{"id":30975,"date":"2026-03-13T15:59:35","date_gmt":"2026-03-13T15:59:35","guid":{"rendered":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=30975"},"modified":"2026-03-14T09:21:57","modified_gmt":"2026-03-14T09:21:57","slug":"the-first-half-mobius-molecule-a-question-about-its-twist","status":"publish","type":"post","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=30975","title":{"rendered":"The first “half-M\u00f6bius” molecule: A question about its twist?"},"content":{"rendered":"
\n

The recent report[1]<\/a><\/span> of what is termed a “half-M\u00f6bius” molecule is generating<\/a> a lot of excitement. It has its origins in a project to make odd-numbered cyclocarbons on STM (scanning tunnelling microscope) surfaces. I had discussed even-numbered cyclocarbons in another post[2]<\/a><\/span>, where I also happened to include several odd-numbered examples, such as C49<\/sub> and C51<\/sub>. In this study[1]<\/a><\/span> they were focussing on C13<\/sub> and a precursor to this was to be C13<\/sub>Cl2<\/sub>. As part of the microscopy, they noticed this latter species was asymmetric (chiral) and so started the story of a “half-M\u00f6bius” molecule (molecules with twists in their topology are of course chiral). I should at this stage say that the concept of a half-M\u00f6bius is quite new and thought provoking. Perhaps the simplest way of explaining why, is that a conventional M\u00f6bius molecule (as with the strip or ribbon) requires two full circuits of the edge of the ribbon to return to the start, whereas this half version requires a full four circuits to achieve the same. More about this later.<\/p>\n

Since STM microscopy is not capable of yielding accurate molecular geometries and bond lengths, the authors proceeded to calculate these – a non trivial undertaking! Basically, because of orbital degeneracies, the wavefunction has important multi-reference character. I thought I would illustrate some of the outcomes here. I actually start with a single reference calculation, using the r2<\/sup>scan-3c<\/strong> method,[3]<\/a><\/span> which I had previously shown[2]<\/a><\/span>,[4]<\/a><\/span> seemed to reliably reproduce geometries of even-numbered cyclocarbons such as C48<\/sub>, and also predicted the onset of bond length alternation (BLA) for rings of about 58 carbon atoms or greater. Applied to C13<\/sub>Cl2<\/sub> r2<\/sup>scan-3c<\/strong> yields a planar molecule (Figure 1) with symmetrically disposed bond lengths.[5]<\/a><\/span>
\n\"\"
\nFigure 1.<\/strong> A r2<\/sup>SCAN-3c\/Def2-mTZVPP optimised geometry for C13<\/sub>Cl2<\/sub>.[5]<\/a><\/span><\/p>\n

The reported[1]<\/a><\/span> calculation using CASPT2\/cc-pVDZ[6]<\/a><\/span> shows different behaviour (Figure 2). The singlet state geometry is asymmetric and non-planar, with interesting BLA in both the C6<\/sub> and C7<\/sub> fragments, but also more “aromatic-looking” values of ~1.40\u00c5 at the chlorine-connected carbon.
\n\"\"
\nFigure 2.<\/strong> A CASPT2\/cc-pVDZ gas phase optimised geometry for A: singlet C13<\/sub>Cl2<\/sub> and b: Triplet.<\/p>\n

A pure CASSCF(12,12) calculation performed here (Figure 3) using the Def2-TZVPP basis set (a triple-\u03b6 basis – the geometry above is for a smaller double-\u03b6 basis) reproduces both the non-planarity and the BLA, but not the aromatic-like bond lengths, confirming that a higher level of theory which includes MP2-like electron correlation perturbation corrections is needed for these deceptively simple molecules.
\n\"\"<\/p>\n

Figure 3<\/strong>. A CASSCF(12,12)\/Def2-TZVPP calculation for singlet C13<\/sub>Cl2.<\/sub><\/p>\n

But what about that twist?<\/h2>\n

Now to the next stage of the story. Using orbitals derived from the wavefunctions, the authors[1]<\/a><\/span> showed that only a 90\u00b0 rotation (\u00bd\u03c0) occurred during a single trip around the edge of the molecular ribbon and hence 4*\u00bd\u03c0 = 2\u03c0 (360\u00b0) was required to achieve a return to the start. A full-M\u00f6bius molecule would achieve a 180\u00b0 rotation or 1\u03c0 for one circuit, and therefore requires only two circuits to achieve 2\u03c0. At this point, my thoughts turned to a well known topological theorem for these types of twisted systems[7]<\/a><\/span>, the C\u00e3lug\u00e3reanu\u2212White\u2212Fuller theorem.[8]<\/a><\/span>. This defines a topological invariant known as the linking number (Lk<\/sub>) which itself is the sum two quantities, the sum of local twists Tw<\/sub> and a writhe Wr<\/sub>. The latter can be thought of as the extent to which coiling of the central curve of the object can relieve local twisting of the ribbon. It is stated as:<\/p>\n

Lk<\/sub> = Wr<\/sub> + Tw<\/sub> <\/strong>(each of which can be expressed in units of \u03c0).‡<\/sup><\/p>\n

One practical example is C14<\/sub>H14<\/sub>,[7]<\/a><\/span>, a molecule not entirely unrelated to C13<\/sub>Cl2<\/sub>. This has a “figure-eight”, lemniscular or “double-M\u00f6bius” topology (Figure 4).
\n\"\"<\/p>\n

Figure 4:<\/strong> A double-twist M\u00f6bius annulene, calculated at the B3LYP\/6-31G level. For the (almost identical) geometry using the more modern r2<\/sup>scan-3c\/Def2-mTZVPP<\/strong>, see the FAIR data archive.[5]<\/a><\/span><\/p>\n

When analysed using the expression above,[7]<\/a><\/span> it shows values of\u00a0Lk<\/sub> = 2\u03c0<\/strong>, Wr<\/sub> = 0.89\u03c0<\/strong> and Tw<\/sub> = 1.11\u03c0<\/strong> (B3LYP\/6-31G* calculation). <\/p>\n

Firstly, a bit of (unrecorded) history. When I discovered this little lemniscular lovely, I did what had been done above, namely I added the total rotation of the orbital basis (the p-\u03c0-orbitals on each carbon) and returned to my starting point in just one circuit. I obtained a rotational sum of ~180\u00b0, or ~\u03c0. I knew that returning to the starting point in one circuit (2\u03c0) meant it was not a conventional M\u00f6bius molecule but a “figure-eight” or double twist topological isomer (which M\u00f6bius, along with Listing<\/a>, had identified!). This form had, not C2<\/sub> symmetry as per a single twist M\u00f6bius, but the higher\u00a0D2<\/sub> chiral symmetry. The correct answer for this twist sum was therefore surely 2\u03c0, not 1\u03c0? I had lost ~1\u03c0\u00a0worth of twist! And this is when I came across the above theorem, which put simply indeed allows any fraction of\u00a0twist to be “lost” by its conversion into writhe, as can be seen from the values shown above.†<\/sup><\/p>\n

So here is my question<\/strong>.\u00a0Might it be possible that the same has happened to C13<\/sub>Cl2<\/sub>? A measured orbital rotation of ~90\u00b0 or ~\u00bd\u03c0 (and hence the term half-M\u00f6bius) would only be correct if the writhe for this molecule – the coiling of the central curve out of a plane – was zero. If instead the writhe also had a value of lets say ~\u00bd\u03c0, then<\/p>\n

Lk<\/sub> = ~1\u03c0<\/strong>, comprising Wr<\/sub> = ~\u00bd\u03c0<\/strong> and Tw<\/sub> = ~\u00bd\u03c0<\/strong><\/p>\n

which would make it a conventional rather than half-M\u00f6bius molecule.<\/p>\n

To conclude:\u00a0the reported interpretation of C13<\/sub>Cl2<\/sub> as a “half-M\u00f6bius” molecule is only correct if it does not “writhe” topologically to any significant extent<\/strong>. Watch this space for updates!<\/p>\n

\n

‡<\/sup>Lk<\/sub> can be both a positive or a negative integer, depending on which enantiomer is used, and hence acts as a chiral descriptor in the manner of the Cahn-Ingold-Prelog convention. †<\/sup>Wr<\/sub> and Tw<\/sub> do not have to have the same sign. Thus the value of Tw<\/sub> can be greater<\/b> than that of Lk<\/sub> if Wr<\/sub> is opposite in sign. For an extreme example of these various effects see here.[9]<\/a><\/span>. The proposed molecular trefoil knot has values of Lk 6\u03c0 = Tw -0.8\u03c0 + Wr +6.8\u03c0<\/b> Not only are the twist and writhe of opposite sign, the knot is composed almost entirely of writhe and no twist! <\/p>\n

\n

This post has DOI: 10.59350\/5q3ka-2ag71<\/a><\/p>\n

References<\/h2>\n
  1. I. Ron\u010devi\u0107, F. Paschke, Y. Gao, L. Lieske, L.A. G\u00f6dde, S. Barison, S. Piccinelli, A. Baiardi, I. Tavernelli, J. Repp, F. Albrecht, H.L. Anderson, and L. Gross, \"A molecule with half-M\u00f6bius topology\", Science<\/i>, 2026. https:\/\/doi.org\/10.1126\/science.aea3321<\/a>\n\n<\/li>\n
  2. H. Rzepa, \"Molecules of the year 2025: Cyclo[48]carbon and others \u2013 the onset of bond alternation and the Raman Activity Spectrum.\", 2025. https:\/\/doi.org\/10.59350\/g4309-gv109<\/a>\n\n<\/li>\n
  3. S. Grimme, A. Hansen, S. Ehlert, and J. Mewes, \"r2SCAN-3c: A \u201cSwiss army knife\u201d composite electronic-structure method\", The Journal of Chemical Physics<\/i>, vol. 154, 2021. https:\/\/doi.org\/10.1063\/5.0040021<\/a>\n\n<\/li>\n
  4. H. Rzepa, \"Quantum crystallography: The structure and C-C bond length alternation of [18]-annulene.\", 2026. https:\/\/doi.org\/10.59350\/k0kjg-hpc66<\/a>\n\n<\/li>\n
  5. H. Rzepa, \"The first "half-M\u00f6bius" molecule: A question about its twist?\", 2026. https:\/\/doi.org\/10.14469\/hpc\/15786<\/a>\n\n<\/li>\n
  6. I. Ron\u010devi\u0107, A. Baiardi, and F. Paschke, \"Supplementary data for A Molecule with Half-Twisted M\u00f6bius Topology\", 2025. https:\/\/doi.org\/10.5281\/zenodo.15495263<\/a>\n\n<\/li>\n
  7. S.M. Rappaport, and H.S. Rzepa, \"Intrinsically Chiral Aromaticity. Rules Incorporating Linking Number, Twist, and Writhe for Higher-Twist M\u00f6bius Annulenes\", Journal of the American Chemical Society<\/i>, vol. 130, pp. 7613-7619, 2008. https:\/\/doi.org\/10.1021\/ja710438j<\/a>\n\n<\/li>\n
  8. G. C\u0103lug\u0103reanu, \"On isotopy classes of three dimmensional knots and their invariants\", Czechoslovak Mathematical Journal<\/i>, vol. 11, pp. 588-625, 1961. https:\/\/doi.org\/10.21136\/cmj.1961.100486<\/a>\n\n<\/li>\n
  9. H. Rzepa, \"Chemistry with a super-twist: A molecular trefoil knot, part 1.\", 2010. https:\/\/doi.org\/10.59350\/j60gh-gzr35<\/a>\n\n<\/li>\n<\/ol>\n\n<\/div> ","protected":false},"excerpt":{"rendered":"

    The recent report of what is termed a “half-M\u00f6bius” molecule is generating a lot of excitement. It has its origins in a project to make odd-numbered cyclocarbons on STM (scanning tunnelling microscope) surfaces. I had discussed even-numbered cyclocarbons in another post, where I also happened to include several odd-numbered examples, such as C49 and C51. 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This indicated, like most thermal pericyclic reactions, an\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.imperial.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2012\/11\/GaussViewScreenSnapz004.jpg?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":2046,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=2046","url_meta":{"origin":30975,"position":3},"title":"Chemistry with a super-twist: A molecular trefoil knot, part 1.","author":"Henry Rzepa","date":"May 31, 2010","format":false,"excerpt":"Something important happened in chemistry for the first time about 100 years ago. A molecule was built (nowadays we would say synthesized) specifically for the purpose of investigating a theory. It was cyclo-octatetraene or (CH)8, and it was made by Willst\u00e4tter and Waser to try to find out if benzene,\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\/2010\/05\/metallatrefoil.jpg?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":26573,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=26573","url_meta":{"origin":30975,"position":4},"title":"The journey from Journal “ESI” to FAIR data objects: An eighteen year old (continuing) experiment.","author":"Henry Rzepa","date":"December 10, 2023","format":false,"excerpt":"Around 1996, journals started publishing what became known as \"ESI\" or electronic supporting information, alongside the articles themselves, as a mechanism for exposing the data associated with the research being reported and exploiting some of the new opportunities offered by the World Wide Web. From the outset, such ESI was\u2026","rel":"","context":"Similar post","block_context":{"text":"Similar post","link":""},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":26601,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=26601","url_meta":{"origin":30975,"position":5},"title":"Molecules of the year: 2023","author":"Henry Rzepa","date":"December 28, 2023","format":false,"excerpt":"The Science education unit at the ACS publication C&EN publishes its list of molecules of the year (as selected by the editors and voted upon by the readers) in December. Here are some observations about three of this year's batch. Diberyllocene with its unusual Be-Be bond has already beeen covered\u2026","rel":"","context":"Similar post","block_context":{"text":"Similar post","link":""},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.ch.ic.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2023\/12\/CF.gif?resize=350%2C200&ssl=1","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\/30975","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=30975"}],"version-history":[{"count":51,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts\/30975\/revisions"}],"predecessor-version":[{"id":31053,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts\/30975\/revisions\/31053"}],"wp:attachment":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=30975"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=30975"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=30975"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fppma_author&post=30975"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}