{"id":21176,"date":"2019-08-30T15:05:56","date_gmt":"2019-08-30T14:05:56","guid":{"rendered":"https:\/\/www.ch.imperial.ac.uk\/rzepa\/blog\/?p=21176"},"modified":"2025-12-18T16:31:43","modified_gmt":"2025-12-18T16:31:43","slug":"cyclo18carbon-the-kekule-vibration-calculated-and-hence-a-mystery","status":"publish","type":"post","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=21176","title":{"rendered":"Cyclo[18]carbon: The Kekul\u00e9 vibration calculated and hence a mystery!"},"content":{"rendered":"
\n

I have discussed the vibration in benzene known as the Kekul\u00e9 mode in other<\/a> posts, the first of which was all of ten years ago<\/a>. 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<\/strong> 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\u00a0\u03c3 rather than the \u03c0-electrons<\/a>! But there are good reasons to believe that as the ring size of the annulene increases, the Kekul\u00e9 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.<\/p>\n

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]<\/a><\/span> This allotrope is also of interest as a “double aromatic” molecule, with 4n+2 electron aromaticity arising from both the\u00a0\u03c0 and the \u03c3 system.<\/p>\n

\"The<\/p>\n

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.<\/p>\n

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 Kekul\u00e9 mode. All the calculations are presented as FAIR data[2]<\/a><\/span>.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Method<\/th>\nKekul\u00e9 mode, cm-1<\/th>\nCC bond length, \u00c5<\/th>\nNumber of -ve force constants<\/th>\n<\/tr>\n
Density functional methods<\/td>\n<\/tr>\n
B3LYP+DG3+BJ\/6-31G(d)<\/td>\n+766<\/td>\n1.284<\/td>\n5, all in-plane bucklings<\/td>\n<\/tr>\n
B3LYP+DG3+BJ\/6-311G(d)<\/td>\n+699<\/td>\n1.278<\/td>\n1, in-plane buckling<\/td>\n<\/tr>\n
B3LYP+DG3+BJ\/Def2-TZVPP<\/td>\n+673<\/td>\n1.276<\/td>\n0<\/td>\n<\/tr>\n
B3LYP+DG3+BJ\/Def2-QZVPP<\/td>\n+649<\/td>\n1.275<\/td>\n0<\/td>\n<\/tr>\n
\u03c9B97X-D\/Def2-TZVPP<\/td>\n-2058<\/td>\n1.273<\/td>\n2, Kekul\u00e9 + in-plane buckling<\/td>\n<\/tr>\n
Double hybrid methods<\/td>\n<\/tr>\n
B2PLYPD3\/Def2-TZVPP<\/td>\n+2598<\/td>\n1.281<\/td>\n0<\/td>\n<\/tr>\n
DSDPBEP86\/Def2-TZVPP<\/td>\n+3470<\/td>\n1.283<\/td>\n0<\/td>\n<\/tr>\n
PBEQIDH\/Def2-TZVPP<\/td>\n+3447<\/td>\n1.277<\/td>\n0<\/td>\n<\/tr>\n
M\u00f8ller-Plesset methods<\/td>\n<\/tr>\n
MP2\/6-31G(d)<\/td>\n+20444<\/td>\n1.295<\/td>\n8, in\/out-of-plane buckling<\/td>\n<\/tr>\n
MP2\/6-311G(d)<\/td>\n+19885<\/td>\n1.293<\/td>\n8, in\/out-of-plane buckling<\/td>\n<\/tr>\n
MP2\/Def2-TZVPP<\/td>\n+19466<\/td>\n1.289<\/td>\n0<\/td>\n<\/tr>\n
MP3\/6-311G(d)<\/td>\n-15637<\/td>\n1.284<\/td>\n11, Kekul\u00e9 + in\/out-of-plane buckling<\/td>\n<\/tr>\n
MP4(SDQ)\/6-31G(d)<\/td>\n-9496<\/td>\n1.287<\/td>\n11, Kekul\u00e9 + in\/out-of-plane buckling<\/td>\n<\/tr>\n
Coupled Cluster methods<\/td>\n<\/tr>\n
CCSD\/6-31G(d)<\/td>\n-4564<\/td>\n1.288<\/td>\n11, Kekul\u00e9 + in\/out-of-plane buckling<\/td>\n<\/tr>\n
CCSD\/6-311G(d)<\/td>\n-4310<\/td>\n1.286<\/td>\n11, Kekul\u00e9 + in\/out-of-plane buckling<\/td>\n<\/tr>\n
CCSD(T)\/6-31G(d)<\/td>\nN\/A<\/td>\n1.276<\/td>\nN\/A<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The conclusions include:<\/p>\n

    \n
  1. The quality of the basis set is crucial. For lower quality bases, in-plane deformations of the ring occur, but the Kekul\u00e9 mode itself is relatively unaffected. The minimum converged basis is Def2-TZVPP, which rather restricts the ability to perform higher level calculations.<\/li>\n
  2. Even within the DFT methods, two popular functionals given diametrically opposite results. The B3LYP procedure predicts no BLA, whereas the alternative \u03c9B97X-D functional suggests strong BLA. I could have proceeded through the functional zoo and added 100s more functionals to this list, but the point I wanted to make is already established!<\/li>\n
  3. Double hybrid methods, which combine exact HF exchange with an MP2-like correlation, sometimes used as the next step up Jacob’s ladder, predict no BLA. With a very high real Kekul\u00e9 mode (2600-3500 cm-1<\/sup>) they provide a first warning sign something is not quite right?<\/li>\n
  4. Moller-Plesset expansions such as MP2 itself gives an unrealistic positive force constant for the\u00a0Kekul\u00e9 mode. The perturbation expansions are a notorious example that for a physically realistic result, the expansion has to converge and moreover is often assumed to converge rapidly. If the expansion is proven non-convergent, then neither the MP method itself, nor other methods which make use of it such as the double hybrid and the coupled cluster methods, can be trusted. For MP2 itself, the Kekul\u00e9 mode has a value that indicates serious issues with the single-reference wavefunction used. The MP3 and MP4 expansions illustrate nicely the oscillation of the method.<\/li>\n
  5. Coupled cluster methods, which also make use of MP expansions, are regarded as the next step up Jacob’s ladder. The computational cost of these scales so quickly that the larger basis sets are not feasible and so the (already proven as deficient) 6-31G(d) basis must be used. Again, we see the CCSD method mirroring the MP2-4 expansions.<\/li>\n<\/ol>\n

    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 Kekul\u00e9 (transition state) mode for C18<\/sub> (shown below) would indeed correspond to an interpretation of the STM observations as BLA.<\/p>\n

    \"\"<\/a><\/p>\n

    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.<\/p>\n

    \"\"<\/a><\/p>\n

    I conclude by pondering why the convergence of these methods is so strange. They are all single-reference methods, and perhaps C18<\/sub> 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?<\/p>\n

    References<\/h2>\n
    1. K. Kaiser, L.M. Scriven, F. Schulz, P. Gawel, L. Gross, and H.L. Anderson, \"An sp-hybridized molecular carbon allotrope, cyclo[18]carbon\", Science<\/i>, vol. 365, pp. 1299-1301, 2019. https:\/\/doi.org\/10.1126\/science.aay1914<\/a>\n\n<\/li>\n
    2. H. Rzepa, \"Cyclo[18]carbon: The Kekule vibration calculated\", 2019. https:\/\/doi.org\/10.14469\/hpc\/6038<\/a>\n\n<\/li>\n<\/ol>\n\n<\/div> ","protected":false},"excerpt":{"rendered":"

      I have discussed the vibration in benzene known as the Kekul\u00e9 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). 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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). 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Yes!","author":"Henry Rzepa","date":"October 19, 2019","format":false,"excerpt":"Increasingly, individual small molecules are having their structures imaged using STM, including cyclo[18]carbon that I recently discussed. The latest one receiving such treatment is Kekulene. As with cyclo[18]carbon, the point of interest was which of the two resonance structures shown below most closely resembled the measured structure. The one on\u2026","rel":"","context":"In "Interesting chemistry"","block_context":{"text":"Interesting chemistry","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=4"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.ch.ic.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2019\/10\/b3-1423-B2u-1024x575.gif?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]},{"id":12329,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=12329","url_meta":{"origin":21176,"position":1},"title":"More (blog) connections spotted. Something new about diphenyl magnesium?","author":"Henry Rzepa","date":"April 17, 2014","format":false,"excerpt":"I have just noticed unexpected links between two old posts, one about benzene, one about diphenyl magnesium\u00a0and\u00a0a link to August Kekul\u00e9.\u2020 The post about benzene dealt with the apparently simple issue of why all the C-C bonds are of equal length. 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The discussion about the norbornyl cation concentrated on whether this species existed in a single minimum symmetric energy well (the non-classical Winstein\/Olah proposal) or a double minimum well connected\u2026","rel":"","context":"In "Interesting chemistry"","block_context":{"text":"Interesting chemistry","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=4"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":12560,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=12560","url_meta":{"origin":21176,"position":3},"title":"Benzene: an oscillation or a vibration?","author":"Henry Rzepa","date":"May 28, 2014","format":false,"excerpt":"In the preceding post, a nice discussion broke out about Kekul\u00e9's 1872 model for benzene. This model has become known as the oscillation hypothesis between two extreme forms of benzene (below). The discussion centered around the semantics of the term oscillation compared to vibration (a\u00a0synonym or not?) and the timescale\u2026","rel":"","context":"In "Historical"","block_context":{"text":"Historical","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=565"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":21925,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=21925","url_meta":{"origin":21176,"position":4},"title":"The singlet and open shell higher-spin states of [4], [6] and [8]-annulenes and their Kekul\u00e9 vibrational modes","author":"Henry Rzepa","date":"March 11, 2020","format":false,"excerpt":"In 2001, Shaik and co-workers published the first of several famous review articles on the topic\u00a0A Different Story of \u03c0-Delocalization. The Distortivity of \u03c0-Electrons and Its Chemical Manifestations. The main premise was that the delocalized \u03c0-electronic component of benzene is unstable toward a localizing distortion and is at the same\u2026","rel":"","context":"In "Interesting chemistry"","block_context":{"text":"Interesting chemistry","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=4"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.ch.ic.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2020\/03\/CBD-Eu.gif?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]},{"id":485,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=485","url_meta":{"origin":21176,"position":5},"title":"Longer is stronger.","author":"Henry Rzepa","date":"June 6, 2009","format":false,"excerpt":"The iconic diagram below represents a cornerstone of organic chemistry. Generations of chemists have learnt early on in their studies of the subject that these two representations of where the electron pairs in benzene might be located (formally called electronic resonance or valence bond forms) each contribute ~50% to the\u2026","rel":"","context":"In "Interesting chemistry"","block_context":{"text":"Interesting chemistry","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=4"},"img":{"alt_text":"The Kekule structures of benzene.","src":"https:\/\/i0.wp.com\/www.ch.ic.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2009\/06\/benzene.jpg?resize=350%2C200","width":350,"height":200},"classes":[]}],"jetpack_likes_enabled":false,"authors":[{"term_id":2661,"user_id":1,"is_guest":0,"slug":"admin","display_name":"Henry Rzepa","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/897b6740f7f599bca7942cdf7d7914af5988937ae0e3869ab09aebb87f26a731?s=96&d=blank&r=g","0":null,"1":"","2":"","3":"","4":"","5":"","6":"","7":"","8":""}],"_links":{"self":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts\/21176","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=21176"}],"version-history":[{"count":41,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts\/21176\/revisions"}],"predecessor-version":[{"id":30487,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts\/21176\/revisions\/30487"}],"wp:attachment":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=21176"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=21176"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=21176"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fppma_author&post=21176"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}