I have discussed the vibration in benzene known as the Kekulé mode in other posts, the first of which was all of ten years ago. It is a stretching mode that lengthens three of the bonds in benzene (a [6]-annulene) and shortens the other three, thus leading to a cyclohexatriene motif (see below). This vibration is real (+ve force constant) in benzene itself, which indicates that distorting the structure from six to three-fold symmetry leads to an increase in energy. Benzene therefore has a symmetrising influence, and it comes as a surprise to most to learn that this is actually due to the σ rather than the π-electrons! But there are good reasons to believe that as the ring size of the annulene increases, the Kekulé vibration will evolve from a real mode into an imaginary (-ve force constant) vibration representing a transition state for mutating the single and double bonds. At some point therefore, the more symmetrical geometry of the annulene in which all the bonds are of equal length will change into one of lower symmetry, in which BLA (bond length alternation) occurs and the symmetrical form becomes a transition state for this process.
With this background, I noticed that a form of [18] annulene in which all the hydrogens have been removed (and is therefore another allotrope of carbon) has recently been synthesized and individual molecules studied on a metal surface using STM (a scanning tunnelling microscope).[1] This allotrope is also of interest as a “double aromatic” molecule, with 4n+2 electron aromaticity arising from both the π and the σ system.
Cyclo[18]carbon, as this form is known, could have either 18-fold symmetry with no bond alternation, or 9-fold symmetry in which alternating short and long bond occur. The STM conclusions pointed to the form of structure with alternating bonds; these are attributed to triple and single in the article, but in fact no actual accurate bond lengths were (or can be) measured to directly support this.
I thought it might be interesting to see how various forms of computational quantum calculation might reflect this new experiment. In order to exploit the higher 18-fold symmetry to reduce calculation time, only the geometry with 18 equal bond lengths is computed here, along with the calculated value of the Kekulé mode. All the calculations are presented as FAIR data at DOI: 10.14469/hpc/6038
Method | Kekulé mode, cm-1 | CC bond length, Å | Number of -ve force constants |
---|---|---|---|
Density functional methods | |||
B3LYP+DG3+BJ/6-31G(d) | +766 | 1.284 | 5, all in-plane bucklings |
B3LYP+DG3+BJ/6-311G(d) | +699 | 1.278 | 1, in-plane buckling |
B3LYP+DG3+BJ/Def2-TZVPP | +673 | 1.276 | 0 |
B3LYP+DG3+BJ/Def2-QZVPP | +649 | 1.275 | 0 |
ωB97X-D/Def2-TZVPP | -2058 | 1.273 | 2, Kekulé + in-plane buckling |
Double hybrid methods | |||
B2PLYPD3/Def2-TZVPP | +2598 | 1.281 | 0 |
DSDPBEP86/Def2-TZVPP | +3470 | 1.283 | 0 |
PBEQIDH/Def2-TZVPP | +3447 | 1.277 | 0 |
Møller-Plesset methods | |||
MP2/6-31G(d) | +20444 | 1.295 | 8, in/out-of-plane buckling |
MP2/6-311G(d) | +19885 | 1.293 | 8, in/out-of-plane buckling |
MP2/Def2-TZVPP | +19466 | 1.289 | 0 |
MP3/6-311G(d) | -15637 | 1.284 | 11, Kekulé + in/out-of-plane buckling |
MP4(SDQ)/6-31G(d) | -9496 | 1.287 | 11, Kekulé + in/out-of-plane buckling |
Coupled Cluster methods | |||
CCSD/6-31G(d) | -4564 | 1.288 | 11, Kekulé + in/out-of-plane buckling |
CCSD/6-311G(d) | -4310 | 1.286 | 11, Kekulé + in/out-of-plane buckling |
CCSD(T)/6-31G(d) | N/A | 1.276 | N/A |
The conclusions include:
So, are the MP and CCSD methods converging to a reliable solution, or are they oscillating too much to make any conclusions? If we think the convergence is approaching, then the Kekulé (transition state) mode for C18 (shown below) would indeed correspond to an interpretation of the STM observations as BLA.
But there might be an alternative explanation, that instead the molecule buckles in the manner illustrated below. This would also lead to 9-fold symmetry.
I conclude by pondering why the convergence of these methods is so strange. They are all single-reference methods, and perhaps C18 depends instead on multi-reference states? This would need a MCSCF (multi-configuration) or VB (valence bond) approach. Given the need for accurate basis sets, this is probably a big ask, but perhaps some group out there can do this and compare with the results here?
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