Henry Rzepa's Blog

In 2001, Shaik and co-workers published the first of several famous review articles on the topic A Different Story of π-Delocalization. The Distortivity of π-Electrons and Its Chemical Manifestations[cite]10.1021/cr990363l[/cite]. The main premise was that the delocalized π-electronic component of benzene is unstable toward a localizing distortion and is at the same time stabilized by resonance relative to a localized reference structure.  Put more simply, the specific case of benzene has six-fold symmetry because of the twelve C-C σ-electrons and not the six π-electrons. In 2009, I commented here on this concept, via a calculation of the quintet state of benzene in which two of the six π-electrons are excited from bonding into anti-bonding π-orbitals, thus reducing the total formal π-bond orders around the ring from three to one. I focused on a particular vibrational normal mode, which is usefully referred to as the Kekulé mode, since it lengthens three bonds in benzene whilst shortening the other three. In this case the stretching wavenumber increased by ~207 cm-1 when the total π-bond order of benzene was reduced from three to one by spin excitation. In other words, each C-C bond gets longer when the π-electrons are excited, but the C-C bond itself gets stronger (in terms at least of the Kekulé mode).^† This behaviour is called a violation of Badger’s rule[cite]10.1063/1.1749433[/cite] for the relationship between the length of a bond and its stretching force constant. 

This blog has come about because I wanted to revisit my original calculations and complete them with a calculation for a heptet state of benzene in which three π-electrons are promoted from bonding into anti-bonding π-orbitals, thus resulting in a total π-bond order of zero. For completness, I here present the results not only for benzene but for some other small-annulene systems, both charged and neutral. These are all done at the coupled-cluster level of theory, both CCSD and CCSD(T), along with two basis set levels (see DOI: 10.14469/hpc/6624 for the whole collection of calculations).^‡

Before discussing the other systems, let your eye drop down the table below to the entries in red. These show the force constants for the singlet, quintet and heptet states of benzene vs the optimized C-C bond length for each (at the same level of theory). These confirm the earlier result in revealing that the quintet state (total ring π-bond order 1) has a longer bond but a stronger force constant for the Kekulé mode than the singlet state (total ring π-bond order 3). The heptet state now has a normal length C-C single bond (total ring π-bond order 0) but a Kekulé distorsion force constant higher than benzene itself! 

Things now start to get more complicated. Firstly, for benzene itself, reducing the remaining π-bond order from 1 to 0 on exciting from quintet to heptet substantially reduces the force constant. So one might conclude that reducing an annulene total π-bond order does not always result in an increase in force constant. Badger’s rule is not always violated and the distortivity of π-electrons may not be a linear phenomenon.

^aUnprojected, with possible Dushinsky coupling ^bProjected from Dushinksky coupling. In all cases, the excited states show -ve force constants for out of plane deformations, but the in-plane Kekule modes are all +ve except for the first entry.

I will now make some short comments about the other ring systems reported above.

1. Cyclobutadiene, dication. The Kekulé mode is very similar to benzene, but based clearly on just two π-electrons rather than six. There are two ways of forming a triplet state by exciting one of the two π-electrons to give a total π-bond order of zero. Both give a C-C distance a little longer than that typical of cyclobutanes (1.56Å). These triplet states however are not equilibrium species but transition states for the dissociation into two molecules of acetylene radical cation, a reaction driven no doubt by the large coulomb repulsions found for a di-cation.
2. Cyclobutadiene. The singlet ground state has Jahn-Teller effects (which by the way are absent from all the other excited states reported here), but the triplet state again has a Kekulé mode is very similar to benzene. Removing all the π-bond orders in the quintet reduces the Kekulé force constant. This is in contrast to benzene itself.
3. Cyclobutadiene, di-anion. Only the singlet state was calculable (the excited states did not converge), and now the Kekulé force constant is distinctly lower than benzene, probably again due to coulombic repulsions of the di-anion coupled with greater Pauli repulsions of the additional electrons. One other vibrational mode is worth showing here, the E_u mode (ν 1267 cm^-1) which shows interesting charge localisation into a carbon-centred anion and a delocalised allylic anion.
4. Cyclo-octatetraene Dication. Although isoelectronic with benzene, it shows very different behaviour. As the spin-multiplicity increases, so the Kekulé force constant decreases and the bond length increases, in accordance with Badger's rule. Again, another vibration (E_3u, ν 1544 cm^-1) shows charge localisation to give a 1,4-separated di-cation.
5. Cyclo-octatetraene. The triplet has a total ring π-bond order 3 (with two electrons in non-bonded orbitals) and a C-C bond length similar to benzene itself. The nonet state total ring π-bond order is reduced to 0, with a C-C length again identical to a single bond. As with the di-cation, the force constant is reduced as the bond length increases, in accordance with Badger's rule.
6. Cyclo-octatetraene di-anion is similar to the neutral system in following Badger's rule.

I do need to insert some caveats here. The original hypothesis[cite]10.1021/cr990363l[/cite] of distortive π-electrons was based on the singlet states (both ground and excited) and the results reported here are based on higher spin states, the assumption being that these are well-described by single reference states or configurations. It may well be that these higher spin states need more complex multi-reference determinants to describe them properly, in which case the coupled-cluster calculations reported here would be inappropriate. Thus for the larger rings, some of the CC calculations either failed to converge for large basis sets or gave some unphysical force constants (i.e. huge). This does tend to suggest that the internal MP expansion performed for coupled-cluster calculations is failing to converge, a well known propensity for systems where a multi-reference determinant is needed.

So one should not conclude too firmly that only benzene itself (in this series; there are many other examples to be found in [cite]10.1021/cr990363l[/cite]) exhibits Badger’s rule violations. Nonetheless, it would be valuable in the future to know whether the concept of distortivity of π-electrons can be applied to the small ring annulenes where the π-bond orders have been progressively reduced down to zero by specifying higher-spin π-states.

^†In 2014, I looked at some of the historical origins of this attribution to Kekulé, and you might also want to read the fascinating discussion by others on this topic.^‡I thank Sason Shaik for his comments on the above results!

This entry was posted on Wednesday, March 11th, 2020 at 1:22 pm and is filed under Interesting chemistry. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.