{"id":485,"date":"2009-06-06T19:21:13","date_gmt":"2009-06-06T18:21:13","guid":{"rendered":"http:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=485"},"modified":"2023-09-16T19:38:27","modified_gmt":"2023-09-16T18:38:27","slug":"longer-is-stronger","status":"publish","type":"post","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=485","title":{"rendered":"Longer is stronger."},"content":{"rendered":"
\n

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 overall wavefunction, and that the real<\/em> electronic description is in effect an average of these two (that is the implied meaning of the double headed arrow). This means that the six C-C bonds in benzene must all be of equal length. The diagrams, everyone knows, do not<\/strong> mean that benzene has three short and three long C-C bonds.<\/p>\n

\"The

The Kekul\u00e9 structures of benzene. Click for 3D.<\/p><\/div>The diagram has much other implied semantics<\/em>. Thus there is no explicit three dimensional information; the molecule looks (and is) flat, and it is tempting to conclude that the electrons are flat and two dimensional as well. Indeed, up to around 1930 (some 105 years after its first discovery), the electrons in benzene were always represented as all lying in the plane of the molecule. This changed when H\u00fcckel announced the principle of \u03c3\/\u03c0 separation. These were the labels he gave to two different symmetries of electrons (actually derived for ethene), one set which did genuinely occupy the plane of the molecule, and a second (\u03c0) set for which this plane represented a node (a region of zero probability for the electron density). The \u03c0 electrons could instead be regarded as occupying the space above and below that plane. H\u00fcckel went on to develop a quantum mechanical theory for benzene based purely on those \u03c0-electrons, of which there are six. This (now called H\u00fcckel) theory predicted that the averaged structure noted above emerged naturally, along with\u00a0another concept known as \u03c0-electron resonance energy<\/em>. This is the difference in energy between the symmetric form of benzene and a structure in which the six \u03c0 electrons do not interact as a whole, but which are localized into three pairs located in the regions of the double bonds. Most people interpret this latter as being equivalent to the two Kekul\u00e9 forms shown above. Symmetrizing the structure (from D3h<\/sub> to the higher D6h<\/sub> symmetry) is accompanied by reducing the \u03c0-energy of the system by that resonance term (often estimated<\/a> as around -152 kJ\/mol of stabilization). For benzene in other words, this is the difference in energy between the symmetric species and a (hypothetical) bond localized cyclohexatriene.<\/p>\n

With such a focus on the\u00a0\u03c0-electrons, it seemed natural to accept that the reason why benzene has six equal C-C lengths is because of the resonance energy gained by the \u03c0-electrons when adopting the six-fold symmetric form. Prior to around 1961, no-one would have dissented from that point of view. The\u00a0first to do so was\u00a0Berry (see DOI:\u00a010.1063\/1.1732256<\/a> ),\u00a0but his was a lone voice at that time. But mysterious and inexplicable observations started to come to light. Perhaps the most direct was a study of the excited state of benzene, in which one \u03c0-electron is promoted from a bonding to a higher energy and antibonding \u03c0-orbital (known as a \u03c0-\u03c0*<\/sup> excitation, see DOI\u00a010.1063\/1.435193<\/a>). A schematic illustration of this process is shown below.<\/p>\n

\"The

The H\u00fcckel Molecular orbital picture for ground and excited states of benzene<\/p><\/div>\n

Diagram (a) shows the normal population of electrons in the (three lowest) energy levels derived using H\u00fcckel’s theory. Diagram (b) shows how this changes in the first excited singlet state, which would be expected to have weaker<\/strong> \u03c0-bonds. The vibrational spectrum of a molecule is one way of measuring how strong the bonds in a molecule are. Berry had already implied that one particular vibrational mode, the so-called Kekul\u00e9 mode (also known as the b2u<\/sub> mode using group theory) seemed unusually low in frequency. In other words, this distorsion was easier than it should have been, and this Berry attributed to the (then almost heretical view) that the \u03c0-electrons did not in fact promote a hexagonal form of benzene. This was instead induced by the \u03c3 electrons, which occupy the plane of the molecule. This effect prevailed over the \u03c0-electrons, which were in fact trying to get benzene to adopt a bond-alternating geometry (managing instead only to lower the energy of the b2u<\/sub> mode). When the vibrational spectrum of the excited state of benzene was analyzed in 1977, it appeared to spectacularly vindicate Berry (DOI 10.1063\/1.435193<\/a>). The Kekul\u00e9<\/em> mode has a value of 1309 cm-1<\/sup> for the normal ground state of benzene, but an exalted<\/em> value of 1570\u00a0cm-1<\/sup> in the excited state. This means that as the bonding due to the \u03c0-electrons is weakened by placing one of them in an antibonding orbital, their overall ability to distort the geometry is also weakened. As a result, the resistance to such distorsion (the Kekul\u00e9 mode<\/em>) is in turn strengthened by an amount corresponding to +261 cm-1<\/sup>. It was evidence such as this, and much else besides that Shaik and his co-workers used to promote the idea of \u03c0-distortivity in benzene (DOI: 10.1021\/cr990363l<\/a>). Despite such advocacy, the idea that all the six bonds in benzene are equal despite<\/strong> rather than because of <\/strong> the \u03c0-electrons is still rarely taught in introductory organic chemistry.<\/p>\n

But the story of excited benzene is not yet quite finished! In 2006, Blancafort and Sola (DOI: 10.1021\/jp064885y<\/a>) reminded us that the (1<\/sup>B2u<\/sub>) excited state of benzene exhibits a type of geometrical distorsion known as pseudo Jahn-Teller<\/em> (PJT), the origins of which have nothing to do with any of the previous arguments. The effect instead arises because the promoted electron emerges from a so-called energy degenerate orbital, and jumps into another degenerate orbital (Figure b above). The exaltation of the b2<\/sub> vibrational mode is in fact strongly coupled with this PJT effect, which complicates disentangling the two effects (PJT and \u03c0-distortivity).<\/p>\n

So another excited state is here proposed which is not susceptible to the PJT effect. Figure (c) above shows a \u03c0-quintet state in which two<\/strong> electrons are both promoted to anti-bonding orbitals. Now the \u03c0-electron bonding has been well and truly weakened! When the vibrational modes are calculated for the (D6h<\/sub>-symmetric) geometry of benzene at the same level of theory (B3LYP\/aug-cc-pvtz) for both singlet ground state and quintet excited state one finds that the b2u<\/sub> vibrational mode has the value of 1332 cm-1<\/sup> for the former and 1524 cm-1<\/sup> for the latter. Significantly, the former mode shows a contribution to the motion from the hydrogen atoms. These, being light, tend to increase the wavenumber of the vibrational mode. The same mode in the quintet state however shows motion of the carbon atoms only (Click on the diagram below to view the b2u<\/sub> mode for the quintet state of benzene, and note how little motion of the hydrogen atoms there is). It is a pure Kekul\u00e9<\/em> mode, whereas that for the ground state is not! If the motion of the hydrogens in the ground state of benzene is suppressed by artificially changing the atomic weight of the hydrogen in the mass-weighting scheme to a large value, the calculated b2u<\/sub> vibrational mode drops to around 1317 cm-1<\/sup>. This means the quintet state mode of benzene is exalted by 207 cm-1<\/sup>, and being PJT-free, it is a truer reflection of the effect of the \u03c0-electrons. Thus the effect first speculated upon by Berry, and championed by Shaik is spectacularly vindicated (again!).<\/p>\n

\"The

The b2u modes in benzene for (a) ground state and (b) quintet state. Click for 3D.<\/p><\/div>But what of the title for this post? Well, the C-C length in the singlet ground state of benzene is 1.391\u00c5. In the quintet state, it becomes longer at 1.454\u00c5 (which is almost exactly the value that Berry originally suggested should be used for the hypothetical cyclohexatriene geometry). Despite this lengthening, the Kekul\u00e9 mode clearly gets stronger. Why is this noteworthy? Well, it is almost always assumed that if a bond is shorter, it means stronger. In this case, we have an example of six bonds each getting shorter and weaker (at least as measured by the b2u<\/sub> mode of vibration), or as the title states, longer and stronger in the quintet state of benzene. Oh, and what about that \u03c0-resonance energy which we started with? Does it play no role after in the symmetric structure of benzene? Well, in fact it does! The answer is that the \u03c0-resonance<\/strong> energy is still at its maximum stabilization at the hexagonal structure of benzene, but it is the total<\/strong> \u03c0-energy that achieves its maximum stability at the non-hexagonal structure. These two energies are quite different beasts, and they each prefer a different geometry!<\/p>\n

\n

Acknowledgments<\/h4>\n

This post has been cross-posted in PDF format at Authorea<\/a>.<\/p>\n\n<\/div> ","protected":false},"excerpt":{"rendered":"

The iconic diagram below represents a cornerstone of organic chemistry. 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Carbenes: a reality check.","author":"Henry Rzepa","date":"September 11, 2016","format":false,"excerpt":"To quote from Wikipedia: in chemistry, a carbene is a molecule containing a neutral carbon atom with a valence of two and two unshared valence electrons. The most ubiquitous type of carbene of recent times is the one shown below as 1, often referred to as a resonance stabilised or\u2026","rel":"","context":"In "crystal_structure_mining"","block_context":{"text":"crystal_structure_mining","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=1745"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":5405,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=5405","url_meta":{"origin":485,"position":1},"title":"Chemistry in an attosecond.","author":"Henry Rzepa","date":"November 4, 2011","format":false,"excerpt":"An attosecond is 10-18s. The chemistry that takes place on this timescale is called electron dynamics. For example, it is the time taken for an electron to traverse the 1s orbit in a hydrogen atom. And chemists are starting to manipulate electrons (and hence chemistry) on this timescale; for example\u2026","rel":"","context":"In "General"","block_context":{"text":"General","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=1"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":1903,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=1903","url_meta":{"origin":485,"position":2},"title":"Carbobenzene: benzene with a difference","author":"Henry Rzepa","date":"April 16, 2010","format":false,"excerpt":"Some molecules, when you first see them, just intrigue. So it was with carbobenzene, the synthesis of a derivative of which was recently achieved by Remi Chauvin and co-workers (DOI: 10.1002\/chem.200601193). Two additional carbon atoms have been inserted into each of the six C-C bonds in benzene. The structure shows\u2026","rel":"","context":"In "General"","block_context":{"text":"General","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=1"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.ch.ic.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2010\/04\/carbobenzene.jpg?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":2183,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=2183","url_meta":{"origin":485,"position":3},"title":"Looking at bonds in a different way: the Laplacian.","author":"Henry Rzepa","date":"July 6, 2010","format":false,"excerpt":"The Cheshire cat in Alice's Adventures in Wonderland comes and goes at will, and engages Alice with baffling philosophical points. Chemical bonds are a bit like that too. In the previous post, we saw how (some) bonds can be tuned to be strong or weak simply by how a lone\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\/07\/ethane-0.3.jpg?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":14492,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=14492","url_meta":{"origin":485,"position":4},"title":"Mesomeric resonance in substituted benzenes: a crystallographic reality check.","author":"Henry Rzepa","date":"August 26, 2015","format":false,"excerpt":"Previously, I showed how conjugation in dienes and diaryls can be visualised by inspecting bond lengths as a function of torsions. Here is another illustration, this time of the mesomeric resonance on a benzene ring induced by an electron donating substituent (an amino group) or an electron withdrawing substituent (cyano).\u2026","rel":"","context":"In "Chemical IT"","block_context":{"text":"Chemical IT","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=2"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":11757,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=11757","url_meta":{"origin":485,"position":5},"title":"Does forming a Wheland intermediate disrupt all aromaticity?","author":"Henry Rzepa","date":"December 6, 2013","format":false,"excerpt":"Text books will announce that during aromatic electrophilic substitution, aromaticity is lost by the formation of a Wheland intermediate (and regained by eliminating a proton). Is that entirely true? I will start by considering the simplest of all such intermediates, the NMR of which was first reported by Olah and\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":"Click for 3D","src":"https:\/\/i0.wp.com\/www.ch.imperial.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2013\/12\/wheland-NNM.jpeg?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\/485","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=485"}],"version-history":[{"count":2,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts\/485\/revisions"}],"predecessor-version":[{"id":26473,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts\/485\/revisions\/26473"}],"wp:attachment":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=485"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=485"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=485"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fppma_author&post=485"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}