The singlet and open shell higher-spin states of [4], [6] and [8]-annulenes and their Kekulé vibrational modes

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[1]. 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[2] 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.

State Method “Kekule”
Mode, cm-1
mass, AMU
length, Å



Cyclobutadiene, dication
1A1g CCSD(T)/Def2-TZVPP 1383a 5.4102 6.5340 1.449 6920
1329b 8.0442 7.7338
3B1g CCSD/Def2-TZVPP -2195a -33.0943 11.6515 1.589 6944
-2211b -17.6740 6.1313
3A1g CCSD/Def2-TZVPP -2171a -32.2296 11.6023 1.593 6933
-2189b -16.0614 5.6844
3A1g CCSD/Def2-SVP 1422a 7.7848 6.5340 1.444 6634
1373b 7.9700 7.1807
CCSD(T)/Def-SVP 1395 7.2647 6.3381 1.449 6643
1345 7.6931 7.2202
CCSD/Def2-TZVPP 1392 6.9173 6.0600 1.438 6671
1342 8.0252 7.5582
CCSD(T)/Def2-TZVPP 1360 7.2647 6.3381 1.449 6672
1310 7.7044 7.6151

5B1g CCSD/Def2-SVP 1192 2.2102 2.6391 1.566 6635
1088 4.9656 7.1153
CCSD(T)/Def-SVP 1176 2.0739 2.5452 1.569 6644
1069 4.7829 7.0983
CCSD/Def-TZVPP 1177 1.8783 2.3023 1.563 6636
1067 5.0294 7.5000
CCSD(T)/Def-TZVPP 1157 1.7555 2.2253 1.568 6678
1045 4.8322 7.5073
Cyclobutadiene Di-anion (isoelectronic with benzene)
1A1g CCSD/Def2-SVP 1283 5.9023 6.0872 1.470 6652
1233 6.3831 7.1172
CCSD(T)/Def2-SVP 1258 5.5888 5.9952 1.475 6653
1209 6.1657 7.1565
CCSD/Def2-TZVPP 1216 4.3487 4.9883 1.467 6676
1165 6.2282 7.7827
CCSD(T)/Def2-TZVPP 1187 4.1223 4.9621 1.473 6679
1138 6.0103 7.8734
1A1g CCSD/Def2-SVP 1337 4.9922 4.7425 1.401 6623
1308 10.7126 10.6244
CCSD(T)/Def2-SVP 1359 6.9812 6.4190 1.405 6647
1339 11.3860 10.7785
CCSD/Def2-TZVPP 1309 3.5714 3.5358 1.392 6646
1273 10.1936 10.6709
CCSD(T)/Def2-TZVPP 1328 5.6130 5.4004 1.398 6710
1306 10.9097 10.8588

5A1g CCSD/Def2-SVP 1600 17.4689 11.5855 1.463 6626
1597 17.4799 11.6274
CCSD/Def2-TZVPP 1572 17.1041 11.7511 1.455 6669
1571 17.1770 11.8198

7B1u CCSD/Def2-SVP 1361 11.0797 10.1574 1.550 6632
1355 12.4274 11.4839
Cyclo-octatetraene Dication (isoelectronic with benzene)
1A1g CCSD/Def2-SVP 1750 21.4531 11.8876 1.414 6648
1750 21.5770 11.9620
CCSD(T)/Def2-SVP 1707 20.4695 11.9226 1.420 6673
1707 20.5570 11.9773

5A1g CCSD/Def2-SVP 1702 20.4349 11.9684 1.444 6663
1702 20.4690 11.9899

7B1g CCSD/Def2-SVP 1460 14.8761 11.8407 1.511 6675
1460 15.0578 11.9890
3B2g CCSD/Def2-SVP 1777 22.3143 11.9956 1.409 6637
1777 22.3176 11.9976

7A1g CCSD/Def2-SVP 1637 18.9335 11.9851 1.484 6639
1637 18.9464 11.9941

9B1g CCSD/Def2-SVP 1441 14.6711 11.9988 1.553 6640
1441 14.6723 11.9998
CCSD(T)/Def2-SVP 1421 13.4795 11.3256 1.555 6677
Cyclo-octatetraene dianion
1A1g CCSD/Def2-SVP 1731 21.1621 11.9903 1.419 6695
1731 21.1668 11.9937

5A1g CCSD/Def2-SVP 1642 19.0214 11.9711 1.461 6698
1642 19.0408 11.9852

9B2u CCSD/Def2-SVP 1517 16.0290 11.8174 1.528 6700
1517 16.1516 11.9186

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 Eu 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 (E3u, 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[1] 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 [1]) 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!


  1. S. Shaik, A. Shurki, D. Danovich, and P.C. Hiberty, "A Different Story of π-DelocalizationThe Distortivity of π-Electrons and Its Chemical Manifestations†", Chemical Reviews, vol. 101, pp. 1501-1540, 2001.
  2. R.M. Badger, "A Relation Between Internuclear Distances and Bond Force Constants", The Journal of Chemical Physics, vol. 2, pp. 128-131, 1934.

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