Imperial College London

Henry Rzepa

The geometry of cyclo-octatetraenes differs fundamentally from the lower
homologue benzene in exhibiting slow (nuclear) valence bond isomerism
rather than rapid (electronic) bond-equalising resonance. In 1992
Anderson and Kirsch\[cite\]10.1039/P29920001951\[/cite\] exploited this
property to describe a simple molecular balance for estimating how two
alkyl substituents on the ring might interact *via* the (currently very
topical) mechanism of dispersion
(induced-dipole-induced-dipole) attractions. These electron correlation
effects are exceptionally difficult to model using formal quantum
mechanics and are nowadays normally replaced by more empirical functions
such as Grimme\'s D3BJ correction.\[cite\]10.1002/jcc.21759\[/cite\]
Here I explore aspects of how the small molecule below might be used to
investigate the accuracy of such estimates of dispersion energies.

[![bu](http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2016/02/bu.svg){.aligncenter
.size-large .wp-image-15672 decoding="async"
width="300"}](http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2016/02/bu.svg){rel="attachment wp-att-15672"}

The concentration of the two forms shown above can be readily estimated
by NMR spectroscopy (the barrier is slow enough to allow peaks for both
isomers to be integrated). This shows that the 1,6 form is present in
greater concentrations than the 1,4 form, equivalent to a difference in
free energy ΔΔG~298~ of 0.39 kcal/mol in favour of the former. Why is
this? Because, it is claimed,  in the 1,6 isomer the two t-butyl groups
are close enough to experience mutual dispersion attractions not
experienced by the 1,4 form. This can be illustrated using the NCI
display below for the two forms.

::: {#attachment_15688 .wp-caption .aligncenter style="width: 410px"}
![Click for 3D. Addition NCI interactions ringed in
red.](http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2016/02/6-nci.jpg){.size-full
.wp-image-15688 decoding="async"
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width="400"
srcset="https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2016/02/6-nci.jpg 712w, https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2016/02/6-nci-300x222.jpg 300w"
sizes="(max-width: 712px) 100vw, 712px"}

Click for 3D. 1,6-isomer: Additional NCI interactions ringed in red.
:::

::: {#attachment_15687 .wp-caption .aligncenter style="width: 410px"}
![Click for
3D](http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2016/02/4-nci.jpg){.size-full
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width="400"
srcset="https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2016/02/4-nci.jpg 751w, https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2016/02/4-nci-300x169.jpg 300w"
sizes="(max-width: 751px) 100vw, 751px"}

Click for 3D, 1,4 isomer.
:::

|                                           |                            |      |          |          |          |                                                                         |
|-------------------------------------------|----------------------------|------|----------|----------|----------|-------------------------------------------------------------------------|
| Method                                    | Equilibrium constant, 298K | ΔΔE  | ΔΔH~298~ | ΔΔS~298~ | ΔΔG~298~ | Source                                                                  |
| Experiment                                | 1.93                       | --   | 1.14     | -2.5     | 0.387    | \[cite\]10.1039/P29920001951\[/cite\]                                   |
| B3LYP/Def2-TZVPP/CDCl3 (no dispersion)    | 1.906^‡^                   | 0.05 | 0.00     | +1.3^‡^  | 0.382^‡^ | \[cite\]10.14469/ch/191875\[/cite\],\[cite\]10.14469/ch/191876\[/cite\] |
| B3LYP/Def2-TZVPP/CDCl3 (gd3bj dispersion) | 8.36                       | 0.75 | 0.66     | +2.0     | 1.25     | \[cite\]10.14469/ch/191874\[/cite\],\[cite\]10.14469/ch/191880\[/cite\] |

^‡^[This contains a contribution of RTLn 2 (= 0.410 kcal/mol = 1.04 in
ΔS]{style="font-size:9px;"}[), where 2 is the symmetry number for a
species with C~2~ rotational symmetry, to the 1,4-isomer
only.]{style="font-size:9px;"}

The interpretation of these results, as is often found, is non-trivial.

1.  The relative concentrations of species in equilibrium equates with
    their relative free energies, ΔG~298 ~and not ΔE (the difference in
    total energy computed using either quantum or molecular mechanics).
2.  ΔG~298 ~ has a component derived from the entropy of the system, and
    this in turn has contributions from symmetry (numbers).  Only the
    1,6-isomer has two-fold rotational symmetry for the lowest energy
    pose of the two t-butyl groups, and this contributes 0.41 kcal/mol
    to ΔG~298~. This aspect is not discussed in the original
    article.\[cite\]10.1039/P29920001951\[/cite\]
3.  The B3LYP/Def2-TZVPP DFT method predicts ΔΔE to be +0.05 kcal/mol
    without the inclusion of the D3BJ dispersion correction but +0.75
    kcal/mol with. One might approximately equate the latter to the
    contributions ringed in red in the NCI distributions shown above.
    The enthalpies (where ΔΔE is corrected for zero point energies) are
    very similar.
4.  Conversion to ΔG~298 ~involves use of the vibrational frequencies to
    obtain the entropy; here one encounters a difference between the two
    double bond isomers. The lowest energy vibration for C~2~-symmetric
    1,4 is 23 cm^-1^, whereas that for the 1,6 is only 7 cm^-1^ (a value
    which also depends on round-off errors and accuracies in the
    calculation). These errors in the RRHO
    (rigid-rotor-harmonic-oscillator) approximations makes meaningful
    calculation of ΔS~298~ and hence~ ~ΔG~298~ problematic at this small
    energy difference level. In both cases, this approach suggests that
    the entropy of the 1,6 form is slightly larger than the 1,4 isomer,
    whereas the reverse is apparently true by experimental measurement.
    It might all boil down to those low-frequency vibrations!

So we may conclude that whereas the dispersion uncorrected method gets
the right answer for the equilibrium constant for probably the wrong
reasons, inclusion of a dispersion correction would get the right answer
were it not for the error in the entropy. Agreement with experiment
would be obtained if the calculated entropy difference were to be -0.9
kcal/mol K^-1^ instead of +2.0. Thus the 1,6 isomer has the two t-butyl
groups weakly interacting (red circle above), which intuition tends to
suggest would reduce the entropy (reduce the disorder) of the system and
not increase it. 

At least in this relatively small molecule, we now have a handle for
estimating these sorts of effects in terms of variables such as the
basis set used, the energy Hamiltonian (e.g. type of functional etc) and
of course the dispersion correction.

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:::


The geometry of cyclo-octatetraenes differs fundamentally from the lower homologue benzene in exhibiting slow (nuclear) valence bond isomerism rather than rapid (electronic) bond-equalising resonance.

A molecular balance for dispersion energy?

By about C17H36, the geometry of “cold-isolated” unbranched saturated alkenes is supposed not to contain any fully anti-periplanar conformations.

By about C~17~H~36~, the geometry of "cold-isolated" unbranched
saturated alkenes is supposed not to contain any fully anti-periplanar
conformations. \[cite\]10.1002/anie.201202894\[/cite\] Indeed, a
(co-crystal) of C~16~H~34~ shows it to have two-gauche
bends.\[cite\]10.1002/chem.200801428\[/cite\]. Surprisingly, the longest
linear alkane I was able to find a crystal structure for,
C~28~H~58~ appears to be fully
extended\[cite\]10.1107/S0108768191011059\[/cite\],\[cite\]10.1107/S0567740876005025\[/cite\]
(an early report of a low quality structure for
C~36~H~74~\[cite\]10.1107/S0365110X5600111X\[/cite\] also appears to
show it as linear).^‡^ Here I explore how standard DFT theories cope
with these structures.

I start with noting the use of a TZVP basis set. In a recent
article\[cite\]10.1039/C3SC53416B\[/cite\] we noted that the
basis-set-superposition-errors for this basis were about a quarter of
that for the standard Pople-type 6-311G(d,p) basis that I tend to use
for modelling in this blog. This matters, since the relative energy of a
folded-conformation *vs* an extended linear one might depend on the
quality of the basis set and its inherent BSSE. The DFT method is the
classical B3LYP. I also modelled C~58~H~118~ as the hydrocarbon as being
well beyond the region anticipated above for folding of the chain to
have started (no, there is no crystal structure). The geometries of
linear and bent forms are shown below.

[![003](http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/003.jpg){.aligncenter
.size-full .wp-image-12216 decoding="async" width="400"
srcset="https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/003.jpg 838w, https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/003-300x17.jpg 300w"
sizes="(max-width: 838px) 100vw, 838px"}](http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/003.jpg)[![001](http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/001.jpg){.aligncenter
.size-full .wp-image-12205 decoding="async" width="400"
srcset="https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/001.jpg 807w, https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/001-300x205.jpg 300w"
sizes="(max-width: 807px) 100vw, 807px"}](http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/001.jpg)

The relative free energy of the V-shaped bent
form\[cite\]10.6084/m9.figshare.978501\[/cite\] emerges as 3.5 kcal/mol
**higher** than the linear
form\[cite\]10.6084/m9.figshare.978502\[/cite\]. Now, to add a Grimme-D3
dispersion correction to the energies. The V-shape of the bent form now
adopts the hairpin mode,\[cite\]10.6084/m9.figshare.978832\[/cite\] and
its energy is now 2.5 kcal/mol **lower** than the linear
form.\[cite\]10.6084/m9.figshare.978833\[/cite\]

[![002](http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/002.jpg){.aligncenter
.size-full .wp-image-12215 decoding="async" width="400"
srcset="https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/002.jpg 1040w, https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/002-300x53.jpg 300w, https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/002-1024x181.jpg 1024w, https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/002-900x159.jpg 900w"
sizes="(max-width: 1040px) 100vw, 1040px"}](http://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2014/03/002.jpg)

Note in the above the very slight strange oscillation (kink) that
appears about 11 atoms away from the hairpin bend. I repeated this with
the wB97XD DFT procedure (in which dispersion is implicit) and found the
same result.

As triple-ζ basis quality modelling of molecules with \>100 atoms
becomes increasingly common, it is worth repeating yet again that the
model should always contain dispersion (and solvent if appropriate)
corrections as default. Indeed, it is probably also worth
re-investigating much early modelling (by this I mean modelling done ten
or more years ago) to see if such corrections significantly influence
the conclusions.\[cite\]10.1039/C3SC53416B\[/cite\]

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

^‡^The searches cannot be carried out according to the formula
C~n~H~2n+2~, but must be done individually for the value of n. I gave up
at C~50~.

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rel="author"}
:::


By about C
<sub>
 17
</sub>
H
<sub>
 36
</sub>
, the geometry of “cold-isolated” unbranched saturated alkenes is supposed not to contain any fully anti-periplanar conformations.

Messaggi di Rogue Scholar

A molecular balance for dispersion energy?

Modelling the geometry of unbranched alkanes.