Imperial College London

Henry Rzepa

A [little while
ago](http://www.ch.imperial.ac.uk/rzepa/blog/?p=2737){target="_blank"
rel="noopener"}, I set out some interpretations of how to push curly
arrows. I also appreciate that some theoretically oriented colleagues
regard the technique as neither useful nor in the least
*rigorous,* whereas towards the other extreme many synthetically minded
chemists view the ability to push a *reasonable* set of arrows for a
proposed mechanism as of itself constituting evidence in its
favour.\[cite\]10.1186/1752-153X-7-94\[/cite\] Like any language for
expressing ideas, the tool needs a grammar (rules) and a vocabulary, and
perhaps also an ability to carry ambiguity. These thoughts surfaced
again *via* a question asked of me by a student: "is the mechanism for
the hydrogens in protonated benzene whizzing around the ring a \[1,2\]
or a \[1,6\] pericyclic sigmatropic shift?". 

![12-v-16](http://www.ch.imperial.ac.uk/rzepa/blog/wp-content/uploads/2013/06/12-v-16.svg){.aligncenter
.size-full .wp-image-10708 decoding="async" width="440"}

I must first explain the "rules" I used to produce the diagram above.

1.  [An arrow comprises an (approximate) electron pair (in the Lewis
    sense)]{style="line-height: 13px;"}
2.  And it operates on individual resonance/valence bond representations
3.  A sigmatropic shift is a pericyclic reaction, and one should be able
    to construct arrows which both conjugate in a cyclic sense, and
4.  can be represented in either a clockwise or anticlockwise cyclic
    direction.
5.  As a thermally allowed pericyclic reaction, we will endeavour to use
    only 4n+2 electrons in our arrow pushing (an aromatic transition
    state for the process). For this molecule, it means either 2 or 6
    electrons (one or three arrows, again in a Lewis sense)
6.  I also use the convention in which the origin of an arrow is the
    midpoint of a (covalent) bond, and its destination the midpoint of a
    forming (covalent) bond, represented above by a dashed line. Here, I
    am guided by the observation that the "coordinates" of the start and
    the end-point of such arrows can be obtained by QTAIM theory, which
    provides them *via* so-called "*bond critical points*".

So for the 2-electron/one arrow process, we should be able to come up
with two schemes, one clockwise, one anticlockwise. It is really easy to
show one of these (rhs for \[1,2\]), but it is a true head-scratching
time to produce the other. I have done so *via* the red arrow in
representation **1** above. More of this in a moment.

For the 6-electron/3-arrow alternative, we start with three distinct
resonance/valence bond forms for the pentadienyl cation, and each of
these again should manifest as arrows drawn in both clock directions.
Five cause no problems, but the sixth again carries a very odd arrow
(red, representation **2** above). 

What does that red arrow mean? Dissected into smaller steps, it consists
of two arrows, joined head to tail in a daisy chain, but still clearly
counting as just a single electron pair. The first component of this
portmanteau arrow represents the formation of benzene and a proton, the
geometry of which in fact resembles the formation of a π-complex between
the two. The second component represents the breakdown of this complex
to reform a classical carbocation. The daisy-chained arrow passes
through the middle of a single bond, which achieves the necessary cyclic
conjugation, and then continues on its way to a final destination at the
mid-point (approximately) of a C-H bond. If it is drawn as a simple
arrow (avoiding passage through the single bond), the mechanism "does
not work"; the wrong carbon gets connected to the shifting hydrogen.

So this little thought experiment has produced a new entry in the
arrow-pushing vocabulary; a daisy-chained arrow, and a new rule which
requires it to pass through (interact with) a bond, but without
stopping. To ask if this invention has any connection to reality, its
time for a full-blown quantum mechanical computation of that transition
state (an IRC for which\[cite\]10.6084/m9.figshare.717183\[/cite\] is
shown below).

![12-16](http://www.ch.imperial.ac.uk/rzepa/blog/wp-content/uploads/2013/06/12-16.gif){.aligncenter
.size-full .wp-image-10717 loading="lazy" decoding="async" width="353"
height="287"}

If you look very carefully at how the representation of the
magenta-coloured  bond (scheme above) changes during the IRC, you will
note that at the transition state it is shorter (1.439Å) than at the
start or the end (1.462Å). The magenta-bond order at the transition
state is 1.17 (Wiberg), and the bond is clearly "involved" in the
process. This does hint that the representation **1** or **2** in the
scheme above may be making a significant contribution. None of the other
forms of arrow pushing in that scheme involve any arrows either starting
or ending at that bond, so they cannot account for the change in bond
length/bond order of that bond. We interpret the daisy-chained arrow as
carrying more information than a normal arrow would regarding the timing
of the process; the path up to the transition state is represented by
the first part of the double-headed arrow, and the path down to the
product by the second component of that arrow. So up to the first
arrow-head, electrons are moving towards the magenta bond, and in the
second phase, they continue their journey by moving onwards and away
from that bond.

Is there any prior art (precedent) for such a process? We actually,
there is a very nice example; the dyotropic rearrangement of a
1,2-dibromoethane (which was recently experimentally shown to be a
concerted process with double inversion of configuration at the carbon
bearing the bromine\[cite\]10.1039/C2CC33676F\[/cite\]). That too
involves a transition state where the central C-C bond is shorter
(1.41Å) than either reactant or product
(1.50Å)\[cite\]10.1002/chem.200501517\[/cite\]. We may represent the
arrows involved in the process in two ways in a manner analogous to
structure **1** above, but this time involving two daisy-chained arrows
(red and blue below). This does not necessarily imply that at the
transition state a full triple bond has formed, merely that as the arrow
"passes through the bond" its bond-order temporarily increases. The
conventional arrow pushing on the right implies no change whatsoever in
that central bond. Like any set of (non-equivalent) resonance
representations, these two sets of arrows *taken together* may be a more
realistic one than either individually for the overall reaction.

![dyotropic](http://www.ch.imperial.ac.uk/rzepa/blog/wp-content/uploads/2013/06/dyotropic.svg){.aligncenter
.size-full .wp-image-10718 decoding="async"}

I conclude with some thoughts about another question: is the process a
\[1,2\] or a \[1,6\] sigmatropic pericyclic reaction? If you regard
pericyclic reactions as manifestations of aromatic transition states,
you could equally well ask; given a choice, does a system prefer
2-electron or 6-electron aromaticity (both of course conform to a 4n+2
rule). To try to cast light on that, I show the computed NICS values at
two points (magenta spheres), the first being the RCP (ring critical
point in a QTAIM analysis of the transition state) of the 6-ring and the
second the BCP of the 3-ring. If we calibrate this to -10 ppm (for
benzene itself), the 6-ring is seen to be only modestly aromatic on this
scale. The 3-ring (corresponding to 2-electron aromaticity) appears to
be highly aromatic, but this is probably due to a large contribution
from local shielding effects; NICS for small rings is not reliable. But
this does constitute a hint that, all other aspects being equal,
2-electron aromaticity may have the edge over 6-electron aromaticity.
Clearly more work is needed on this aspect.

![12-16](http://www.ch.imperial.ac.uk/rzepa/blog/wp-content/uploads/2013/06/12-16.jpeg){.aligncenter
.wp-image-10722 loading="lazy" decoding="async" width="331"
height="224"}

Whether or not you believe in the theoretical rigour of arrow pushing
(or indeed its absence of rigour), I would suggest that it has proved a
useful tool, a mechanism if you like, for helping to think about how
reactions proceed. Certainly I also consider it desirable, if arrow
pushing is to continue as a useful tool up to its [100th
birthday](http://www.ch.imperial.ac.uk/rzepa/blog/?p=7234 "The first ever curly arrows."){target="_blank"
rel="noopener"}, that some effort should be devoted to updating it for
21st century chemistry.

\[cite\]http://hdl.handle.net/10042/24720\[/cite\]

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


A little while ago, I set out some interpretations of how to push curly arrows.

Messaggi di Rogue Scholar

Mechanistic arrow pushing. A proposed addition to its rules.