Posts Tagged ‘conformational analysis’

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Are close H…H contacts bonds? The dénouement!

Monday, October 10th, 2011

I wrote earlier about the strangely close contact between two hydrogen atoms in cis-butene. The topology of the electron density showed characteristics of a bond, but is it a consensual union? The two hydrogens approach closer than their van der Waals radii would suggest is normal, so something is happening, but that something need not be what chemists might choose to call a “bond“. An NCI (non-covalent analysis) hinted that any stability due to the electron topologic characteristics of a bond (the BCP) might be more than offset by the repulsive nature of the adjacent ring critical point (RCP). Here I offer an alternative explanation for why the two hydrogens approach so closely.

One of four C-H NBO donor orbitals. Click for 3D

The empty C=C π* orbital. Click for 3D to show both orbitals superimposed.

We need to try to “concentrate” or “focus” the effect into the bonds of the molecule, and a good way of doing this is to calculate the NBO (natural bond orbitals). The first we focus on is localised onto one (of four) C-H bonds of the methyl group; the other is the anti bonding π* orbital of the alkene. NBO theory allows us to calculate how these two orbitals perturb each other, in the sense of the occupied orbital donating to the empty orbital. This energy is known as E(2), and for any of the four (equivalent) interactions above, it is computed at 5.17 kcal/mol. If you look closely at the orbitals, the C-H bond is leaning away from the centre, but so is the π* acceptor orbital (that is the nature of anti bonding orbitals). These characteristics improve the overlap of the orbitals, and hence tend to increase the value of E(2).

What about an alternative conformation of cis-butene in which the close contact of the H…H atoms is removed by rotation? Well, the C-H NBOs now rotate in, but the anti bonding π* orbital still tilts out. The overlap between them is no longer quite so good, and indeed the E(2) energy decreases to 4.43 kcal/mol.

Donor C-H bond in rotated isomer of cis-butene. Click for  3D

NBO C=C p* orbital in rotated isomer of cis-butene. Click for  3D

How many more orbitals should be considered? Well, the NBO technique in effect concentrates these effects into a relatively small number of orbitals (those separated by the smallest energy gap). We can also add in the four interactions between the bonding π orbital and the anti-bonding C-H* NBO. The totals for the first conformation come to 34.08 and for the second 30.44.

So we can conclude by observing that cis-butene makes a sacrifice for its greater good. Rotating the methyl groups means that the overlap of four C-H bonds with the alkene is optimised, but an undesired side effect is to induce two hydrogens to get close to each other. They would not normally be happy doing so, but the gain from the first effect is greater than the loss from the second. Whilst they may be close, chemists would prefer not to call the H-H approach a bond, even though the topology of the electron density might say it is.

By the way, this is my 150th post. I had little idea when I started that I might reach this milestone.

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Are close H…H contacts bonds?

Friday, October 7th, 2011

The properties of electrons are studied by both chemists and physicists. At the boundaries of these two disciplines, sometimes interesting differences in interpretation emerge. One of the most controversial is that due to Bader (for a recent review, see DOI: 10.1021/jp102748b) a physicist who brought the mathematical rigor of electronic topology to bear upon molecules. The title of his review is revealing: “Definition of Molecular Structure: By Choice or by Appeal to Observation?”. He argues that electron density is observable, and that what chemists call a bond should be defined by that observable (with the implication that chemists instead often resort to arbitrary choice). Here I explore one molecule which could be said to be the focus of the differences between physics and chemistry; cis-but-2-ene.

Two possible conformations of cis but-2-ene.

The structure of this system has been determined by electron diffraction to exhibit a H…H distance of ~2.1Å as in (a), DOI: 10.3891/acta.chem.scand.24-0043. Why is this of interest? Because a rotational alternative, shown as (b) could result in a significantly longer H…H distance (~ 2.6Å). Now bear in mind that the van der Waals radius of hydrogen is estimated at ~1.2Å, and that two hydrogens will be most strongly attracted by dispersion forces when separated by ~2.4Å. As they get closer, that attraction will be counterbalanced by a repulsion, which will eventually win out. Structure (b) does not benefit from H…H dispersion attractions, but are the hydrogens in structure (a) too close to do so as well?

Well, let us adopt Bader’s approach, and look at the topology (QTAIM) of the electronic distribution in structure (a). The features to concentrate on are the purple dots, which in this analysis have been named bond critical points (BCP)  and the single yellow sphere, which is a ring critical point (RCP). There are two other types of critical points, which I will call nuclear attractors (NACP) and cage points (CCP). The total occurrences of all these critical points is determined by a topological theorem (Poincare-Hopf), which states that NACP-BCP+RCP-CCP=1. You can see below that the H…H region is indeed connected by a bond critical point (none of the other purple dots are controversial). To a physicist, this is a real feature of the (observable or in this instance the calculated) electron density. In effect, it is a topological bond. Unfortunately, chemists like to think of bonds as entities which result in stability; a bond contributes to the stability of a molecule (and an anti-bond, should such exist, to instability). Hence aromaticity (=stability) vs anti-aromaticity (=instability). Chemists, by choice, might prefer not to call the H…H region a bond because it is probably not contributing to the stability of the molecule. Of course, the two camps are arguing about different things; the topology of the electron density ρ(r) vs the energy of the molecule.

QTAIM topological analysis of cis but-2-ene. Click for 3D.

Somewhat ignored in this discussion up to this point has been the yellow dot, the RCP. Firstly, notice how close it is to the H…H BCP. Secondly, notice from the Poincare-Hopf relationship that if you remove BOTH of these points, the theorem is still satisfied; a BCP and a RCP are said to be capable of annihilating each other (much like and electron and a positron might). So perhaps a chemical picture might emerge if you choose to consider BOTH points together, rather than just the BCP?

I have done so below using the NCI (non-covalent interaction) procedure (which I have commented on in many other posts here). The NCI surface is shown below, embedded within the NCI surface as purple dots are the BCP and RCP discussed above. Now, the NCI method cleverly attempts to ascertain whether a region is attractive or repulsive, and it colour codes the surface accordingly. In the below, blue is deemed attractive, and red repulsive. We immediately see that the H…H BCP is embedded in an attractive region, but the adjacent RCP is embedded in a repulsive region. Whatever attraction the BCP might be experiencing is negated by the repulsion the RCP has. The two cancel (annihilate). Taken together, the H…H region is probably repulsive (ways of quantifying how NCI regions integrate are being investigated and will be discussed in a future post). Perhaps this manner of looking at it satisfies both the physicists and the chemists?

The NCI surface for cis but-2-ene. Click for 3D.

Well, not quite. A chemist would still ask why structure (a) is preferred over structure (b), if the wider H…H region is not deemed attractive (I call it a region rather than a bond in a probably futile attempt to avoid controversy). Actually, the answer might be in how the two methyl groups each interact with the other part of the molecule, the alkene, and how that might depend on their orientation with respect to the alkene. But that analysis is for another post!

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Conformational restriction involving formyl CH…F hydrogen bonds.

Tuesday, May 31st, 2011

The title of this post paraphrases E. J. Corey’s article in 1997 (DOI: 10.1016/S0040-4039(96)02248-4) which probed the origins of conformation restriction in aldehydes. The proposal was of (then) unusual hydrogen bonding between the O=C-H…F-B groups. Here I explore whether the NCI (non-covalent-interaction) method can be used to cast light on this famous example of how unusual interactions might mediate selectivity in organic reactions.

Crystal structure of RUGYEX. Click for 3D

The crystal structure revealed a fixed syn orientation between one B-F bond and the formyl C-H bond, with a distance of ~2.36Å (using un-normalised C-H lengths), compared with a value of 2.55Å for the vdW sum. Since hydrogen bonds to fluorine are quite rare, it seems worth applying the NCI method (DOI for calculation).

NCI surfaces for RUGYEX. click for 3D.

The result indeed shows a strongly blue (= attractive) interaction surface for the CH…F region, and only a tiny one for the adjacent  H…H region. A welcome surprise however is the  O…O region, which also shows a distinct (strong) attraction. Perhaps this is due to the presence of the electron withdrawing BF3 group on one of the oxygens. This could also be regarded as a through-space anomeric effect, from one oxygen donating into the  O-B σ* orbital. Such through spaceanomeric effects (they are normally propagated through bonds, as in sugars) are probably more common than we believe.

This result nicely illustrates the utility of the NCI method; it brings an immediate visual impact, which serves to highlight interactions which otherwise might be ignored.

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The inner secrets of the DNA structure.

Wednesday, May 18th, 2011

In earlier posts, I alluded to what might make DNA wind into a left or a right-handed helix. Here I switch the magnification of our structural microscope up a notch to take a look at some more inner secrets.

A fragment of a single chain of DNA, taken from a Z-helix. Click for 3D.

The 3D coordinates of this fragment were obtained by taking a crystal structure of a Z-d(CGCG)2 containing oligomer, editing (to remove water, and superfluous bases) and subjecting it to ωB97XD/6-311G(d,p)/SCRF=water geometry refinement. This should be accurate enough to recover dispersion attractions, and various electronic and electrostatic interactions. Z-d(CGCG)2 was then reduced to the fragment you see above, which is large enough to capture the essence of the Z-helical wind, but small enough to be able to spot things which a larger fragment might overwhelm.

  1. The 3D model (click on above to obtain it) reveals that the oxygen of one of the five-membered (tetrahydrofuran) rings has a close contact to the guanine base of 2.85Å. This is some 0.3Å shorter than the combined van der Waals radii, and very typical of oxygen…electrophilic carbon interactions (see discussion here for more details). We can reasonably assume its real. It is supported by a small NBO perturbation term (~1.1 kcal/mol) corresponding to donation of the oxygen lone pair into a C-N π* orbital.
  2. The next interaction detected comes from a furanose C-H bond, in which the hydrogen approaches to within 2.48Å of the oxygen on the furanose the other side of the phosphate. This is ~0.14Å shorter than the combined van der Waals distances (remember, even at the actual vdW sum, the attraction is still attractive). Why would an apparently inert C-H bond do that? Such bonds are not normally considered in such analyses. Well, this one is special.
  3. It may well be (slightly) more acidic than normal due to a C-Hσ/C-Oσ* anti-periplanar interaction (E2 5.8, magenta bonds in 3D model) into the CH2-OH bond of the furanose. Hence the acidified H can form a weak(ish) hydrogen bond to the oxygen. The NBO E2 energy is 1.4 from the Olp to the C-H* bond (larger E2 interactions normally occur through bonds, but this one is through space, which is why it is smaller).
  4. These two interactions in turn set up a good orientation for the guanine to create a strong anomeric effect between it and the ribose; NLp/C-Oσ*; E2 11.6 (violet bond in 3D model, blue bond b in above diagram). To calibrate this interaction, anomeric effects in sugars are of the order of 14-16 kcal/mol. These stereoelectronic effect helps to slightly rigidify the relationship between the guanine base and the furanose it is connected to.
  5. In contrast, the cytosine-furanose link avoids the classical anomeric effect, and instead sets up a weaker one with a C-C bond instead (E2 6.8, indigo bond, blue bond a in above diagram). The Nlp is not as good a donor, because it is sequestered into the adjacent carbonyl group on the cytosine. The guanine has no such adjacent group, and so its Nlp is a better donor. The outcome of all of this is that the two bases, C and G end up having a different geometrical relationship to the furanose they are each connected to.
  6. Notice the gauche-like conformation of the ethane-1,2-diol fragment (gold bond in 3D model), which is again due to stereoelectronic alignments.
  7. Notice the Nlp …H-C distance of 2.51Å, which like 2 above, is around the sum of the vdW radii. It might be slightly more than just a dispersion attraction, since the NBO E2 Nlp/C-H* through space interaction is ~2 kcal/mol.
  8. There are some other relatively close atom-atom approaches, but I do not list them here. Do explore them yourself (they are labelled 8 in the diagram below).

To complete the present analysis, I include an ELF diagram. This can locate lone-pairs (as monosynaptic basins) as well as bond pairs (disynaptic basins) and so is useful for visualising the anti-periplanar anomeric effects between a lone pair and a bond (connecting a mono and a disynaptic basin if you like). Some of the interactions described in the list above are shown below with dotted lines (note that some of the lone pairs appear as two basins, distributed either face of the aromatic base).

ELF analysis. Click for 3D.

Well, cranking up the magnification on a microscope will always reveal new details. You might ask if these new details matter? Well, since DNA is such a very long polymer, repeating even a very weak (but predictable) interaction millions of times is bound to have some sort of cumulative effect. Who knows which of the ones above might play an important role in the super-winding of DNA, or its packing into a cell, or interaction with proteins, and so on. I do wonder how many of the terms I have identified above have been previously considered for such roles. Anyone know?

Postscript: Shown below is a  non-covalent-analysis (NCI,  see earlier post). A reminder that the interaction surface is colour coded with orange or red tinge if repulsive, blue if attractive, and green for weaker interactions. These surfaces pretty much recapitulate what it itemised above, adding also other interactions not listed above (labelled 8 in diagram).

NCI analysis for Z-CG fragment. Click for 3D.

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Updating a worked problem in conformational analysis. Part 2: an answer.

Tuesday, May 17th, 2011

The previous post set out a problem in conformational analysis. Here is my take, which includes an NCI (non-covalent interaction) display as discussed in another post.

The lowest energies of the four diastereomers A-D, each in two conformations (1/2) were calculated at the ωB97D/6-311G(d,p)/SCRF=ethanol level, and are shown here relative to A1 (kcal/mol) as free energies. The values of ΔΔG of each pair (relative to the lower energy conformation) are converted to relative concentrations using ΔΔG = -RT Ln K and these are shown in blue. Those conformations where a C-H bond is aligned anti-periplanar (app) with the C-NMe3 bond are highlighted in red. Various aspects of the questions posed include:

  1. Those conformations with two app combinations are likely to result in a mixture of two alkenes E and F (shown in the previous post), and likewise those with only one app alignment will give a single alkene.
  2. The concentration of the neomenthyl system (C) favours the conformer C1, which is also the one with app allignments. This will react readily. The menthyl system A has only a low relative concentration of A2, and so will react slowly. The experiments show the rate is about 10 times less than C, whereas the ratio of relative concentrations is larger. This may be due to difference in the transition state for the elimination, or possibly effects due to the presence of a counterion.
  3. The concentration of D1 is the lowest of the four pairs of conformational isomers. This might be due to buttressing, in which the relative proximity of the NMe3 and iPr groups forces the former into closer proximity with the axial methyl than is found in A2, the other conformer with an unfavourable 1,3-diaxial interaction.
  4. The relative energies of most of the pairs of combinations of conformers/diasteromers are interesting, and I leave it to you to play with them and draw conclusions (or detect mysteries).

I thought I would finish by adding an NCI analysis using a wavefunction computed for ωB97D/6-31G(d,p)/SCRF=ethanol. The green surfaces are indicative of regions where non-covalent interactions are occurring (which include electrostatic and van der Waals effects). The colour scheme is set to blue(ish) to indicate more attractive NCIs (such as strong hydrogen bonds), green to indicate weak(ish) NCIs, with a red or orange tinge indicating a repulsive  NCI.

A2 diaxial. Click for 3D.

B2 Axial-equatorial. click for 3D.

A1. Di-equatorial. Click for 3D.

B1.Di-equatorial. click for 3D.

C2. Equatorial-axial. Click for 3D.

D2. Equatorial-axial. Click for 3D.

C1 SRR. Axial-equatorial. Click for 3D.

D1 SSS. Axial-equatorial. Click for 3D.

Amongst the many features visible in these plots is the strong 1,3-diaxial NCIs seen for A2 and particularly D1 and the conformational variability in the NCI between the iPr and NMe3 groups.

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Updating a worked problem in conformational analysis. Part 1: the question.

Friday, May 13th, 2011

Conformational analysis comes from the classical renaissance of physical organic chemistry in the 1950s and 60s. The following problem is taken from E. D. Hughes and J. Wilby J. Chem. Soc., 1960, 4094-4101, DOI: 10.1039/JR9600004094, the essence of which is that Hofmann elimination of a neomenthyl derivative (C below) was observed as anomalously faster than its menthyl analogue. Of course, what is anomalous in one decade is a standard student problem (and one Nobel prize) five decades later.

Hofmann elimination from a family of cyclic systems.

One can pose two questions about these systems:

  1. What is the expected product formed by reaction of A-D; E or F or both?
  2. Can the four reactions be ranked in the order fastest to slowest (the hint that C is anomalously fast may or may not be given the students!)

Its a problem that simply requires a model to be built for its solution. And probably some hints. I give the students two:

  1. Each of the reactants can have two alternative chair conformations; let us call them A1 and A2 etc (although a really adventurous student might ask if any twist-boats are also possible). In general, only one of these two can react to eliminate the trimethylammonium group to form an alkene. The task is to determine which, and whether that also reveals whether E or F or both might be formed.
  2. The rate of reaction (all other things being equal, which they might not be) will be related to the concentration of the active conformation compared to the inactive one. So one has to decide which of the two conformations is likely to be lower in energy, and by how much. Here one can bring as many rules as you might find in the texts books (or lecture courses) to bear whilst you decide. If you are really keen, you can try building a model using suitable molecular modelling software.

So that I do not spoil your fun, I will not reveal (my) answers here, but in the next post. Try writing down answers to these two questions, and see if they agree with mine!

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Why are α-helices in proteins mostly right handed?

Saturday, April 9th, 2011

Understanding why and how proteins fold continues to be a grand challenge in science. I have described how Wrinch in 1936 made a bold proposal for the mechanism, which however flew in the face of much of then known chemistry. Linus Pauling took most of the credit (and a Nobel prize) when in a famous paper in 1951 he suggested a mechanism that involved (inter alia) the formation of what he termed α-helices. Jack Dunitz in 2001 wrote a must-read article on the topic of “Pauling’s Left-handed α-helix” (it is now known to be right handed).  I thought I would revisit this famous example with a calculation of my own and here I have used the ωB97XD/6-311G(d,p) DFT procedure to calculate some of the energy components of a small helix comprising (ala)6 in both left and right handed form.

Firstly, it is important to note that Pauling was apparently not aware of the absolute handedness of amino acids (which are (S) in CIP terminology). This had in fact only been established a few months before Pauling’s publication by Bijvoet, and news of this might not have reached Pauling. So Pauling guessed (or perhaps, he had already built his models, and did not have time to reconstruct them) and his famous α-helix diagram turned out to be the enantiomer of the real McCoy. As with DNA itself, the helix bears a diastereomeric relationship to the chirality of the amino acids; both have to be inverted to get the proper enantiomer (which is what Pauling did). The secret that Pauling had discovered was hydrogen bonding, and particular, weak N-H…O=C interactions (Wrinch had thought it was strong covalent N-C-OH bonding instead). Of course, there are other effects at work, which include van der Waals or dispersion interactions, electrostatic effects resulting from the large dipoles in peptides (not least due to the zwitterionic character), the planarity of the peptide bond itself, the potential for other types of hydrogen bond (e.g. C-H…O) and entropic effects. I have split some of these down for left and right handed forms of DNA in another post.

It turns out calculating most of these effects on an even-handed basis is not that easy. Only the recent advent of dispersion-corrected DFT procedures, together with solvation algorithms that allow for accurate geometry optimisation and subsequent evaluation of free energies allows such a calculation to be performed. Hitherto, it has been mostly molecular mechanics that has been used (which itself relies on many parameters from quantum mechanics, such as atom charges, and explicitly identifying interactions for hydrogen bonding). By returning to a quantum-mechanical model, some of these assumptions inherent in the mechanics method need not be made.

We showed in 1991 that an effective solvation treatment required for the zwitterionic form of amino acids in aqueous solutions would ideally comprise not only a self-consistent-reaction-field, but also explicit water molecules as solvent. Here only the former solvation term is included, but expanding the model to include water is certainly possible. Both the zwitterionic and the neutral forms of (ala)6 are included below, so that the effect of a large dipole on the structure and relative helical stability can be estimated. One notes that (even in a dielectric cavity corresponding to water), the extended zwitterions are high energy species.  In a protein, they of course would be stabilized by the immediate environment of the ions. The right-handed helix clearly comes out as more stable (by about 1 kcal/mol per residue, see also DOI: 10.1021/ja960665u),  but this is not really due to either dispersion effects or entropy and must therefore arise largely from the hydrogen-bond like interactions. Ionizing the termini to form a zwitterion increases the propensity for a right handed helix slightly.

Relative thermodynamic energies (kcal mol-1) of (ala)6 α-helices
System Total energy Dispersion ΔΔH298 Δ(T.ΔS298) ΔΔG298
Left, neutral 0.0
 0.0 0.0 0.0 0.0
Right, neutral -4.0
 +0.2 -4.0 0.9 -4.9
Left, zwitterion 0.0 0.0 0.0 0.0 0.0
Right, zwitterion -7.1 0.1 -6.3 1.7 -8.0

Shown below are the calculated structures. The chains have (inter alia) unusual bifurcated hydrogen-bonding interactions, between one carbonyl group and two N-H groups (show as atom with halo). These are not quite the models that Linus Pauling built!

Left handed. Click for 3D

Right handed. Click for 3D

Left handed zwitterion. Click for 3D

Right handed zwitterion. Click for 3D

For a more objective analysis of the interactions within the system, a QTAIM analysis is shown below.

Left helix. Bond critical points in green. Click for 3D.

Right helix. Click for 3D

Whilst the overall conclusion is that theory agrees well with the experimental observation that peptide sequences tend to coil into right rather than left handed helices,  the reasons they do so is a little more subtle than simple model building alone can reveal.  As the AIM shows, a plethora of unusual and weaker interactions occur within these helices, a full analysis of which must await presentation elsewhere.

An NCI analysis reveals strong hydrogen bonds as blue-shaded surfaces.

NCI surface. Click for 3D.

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A comparison of left and right handed DNA double-helix models.

Saturday, January 1st, 2011

When Watson and Crick (WC) constructed their famous 3D model for DNA, they had to decide whether to make the double helix left or right handed. They chose a right-handed turn, on the grounds that their attempts at left-handed models all “violated permissible van der Waals contacts“. No details of what these might have been were given in their original full article (or the particular base-pairs which led to the observation). This follow-up to my earlier post explores this aspect, using a computer model.

One half of a (CGCG) DNA strand

The DNA model used here is shown above; in shorthand it is d(CGCG)2. A crystal structure reveals it to form a (non-Watson-Crick) left-handed helix. If you open the 3D model below (based on a ωB97XD/6-31G(d)/SCRF=water optimisation), some of the short van der Waals contacts are measured. Most are around 2.25Å and the shortest is 2.1Å. It is worth noting that WC note in their article that a distance of 2.1Å for the B-form is acceptable (p92, bottom) and not a violation. All twelve hydrogen bond lengths H…O or H…N are normal, with lengths around 1.8Å. Given that a H…H distance is at its most attractive at ~2.4Å, and plenty of H…H distances of ~2.1Å are known from the crystal structures of organic molecules, one might conclude that (for the CG base pair), their hypothesis that the Z-form could be eliminated was wrong.

The DNA duplex d(CGCG) showing a left handed helix with short H...H contacts shown. Click for 3D

But might the original WC-right handed form for this system be at least competitive? There is one H…H of 2.05Å and quite a few at ~2.5Å (3D model below). The “violation” of van der Waals contacts is if anything slightly worse than with the left-handed helix. The total difference in the dispersion energy is a rather astonishing ~12 kcal/mol in favour of the Z-form. I will update this post (as a comment) when the relative free energies of the two forms are available (this calculation takes a while), but there is little doubt that the Z-form is indeed the more stable.

The DNA duplex d(CGCG) showing a right handed helix with short H...H contacts shown. Click for 3D

What can also be said about the Watson-Crick right handed form is that the hydrogen bonding is not so optimal. One of the twelve interactions between a (terminal) CG pair has some signs of being “unzipped“, with an N-H…O=C distance of ~1.9Å (there is no sign of similar unzipping in the Z-form). One must wonder whether this difference in the Z- and B-helices for the CG pair has been exploited in nature.

 

One crucial aspect of DNA is the local conformation about the bond connecting the base and the ribose, N9-C8 in the diagram below(green arrow).

Conformation of the base-ribose unit

An analysis of this bond can be expressed in terms of NBO theory. This clearly shows a strong interaction energy (E2) between the lone pair on N9 and the C8-O4 antibonding orbital of 13.3 kcal/mol, a classical anomeric effectin fact. In this case, it promotes the local conformation of this unit, which has a significant effect on the final model.

What else can analysis of the wavefunction tell us? Well, curiously, the optical rotation of this particular small oligomer has never been reported in the literature, and an intriguing question is whether it might have proved useful to distinguish between B- and Z-forms of the duplex? To do this, one needs a reasonably reliable way of computing [α]D for both isomers. This is because optical rotations are not reliably additive, and it is difficult to estimate them accurately based purely on the fragments present in the molecule. In 2011, is is now perfectly possible to calculate this quantity quantum mechanically, even for 250 atoms, using a reasonable basis set and making allowance for solvation (which is known to affect the calculated rotation). The values (CAM-B3LYP/6-31G(d)/SCRF=water) for the Z-isomer are 66° and 32° for the B-isomer. Of course the model is not complete, lacking a counterion for the phosphate and explicit water molecules, but even so, it might appear that the reason optical rotations are not reported is that they truly are not useful!

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The conformation of 1,2-difluoroethane

Tuesday, April 6th, 2010

Here I offer another spin-off from writing a lecture course on conformational analysis. This is the famous example of why 1,2-difluoroethane adopts a gauche rather than antiperiplanar conformation.

The gauche and antiperiplanar conformations of 1,2-difluoroethane

One major contribution to the greater stability of the gauche is the stereoelectronic interactions, and this is best probed using the NBO (Natural Bond Orbital) approach of Weinhold (DOI: 10.1021/ja00501a009). The process is approximately described as first reducing the wavefunction down to a set of orbitals which have been localized (using appropriate algorithms) down to two or one centres (corresponding to two-centre covalent bonds, or one-centre electron lone pairs). Perturbation theory is then used to evaluate the interaction energy between any filled and any empty combination. For the molecule above, six such combinations are inspected, involving any one of the six filled C-H or C-F σ-orbitals, and the best-overlapping σ* orbital which turns out to be located on the C-H or C-F bond anti-periplanar to the filled orbital.

Filled C-H NBO orbital. Click for 3D to superimpose empty C-F anti bonding orbital.

Empty C-F antibonding NBO orbital. Click for 3D

A filled C-H orbital is shown above on the left, accompanied by an empty C-F σ* orbital on the right which is anti-periplanar to the first. This alignment allows the phases of the two orbitals to overlap maximally (blue-blue on the top, red-red beneath).

The interaction energy between this pair is determined not only by the efficacy of the overlap, but by the energy gap between the two. The smaller the gap, the better the interaction energy (referred to as E2, in kcal/mol). For the gauche conformation, the six pairs of orbitals have the following interaction energies; two σC-H/σ*C-F interactions (illustrated above), 4.9; two σC-H/σ*C-H 2.6 and two σC-F/σ*C-H 0.8 kcal/mol. For the anti-periplanar conformation, the terms are four σC-H/σ*C-H 2.5 and two σC-F/σ*C-F 1.8 kcal/mol. The two totals (16.6 vs 13.6) indicate that gauche is stabilized more by such interactions.

There is of course a bit more to this story, but I have documented the above here, since I can include an explicit (and rotatable) illustration of the orbitals involved (which  I have not seen elsewhere). If you want a recipe for generating these orbitals, go here.

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Conformational analysis of biphenyls: an upside-down view

Friday, April 2nd, 2010

One of the (not a few) pleasures of working in a university is the occasional opportunity that arises to give a new lecture course to students. New is not quite the correct word, since the topic I have acquired is Conformational analysis. The original course at Imperial College was delivered by Derek Barton himself about 50 years ago (for articles written by him on the topic, see DOI 10.1126/science.169.3945.539 or the original 10.1039/QR9561000044), and so I have had an opportunity to see how the topic has evolved since then, and perhaps apply some quantitative quantum mechanical interpretations unavailable to Barton himself.

The example I have chosen to focus on here is biphenyl (a derivative of which also happens to be the first structure shown by Barton in his 1970 Science article noted above), but modified with iso-electronic B/N substitution for carbon for a particular reason.

biphenylFour hydrogen atoms are highlighted in the above drawings by virtue of how close they might approach each-other, and what impact this will have on the conformation of each species. Such close approaches are normally defined with reference to the so-called van der Waals radius of the element concerned. For hydrogen, this radius is either 1.2Å (if the contact is to another hydrogen) or 1.1Å (if its to any other element, see DOI: 10.1021/jp8111556). An interpretation of this value is that the van der Waals attraction due to to dispersion or long range correlation effects reaches a maximum for two non-bonded hydrogen atoms at ~2.4Å. Significantly, a slightly closer approach than this value might still be mildly attractive, but it would be generally agreed that any distance less than ~2.1Å now represents a genuine repulsion between the hydrogens (see also this post). This represents a somewhat more quantitative judgement on what used to be called steric interactions.

With the scene set, let me introduce the results of a calculation (wB97XD/6-31G(d,p), a DFT method selected because it treats the long range correlation effects with a specific correction)

Conformational analysis of biphenyl 1

One can see here minima at ~45, 135, 225 and 315° for 1 (see DOI 10042/to-4853). Due to symmetry, the first and last are identical as are 2nd and 3rd, and the 1st and 2nd minima are in fact enantiomers of each other (the symmetry is D2, which is chiral). Two different transition states connect these minima, one with angles of 0/180 and the other slightly lower energy at 90/270°.

The non-bonded H…H distance are as follows: 1.95Å@0°, 2.39Å@45° and 3.54Å@90°. We may conclude that the first of these is repulsive, the second attractive and the third non interacting. Counterbalancing this effect is of course resonance due to π-π-overlaps across the central bond, which decreases to zero as the angle moves to 90°. The conformational minimum @45° is such because of the maximal H…H dispersion attraction and the still significant π-π-overlap. This brief analysis suggests however that these two effects are finely balanced, and so the next question is whether one might be able to perturb the system to distort the balance. The perturbation chosen is to replace one or two pairs of carbon atoms with the iso-electronic combination B+N.

The first perturbation is to replace the central rotating bond by a B-N combination 2 (DOI: 10042/to-4854).

Rotation about the B-N bond in 2

For this species, the H…H distances are 2.02Å@0°, 2.36Å@45° and 3.61Å@90°, the only significant difference with 1 emerging as the 0° conformation being around 1 kcal/mol lower relative to the other two. It is tempting to attribute this to the longer H…H separation for this rotamer in 2 due to the B-N bond being longer (1.562Å) than the C-C bond it replaced (1.496Å)

The next perturbation is to relocate the N/B pair as in 3 (DOI: 10042/to-4855). If one imagines that this will be a minor perturbation, take a look at the profile below.

Rotation about central C-C bond in 3.

The world has been turned upside down. What were transition states @0° and @180° are now minima and the reason is easy to find. The central C-C bond is now only 1.400Å long, having acquired substantial double bond character, and being accordingly very much more difficult to twist (the barrier being ~30 kcal/mol). The π-π-overlap has won out completely, and in the process has forced the H…H distance down to a presumably repulsive 1.918Å. The penalty for this is that the overall energy of 3 is some 22.8 kcal/mol higher than 2.

Added in proof (as the expression goes): If the above profile is conducted with full geometry optimization in a solvent field (water), which helps stabilise charge separations, the profile changes to the below. The solvation reduces the barrier to rotation considerably, the energy maxima now reveal a proper stationary point (rather than the cusp), the minima are very slightly non-planar, but the basic inversion of the potential energy surface compared to 1 or 2 is still observed.

Rotation about the C-C bond for 3, with solvation correction

The final perturbation is 4 (DOI: 10042/to-4856) with the following rotational profile. Another surprise:

Rotation about the central C-C bond in 4.

The H…H distances are 1.930Å@0°, 1.789/2.275Å@180°. The difference from 1 is that the hydrogens now have opposite polarity for the N-H (which is positive) and the B-H (which is negative). At the rotation angle of 0°, two H(+)…(-)H style dihydrogen bonds (see also this post) are established (these are presumed to be very attractive); at an angle of 180°, the H(+)…(+)H and H(-)…(-)H interactions are presumed to be very repulsive. The difference between the two is ~18 kcal/mol.

We have learnt that conformational analysis for molecules such as these is a fight between π-π-overlaps, which themselves can have unexpected outcomes, weak van der Waals dispersion interactions between “neutral” non-bonded hydrogen atoms, and strong electrostatic attractions and repulsions between “ionic” hydrogens. Now perhaps the reason for the choice of the wB97XD DFT method can be seen; it is capable (at least in theory) of balancing these forces properly.

So the world of conformational analysis can be turned upside down, and analysing what happens from this topsy-turvy viewpoint can teach a lot!

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