energy « Henry Rzepa's blog

Posts Tagged ‘energy’

Bio-renewable green polymers: Stereoinduction in poly(lactic acid)

Saturday, July 24th, 2010

Lactide is a small molecule made from lactic acid, which is itself available in large quantities by harvesting plants rather than drilling for oil. Lactide can be turned into polymers with remarkable properties, which in turn degrade down easily back to lactic acid. A perfect bio-renewable material!

Lactide

The starting point for ring opening polymerisation is racemic lactide, or rac-LA. This is an equal mixture of the R,R and S,S enantiomers, and it is now treated with a catalyst based on a metal M. If M=Mg, there is a rather remarkable stereochemical outcome for the resulting polymer. The catalyst selects alternating enantiomers for the assembly, resulting in a chain (R,R),(S,S),(R,R),(S,S), etc, the name for which is a heterotactic polymer. It could instead have created a blend of equal proportions of (R,R),(R,R),(R,R) and (S,S),(S,S),(S,S) which is an isotactic polymer. Needless to say, these two polymers have quite different properties, and it very much matters which is formed. Without such a catalyst, a random atactic polymer is created rather than a stereoregular arrangement.

Poly (lactic acid)

The question is how does the catalyst manage to assemble the polymer with such stereoinduction? The origins of this depend on a detailed understanding of the mechanism of the reaction, and in 2005 we suggested one which offered an explanation for the stereospecificity (see E. L. Marshall, V. C. Gibson, and H. S. Rzepa, DOI: 10.1021/ja043819b and an interactive storyboard).

Mechanism for stereoregular polymerisation

The key features of this rational were:

Two possible transition states may control the reaction, TS1 and TS2. Which one depends on which is the higher in energy.
The smallest model for this process involves loading two molecules of lactide onto the catalyst. The first has already been ring opened, and will control the stereochemistry of the second, which is the one suffering the ring opening bond formations/breakings shown above (the first is lurking in the group R).
This leads to four different possibilities, (R,R)-(R,R)*, (S,S)-(S,S)*, (R,R)-(S,S)*, and (S,S)-(R,R)* (where the * denotes the reacting lactide, as in the diagram above). These are all diastereomers, and hence will be different in energy. If one of the first two is the lowest, then isotactic polymer will result; if the latter two then a heterotactic polymer.

Back in 2004, we had constructed a model based on B3LYP and of necessity a mixed basis set, being 6-311G(3d) on the Mg, 6-31G on the lactide and only STO-3G on the catalyst. This was done because the complete system was actually rather large. Even so, a transition state calculation would regularly take at least 10 days to find using the fastest computers available to us at that time. Using this procedure, we found that the rate limiting kinetic step was in fact TS2 for all four possibilities noted above. Of these, the (R,R)-(S,S) transition state turned out to represent the lowest energy pathway, thus confirming the observed heterotacticity for this particular catalyst.

Well, times have moved on:

Six years later, computers are around 20 times faster! We can now afford to improve the basis set to 6-31G(d,p) on all the atoms, including the catalyst (the Mg stays at 6-311G(3d) however; improving it to 6-311G(3d,2f) makes little difference).
We can now include the solvent (thf) as a continuum field.
In the last five years the B3LYP functional has been shown to underestimate the energies of globular molecules. A modern functional such as ωB97XD, which includes dispersion energy corrections, should be expected to do much better.

It is the purpose of this blog to report an update to the modelling. Quoting relative free energies (including the solvation correction), the results come out as;

(R,R)-(S,S) 0.0 kcal/mol for the TS1 geometry (see DOI: 10042/to-4950)
(S,S)-(S,S) 1.8 for the TS2 geometry
(S,S)-(R,R) 5.5 for the TS1 geometry
(R,R)-(R,R) 9.1 for the TS1 geometry.

Well, there are surprises! Using the gas phase B3LYP model the key transition state was TS2; now its TS1 (for in fact three of the four possible transition states). The bottom line (almost) is that the same stereoisomer as before comes out the winner! The take home lesson is that in six years of progress, modelling can now encompass solvent and dispersion corrections. Many mechanisms with > ~100 atoms investigated in the past without inclusion of these effects could probably do with a re-investigation, especially if the transition states are “globular” in nature. Any by now you are probably wondering what the transition state looks like. Well, here it is (and see it in all its glory by clicking on the diagram below).

(R,R)-(S,S) Transition state for stereoregular lactide polymerisation. Click for animation

And if you are also wondering how one might proceed to analyse the origins of the stereoinduction, the NCI interaction surfaces (as described in this post) are shown below. Note how the extensive degree of green interaction surface is associated with the globular nature referred to above.

Non-covalent interaction (NCI) surfaces for the (R,R)-(S,S) transition state. Click for 3D

Tags:3g, animation, catalysis, chiroptical, dispersion energy corrections, E. L. Marshall, energy, gas phase, General, H. S. Rzepa, Interesting chemistry, Julia Contreras-Garcia, lowest energy pathway, oil, polymerisation, ring opening, V, V. C. Gibson
Posted in General, Interesting chemistry | 3 Comments »

The weirdest bond of all? Laplacian isosurfaces for [1.1.1]Propellane.

Wednesday, July 21st, 2010

In this post, I will take a look at what must be the most extraordinary small molecule ever made (especially given that it is merely a hydrocarbon). Its peculiarity is the region indicated by the dashed line below. Is it a bond? If so, what kind, given that it would exist sandwiched between two inverted carbon atoms?

1.1.1 Propellane

One (of the many) methods which can be used to characterize bonds is the QTAIM procedure. This identifies the coordinates of stationary points in the electron density ρ(r) (at which point ∇ρ(r) = 0) and characterises them by the properties of the density Hessian at this point. At the coordinate of a so-called bond critical point or BCP, the density Hessian has two negative eigenvalues and one positive one. The sum, or trace of the eigenvalues of the density Hessian at this point, denoted as ∇²ρ(r), provides in this model a characteristic indicator of the type of bond, according to the following qualitative partitions:

ρ(r) > 0, ∇²ρ(r) < 0; covalent
ρ(r) ~0, ∇²ρ(r) > 0; ionic
ρ(r) > 0, ∇²ρ(r) > 0; charge shift

The third category of bond was first characterised by Shaik, Hiberty and co. using valence-bond theory¹ and they went on to propose [1.1.1] propellane (above, along with F₂) as an exemplar of this type.²Matching the conclusions drawn from VB theory was the value of the Laplacian. As defined above, for the central C-C bond, both ρ(r) and ∇²ρ(r) have been calculated to be positive, supporting the identification of this interaction as having charge-shift character.³

The Laplacian represents one of those properties where quantum mechanics meets experiment, in that its value (and that of ρ(r) itself) can be measured by (accurate) X-ray techniques.⁴ This was recently accomplished for propellane,⁵with the same conclusion that the Laplacian in the central C-C region has the significantly positive value of +0.42 au. The electron density ρ(r) at this point was measured as 0.194 au. Calculations⁵ at the B3LYP/6-311G(d,p) level report ρ(r) as ~0.19 and ∇²ρ(r) as +0.08 au. Whilst the former is in good agreement with experiment, the latter is calculated as rather smaller than expected. This was originally interpreted as indicating that the “the experimental bond path has a stronger curvature [in ρ(r)] than the theoretical” although more recent thoughts are that both experimental and theoretical uncertainty may account for the discrepancy.^5,6 An experiment worth repeating?

A hitherto largely unexplored aspect of characterising a bond using the Laplacian is whether the value at the bond critical point is fully representative of the bond as a whole. The Laplacian is related to two components of the electronic energy by the Virial theorem;

2G(r) + V(r) = ∇²ρ(r)/4; H(r) = V(r) + G(r)

where G(r) is the kinetic energy density, V(r) is the potential energy density and H(r) the energy density. Charge-shift bonds exhibit a large value of the (repulsive) kinetic energy density, a consequence of which is that ∇²ρ(r) is more likely to be positive rather than negative. The relationships above hold not just for the specific coordinate of a bond critical point, but for all space. Accordingly, another way therefore of representing the Laplacian ∇²ρ(r) is to plot the function as an isosurface, including both the negative surface (for which |V(r)| > 2G(r)) and the positive surface [for which |V(r)| < 2G(r)].

Such an analysis is the purpose of this post, using wavefunctions evaluated at the CCSD/aug-cc-pvtz level (see DOI: 10042/to-5012). The values of ρ(r) and ∇²ρ(r) at the bcp for the central bond are 0.188 and +0.095 au, which compares well with previous calculations. The values for the wing C-C bonds are 0.242 and -0.491 respectively (and were measured⁵ as 0.26 and -0.48). Laplacian isosurfaces corresponding to ± 0.49 (the value at the wing C-C bcp), ± 0.47 and ± 0.2 (which reveals prominent regions of +ve values for the Laplacian) can be seen in the figures below (and can be obtained as rotatable images by clicking).

Laplacian isosurface contoured at ± 0.49

–

Laplacian isosurface contoured at ± 0.47. Red = -ve, blue= +ve.

Laplacian isosurface contoured at ± 0.20

A significant feature is the isosurface at -0.47, which corresponds to the lowest contiguous Laplacian isovalued pathway connecting the two terminal carbon atoms (and which coincidentally is similar in magnitude to that reported⁵ as measured for these two atoms). Three such bent pathways of course connect the two carbon atoms. The energy density H(r) shows a minimum value of -0.21 au along any of these pathways. It is significantly less negative (-0.13) for the direct pathway taken along the axis of the C-C bond.

Energy density H(r) @-0.21

Energy density H(r) @-0.13

ELF isosurface @0.7

A useful comparison with this result is the ELF isosurface. This too is computed at the correlated CCSD/aug-cc-pVTZ using a new procedure recently described by Silvi.⁷ Contoured at an isosurface of +0.7, the ELF function is continuous between the two terminal atoms, much in the manner of Laplacian. Significantly, the ELF function at the bcp appears at the very much lower threshold value of 0.54, and forms a basin with a tiny integration for the electrons (0.1e). Since both methods provide a measure of the Pauli repulsions via the excess kinetic energy, the similarity of the Laplacian to the ELF function is probably not coincidental.

The issue then is whether a bond must be defined by the characteristics of the electron density distribution along the axis connecting that bond, or whether other, non-least-distance pathways can also be considered as being part of the bond.⁸ The former criterion defines a pathway involving a positive Laplacian (+0.095) and would be interpreted as indicating charge shift character for that bond. The latter involves three (longer) pathways for which the Laplacian is strongly -ve, and which would therefore per se imply more conventional covalent character for the interaction. Considered as a linear (straight) bond, it has charge shifted character; considered as three “banana” bonds, it may be covalent. Weird!

Shaik, S.; Danovich, D.; Silvi, B.; Lauvergnat, D. L.; Hiberty, P. C., “Charge-Shift Bonding – A Class of Electron-Pair Bonds That
Emerges from Valence Bond Theory and Is Supported by the Electron Localization Function Approach,” Chem. Eur. J., 2005,
11, 6358-6371, DOI: 10.1002/chem.200500265 and references cited therein.
W. Wu, J. Gu, J. Song, S. Shaik, and P. C. Hiberty, “The Inverted Bond in [1.1.1]Propellane is a Charge-Shift Bond,” Angew. Chem. Int. Ed., 2008,
DOI: 10.1002/anie.200804965; 10.1002/cphc.200900633
S. Shaik, D. Danovich, W. Wu & P. C. Hiberty, “Charge-shift bonding and its manifestations in chemistry”, Nature Chem, 2009, 1, 443-3439. DOI: 10.1038/nchem.327
P. Coppens, “Charge Densities Come of Age”, Angew. Chemie Int. Ed., 2005, 44, 6810-6811. DOI: 10.1002/anie.200501734
M. Messerschmidt, S. Scheins, L. Grubert, M. Pätzel, G. Szeimies, C. Paulmann, P. Luger. “Electron Density and Bonding at Inverted Carbon Atoms: An Experimental Study of a [1.1.1]Propellane Derivative, Angew. Chemie Int. Ed., 2005, 44, 3925-3928. DOI: 10.1002/anie.200500169
L. Zhang, W. Wu, P. C. Hiberty, S. Shaik, “Topology of Electron Charge Density for Chemical Bonds from Valence Bond Theory: A Probe of Bonding Types”, Chem. Euro. J., 2009, 15, 2979-2989. DOI: 10.1002/chem.200802134
F. Feixas , E. Matito, M. Duran, M. Solà and B. Silvi, submitted for publication. See also this abstract.
See for example the work of R. F. Nalewajski

Rzepa, Henry S. The weirdest bond of all? Laplacian isosurfaces for [1.1.1]Propellane. 2010-07-21. URL:http://www.ch.ic.ac.uk/rzepa/blog/?p=2251. Accessed: 2010-07-21. (Archived by WebCite^® at http://www.webcitation.org/5rOFp6EuM)

Tags:electronic energy, energy, energy density, excess kinetic energy, Hiberty and co, Hypervalency, Interesting chemistry, kinetic energy density, potential energy density, representative, X-ray
Posted in Hypervalency, Interesting chemistry | 4 Comments »

Anatomy of an asymmetric reaction. The Strecker synthesis, part 1.

Monday, May 24th, 2010

The assembly of a molecule for a purpose has developed into an art form, one arguably (chemists always argue) that is approaching its 100th birthday (DOI: 10.1002/cber.191104403216) celebrating Willstätter’s report of the synthesis of cyclo-octatetraene. Most would agree it reached its most famous achievement with Woodward’s synthesis of quinine (DOI: 10.1021/ja01221a051) in 1944. To start with, the art was in knowing how and in which order to join up all the bonds of a target. The first synthesis in which (relative) stereocontrol of those bonds was the primary objective was reported in 1951 (10.1021/ja01098a039). The art can be taken one step further. It involves control of the absolute stereochemistry, involving making one enantiomer specifically (rather than the mirror image, which of course has the same relative stereochemistry). Nowadays, a synthesis is considered flawed if the enantiomeric excess (of the desired vs the undesired isomer) of such a synthesis does not achieve at least ~98%. It is routine. But ask the people who design such syntheses if they know exactly the reasons why their reaction has succeeded, you may get a less precise answer (or just a lot of handwaving; chemists also like to wave their hands as well as argue).

Here I set out one such asymmetrically stereospecific scheme, which is the first part of a reaction used to make both natural and un-natural configurations of aminoacids; the Strecker synthesis.

The asymmetric synthesis of an S(S) sulfoxide. Click for 3D model

It makes use of a natural product based on the camphor ring system which nature provides as a single enantiomer. It is converted to an oxaziridine, and this reagent is now used to transfer one oxygen atom to an imino-thioether (DOI: 10.1021/ja00030a045). The result is the formation of a single S(S) enantiomer (the enantiomeric excess is > 98%) of a sulfoxide. In the second stage, cyanide is then delivered asymmetrically (i.e. to one face rather than the other) of the C=N group, the precursor to forming a pure enantiomer of an amino acid. Here we will probe why the first reaction, the asymmetric oxygen atom delivery, is so specific. It would be fair to say that this reaction was probably originally designed with no fundamental understanding of how it might achieve its magic asymmetric delivery. For example, those two chlorine atoms on the camphor ring look as if they were selected by trial-and-error. What indeed IS their role? Steric? Electronic? Other?

If you click on the diagram above, a rotatable 3D model should appear (a static version is shown below). This is an AIM (atoms-in-molecules) analysis of the curvature of the electron density in this transition state (see DOI: 10042/to-4929). To help you navigate, arrow 1 is pointing to the small purple sphere representing the BCP (bond critical point) for the forming S…O bond. Three more purple spheres are highlighted with a halo. One of these is pointed to by arrow 2 below (to see the other two, you really will need the 3D model). This represents a BCP which appears between the hydrogen of the N=CH group and one of the oxygen atoms of the sulphone group. The label indicates the electron density at that point (0.017 au). This is characteristic of a hydrogen bond, albeit an unusual C-H…O type (a type that is too rarely invoked when explanations of stereospecificity are sought), and the density indicates its a reasonably strong one!

AIM analysis of Transition state for oxygen transfer

In fact, two more BCPs can be located between this H and other groups, and they too are marked with halos. The first leads to the oxygen atom being transferred, and the second to specifically one of the two chlorine atoms (there are other interactions to the chlorines as well). Now, it turns out that these interactions are largely absent for the alternative transition state (which would form the enantiomeric R(S) sulfoxide). Since a C-H…O hydrogen bond can easily be worth ~2 kcal/mol, it is no surprise to find that the energy of the favoured transition state is overall 2.4 kcal/mol lower in free energy compared to the isomer not formed. This represents (@300K) a ratio of 60:1 in the predicted ratio of the two species, or indeed an ee ~98%.

Armed with this insight, one could design further experiments to test the hypothesis. For example, it appears only one of the two chlorines plays an active role. Replacing the passive chlorine with e.g. hydrogen might make little difference. Suppressing the hydrogen bonds by changing the N=CH to e.g. N=CF should have a big effect. The two oxygens of the sulfone also do not play equal roles. Perhaps this can be tested with a sulfoxide in place of the sulfone? All these hypotheses can of course first be tested with calculations. Of course, coming up with synthetic strategies for these new molecules might be tricky. But these experiments may give confidence (or demolish it) in the AIM technique used here to analyse the stereospecificity of this reaction.

So the next time you hear a synthetic chemist proudly announce a new enantioselective synthesis, ask them what their deeper understanding of why their reaction works is. And be prepared to run away fast if they growl at you!

Tags:energy, free energy, Interesting chemistry, natural product, synthetic chemist
Posted in Interesting chemistry | No Comments »

Semantically rich molecules

Sunday, May 2nd, 2010

Peter Murray-Rust in his blog asks for examples of the Scientific Semantic Web, a topic we have both been banging on about for ten years or more (DOI: 10.1021/ci000406v). What we are seeking of course is an example of how scientific connections have been made using inference logic from semantically rich statements to be found on the Web (ideally connections that might not have previously been spotted by humans, and lie overlooked and unloved in the scientific literature). Its a tough cookie, and I look forward to the examples that Peter identifies. Meanwhile, I thought I might share here a semantically rich molecule. OK, I identified this as such not by using the Web, but as someone who is in the process of delivering an undergraduate lecture course on the topic of conformational analysis. This course takes the form of presenting a set of rules or principles which relate to the conformations of molecules, and which themselves derive from quantum mechanics, and then illustrating them with selected annotated examples. To do this, a great many semantic connections have to be made, and in the current state of play, only a human can really hope to make most of these. We really look to the semantic web as it currently is to perhaps spot a few connections that might have been overlooked in this process. So, below is a molecule, and I have made a few semantic connections for it (but have not actually fully formalised them in this blog; that is a different topic I might return to at some time). I feel in my bones that more connections could be made, and offer the molecule here as the fuse!

Two chair conformations of the molecule DULSAE. Click here for 3D. Note the (attractive) short H...H contacts.

To list all the likely semantics that a chemist would perceive in the graphic above would take far too long (by the time one would have finished, a text book would have been written). So here is a very very short summary in the context of conformational analysis.

The molecule has a six membered ring as its backbone
which can adopt two possible chair conformations
which can interconvert by exchanging the axial and equatorial group pair for each of the four carbon atoms in the ring.
An organic chemist will immediately notice a very unusual group, Fe(CO)₂Cp, which itself is a semantic goldmine,
but for the purposes here we will regard merely as a C-Fe bond!

The (semantic) question to be posed is “which of the two conformations shown above is the most stable“? That too of course has an abundance of implicit semantics, but most human chemists will probably know that this refers to asking which of the two geometries represents the lowest thermodynamic free energy (and we leave aside the issue of what medium the molecule is in, i.e. solid, solution or gas). A far trickier question is “why”?

So to (some interim) answers. Well, a ωB97XD/6-311G(d) calculation (wow, think of what is implied in that concise notation) predicts conformation (a) to be more stable by 2.3 kcal/mol (2.1 in ΔG, see DOI: 10042/to-4911). Now to the why. What connections would someone well versed in conformation analysis spot?

The molecule has two methyl groups on adjacent atoms. They may prefer to be di-axial rather than di-equatorial to avoid excessive steric repulsions (whatever we mean by that!). That might prefer (b).
The molecule has one carbon with both a cyano and an ether linkage. Well, that is susceptible to an anomeric effect (although, as I pointed out in an earlier post here, this connection has in fact often NOT been made in the literature). Only in conformation (a) is one of the oxygen lone pairs aligned anti-periplanar to the axis of the C-CN bond. The reasons why this is important are outlined in my Lecture course.
Having spotted the last, the human might ask whether there is any possibility of an anomeric effect between an oxygen lone pair and the axis of the C-Fe bond? Well, I rather think that not a single human ever has asked that question! (I cannot know that of course, and perhaps someone has speculated upon this in the literature; this is where a full semantic web would help. That question could be posed of it! The reason I suspect the connection might not have been made is that the anomeric effect is the domain of the organic chemistry, and C-Fe bonds are those of the organometallic chemist. They do tend to see the chemical world rather differently, these two groups of chemists). If there was such an effect, it would favour (a).
Then we have an X-C-C-Y motif. Depending on the nature of X and Y, the molecule might actually prefer a gauche conformation, i.e the dihedral angle XCCY would be around 60°. There are several such motifs one can detect; X=Y=O (twice). It might be that other permutations such as X=CN, Y=Fe(CO)₂Cp, favour anti-periplanar. There are other permutations whose orientational preference may not even be recorded (in text books). Suddenly its gotten complicated!
There are a number of short (~2.4Å) H…H contacts
We are starting to understand that to unravel the conformation of this molecule, one may have to identify quite a number of different “rules”, and then to quantify each, and add up the numbers to get the final result. That energy of 2.3 kcal/mol may be composed of the result of applying quite a number of different rules. Hence the title of this post, a semantically rich molecule!

Well, I will leave it here for this post, without giving answers to the six points listed above, or really answering my main question posed above. That would make the post too complex (but I will follow this up!). I do want to end by planting the idea that answering this question involves making a great many chemical connections about the properties of this molecule, and then identifying quantitative ways (algorithms) in which an answer can be formulated. The molecule above is presented as a challenge for the Semantic Web to address!

Tags:chair, chemical connections, Chemical IT, chemical world, chemist, energy, Fe, General, Interesting chemistry, lowest thermodynamic free energy, organic chemist, organometallic chemist, Peter Murray-Rust, semantic web, unusual
Posted in Chemical IT, General, Interesting chemistry | 2 Comments »

The conformational analysis of cyclo-octane

Sunday, January 31st, 2010

In the previous post, I suggested that inspecting the imaginary modes of planar cyclohexane might be a fruitful and systematic way in which at least parts of the conformational surface of this ring might be probed. Here, the same process is conducted for cyclo-octane. The ring starts with planar D_8h symmetry, and at this geometry (B3LYP/6-311G(d,p), DOI: 10042/to-3742) five negative force constants (corresponding to imaginary modes) are calculated. The most negative is non-degenerate, and gives directly the crown conformation of D_4d symmetry (DOI: 10042/to-3738). The remaining four modes comprise two degenerate pairs. Following either of the E_2u eigenvectors downhill leads to another conformation, D_2d (DOI: 10042/to-3741), with a geometry which is noteworthy for exhibiting a pair of unusually close non-bonded H…H contacts (1.908Å). This value is about 0.3Å shorter than the sum of the Wan der Waals radii (DOI: 10.1021/jp8111556). We can debate whether such a close approach or inter-penetration of two hydrogens is a bond or not (an AIM analysis appears at the bottom of this post).

D8h, +82.8 kcal/mol
Follow B_2u 467i	Follow E_3g 404i	Follow E_2u 230i
to D_4d +0.8	to C_i 131i (A_u), +7.5	to D_2d +3.6
B2u	E3g	E2u
C_s0.0	C₂ +1.6	–

Following the remaining E_3g mode leads to a stationary point of C_i symmetry (DOI: 10042/to-3743). This is a valley-ridge potential, since this point turns out to be a transition state itself, and following the A_u imaginary mode at this point results in another, this time stable conformation, of chiral C₂ symmetry (DOI: 10042/to-3744). This has a calculated optical rotation [α]_D of 72° (at 589nm in chloroform).

Are these three conformations all there are? Well, a thorough analysis of the conformational space has in fact identified six minima (DOI: 10.1002/(SICI)1096-987X(19980415)19:5<524::AID-JCC5>3.0.CO;2-O), of which the most stable has C_s symmetry (the so-called chair-boat conformation, and the one most frequently found in crystal structures of cyclo-octanes). Where is that one in the above analysis? It arrives by a distortion of the D_4d form (DOI: 10042/to-3747) via a transition state of no symmetry (DOI: 10042/to-3752)

Whilst the full potential surface clearly has many more features, following the modes of the planar conformation of cyclo-octane is a simple and rapid way of establishing four of the six limiting stable conformations (the two remaining forms have D₂ and S₄ symmetry, see DOI 10.1016/0166-1280(88)80008-3).

AIM analysis of D2d cyclo-octane.

Finally as promised, the AIM analysis of the D_2d conformer (above). The ρ(r) value at the interesting H…H critical point is 0.015, which is pretty high in comparison to most normal hydrogen bonds, and would be conventionally taken to indicate attraction. The Laplacian ∇²ρ(r) is +0.05. The “bond” ellipticity ε has a value of 0.29. Single bonds are close to zero, and C=C double bonds are ~0.4, so this is pretty high (see also DOI: 10.1002/anie.200805751).

The two highest C-H stretching vibrations for this conformation are well separated from all the others (ν 3095, 3103 cm^-1 for the symmetric A₁ and antisymmetric B₂ combinations, below for animations). These vibrations serve to both decrease and increase the H…H distances as part of the atomic (harmonic) displacements, and clearly doing so takes more energy than vibrating any of the other C-H bonds. It seems unlikely that the C-H bonds are themselves stronger, so does that mean that the H…H interaction is attractive or is it repulsive? In this context, it is worth noting that the symmetric vibration (both H…H distances decrease/increase at the same time) is lower in wavenumber than the mode which decreases one and increases the other.

Tags:3g, conformational analysis, cuyclooctane, energy, General
Posted in General | 1 Comment »

The conformation of cyclohexane

Thursday, January 28th, 2010

Like benzene, its fully saturated version cyclohexane represents an icon of organic chemistry. By 1890, the structure of planar benzene was pretty much understood, but organic chemistry was still struggling somewhat to fully embrace three rather than two dimensions. A grand-old-man of organic chemistry at the time, Adolf von Baeyer, believed that cyclohexane too was flat, and what he said went. So when a young upstart named Hermann Sachse suggested it was not flat, and furthermore could exist in two forms, which we now call chair and boat, no-one believed him. His was a trigonometric proof, deriving from the tetrahedral angle of 109.47 at carbon, and producing what he termed strainless rings.

Whilst the chair form of cyclohexane now delights all generations of chemistry students, the boat is rather more mysterious. Perhaps due to Sachse, it is still often referred to as a higher energy form of the chair (Barton, in the 1956 review that effectively won him the Nobel prize, clearly states that the boat is one of only two conformations free of angle strain, DOI: 10.1039/QR9561000044). Over the last 30 years or so, and especially with the advent of molecular modelling programs, the complexity of the conformations of cyclohexane has become realised. A nice recent illustration of that complexity is by Jonathan Goodman using commercial software. Here a slightly different take on that is presented.

The starting point is the flat Baeyer model for cyclohexane. Like benzene, it has D_6h symmetry. When subjected to a full force constant analysis using a modern program (in this instance Gaussian 09), this geometry is revealed (DOI: 10042/to-3708) to have three negative force constants, which in simple terms means it has three distortions which will reduce its energy. The eigenvectors of these force constants are shown below, and each set of vectors acts to reduce the symmetry of the species. Such symmetry-reduction is a well known aspect of group theory, and its analysis in the Lie symmetry groups is used in many areas of physics and mathematics, but it is a less used in chemistry.

348i cm^-1(B_2g)	244i (E_2u)	244i (E_2u)
D6h to C2h for cyclohexane. Click for animation.	D6h to D2. Click for animation.	D6h to C2v. Click for animation.

The first of these symmetry-reducing vibrations (the B_2g mode) converts the geometry immediately to the chair conformation of cyclohexane. So in some ways, this use of symmetry is a modern equivalent of the trigonometry used by Sachse to prove his point.

The next two modes are degenerate in energy, and the first of these reduces the symmetry to D₂. The result is what we now call the twist-boat. It is interesting, because the D₂group is one of the (relatively few) chiral groups, and the twist-boat exhibits disymmetric symmetry. In other words, following the vibrational eigenvectors in one direction leads to one enantiomer of the twist boat, and in the other direction to the other enantiomer. So (in theory only), one might actually be able to produce chiral cyclohexane (the experiment and resolution would have to be done at very low temperatures!). It is also interesting that theory nowadays could quite reliably calculate the optical rotation of this species (and its circular dichroism spectrum), so we certainly would know what to look out for.

The second component of the degenerate E imaginary mode leads directly to a species of C_2v symmetry, which we recognize as Sachse’s second possible form of cyclohexane. The symmetry-reductions of D_6h to C_2h, D₂ and C_2v all have paths on the grand diagram of the 32 crystallographic point groups and their sub groups, and is an interesting application of group theory to a mainstream topic in organic chemistry.

But the story is not quite complete yet. The C_2v boat is not the final outcome of the last distortion! It too is a transition state, connecting again the two D₂ forms. So the path from D_6h to C_2v is NOT a minimum energy reaction path, but a rather different type of path known as a valley-ridge inflection path. An example of such a surface can be seen for the dimerisation of cyclopentadiene (DOI: 10.1021/ja016622h) and effectively it connects one transition state to a second transition state, without involving any intermediates on the pathway. At some stage, the dynamics of the system takes over, and the symmetry breaks without the system ever actually reaching the second transition structure. This final aspect of the potential energy surface of cyclohexane was not discussed by Jonathan Goodman in his own article on the topic.

So symmetry-breaking is the topic of this blog, and its connection to physics and mathematics. And, I might add that the same approach can be taken for looking at the conformations of cyclobutane, pentane, heptane and octane. But that will be left for another post.

Postscript. See this more recent post.

Tags:Adolf von Baeyer, animation, C, chair, conformational analysis, energy, General, Hermann Sachse, higher energy form, Interesting chemistry, Jonathan Goodman, minimum energy reaction path, model for cyclohexane, potential energy surface, symmetry breaking, Tutorial material
Posted in General, Interesting chemistry | 17 Comments »

Chemical intimacy: Ion pairs in carbocations

Monday, January 11th, 2010

The scheme below illustrates one of the iconic reactions in organic chemistry. It is a modern representation of Meerwein’s famous experiment from which he inferred a carbocation intermediate, deduced from studying the rate of enantiomerization of isobornyl chloride when treated with the Lewis acid SnCl₄.

The isomerisation of iso-bornyl chloride

Meerwein himself suggested (in effect, since he lacked the modern terminology used here) that the reaction proceeded via a hydride shift 3, which was acting as the mirror in reflecting 1 onto 1‘. A few years later, isotopic labelling studies demonstrated that another pathway occurs, at more or less the same rate. This alternative proceeds via a series of [1,2] carbon shifts, with the mirror now being 8 rather than 3. I have documented the story in detail in an article that will shortly appear in the J. Chemical Education (DOI: 10.1021/ed800058c). There, calculations reveal that the two transition states, 3 and 8 (which the experiments above suggest should be almost equal in energy) in fact differed by ~8 kcal/mol in favour of the latter for a gas-phase model which does not include the counterion. These calculations were done at a level (B3LYP/cc-pVQZ) which indicates that 8 kcal/mol represents a real discrepancy not so much in the calculation as in the model used for that calculation. I suggested that perhaps the discrepancy might be due to tunneling effects in the hydride transfer reaction, accelerating that pathway compared to methyl transfer.

What was missing from that particular model was the counter-ion, which is supposed to form an intimate ion-pair with the carbocation in moderately polar solvents. How much does the presence of such an object perturb the transition states? To find out, we need calculate such systems (which by definition have very large dipole moments) with inclusion of solvation corrections. Now that new algorithms for computing transition states with solvation have made this a routine calculation, I can report an update to these results. This was done at the B3LYP/cc-pVTZ (aug-cc-pVTZ-pp for the Sn) level, using dichloromethane as a continuum solvent. Without the SnCl5 counterion, 3 and 8 differ by 5.4 kcal/mol in free energy (this difference now includes all the solvation free energy terms), and in the presence of the counter-ion this remains unchanged at 5.4 kcal/mol (see DOIs 10042/to-3668 and 10042/to-3667 without SnCl5 and 10042/to-3670 and 10042/to-3665 with). The free energy of activation with SnCl5 (see DOI: 10042/to-3695 for starting material) is 16.6 kcal/mol (for the [2,6] H shift) and 11.2 kcal/mol (for the [1,2] Me shift), which indicates a facile room temperature reaction (as indeed is the case).

TS H-transfer. Click for animation

TS 1,2 Methyl shift. Click for animation

What are the implications for this result?

Modelling an (intimate) ion-pair is different from that of covalent compounds in one respect. Whereas the geometry at covalent atoms is very well established and largely predictable, ion-pairs are potentially much more flexible. In other words, it is nowhere near as obvious where to place the counter-ion. In the above diagrams, the SnCl₅ is located at a reasonable position, but there are other positions where it could be. Although what is shown is an energy optimized structure, a full search of all the possible positions that the SnCl₅ could adopt has not been undertaken, and the possibility must remain that another pose of the ion might be lower in energy, for either of the two transition states. Indeed, if it turns out there are many positions for the ion of very similar in energy, then the entropy of the system would have to be corrected for these microstates.
Nevertheless, one can draw insight from the two structures shown above (click to animate the transition mode). The counter-ion for the hydride transfer does approach the transferring hydrogen quite closely, and does appear to establish a H-bond between two hydrogens and one chlorine. This would stabilize that structure relative to the methyl shift transition state, where such hydrogen bonds do not appear to form. In this case however, these interactions do not change the relative stabilties.
These ion-pairs do have very large dipole moments (~23D for 3, ~27D for 8), which suggests that the result might in fact be sensitive to the nature of the solvent (and presumably the counter-ion itself).

Many reactions do take place in which intimate ion-pairs are formed (including a fair number of catalytic systems involving metals). We cannot generalise from the result above, but it may well be that the perturbation induced by such counter-ion may play significant roles in deciding selectivities. I would venture to suggest that increasingly modelling such as reported here will play a significant role in establishing mechanisms accounting for the selectivity of catalytic reactions.

Tags:animation, catalytic systems, Chemical intimacy, energy, free energy, gas phase model, Interesting chemistry, Meerwein, solvation free energy terms, watoc11
Posted in Interesting chemistry | No Comments »

The SN1 Reaction- revisited

Wednesday, November 11th, 2009

In an earlier post I wrote about the iconic S_N1 solvolysis reaction, and presented a model for the transition state involving 13 water molecules. Here, I follow this up with an improved molecule containing 16 water molecules, and how the barrier for this model compares with experiment. This latter is nicely summarized in the following article: Solvolysis of t-butyl chloride in water-rich methanol + water mixtures, which (for pure water) cites the following activation parameters

ΔH^†₂₈₃ = 23.0 kcal/mol
ΔG^†₂₈₃ = 19.7 kcal/mol
ΔS^†₂₈₃ = +11.1 cal/mol/K

But first, a word about how this new transtion state has been obtained. The DFT treatment used is quite standard (B3LYP/6-31G(d) ), and one can indeed locate a transition state using just this approach (this is how the previous model was obtained). One has to work very hard to orient the starting guess for the geometry so that as many hydrogen bonds between the waters themselves, and to the substrate, are created. The previous model took quite a few guesses and attempts! The solvent in such a model is simulated by the explicit water molecules themselves. Of course, the quality of the solvent then depends on how many water molecules are used. A proper solvent field using explicit water molecules is thought to require 100s of water molecules! But a reasonable approximation/compromise may well be 13.

So how can the model be improved? Well, in many ways, some of which include treating the dynamics of the system. But I will stick just to two.

Firstly, we assume that the water molecules are used to form a bridge between the incoming nucleophile (another water) and the leaving group (the chloride). In the previous model, two such bridges were constructed using the 13 water molecules. But in fact, there is still space between two of the methyl groups to construct a third bridge. This takes the total solvent molecules to 16.
Solvent can also be modelled as a continuum, in which a cavity which the substrate occupies is surrounded by a field generated by the continuum solvent. The problem with these cavity approaches in the past has been that it is not easy to optimize the geometry of the molecule contained within the cavity. Because the cavity was constructed by tesselation, the first derivatives of the energy of the molecule within the cavity were not regular, and as a result, geometry optimization (and particularly transition state optimization) would frequently meander and fail to converge. Darrin York and Martin Karplus came to the rescue (some time ago, it has to be said, DOI: 10.1021/jp992097l) by formulating a smoothed out solvation cavity where the first (and second derivatives) are stable and well behaved. This new algorithm has now been implemented in Gaussian09, and it now allows really easy transition state location within a solvent cavity

The result of this optimization is shown below (and can be seen in original form at the following DOI: 10042/to-2894).

Transition state for Sn1 solvolysis of tert-butyl chloride. Click for animation.

The model has not changed that much compared to before. The reaction (imaginary) mode still clearly shows formation of the C-O bond and cleavage of the C-Cl bond. Also as before, there is a lot of motion of the methyl groups, as the forming cation induces stereo-electronic alignment with the adjacent C-H bonds (and which explains the large secondary deuterium isotope effects measured for this reaction, k_H/k_D (298) = 2.39, see DOI: 10.1021/ja01080a004). The hydrogen bonding pattern is also retained (despite the surrounding solvent field!). But what of the predicted activation parameters

ΔH^†₂₉₈ = 17.4 kcal/mol
ΔG^†₂₉₈ = 18.7 kcal/mol
ΔS^†₂₉₈ = -4.4 cal/mol/K

The overall free energy is in great agreement with experiment! But the entropy is the wrong sign!! The calculation is predicting that the transition state is more rigid than the reactant. One can see how this might happen, since the greater ionic character produces very much stronger hydrogen bonds, which strengthen the three solvent bridges. It may be simply that the rigid-rotor-harmonic-oscillator approximation breaks down horribly for the entropy in this calculation. But it is encouraging that the activation barrier is reproducing experiment, which suggests the model cannot be completely wrong!

Tags:animation, Darrin York, energy, Historical, Interesting chemistry, Martin Karplus, Organic reaction mechanism, overall free energy
Posted in Interesting chemistry | 7 Comments »

Towards the ultimate bond!

Monday, August 24th, 2009

The 100th anniversary of G. N. Lewis’ famous electron pair theory of bonding is rapidly approaching in 2016 (DOI: 10.1021/ja02261a002). He set out a theory of bond types ranging from 1-6 electrons. The strongest bond recognized by this theory was the 6-electron triple bond, a good example of which occurs in dinitrogen, N₂. In terms of valence electrons, nitrogen has an atomic configuration of 2s², 2p³. Each atom has five electrons in total, some or all of which in principle could be used for forming bonds. An exploration of this motif across the entire periodic table is presented in part one of this blog.
Elements in Groups 5/15 of the Periodic Table.
Nitrogen is in the main group 15, and the element at the bottom of this group is Bismuth (also with the same atomic configuration). We can then move to the corresponding column of the transition series, this time occupying group 5. The first examplar in this set, Vanadium has an atomic configuration of 3d³, 4s², again five valence electrons, but now utilizing the d- rather than the p-shell of valence atomic orbitals (AOs). The final forage across the period table would land us with Pr and Pa, which occupy the lanthanide and actinide series respectively, and which have atomic configurations of 4f³, 6s² and 5f², 6d¹ and 7s² respectively. You can now see the theme developing; how does the bonding develop between two atoms that between them have ten valence electrons occupying molecular orbitals constructed from s, and then either p, d or f atomic orbitals. The next in that series, g atomic orbitals, are thought unlikely to have any chemical significance in the presently known periodic table.

We are in fact going to study the diatomic molecule comprising two atoms of each element, and further we are going to protonate this species on one of these atoms resulting in the molecule HX₂⁺. So let us start with the two systems HN₂⁺ (Figure 1, 1a-1e) and HBi₂⁺ (Figure 1, 2a-2e).

Figure 1. The molecular orbitals of HN2(+) and HBi2(+)

The two most stable valence molecular orbitals (MOs) for each system (1a and 2a) are the symmetric and antisymmetric combinations of (assumed pure) s-AOs, each populated with two electrons. For Bi (2a/2b) it is fairly clear that this bonding and anti-bonding combination cancel almost exactly. The bond order resulting from these four valence electrons is therefore close to zero. But N is in fact rather odder (and in part, the reason for protonating the systems was to tease out this oddity)! The apparently antibonding 2s-2s combination (1b) actually has electron density along the N-N bond, and the node occurs not along the bond, but at the actual nitrogen atom. So for N, the bonding and anti-bonding σ-bond combinations do not cancel, and the sum of these two may actually lead to a non-zero bond order.

With Bi, the next most stable MO results from the overlap of two 6p-AOs end on (2c); the anti-bonding combination of these AOs is in fact the 2e (with no occupying electrons). Result: bond order of +1, and this bond is called a σ-bond. The final MO shown (2d) is in fact one of a pair of MOs (only one of which is shown here), resulting from parallel overlap of two 6p-AOs. Result: bond order of +2, and we call these two p-π-bonds. The total bond order is +3, comprising 1 σ-bond and two p-π-bonds. No surprises here yet! For N, the relative order of the π- (1c) and the σ-bonds (1d) is swapped compared to Bi (which might be due to relativistic effects, see below), but one can again argue that these three orbitals together contribute 1 σ-bond and two p-π-bonds to the total bonding. So, to summarize, HBi₂⁺ exhibits a classical triple Bi…Bi bond; HN₂⁺ in contrast may actually exceed the bond order of +3, and a case could be made for arguing it is an abnormally strong bond (there is evidence that the N…N stretching frequency in the protonated species is significantly higher than the simple diatomic nitrogen gas).

Let us now move to the combination HV₂⁺ (3) and HTa₂⁺ (4, Figure 2). There are two major differences for V compared to N. Firstly, a 3d rather than 2p -AO is used for the bonding. Secondly, the 3d-AO is actually lower in energy than the 4s-AO, the reverse of the 2s/2p order for eg nitrogen. So what effect does this have on the resulting molecular orbitals?

Figure 2. Valence molecular orbitals for HV2(+) and HTa2(+). Click image for 3D of Delta orbital

The difference between N and V turns out to be spectacular, in several regards! The most stable MO (3a) now turns out to be composed of end-on overlaps of two 3d-AOs. There are in fact two of these orbitals (only one is shown in Figure 2), and together they form two 3d-π bonds. The next MO (3b) involves the end-on overlap of a 3d_z2-AO, this time forming one 3d-σ-bond. Finally, the 4s-AOs get in on the act, forming now one bonding s-σ-bond (3c). So far, eight of the ten valence electrons have been consumed; two more to go. But now we have a problem. The next MO is formed by parallel overlap of two 3d-AOs (Figure 2, 3d), and in fact there is a pair of these combinations (again, only one is shown in the figure), which are in fact equal (degenerate) in energy. Because they are equal in energy, they must both be populated by an equal number of electrons, but since there are only two valence electrons left, we end up with one electron in each of these two orbitals, resulting in a triplet spin state. Exactly the same phenomenon is responsible for diatomic oxygen also adopting a triplet rather than singlet spin state. This parallel overlap of two 3d-AOs is said to form a 3d-δ-bond. The total bond order in molecule 3 therefore comprises two 3d-π-bonds, one 3d-σ-bond, one 4s-σ and two half 3d-δ-bonds, i.e. five in total and enforces a triplet spin state. So in the sense of formal bond order, the V-V bond in 3 is greater (and perhaps even stronger) than in 1. What a difference from nitrogen!

How about Ta? This has an atomic electronic configuration of 5d³, 6s². The molecular orbitals are shown in Figure 2 (right). The significant difference in this region of the periodic table is that so-called relativistic effects start to influence the relative ordering of the atomic orbitals. This so-called relativistic contraction impacts upon s-AOs far more than p, or d or f. Thus orbital 4a (Figure 2, right) comprising a symmetric combination of Ta 5s-AOs is more compact than the corresponding V orbital (3c), and relatively more stable. Next come the end-on overlaps of two 5d-AOs (4b), of which there are two (degenerate) combinations (only one is shown in Figure 2). MOs 4c and 4d are again best described as originating from the 6s Ta AO, but with significant contributions from the 5d-AOs (sd-hybrids if you will). Significantly, the relativistic effect means that the δ-bond formed by parallel overlap of two 5d-AOs (4e) is left vacant. Thus the bonding in 4 comprises an 6s-σ-bond, two 5d-π-bonds, and two sd-σ-bonds (one of which does look rather anaemic) but NO 5d-δ-bond! The consequences of the relativistic effect is the relegation of the δ-bond to unoccupied status and the formation of a singlet rather than a triplet spin ground state.

Given the big differences in bonding which occur upon changing a 2p-valence atomic orbital to 3-5d-AOs, one wonders what will happen with 4-5f-AOs? A π-bond can be formed by the parallel overlap of two p-AOs, or end-on overlap of two d-AOs. A δ-bond can be formed by the parallel overlap of two d-AOs. Could then a δ-bond also be formed by the end-on overlap of two f-AOs? Would a φ-bond be formed by the parallel overlap of two f-AOs? The latter might look something like that shown in Figure 3, shown in two different orientations (these diagrams were obtained by inspecting the unfilled MOs in the V/Ta examples shown here; notice that although the φ bond appears to be bonding, the system has chosen not to occupy it with any electrons!).

Figure 3. Two views of a φ bonding MO. Click image for 3D

Figure 4. A φ* antibonding MO. Click image for 3D

This blog will end by posing the question “can any molecule be devised which supports one or more φ-bonds”, or will the relativistic contraction always scupper such efforts by depriving such bonds of electrons (e.g. Ta above)? Are the systems reported in DOI: 10.1021/ja067281g examples of a 5f-φ bond for uranium (the claim is made in a very low key manner)? This will be investigated in the follow up to this post!

(For serious geeks/computational chemists only, N was computed at the B3LYP/6-31G(d) level, V at the ROHF/6-31G(d) level, and Bi/Ta at a triple-ζ-pseudopotential level (which incorporates some of the relativistic effects).

Tags:chemical significance, diatomic nitrogen gas, energy, Interesting chemistry, pence, stable MO, The 100th anniversary of G. N. Lewis' famous electron pair theory of bonding, Vanadium
Posted in Interesting chemistry | 3 Comments »

Longer is stronger.

Saturday, June 6th, 2009

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 benzene

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!

Tags:energy, energy levels, higher energy, Interesting chemistry, pseudo Jahn-Teller, resonance energy, so-called energy degenerate orbital, π-electron resonance energy, π-energy, π-resonance energy
Posted in Interesting chemistry | 16 Comments »

Henry Rzepa's blog