Henry Rzepa's blog

The theme of the last three posts derives from the recently reported claimed experimental observation of bond length alternation (BLA) in cyclo[18]carbon, a ring of just 18 carbon atoms.[1] Having found that different forms of quantum calculation seem to find this property particularly difficult to agree upon, not only for cyclocarbon but for twisted lemniscular annulenes (which contain CH rather than just C units), I thought it might be time to look at some more experimental data and my chosen system is a class called the hexaphyrins, of which there are a number of experimental crystal structures.

The general form these molecules take is shown below. Here, QA can be N or S and the hashed bonds are defined as any type. T3 indicates an atom designated as having just three substituents. For each of the six meso carbons to which a non-metal (NM) is attached, a pair of C-C distances is specified as the search variable. These will define any bond length alternation.

A search of the current crystal structure database specifies no errors and no disorder (see FAIR DOI: 10.1021/ja801983d). For any temperature, 64 hits are obtained. Specifying a temperature of < 95K and an R factor of <7.5% reduces the number to 18. The six pairs of distances are then processed for each molecule as follows;

1. The Abs() operator is chosen and the absolute value of the difference in values between each of the six pairs is calculated, using the minus operator.
2. Then using the Least() and Greatest() operators, the corresponding values for each of the six differences is calculated.
3. Finally, a Heat plot of the least vs the greatest values is constructed for each of the molecules.

The result is shown below for the first search

There are two main clusters.

1. The cluster with the hotspot (red) shows that the minimum and maximum BLA (bond length alternations) at any of the six meso carbon atoms is < 0.01Å.
2. There is a second more diffuse cluster for which a greater minimum BLA at any meso atom of ~0.045Å and the maximum ~0.10Å are recorded.
3. The most diffuse cluster is where the minimum distance is 0.01Å but the maximum is again ~0.10Å.

This result suggests that BLA may be sensitive to the nature of the substituents on the ring. Cluster 1 above, with the least BLA, may also arise because the crystal structure is actually the average of two BLA forms. We can reduce the temperature of the measurements to e.g. < 95K to see the effect this might have on any dynamically averaging effect on the distances. A very similar distribution can be seen (below).

Next, an equivalent search for octaphyrins. Although there are fewer examples, the same clustering is seen.

The compound (code EGIHUY, a [26]phyrin, where 26 = 4n+2 = aromatic, DOI: 10.5517/ccrts0b[2]) in the red hot spot in the top diagram, with the smallest BLA was chosen for computation (FAIR data DOI: 10.14469/hpc/6179). This has C_i symmetry and so only three pairs of distances need to be considered.

Click image for 3D model

The results show that both B3LYP and ωB97XD DFT methods actually get pretty good geometries, reproducing the non-bond-alternating characteristics of this hexaphyrin. This in turn suggests that the absence of BLA may be real rather than a crystallographic averaging artefact.

What has been achieved? Well, the crystal structures show that whilst some hexaphyrin molecules have no bond alternation, most in fact do. DFT calculations reproduce the lack of bond alternation in one molecule. The next step is to show whether they also reproduce those which do have BLA. The story is not ended yet!

References

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, vol. 365, pp. 1299-1301, 2019. https://doi.org/10.1126/science.aay1914
2. J. Sankar, S. Mori, S. Saito, H. Rath, M. Suzuki, Y. Inokuma, H. Shinokubo, K. Suk Kim, Z.S. Yoon, J. Shin, J.M. Lim, Y. Matsuzaki, O. Matsushita, A. Muranaka, N. Kobayashi, D. Kim, and A. Osuka, "Unambiguous Identification of Möbius Aromaticity for<i>meso</i>-Aryl-Substituted [28]Hexaphyrins(1.1.1.1.1.1)", Journal of the American Chemical Society, vol. 130, pp. 13568-13579, 2008. https://doi.org/10.1021/ja801983d

In the previous posts, I tried to track down the onset of bond length alternation (BLA) as a function of ring size in aromatic cyclocarbons, finding the answer varied dramatically depending on the type of method used to calculate it. So here I change the system to an unusual kind of aromatic ring, the leminiscular or figure-eight annulene series.^♥ I explore the Kekulé vibration for such species for which a 4n+2 π electron count means they are cyclically Möbius aromatic.[1]

The advantage of using a lemniscular motif compared to the untwisted annulene is that perturbations due to trans-annular steric interactions between inward facing substituents are minimised. Before introducing the results for this type of molecule, I should also explain why the series C_nF_n is used rather than C_nH_n. This is because the C-C-H bending vibration is very similar in energy to the Kekulé C-C stretches, causing them to mix significantly and obscure the results. Substitution with F produces “clean” Kekulé modes. You can see this below:

One consequence of introducing two half-twists into the π-system is that this topology gets partitioned into true twist (T_w) and into a different property known as writhe (W_r),[2] the overall effect of which is to reduce the p-π/p-π overlaps of adjacent carbon atoms from suffering two twists to about half this. This in turn may affect the distortive tendency of the π-electrons to induce BLA in the ring.[3]  Let us now see what this change of molecule does to the value (and sign) of the Kekulé vibration. Included are conformations for the larger rings which vary in the total number of trans F-CC-F units in the ring. All the FAIR data for these calculations is at DOI: 10.14469/hpc/6139

Two density functional methods have been used, at opposite ends of the spectrum revealed in the previous posts, together with a reasonable Def2-TZVPP basis set.^‡ Each ring size can have different isomers, depending on the total number of transoid motifs present. For smaller rings (n=6, 10), the B3LYP+GD3BJ and ωB97XD functionals give very similar results. By n=18 however, a clear divergence has occurred, with the Kekulé modes being real (+ve force constant) for the former and almost equally imaginary (-ve force constant) for the latter. 

Overall, the B3LYP+GD3BJ series shows a slow and reasonably regular decline in the value of the Kekulé modes. As the ring sizes gets larger, the double bond configurations of the rings start to become unstable, with C=C bond rotations occuring during geometry optimisations. At n=26 using B3LYP, we seen a sudden change from a real Kekulé mode mode for the 8-trans isomer to an imaginary one for the 10-trans isomer. As noted above, this abrupt change occurs much earlier with the ωB97XD functional at n=14, with a discontinuity between a conformation with two transoid motifs and the ones with four and six. I can offer no explanation (yet) for this strange abrupt onset of an imaginary Kekulé mode, except perhaps to speculate that it might be related to the T_w and W_r partitioning noted above, given that T_w and BLA might themselves be related. As with the cyclocarbons, the BLA phenomenon seems to be peculiarly sensitive to the method used to compute it, more so than most other molecular properties.

Something deep and important is clearly happening and the underlying cause does need to be identified. It reminds in one sense also of the discontinuous transition between planar aromatic (bond equal) and  non-planar non-aromatic (BLA)  isomers of 10-π-Dihetero[8]annulenes[4]  Who might have imagined that simple aromatic rings could be so tantalisingly complex!

^♥ The first example of this was identified in 2005[5] and is characterised by a topological chiral property known as a linking number or Lk. For lemniscular molecules, Lk = 2π, or in plainer english it contains a double half twist in the π system around the ring. ^‡This investigation is a perfect example of the benefits of using a data repository. Many of these species were originally included in an article published in 2009[6] with the calculations being deposited in 2007. So all the starting geometries for the present investigation were quickly obtained from that source. †Bonds rotate to 10-trans isomer.

References

1. P.L. Ayers, R.J. Boyd, P. Bultinck, M. Caffarel, R. Carbó-Dorca, M. Causá, J. Cioslowski, J. Contreras-Garcia, D.L. Cooper, P. Coppens, C. Gatti, S. Grabowsky, P. Lazzeretti, P. Macchi, �. Martín Pendás, P.L. Popelier, K. Ruedenberg, H. Rzepa, A. Savin, A. Sax, W.E. Schwarz, S. Shahbazian, B. Silvi, M. Solà, and V. Tsirelson, "Six questions on topology in theoretical chemistry", Computational and Theoretical Chemistry, vol. 1053, pp. 2-16, 2015. https://doi.org/10.1016/j.comptc.2014.09.028
2. S.M. Rappaport, and H.S. Rzepa, "Intrinsically Chiral Aromaticity. Rules Incorporating Linking Number, Twist, and Writhe for Higher-Twist Möbius Annulenes", Journal of the American Chemical Society, vol. 130, pp. 7613-7619, 2008. https://doi.org/10.1021/ja710438j
3. 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. https://doi.org/10.1021/cr990363l
4. H.S. Rzepa, and N. Sanderson, "Aromaticity on the edge of chaos: An ab initio study of the bimodal balance between aromatic and non-aromatic structures for 10π-dihetero[8]annulenes", Phys. Chem. Chem. Phys., vol. 6, pp. 310-313, 2004. https://doi.org/10.1039/b312724a
5. H.S. Rzepa, "A Double-Twist Möbius-Aromatic Conformation of [14]Annulene", Organic Letters, vol. 7, pp. 4637-4639, 2005. https://doi.org/10.1021/ol0518333
6. C.S. Wannere, H.S. Rzepa, B.C. Rinderspacher, A. Paul, C.S.M. Allan, H.F. Schaefer, and P.V.R. Schleyer, "The Geometry and Electronic Topology of Higher-Order Charged Möbius Annulenes", The Journal of Physical Chemistry A, vol. 113, pp. 11619-11629, 2009. https://doi.org/10.1021/jp902176a

In the previous post, I looked at the so-called Kekulé vibration of cyclo[18]carbon using various quantum methods and basis sets.  Because some of these procedures can take a very long time,  I could not compare them using the same high-quality consistent atom basis set for the carbon (Def2-TZVPP). Here I try to start to do this using the smaller six and ten carbon rings to see what trends might emerge. FAIR data are at DOI: 10.14469/hpc/6069

The conclusions can be summarised as:

1. For six carbons, all the methods agree that the Kekulé vibration is real (+ve force constant), but there are distinct signs that the MP expansion may  not be fully converged, with  MP2 and MP3 differing significantly
2. For ten carbons, we already see that  MP3 and MP4 differ from  MP2 in predicting a -ve force constant.

Multi-configuration calculations are problematic with these species.  For C₁₀ for example, 20 electrons (10 π and 10 σ) in an active orbital space of 20 is required; a CASSCF(20,20) is beyond the scope of most quantum programs. And there is a need to evaluate the second derivatives of such a wavefunction in order to get the force constant for the required vibration. 

So the onset of bond length alternation in these cyclo-carbons could be as early as 10 atoms. But until higher level calculations can be performed in a systematic manner on these rings, the jury should perhaps still remain out as to when bond length alternation starts in terms of ring size.

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

The conclusions include:

1. The quality of the basis set is crucial. For lower quality bases, in-plane deformations of the ring occur, but the Kekulé mode itself is relatively unaffected. The minimum converged basis is Def2-TZVPP, which rather restricts the ability to perform higher level calculations.
2. Even within the DFT methods, two popular functionals given diametrically opposite results. The B3LYP procedure predicts no BLA, whereas the alternative ωB97X-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!
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 Kekulé mode (2600-3500 cm^-1) they provide a first warning sign something is not quite right?
4. Moller-Plesset expansions such as MP2 itself gives an unrealistic positive force constant for the Kekulé 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 Kekulé 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.
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.

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?

References

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, vol. 365, pp. 1299-1301, 2019. https://doi.org/10.1126/science.aay1914

The World-Wide-Web is currently celebrating its 30th anniversary; you can get the T-shirt in the CERN visitor centre!  Five years on, in May 1994, the first Web conference took place (WWW94) at CERN and now celebrating its own 25th anniversary. That 1994 conference also had various break-out sessions, one of which summarised the state of chemistry on the web at the time. You can see my general but entirely personal impressions written after the workshop (DOI: 10.14469/hpc/5850), with a chemistry specific version at DOI: 10.14469/hpc/5851.  A real trip down memory lane and an indication of how much has happened in 25 years!

I was lucky enough to be able to visit CERN a few days ago, primarily to view the CMS (Compact Muon Solenoid) detector and I also took the opportunity to remember WWW94. Here are a few photos of the visit. Note the chemistry seen in the final photo, taken in the control room (hint: look on the shelf at the back).

You can find a full report of the visit at https://www.imperial.ac.uk/news/191569/imperial-celebrates-towering-intellectual-achievement-cern/ with more photos. It is noteworthy that as well as celebrating the achievements of the  LHC, we also find that “CERN’s work often leads to paradigm-shifting technologies, … The most famous example is the World Wide Web, which was invented by British scientist Tim Berners-Lee while working at CERN in 1989.” Hence the link between WWW94 and the CMS.

I end with some chemistry. The design of the CMS detector (one of two that detected the Higgs Boson) took 27 years to complete! It started in the mid 1980s at a point when some of the science required for the detector was still unknown! It was a stupendous act of faith that during the time it took to design the detector, science would provide a solution. One such solution took the form of Lead tungstate (PbWO₄), which scintillates when struck by high energy particles. Some 75,848 single crystals of lead tungstate were used in the detector, each being 23 cm long and weighing in total 100 tonnes! The technology to purify and then crystallise this material to optical transparency was developed in Russia and China.

The CMS is now undergoing experiments to find the missing mass in the universe, the so-called dark matter. Maybe another visit in 25 years time is called for to hear where that missing mass is!

In August 2016, the launch of a chemistry pre-print service ChemRxiv was announced. I was phoned a day or so later by a staff journalist at C&E News for my opinion. The only comment that was retained for their report was my instantaneous feeling that “the community needed a chemistry pre-print server like one needed a hole in the head“. I had been there before you see, recollecting a pre-print server launched by the ChemWeb service around 1996 or 1997 and which lasted only about two years before being withdrawn due to the low quality of the preprints. So what do I think of ChemRxiv now in 2019?

Let me set the scene first. Nowadays, many journals offer open access options, most upon payment of an APC (article processing charge). One can sometimes get a grant for this fee from institutional libraries. Mine for example has a policy that to apply for an APC, one has to deposit a “final author version” (FAV) of a manuscript in our local institutional repository (Spiral). Thus the final outcome is two versions of open access articles, one the FAV and then a version-of-record (VOR) held by the publisher. ChemRxiv can now add a third version to the process, since the expectation is that after some life as a pre-print, the manuscript can then be submitted to a peer-reviewed journal. Because the pre-print is allocated a persistent identifier (a DOI), the expectation is that the pre-print will indeed be persistent, with no expiration. Three versions of any given article are therefore now likely to be around, in effect permanently (or what goes for permanence nowadays). Importantly, there is no clear protocol for indicating how these three versions might differ, if they do. Even the FAV and the VOR may contain differences such as errors found in galley-proofing which will appear in the VOR but may not be propagated to the FAV. The congruence between the pre-print and any VOR is even less obvious.

All this came to a head as a result of the pre-print I noted in my previous two posts.[1] Unlike the topic of an earlier post of mine, where the VOR article[2] (not a preprint) allows readers to comment (see e.g. https://www.nature.com/articles/s41586-019-1059-9#article-comments) I have not been able to identify a mechanism to post any comment about pre-prints. After all, that did seem to me to be a primary reason for exposing a pre-print, which is to invite insights from the community, perchance to improve the science or make suggestions related to it. What I have spotted however was an altmetric index. Hover over that and you get social media metrics. For this pre-print[1], these put it in the top 5% of all outputs, so it is clearly attracting much interest. This interest includes (currently) 1955 views, 539 downloads and commentary via two blog posts (www.altmetric.com/details/59250193/blogs) and 40 tweets (www.altmetric.com/details/59250193/twitter). You would have to work quite hard to visit all the blog posts and read all the tweets to assess overall how the community was responding to any specific pre-print. 

So what is the purpose of posting (or should I use the term publishing?) a ChemRxiv pre-print? Is it primarily to gather commentary via social media such as blog and Twitter posts and to use this to improve the final VOR based on such feedback? A colleague I discussed this with suggested that in some very competitive areas of science/chemistry, it might also serve to acquire a date-stamp for the research (part of the metadata associated with a DOI) and hence to claim priority, a stamp which would thus pre-date that obtained from VOR publication by a few months. This might be perceived as making all the difference in a competitive area in terms of gathering evidence of esteem, inclusion in grant proposals etc, especially for early career researchers. There may be other reasons which I have not thought of and comments here for these are most welcome.

I will end with noting the following project: en.wikiversity.org/wiki/WikiJournal_of_Science,[3] being part of the WikiVersity. Here, the APC is dispensed with (no publication costs, at least to the authors), a DOI is again allocated and each article is subjected to both public peer review (en.wikiversity.org/wiki/WikiJournal_of_Science/Peer_reviewers) and can also carry post-publication review comments and even direct edits in the manner of Wikipedia. The other infra-structures of the Wiki ecosystem are available, including access to WikiData, which is high quality reference data.

So I think it is going to be an interesting debate about how the publication of primary research articles is going to evolve. Is a Triad of articles (the pre-print, the FAV and the VOR) the future? Or could it be e.g. the Wiki Journal of Science (extended perchance in the future to Wiki Journal of chemistry?) showing an interesting alternative way? Or is it all just getting too fragmented and confusing?

References

1. K. Miyamoto, S. Narita, Y. Masumoto, T. Hashishin, M. Kimura, M. Ochiai, and M. Uchiyama, "Room-Temperature Chemical Synthesis of C2", 2019. https://doi.org/10.26434/chemrxiv.8009633.v1
2. J. Lee, K.T. Crampton, N. Tallarida, and V.A. Apkarian, "Visualizing vibrational normal modes of a single molecule with atomically confined light", Nature, vol. 568, pp. 78-82, 2019. https://doi.org/10.1038/s41586-019-1059-9
3. . , and T. Shafee, "The aims and scope of WikiJournal of Science", WikiJournal of Science, vol. 1, pp. 1, 2018. https://doi.org/10.15347/wjs/2018.001