Tag: higher energy

Imaging normal vibrational modes of a single molecule of CoTPP: a mystery about the nature of the imaged species.

Previously, I explored (computationally) the normal vibrational modes of Co(II)-tetraphenylporphyrin (CoTPP) as a “flattened” species on copper or gold surfaces for comparison with those recently imaged[cite]10.1038/s41586-019-1059-9[/cite]. The initial intent was to estimate the “flattening” energy. There are six electronic possibilities for this molecule on a metal surface. Respectively positively, or negatively charged and a neutral species, each in either a low or a high-spin electronic state. I reported five of these earlier, finding each had quite high barriers for “flattening” the molecule. For the final 6th possibility, the triplet anion, the SCF (self-consistent-field) had failed to converge, but for which I can now report converged results.^†

charge	Spin Multiplicity	ΔG, Twisted Ph, Hartree	ΔG, “flattened”, Hartree	ΔΔG, kcal/mol
-1	Triplet	-3294.68134 (C₂)	-3294.64745 (C_2v)	21.3
			-3294.616684 (C_2v)	40.6
			-3294.37012 (D_2h)	195.3
	Singlet	-3294.67713 (S₄)	-3294.39418 (D_4h)	175.6
			-3294.39321 (D_2h)	178.2
			-3294.56652 (D₂)	69.4
FAIR data at DOI:10.14469/hpc/5486 FAIR data version of the tables in this and previous post at DOI:10.14469/hpc/5561

I am exploring the so-called “flattened” mode, induced by the voltage applied at the tip of the STM (scanning-tunnelling microscope) probe and which causes the phenyl rings to rotate as per above. This rotation in turn causes the hydrogen atom-pair encircled above to approach each other very closely.^‡ To avoid these repulsions, the molecule buckles into one of two modes. The first causes the phenyl rings to stack up/down/up/down. The second involves an all-up stacking, as shown below. Although these are in fact 4th-order saddle points as isolated molecules, the STM voltage can inject sufficient energy to convert these into apparently stable minima on the metal surface.

The up/down/up/down “flattened” form (below) shows a much more modest planarisation energy than all the other charged/neutral states reported in the previous post, whereas the all-up isomer (which on the face of it looks a far easier proposition to come into close contact with a metal surface) is far higher in free energy.

The caption to Figure 3 in the original article[cite]10.1038/s41586-019-1059-9[/cite] does not explicitly mention the nature of the metal surface on which the vibrations were recorded, but we do get “The intensity in the upper right corner of the 320-cm⁻¹ map is from a neighbouring Cu–CO stretch” which suggests it is in fact a copper surface. Coupled with the other observation that in “contrast to gold, the Kondo resonance of cobalt disappears on Cu(100), suggesting that it acquires nearly a full electron from the metal (see Extended Data Fig. 2),” the model below of a triplet-state anion on the Cu surface seems the most appropriate.

Syn/anti mode, Triplet anion with C2v symmetry

There is one final remark made in the article worth repeating here: “This suggests that the vibronic functions are complex-valued in this state, as expected for Jahn–Teller active degenerate orbitals of the planar porphyrin.²⁶” Orbital degeneracy can only occur if the molecule has e.g. D_4h point group symmetry, whereas the triplet anion stationary-point shown in the figure above has only C_2v symmetry for which no orbital degeneracies (E) are expected. Enforcing D_4h symmetry on Co(II) tetraphenylporphyrin results in eight pairs of H…H contacts of 1.34Å,^‡ which is an impossibly short distance (the shortest known is ~1.5Å). Moreover this geometry has an equally impossible free energy 176 kcal/mol above the relaxed free molecule. Visually from Figure 3, the H…H contact distance looks even shorter (below, circled in red)! A D_2h form (with no E-type orbitals) can also be located.

Singlet, Calculated with D_4h symmetry. Click for vibrations.	Singlet, Calculated with D_2h symmetry. Click for vibrations.
Taken from Figure 3 (Ref 1).

These totally flat species are calculated to be at 13 or 12th-order saddle points, with the eight most negative force constants having vectors which correspond to up/down avoidance motions of the proximate hydrogen pairs encircled above and the remaining being buckling modes of the entire ring.

So to the mystery, being the nature of the “flattened” CoTPP on the copper metal surface, as represented in Figure 3 of the article.[cite]10.1038/s41586-019-1059-9[/cite] Is it truly flat, as implied by the article? If so, the energy of such a species would be beyond the limits of what is normally considered feasible. Moreover, it would represent a species with truly mind-blowing short H…H contacts. Or could it be a saddle-shaped geometry, where the phenyl rings are not lying flat in contact with the metal but interacting via the phenyl para-hydrogens? That geometry has not only a much more reasonable energy above the unflattened free molecule, but also acceptable H…H contacts (~2.0.Å) However, would such a shape correspond to the visualised vibrational modes also shown in Figure 3? I have a feeling that there must be more to this story.

^†These convergence problems were solved by improving the basis set via adding “diffuse” functions, as in (u)ωB97XD/6-311+G(d,p). Convergence to the lowest energy electronic state (³B₂) is achieved using a Huckel initial guess rather than the default Harries, which gives the higher energy ³A₂. ^‡If the crystal structure for these species is flattened without geometry optimisation, the H-H distance is around 0.8Å ^♥This blog has a DOI: 10.14469/hpc/5559.

April 25, 2019

Ritonavir: a look at a famous example of conformational polymorphism.
Here is an inside peek at another one of Derek Lowe’s 250 milestones in chemistry, the polymorphism of Ritonavir.[cite]10.1023/A:1011052932607[/cite] The story in a nutshell concerns one of a pharma company’s worst nightmares; a drug which has been successfully brought to market unexpectedly “changes” after a few years on market to a less effective form (or to use the drug term, formulation). This can happen via a phenomenon known as polymorphism, where the crystalline structure of a molecule can have more than one form.[cite]10.1021/ar00052a005[/cite],[cite]10.1002/anie.201410356[/cite],[cite]10.1039/D1SC06074K[/cite] In this case, form I was formulated into soluble tablets for oral intake. During later manufacturing, a new less-soluble form appeared and “within weeks this new polymorph began to appear throughout both the bulk drug and formulation areas“[cite]10.1023/A:1011052932607[/cite]

The structure of the original form I is shown below (3D DOI: 10.5517/CCRVC75). The compound has three HN-CO peptide linkages, all of which are in the stereoelectronically favoured s-cis form, with a dihedral angle of 180° across the H-N and C=O vectors.

Click for 3D

To show how favourable this s-cis form is, here is a search of the Cambridge structural database for acyclic HN-C=O bonds; of the ~8200 examples, only 5 have an s-trans torsion of ~180°. It is I feel statistically not entirely correct to convert this ratio of K=1640 to a free energy, but if one does, then at 298K, RTlnK works out to 4.4 kcal/mol. Note also that two compounds show an angle of ~90° (artefacts?).

The new type-II form that emerged has only two s-cis peptide linkages, and the third has isomerised to this higher energy s-trans form (3D DOI: 10.5517/CCRVC97)

Click for 3D

This has various knock-on effects on the conformation of the actual molecule itself.
1. The cis-trans isomerisation of a peptide or amide bond is a relatively high energy process, since the C=N bond order is higher than 1. For example, in the ¹H NMR spectrum of N,N-dimethyl formamide at room temperature, one can famously observe two methyl resonances and it is only at higher temperatures that the two signals coalesce due to more rapid rotation about the C=N bond.
2. A pedant might query whether this isomerism is correctly termed a conformational or a configurational change? High-energy rotations that result in cis/trans isomerisms are normally referred to as a configurational changes, whereas low energy rotations about e.g. single bonds are known as conformational changes (thus the conformational changes in cyclohexane). There is a grey region such as this one, where the boundary between the two terms is encountered.
3. This isomerism has the knock-on effect of inducing a much lower energy rotation of a C-C single bond (on the left hand side of the representations above), rotating from a dihedral angle of +193 in form I to +51 in form II.
4. More minor affects are seen in the conformation of the central benzyl group and the S/N heterocyclic ring on the right hand side.
5. All these low energy conformational effects occur because a better hydrogen bonding network can then be set up in the crystal lattice, something not easily predictable from the diagrams of the single molecules shown above.
6. Overall, the free energy of the lattice is lower, despite the higher energy of the s-trans peptide bond.
7. Clearly, the dynamics of crystallisation initially favoured form I (despite the higher energy of the crystallised outcome), but if a tiny seed of form II is present (or perhaps other impurities) this can dramatically (but unpredictably) change these crystallisation dynamics.
I suspect that since 1998 when this story unfolded, all new drugs in which one or more s-cis peptide bonds are present have caused anxiety. In the system above for example, one might ask whether cis/trans isomerisation of instead either of the other two peptide bonds present might have similar results? Or hypothesize whether inhibiting the associated rotation of the C-C single bond noted above by appropriate “tethering” might prevent form I from converting to form II. Since 1998, I am sure trying to predict the solid form of an organic molecule from its isolated structure using computational methods has dramatically increased, although I have not found in SciFinder any reported instances of such modelling for Ritonavir itself.[cite]10.1021/op000023y[/cite] Perhaps, if such a method were found, it might be too commercially valuable to share?
January 2, 2017
An unusual [1,6] shift in homotropylium cation exhibiting zones of aromaticity.

One thing leads to another. Thus in the previous post, I described a thermal pericyclic reaction that appears to exhibit two transition states resulting in two different stereochemical outcomes. I noted that another such reaction appeared to be a [1,6] carousel migration in homotropylium cation,[cite]10.1134/S1070428007080076[/cite] where transition states for both retention and inversion of the configuration of the migrating group (respectively formally allowed and forbidden) were reported (scheme below). Here I explore this system further. Firstly, the pathway leading to inversion.[cite]10.6084/m9.figshare.1134556[/cite] The reaction path (ωB97XD/6-311G(d,p)/SCRF=chloroform) has got a very odd (table-top mountain) shape, whereby the region of the transition state (IRC = 0.0) is very flat, and the region close to reactant and (identical) product is very steep. The gradient norm shows this best, with sharp spikes at IRC ± 4.2. Something clearly is happening here to cause this behaviour. Before moving on to analyze this, I want you first to observe the methyl groups below. Note how one of them rotates at the start of the process, and the other at the end. I have elsewhere called this behaviour the methyl flag, and it is due to stereoelectronic re-alignments of the C-H groups accompanying the changes in the conjugated array. The homotropylium cation is said to be homoaromatic, indicating that cyclic conjugation can be maintained across a ring in which the σ framework is interrupted at one point. A NICS probe placed at the ring critical point of this molecule reveals a chemical shift of -11.3 ppm[cite]10.6084/m9.figshare.1135694[/cite], very similar to eg that obtained for benzene itself. The three highest doubly occupied NBOs (below) show two normal π-type orbitals and one rather different one that spans the homo-bond (the MOs, before you ask, are a bit of a mess, with lots of mixed contributions from other parts of the σ framework).

HONBO (two) HONBO-2

Click for 3D

Click for 3D

At the transition state for the [1,6] migration, the same NICS probe registers a value of +2.6 ppm[cite]10.6084/m9.figshare.1135695[/cite], now firmly in the non-aromatic zone. So this reaction is characterised by two zones, ring-aromatic ones at the start and the end of the process, and a higher energy non-aromatic one in the middle of the reaction pathway as ~enclosed by the region of IRC ± 4.2. The homo-bond in the aromatic zone starts with a length of 1.74Å, reduces to 1.53Å at the transition state and ends up as a normal aromatic bond of length 1.41Å. Meanwhile, the relocated homo-bond changes in the opposite sense, starting as a normal aromatic length of 1.41Å, becoming 1.53Å at the transition state and ending as a homo-length of 1.74Å. Presumably, virtually full strength homoaromaticity can be sustained for a homo-bond of 1.74Å, but as that bond mutates to a σ-bond of 1.53Å, the cyclic conjugation falls off the edge of the cliff, to be restored only at the end. Pericyclic reactions are themselves said to sustain aromatic transition states,[cite]10.1021/ed084p1535[/cite] and so a simplistic way of looking at this is that the “aromaticity” relocates (or morphs) from the reactant to the transition state, and then back again during the course of the migration. A reaction path from which one can indeed learn a lot.

Now to the pathway in which the migrating group retains configuration. This is no longer a single step concerted reaction,[cite]10.6084/m9.figshare.1135668[/cite] since at the half-way point we no longer have a transition state but a shallow intermediate (~IRC +2, [cite]10.6084/m9.figshare.1134559[/cite]). It (formally at least) becomes a two-step non-concerted process, and the overall barrier is ~5 kcal/mol lower than for the inversion path. The aromaticity changes in a similar manner to before (i.e. IRC ~-5).

So this emerges as not quite the example I thought it was, but nonetheless unusual with the “forbidden” pathway being concerted and the “allowed” pathway being non-concerted. Molecular dynamics on these two systems would indeed be interesting to see what proportion of the trajectories go via each pathway.

August 12, 2014
A connected world (journals and blogs): The benzene dication.

Science is rarely about a totally new observation or rationalisation, it is much more about making connections between known facts, and perhaps using these connections to extrapolate to new areas (building on the shoulders of giants, etc). So here I chart one example of such connectivity over a period of six years.

The story starts with this article[cite]10.1002/anie.200902125[/cite], a preview talk about which (Hypervalent Carbon Atom: “Freezing” the S_N2 Transition State) I actually saw at an ACS conference a year or so earlier. When the article was published, Steve Bachrach blogged about it, noting the claim for pentavalent carbon. The semantics of a valency vs a coordination are subtle, and I was not convinced that this frozen transition state deserved its elevation from penta-coordinate to pentavalent. After some discussion on Steve’s blog, I built upon these ideas with a few thoughts of my own on the present blog and then wondered whether they could be finally distilled into a more formal publication (testing the precedent in some ways of whether collaborative and public discussions of ideas could be published formally, or whether they would be rejected as having been already “published”). Well, these final distilled thoughts were indeed published in 2010[cite]10.1038/nchem.596[/cite], including their genesis in Steve’s blog (I wanted to put blogs more firmly into the acceptable scientific circle). This article included one species (numbered 5 in that article in 2010[cite]10.1038/nchem.596[/cite]) and pointed out an analogy to replacing CH²⁺ by e.g the isoelectronic BH¹⁺, in as much as an example of the latter is indeed known as a stable crystalline compound.[cite]10.1016/0022-328X(94)05089-T[/cite]. Iso-electronics is a very fruitful source of connections in chemistry!

Matters rested there until yesterday, when I spotted this on Steve’s blog where he discusses this recent article on the structure of the benzene dication.[cite]10.1021/ja412109h[/cite] Hey presto, there is that molecule again, but now there is firm experimental evidence of its existence! It was I think rather too much to expect the authors of this article to have spotted the connection to mine (although as it happens, both address the issue of complexes to He). The relationship between CH²⁺ and BH¹⁺ is a little more subtle. From my point of view, it is always worth trawling through the crystal structure database in favour of evidence for hypothetical species (or their isoelectronic substitutions), and so it proved in this case!

There are other connections possible. Thus the dication of benzene has a (higher energy) isomer which is in fact a 4π antiaromatic species which avoids this antiaromaticity by a geometric distortion, with two C-H bonds bending above and below the ring. Such avoided antiaromaticity has been noted elsewhere here.

There is one final connection for me to make. My 2010 article[cite]10.1038/nchem.596[/cite] contained one of my interactive tables containing the data for the various structures (yes, although its data, you will need to have a subscription to the journal to access it). As it happens, last year we wished to reprise this style of publication, but as I blogged at the time, the journal had changed its production processes, and they could no longer offer me that opportunity. Some quick thinking came up with a replacement, which we now use extensively.[cite]10.1039/C3SC53416B[/cite] So the chain of connections resulting from that original talk some six years ago continues.

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p>As for that chain, it arose distressingly randomly. I do not routinely read the entire ToC of JACS and so would not have discovered[cite]10.1021/ja412109h[/cite] the connection by that route. Fortunately, Steve Bachrach does and helped me make that connection to the molecule shown above. Although I did spend a few minutes thinking to myself “does that structure ring any bells?”. Fortunately, one did (eventually) ring. But for every connection made in this wonderfully human manner, I cannot help but think how many are not! However, if connections were much easier to make, could we as humans cope with the overwhelming deluge of new ideas?

April 10, 2014
The mechanism of the Benzidine rearrangement.
The benzidine rearrangement is claimed to be an example of the quite rare [5,5] sigmatropic migration[cite]10.1021/ja00335a035[/cite], which is a ten-electron homologation of the very common [3,3] sigmatropic reaction (e.g. the Cope or Claisen). Some benzidine rearrangements are indeed thought to go through the [3,3] route[cite]10.1021/ja00309a041[/cite]. The topic has been reviewed here[cite]10.1002/poc.610020702[/cite].

In this post, I offer a calculated transition state and IRC for this reaction, to see what insights might accrue. How was this obtained?
1. At the ωB97XD/6-311G(d,p)/SCRF=water level. This procedure would allow for any dispersion-like effects to be allowed for in the π-π-stacking.
2. The rearrangement is normally promoted by acid, and the active species is thought to be diprotonated[cite]10.1002/ejic.201101115[/cite]^‡ (although monoprotonated catalysis is also observed[cite]10.1021/ja00335a035[/cite]. Here I report just the diprotonated route, together with chloride anions to balance the charges, and have added a continuum water field to allow this double ion-pair to be at least partially stabilised.
3. The rate determining step is the N-N cleavage/C-C bond formation. This is followed by presumed rapid proton transfers, which are not modelled here.
The [5,5] transition state for the benzidine rearrangement. Click for 3D.

This [5,5] transition state is 2.9 kcal/mol lower in ΔG^‡₂₉₈ than the transition state for the isomeric [3,3] rearrangement. The NCI (non-covalent-interactions) shows the forming C-C bond to be on the border of covalent, and non-covalent (blue), but that the π-π-stacking region is all weakly attractive (green). You can also observe the strong hydrogen bonds between the chloride anion and an N-H group (blue), and the weak attractive zones between the two nitrogen centres, between the chloride and the ortho-C-H hydrogens, and even between the two chloride anions (blue-green or green). I should point out that the initial position for these anions was over the aryl ring, but they migrated to the NH region during optimisation of the transition state.

NCI surface. Click for 3D.

The molecular electrostatic potential (isosurface = 0.11 au) shows both aryl rings as a single unit attracted by a positive potential (blue)

Calculated electrostatic potential. Click for 3D.

The highest-occupied molecular orbital shows the two bonds involved in the [5,5] shift (N-N and C-C) are both bonding, but more significantly, the central region of the two stacked aryl rings is also bonding. This is a clear manifestation of a π-complex, which the benzidine rearrangement has often (and it has to be said controversially) described as, and which elevates this particular reaction from that of a simple bond forming/bond cleaving sigmatropic. Another way of looking at it is that secondary orbital interactions (such as often invoked in Diels-Alder cycloadditions) are exceptionally important here.

HOMO for 5,5 benzidine rearrangement. Click for 3D.

The LUMO is strongly antibonding in that region; indeed adding two electrons to form a 12-electron process would be strongly destabilising. In this regard, this unusual sigmatropic reaction follows the same 4n+2 electron rule as more conventional ones.

LUMO. Click for 3D.

The next two diagrams illustrate the competing (higher energy) [3,3] shift, which also has some π-complex character.

A [3,3] alternative to the benzidine rearrangement. Click for 3D.

NCI surface for 3,3 rearrangement. Click for 3D.

I will end with three autobiographical notes.
1. The benzidine rearrangement was one of the earliest reactions I did in my home laboratory, at the age of about 13. As I recollect, I prepared about 1.5 grams (blissfully ignorant of how carcinogenic it is), and used it via diazotization to couple to phenol. My fascination with chemistry most certainly started with colour (and how to express the bonding in nitric oxide).
2. About eight years later, I was about to commence my Ph.D. studies. The objective was to use kinetic isotope effects to infer the structure of transition states. In my case (proton transfers to indoles) I never did achieve this objective. But it is noteworthy that the mechanism of the benzidine rearrangement was largely unravelled using such isotopic studies.
3. By 1974 as a post-doctoral researcher, I had moved on to studying mechanisms using quantum theory and had decided that it was easier to invert the use of isotope effects by predicting a transition structure using this method, and then seeing if the computed isotope effects matched the experiment. We did this for the Diels-Alder reaction[cite]10.1021/ja00486a013[/cite] and more generally[cite]10.1021/ja00493a008[/cite], and then for some gas-phase eliminations[cite]10.1039/C39810000939[/cite], this latter being my first entirely independent publication.
4. So, putting all this together, one might infer that armed with a computed transition state structure for the benzidine rearrangement, it is trivial to compute the kinetic isotope effects and hence to see if they correspond to those measured. You might expect a report on this in a future post here.
^‡ Crystal structures of diprotonated dimethyl hydrazines[cite]10.1002/ejic.201101115[/cite] show a N-N bond length of ~1.45Å (typical counter-anions being nitrate, perchlorate or sulfate). That calculated for the diprotonated diphenyl hydrazine is ~2.5Å, which suggests that with the phenyl group, electrons from the N-N region may be borrowed to contribute to the π-π-complex.
January 6, 2013
Shorter is higher: the strange case of diberyllium.

Much of chemistry is about bonds, but sometimes it can also be about anti-bonds. It is also true that the simplest of molecules can have quite subtle properties. Thus most undergraduate courses in chemistry deal with how to describe the bonding in the diatomics of the first row of the periodic table. Often, only the series C₂ to F₂is covered, so as to take into account the paramagnetism of dioxygen, and the triple bonded nature of dinitrogen (but never mentioning the strongest bond in the universe!). Rarely is diberyllium mentioned, and yet by its strangeness, it can also teach us a lot of chemistry.

The diagram below is what many textbooks show. The diagram can vary (and hence confuse) slightly, in regard to the relative ordering of the σ and π energy levels originating from the overlap of the 2p orbitals. It depends on the atom, and for Be, the σ comes out higher than the π. The other key ordering is that the σ* antibonding orbital resulting from out of phase overlap of the two 2s orbitals is actually lower in energy than the π bonding orbital resulting from in-phase overlap of the 2p orbitals. Yes, an antibonding orbital is more stable than a bonding orbital!

Molecular orbital diagram for Be2

Well, the diagram shows that the pair of occupied molecular orbitals resulting from the two (symmetric and antisymmetric, or g and u) combinations of the 1s orbitals cancel each other, as do the 2s combinations, and we conclude the bond order for this molecule is zero! Actually, if a quantum mechanical calculation is performed (at the ωB97XD/6-311G(d,p) level), the bond length emerges as 2.81Å and a vibrational wavenumber of 167 cm^-1 is predicted. Despite the zero bond order, a weak bond IS predicted, and this is the van der Waals or dispersion bond.

Let us now pump this molecule up to a higher energy state by a double excitation of the two electrons in the 2s σ* electrons. We have to split them up, one each, into the next available orbital, which is the π, to form a triplet state (just like di-oxygen).

The doubly excited state of diberyllium

Well, this (higher energy) state is certainly shorter (a contrast with my item on longer being stronger). The length is now 1.78Å, which is more than 1Å shorter than the original state, despite being ~ 45 kcal/mol higher in energy. The Be-Be stretching wavenumber goes up to 917 cm^-1. With four electrons in bonding orbitals, diberyllium has a double bond! One can also pair the π electrons up to form an open shell (excited) singlet, which is ~ 51 kcal/mol higher than the closed shell (unbonded) singlet. This also has a length of 1.78Å and a marginally lower stretch of 909 cm^-1. Read more about the doubly excited state of this molecule[cite]10.1139/v96-111[/cite].

One might be tempted to make an analogy between physics, and its particles and antiparticles. Yes, electrons can occupy antibonding as well as bonding orbitals. But the overall bond order will be reduced to zero if the total numbers of each are equal. And one can be pretty certain that there is no molecule at all in which the number of antibonding electrons exceeds the bonding ones! Or, if anyone is aware of such an example, do tell!

January 21, 2011
Longer is stronger.

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 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 mean that benzene has three short and three long C-C bonds.

The Kekulé structures of benzene. Click for 3D.
The diagram has much other implied semantics. 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 Hückel announced the principle of σ/π 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 (π) set for which this plane represented a node (a region of zero probability for the electron density). The π electrons could instead be regarded as occupying the space above and below that plane. Hückel went on to develop a quantum mechanical theory for benzene based purely on those π-electrons, of which there are six. This (now called Hückel) theory predicted that the averaged structure noted above emerged naturally, along with another concept known as π-electron resonance energy. This is the difference in energy between the symmetric form of benzene and a structure in which the six π 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 Kekulé forms shown above. Symmetrizing the structure (from D_3h to the higher D_6h symmetry) is accompanied by reducing the π-energy of the system by that resonance term (often estimated 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.

With such a focus on the π-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 π-electrons when adopting the six-fold symmetric form. Prior to around 1961, no-one would have dissented from that point of view. The first to do so was Berry (see DOI: 10.1063/1.1732256 ), but 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 π-electron is promoted from a bonding to a higher energy and antibonding π-orbital (known as a π-π^* excitation, see DOI 10.1063/1.435193). A schematic illustration of this process is shown below.

The Hückel Molecular orbital picture for ground and excited states of benzene

Diagram (a) shows the normal population of electrons in the (three lowest) energy levels derived using Hückel’s theory. Diagram (b) shows how this changes in the first excited singlet state, which would be expected to have weaker π-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 Kekulé mode (also known as the b_2u 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 π-electrons did not in fact promote a hexagonal form of benzene. This was instead induced by the σ electrons, which occupy the plane of the molecule. This effect prevailed over the π-electrons, which were in fact trying to get benzene to adopt a bond-alternating geometry (managing instead only to lower the energy of the b_2u 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). The Kekulé mode has a value of 1309 cm^-1 for the normal ground state of benzene, but an exalted value of 1570 cm^-1 in the excited state. This means that as the bonding due to the π-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 Kekulé mode) is in turn strengthened by an amount corresponding to +261 cm^-1. It was evidence such as this, and much else besides that Shaik and his co-workers used to promote the idea of π-distortivity in benzene (DOI: 10.1021/cr990363l). Despite such advocacy, the idea that all the six bonds in benzene are equal despite rather than because of the π-electrons is still rarely taught in introductory organic chemistry.

But the story of excited benzene is not yet quite finished! In 2006, Blancafort and Sola (DOI: 10.1021/jp064885y) reminded us that the (¹B_2u) excited state of benzene exhibits a type of geometrical distorsion known as pseudo Jahn-Teller (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 vibrational mode is in fact strongly coupled with this PJT effect, which complicates disentangling the two effects (PJT and π-distortivity).

So another excited state is here proposed which is not susceptible to the PJT effect. Figure (c) above shows a π-quintet state in which two electrons are both promoted to anti-bonding orbitals. Now the π-electron bonding has been well and truly weakened! When the vibrational modes are calculated for the (D_6h-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 b_2u vibrational mode has the value of 1332 cm^-1 for the former and 1524 cm^-1 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 b_2u mode for the quintet state of benzene, and note how little motion of the hydrogen atoms there is). It is a pure Kekulé 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 b_2u vibrational mode drops to around 1317 cm^-1. This means the quintet state mode of benzene is exalted by 207 cm^-1, and being PJT-free, it is a truer reflection of the effect of the π-electrons. Thus the effect first speculated upon by Berry, and championed by Shaik is spectacularly vindicated (again!).

The b2u modes in benzene for (a) ground state and (b) quintet state. Click for 3D.
But what of the title for this post? Well, the C-C length in the singlet ground state of benzene is 1.391Å. In the quintet state, it becomes longer at 1.454Å (which is almost exactly the value that Berry originally suggested should be used for the hypothetical cyclohexatriene geometry). Despite this lengthening, the Kekulé 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 b_2u mode of vibration), or as the title states, longer and stronger in the quintet state of benzene. Oh, and what about that π-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 π-resonance energy is still at its maximum stabilization at the hexagonal structure of benzene, but it is the total π-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!

Acknowledgments

This post has been cross-posted in PDF format at Authorea.

June 6, 2009

The [5,5] transition state for the benzidine rearrangement. Click for 3D.

Tag: higher energy

Acknowledgments