1.34Å. That characteristic of benzene,
    in having all its (C-C) bonds equal, seems true here as well (at
    least at the B3LYP/6-311G(d,p) level, see
    e.g.[10042/to-2109](http://hdl.handle.net/10042/to-2109). There are
    reasons for thinking that in fact the B3LYP method does predict this
    behaviour more or less realistically). By the way, a molecule with a
    π-linking number of six is indeed classified by the same selection
    rule as benzene, ie 4n+2 (odd numbers of  Lk are governed by a 4n
    rule instead).\[cite\]10.1016/j.comptc.2014.09.028\[/cite\]

It is tempting to conclude that perhaps the extended conjugation of this
molecule (shown by the bond length equality) is somehow connected to the
dominance of writhe over twist in this trefoil.

I will follow this post up with another relating to the predicted
chiro-optical properties. For now, I leave its synthesis to be
contemplated by a present day Willstätter or Waser.

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

Webcite archive:. Chemistry with a super-twist: A molecular trefoil
knot, part 1. . 2010-06-01.
URL:http://www.ch.ic.ac.uk/rzepa/blog/?p=2046. Accessed: 2010-06-01.
(Archived by WebCite® at
[http://www.webcitation.org/5qA81X8qW](http://www.webcitation.org/5qA81X8qW){target="_blank"
rel="noopener noreferrer"})

::: {.wp_orcid_field}
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rel="author"}
:::


Something important happened in chemistry for the first time about 100 years ago. A molecule was built (nowadays we would say synthesized) specifically for the purpose of investigating a theory.

Messaggi di Rogue Scholar

Chemistry with a super-twist: A molecular trefoil knot, part 1.