Posts Tagged ‘interaction energy’

A periodic table for anomeric centres, this time with quantified interactions.

Monday, August 8th, 2016

The previous post contained an exploration of the anomeric effect as it occurs at an atom centre X for which the effect is manifest in crystal structures. Here I quantify the effect, by selecting the test molecule MeO-X-OMe, where X is of two types:

  1. A two-coordinate atom across the series B-O and Al-S, and carrying the appropriate molecular charge such that X carries two lone pairs of electrons (thus the charge is 0 for O, but -3 for B).
  2. A four-coordinate atom across the series B-O and Al-S, with X-H bonds replacing the lone pairs on this centre in the previous example, and again with appropriate molecule charges (e.g. +2 for  SH2).

The donor in the anomeric interaction always originates on the oxygen of the MeO group attached to X. The acceptor is always the X-O σ* empty orbital. The results (table below, ωB97XD/Def2-TZVPP calculation, NBO E(2) in kcal/mol) confirm that as X gets more electronegative, the X-O σ* empty orbital becomes a better acceptor, and so the NBO E(2) interaction energy which quantifies the anomeric interaction gets larger. Eventually (with X=OH2) the donation of electrons into the X-O σ* empty orbital becomes so effective that the X-O bond (in this case O-O) dissociates fully and the NBO perturbation cannot be computed. Also for reference, a “normal” anomeric interaction (such as is found in e.g. sugars) is around 18 kcal/mol. Anything larger than this could be considered especially strong, and anything less than ~10 kcal/mol would be regarded as weak. 

X[1]*
BH2 CH2 NH2 OH2
12.5 17.7 18.5 dissociates
AlH2 SiH2 PH2 SH2
6.9 12.9 21.9 31.3
B C N O
8.3 11.7 12.9 14.2
Al Si P S
4.8 6.6 11.2 18.2

For the entry X=S, the E(2) term is actually larger than for the oxygen. I should note that the Me group itself is not passive in this process. The C-H bonds can also act as significant electron donors, but here I am not going to analyse this additional complexity.

This table reveals that there is nothing special about carbon as an anomeric centre, and here also the normal intimate association with the term anomeric and heterocyclohexanes such as found in sugars.

* Here I introduce a refinement to my normal process of citing the data produced for any specific calculation. Rather than including 16 individual citations for each cell in the table, I have gathered all these calculations into a collection and cite here only the DOI of that collection. When resolved, the individual members of that collection can then be inspected for the actual data.

References

  1. H. Rzepa, "Anomeric interactions at atom centres", 2016. https://doi.org/10.14469/hpc/1221

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A periodic table for anomeric centres.

Saturday, August 6th, 2016

In the last few posts, I have explored the anomeric effect as it occurs at an atom centre X. Here I try to summarise the atoms for which the effect is manifest in crystal structures.

The effect is defined by X bearing two substituents, one of which Z is a centre bearing a “lone pair” of electrons (or two electrons in a π-bond), and another Y in which the X-Y bond has a low-lying acceptor or σ* empty orbital into which the lone pair can be donated. This donation will only occur if the Z-lone pair and the X-Y bond vectors align antiperiplanar. Here is the summary so far.

X Blog entry
B 16601
C 14508,8898
N this one
O 16646
Si 16601
P 16601
S this one

As required of a good periodic table, it has gaps that need completing, in this case X=N and X=S. Firstly N, for which the small molecule below is known (FUHFAP).

FUHFAP

A ωB97XD/Def2-TZVPP calculation[1] yields an electron density distribution, which can be collected into monosynaptic basins using the ELF technique. There are two oxygen lone pairs (17 and 20) that are close to antiperiplanar to the adjacent N-O bond, having E(2) interaction energies obtained using the NBO method of 15.1 and 15.8 kcal/mol, typical of the anomeric range. The basin labelled 13 on X=N1 below is also perfectly aligned antiperiplanar with the adjacent O3-C2 bond, but its E(2) interaction energy is only 7.3 kcal/mol. Thus a strong anomeric interaction on the anomeric atom itself does not seem to occur. The same effect was noted for X=O in the previous post; the explanation remains unidentified.

FUHFAP

With the X=S gap, we have lots of opportunity with polysulfide compounds, a good example of which is the C2-symmetric and helical S8 dianion TEGWAF[2]

TEGWAF

Each of the 8 sulfur atoms exhibits antiperiplanar orientation of an S lone pair with an adjacent S-S acceptor σ* orbital;
1:2-3=23.7 kcal/mol;
2:3-4=18.5;
3:4-8=11.7, 3:2-1=7.4;
4:8-7=11.4, 4:3-2=9.2.

This just surveys the central main group elements, and it is possible that this little mini-periodic table may yet grow.

References

  1. H.S. Rzepa, "C 2 H 7 N 1 O 2", 2016. https://doi.org/10.14469/ch/195294
  2. Rybak, W.K.., Cymbaluk, A.., Skonieczny, J.., and Siczek, M.., "CCDC 880780: Experimental Crystal Structure Determination", 2012. https://doi.org/10.5517/ccykj88

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The conformational preference of s-cis amides.

Sunday, February 10th, 2013

Amides with an H-N group are a component of the peptide linkage (O=C-NH). Here I ask what the conformation (it could also be called a configuration) about the C-N bond is. A search of the following type can be defined:

cis-amide

The dihedral shown is for H-N-C=O (but this is equivalent to the C-C-N-C dihedral, which is also often called the dihedral angle associated with the peptide group). I have also added a distance, from a C-H to the carbonyl oxygen. Other search constraints include T ≤ 175K, R < 0.05, no disorder, no errors, that neither N-C bonds are part of a ring and that the two carbons marked T4 both have four connected bonds. The search results in 619 hits (January 2013 version of the CCDC database), and these are displayed below.

cis-amide-search-heat

The horizontal axis reveals the highest concentration (red) at ~2.4Å due to a syn-co-planar alignment of the C-H bond with the plane of the C=O bond in the s-cis conformer (the significantly smaller hot-spot at ~3.9A may be due to an anti-co-planar alignment of this C-H bond).

s-cis-amide

The vertical axis shows a clear preference for a dihedral of 179° (in fact no hits with a dihedral of less than 14o° were found) and this can only arise from the s-cis conformation in which the H-N bond is oriented antiperiplanar to the axis of the C=O bond. This preference can be rationalised by filled/empty NBO-orbital interactions, which include:

  1. Antiperiplanar interaction between the N-H as donor and the C=O as a σ-acceptor (E(2) = 4.1 kcal/mol)
  2. Antiperiplanar interaction between the N-H as acceptor and C-H as donor (E(2) = 4.7 kcal/mol)
Click for 3D

H-N/C=O. Click for 3D

 

Click for 3D.

Click for 3D.

This latter overlap conspires to bring the C-H hydrogen close to the oxygen (~2.35Å, DIST1 in the diagram above). So one might be entitled to ask: is this a hydrogen bond? There are (at least) two ways of testing this.

  1. The NBO E(2) interaction energy between the oxygen in-plane lone pair and the H-C as acceptor is 0.8 kcal/mol. For hydrogen bonds, such E(2) energies more or less resemble the actual H-bond strengths, i.e. a strong H-bond has an E(2) energy of ~ 8 kcal/mol; and a medium O…H-C hydrogen bond weighs in at around 3 kcal/mol.  So this one is very weak. This is due to poor overlap resulting from the small ring size (5).
  2. The NCI (non-covalent-interaction) surface does reveal a feature in the CH…O region, but the colour coding (which indicates how attractive/repulsive this is) is both pale blue (attractive) and yellow (repulsive). Again this is only consistent with a very weak overall H-bond.
NCI surface. Click for 3D.

NCI surface. Click for 3D.

I end by reminding that the s-cis H-N-C=O conformation is a very common feature in peptides (the CCDC database comprises mostly small molecules, not larger peptides and proteins) arising from really quite subtle orbital interactions.

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Hydrogen bond strength as a function of ring size.

Thursday, January 3rd, 2013

One frequently has to confront the question: will a hydrogen bond form between a suitable donor (lone pair or π) and an acceptor? One of the factors to be taken into consideration for hydrogen bonds which are part of a cycle is the ring size. Here I explore one way of quantifying the effect for the series below, n=1-5 (4-8 membered rings).h-bond

I will use the NBO approach. To remind, this reduces the wavefunction for a molecule to a set of localised orbitals, referred to as natural bond orbitals. The perturbation interaction energy E(2) between any (doubly occupied, i.e. donor) orbital and an (unoccupied) acceptor orbital establishes the strength of that interaction. For a hydrogen bond, this can be expressed as the NBO corresponding to the (in this case oxygen) lone pair (shown in orange and purple below) and the corresponding H-O σ* empty orbital (shown as red and blue below). E(2) is a function both of how close in energy this pair of orbitals is (the smaller the energy gap the better) and how well they overlap (the relevant overlap in this case is the positive one between purple and blue). This latter attribute is shown below for the series n=2,3,4,5 (n=1 does not form any discernible hydrogen bond), at the ωB97XD/6-311G(d,p) computational level.

NBO interaction for 5-ring H-bond. Click for 3D.

NBO interaction for 6-ring H-bond. Click for 3D

NBO interaction for 7-ring H-bond. Click for 3D.

NBO interaction for 8-ring H-bond. Click for 3D.

The interaction energies E(2) are collected below, together with the computed lengths. To put E(2) into context, it is around 16 kcal/mol for a strong anomeric interaction, and about 6 kcal/mol for the stereoelectronic influence in di-fluoroethane. One can see that by the time the angle subtended at the hydrogen has increased to ~150°, the interaction energy has reached a respectable value.

n  E(2), kcal/mol  O…H length, Å  Angle, °
1 ~0.0  –  81.8
2 0.75 2.294  109.7
3 3.56 1.984  139.6
4 6.24 2.017  146.5
5 8.35 1.957  153.4

So the simple trick of looking at the donor-acceptor NBO interaction in a cyclic hydrogen bond can give us a straightforward way of quantifying how the size of the ring and hence the orbital overlap (one presumes that the Lp/C-O σ* energy gap is similar for all the systems) affects the strength of the interaction. One might also explore this by looking at structures in the Cambridge crystal database. But note from the above that whilst the  E(2) energies follow ring size, this does not appear to happen for the H…O lengths! The analysis reveals that the maximum number of structures for the span 5 to 8-rings occurs at ~2.15, 1.85, 1.65 and 1.85Å respectively. 

Crystal data for 5-rings

Crystal data for 5-rings
Crystal data for 6-rings.

Crystal data for 6-rings.
Crystal data for 7-rings

Crystal data for 7-rings
Crystal data for 8-rings

Crystal data for 8-rings

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Spotting the unexpected. The trifluoromeric effect in the hydration of the carbonyl group.

Friday, March 9th, 2012

The equilibrium for the hydration of a ketone to form a gem-diol hydrate is known to be highly sensitive to substituents. Consider the two equilibria:

For propanone, it lies almost entirely on the left, whereas for the hexafluoro derivative it is almost entirely on the right. The standard answer to this is that electron-withdrawing substituents destabilize the carbonyl compound more than the hydrate. But could there be more to it than that? Might the converse also be true, that electron-withdrawing substituents stabilise the hydrate more than the carbonyl compound? To answer this last question, consider the anomeric interactions possible in the diol.

  1. There is the standard anomeric effect operating between the two hydroxy groups, whereby a lone pair donor on one oxygen interacts with the C-O acceptor bond of the other oxygen, and vice versa, a total of two stabilising interactions.
  2. But what if the C-CF3 group could also act as an acceptor instead of the C-O? That would give the trifluoromethyl system a total of four anomeric interactions, each of them stabilising, compared to only two for the methyl system.

Garnering evidence, firstly we compute (ωB97XD/6-311G(d,p) ) the free energy difference for the two equilibria above. These turn out to be +3.3 kcal/mol for the top equilibrium, and -9.0 kcal/mol for the bottom, which agrees with the assertions made earlier. The computed geometry looks as below.

Geometry of hydrate. Click for 3D.

We must now go hunting for anomeric interactions, and this is done using an NBO analysis. We look for large interactions between a donor (a lone pair on either oxygen) and an acceptor (which is conventionally the C-O anti-bonding NBO, but can now also be the C-CF3 anti-bonding NBO). Indeed exactly four large interactions are found, in pairs of E(2) = 17.5 and 9.8 kcal/mol. The former is common to both the systems above, but the latter is larger for the trifluoromethyl substituted equilibrium than the methyl system (for which E(2) is 6.2 kcal/mol), and therefore constitutes additional stabilisation by the electron-withdrawing groups of the diol.

Each oxygen has two lone pair NBO orbitals. The initial hypothesis is surely that it uses one of these to align with a C-O anti bonding acceptor, and the other to align with the C-CF3 anti bonding acceptor. The first of these is shown below.

The interaction between an O(Lp) and a O-C BD* orbital. Click for 3D.

  1. The colour code is that the two phases of the oxygen lone pair (Lp) are shown as purple/orange.
  2. These are superimposed upon the C-O anti bonding NBO (referred to as BD* in the output), which has the colours red and blue.
  3. I advise you now to click on the graphic above to load the 3D model and the orbital surfaces. You should spot the node along the C-O bond with a blue-red boundary.
  4. You will also spot that the orange phase of the Lp overlapping with the red phase of the C-O BD*. This is defined as a positive (stabilizing) overlap.
  5. Likewise the purple phase of the Lp overlaps with the blue phase of the C-O BD*. In other words orange=red, and purple=blue. I have made orange and red, and purple and blue deliberately different so that the origins of each NBO can be spotted.
  6. This combination therefore has good overlap, and this gives rise to the large E(2) interaction energy of 17.5 kcal/mol.
Now for the interaction with the C-CF3 BD*, the one with E(2) = 9.8 kcal/mol.

The interaction between an O(Lp) and a C-CF3 BD* orbital. Click for 3D.

  1. You can see the blue-red node along the C-CF3 bond quite clearly.
  2. But hang on, the O Lp orbital is the same as before! It is overlapping with BOTH the C-O and the C-CF3 BD* orbitals.

The other O Lp is shown below (viewed along the axis of the C-CF3 bond). Note how an equal proportion of the orange phase and the other purple phase of the O Lp overlap equally with the blue phase of the C-CF3 bond. In other words, one cancels the other.

The interaction between the other O(Lp) and a C-CF3 BD* orbital. Click for 3D.

So we have found that just one (of the two lone pairs) on each oxygen overlaps with both the C-O and the C-CF3 anti bonding NBOs, the latter giving a stabilisation not present when the group is instead C-CH3. We can attribute this to the far greater acceptor properties of the C-CF3 BD* because of the electronegative character of the fluorines.

This is an anomeric effect with a difference. The CF3 group is not normally associated with inducing such an effect (just as the CN group is not, see this post or this post where an alkene acts the donor instead of a lone pair). Also unusual (more accurately, I have not encountered it before) is the (apparent) use of the SAME donor lone pair to induce TWO quite different anomeric interactions. Before getting too excited by this unexpected effect, it it is worth taking a look at another technique for analysing lone pairs. The ELF (electron localisation function) can provide the centroid of what is referred to as a monosynaptic basin (a lone pair in other words).

EKF function, showing O Lone pairs (in yellow). Click for 3D.

You can see in yellow the oxygen lone pairs. Note how one of them aligns with the C-O bond, and the other with the C-CF3 bond. Unfortunately, the ELF method does not allow the strength of the interaction to be quantified, which is why the NBO analysis is preferred.

So we can conclude that not only might electron-withdrawing substituents destabilize the carbonyl compound more than the hydrate, but they certainly also stabilise the hydrate more than the carbonyl compound.

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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 mysteries of stereoinduction.

Thursday, July 1st, 2010

Stereo-induction is, lets face it, a subtle phenomenon. The ratio of two stereoisomers formed in a reaction can be detected very accurately by experiment, and when converted to a free energy difference using ΔG = -RT Ln K, this can amount to quite a small value (between 0.5 – 1.5 kcal/mol). Can modelling reproduce effects originating from such small energy differences? Well one system that has been argued about now for several decades is shown below as 1.

Norbornene systems

Way back in 1992, we thought that the explanation for attack by an electrophile on the alkene from the syn face was electrostatic (although it did depend on the nature of the electropile; thus we concluded that attack by Hg(OH)2 was electrostatic, but by BH3 was orbital controlled). Recently, a different explanation has emerged, relating to how the fundamental normal vibrational modes of the molecule interact with the transition normal mode for the reaction. A new example of this, relating to reaction of the isomeric 2 with a peracid has recently been discussed on Steve Bachrach’s blog. Here, the peroxide of the peracid is thought to act as an electrophile (although one must bear in mind that it does bear two electron lone pairs!). The conclusion was pretty clear cut; the experimental preference for syn (92%) over the anti isomer (8%, ΔΔG = 1.4 kcal/mol) was NOT due to electrostatic effects, but due to distorsional asymmetry in the vibrational mode that couples/forms the transition state mode.

I could not resist revisiting this system. As in 1992, a molecular electrostatic potential was calculated for 2. The method used was wB97XD/aug-cc-pvdz, and if you want the archive of this calculation to evaluate it yourself, see here).

MEP for 2. Click on diagram for 3D.

A very clear electrostatic bias for syn attack emerges (orange = attractive to a proton=electrophile). This electrostatic picture is not directly related to any distortional asymmetry, although of course both could arise from the same electronic factors. They may indeed be different manifestations of the same underlying nature of the wavefunction. But I would claim here that to make the clear statement that electrostatic effects are NOT responsible for the discrimination in this reaction is perhaps too simplistic (electrostatic potentials were not reported in the original article). The control experiment is 3, which is known to be far less selective. The calculated electrostatic potential likewise shows much less discrimination.

The norbornene with a four-membered ring

Is there another take on 2? Well, how about an NBO (natural bond order) analysis? The interaction energy between the filled C1-C2 orbital and the antibonding C15-C16 π* bond is 3.24. This could be regarded as pushing electrons into the anti-periplanar syn face of the alkene. The corresponding C2-C9/C15-C16 interaction resulting in an anti-preference is less at 2.55 kcal/mol. This effect arises because the C1-C2 bond (localised as an NBO) is a better donor (E=-17.8eV) than C2-C9 (E=-18.1eV). Because C2 is common to both, it must be the difference between C1 and C9 (i.e. the hybridization of each). Perhaps it’s an orbital effect after all?

Norbornene electrostatic potential

I would conclude by saying that it can be ferociously difficult to identify the fundamental origins of stereo-induction. But I leave the argument in the hands of the reader now. What do you think?

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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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How do molecules interact with each other?

Sunday, April 12th, 2009

Understanding how molecules interact (bind) with each other when in close proximity is essential in all areas of chemistry. One specific example of this need is for the molecule shown below.

The Pirkle reagent

The Pirkle reagent

This is the so-called Pirkle Reagent and is much used to help resolve the two enantiomers of a racemic mixture, particularly drug molecules. The reagent binds to each enantiomer of a racemic drug differently, and this difference can be exploited by using e.g. column chromatography to separate the two forms. The conventional wisdom is that such chiral recognition occurs via a three-point binding model. In other words, at least three different interactions must occur between the Pirkle reagent and the drug to allow such chiral recognition.

So how do we identify where these bindings might occur? A good place to start is to look at the self-binding of the Pirkle reagent, in other words, how does it interact with itself in the crystal state? An X-ray structure of the pure enantiomer of the Pirkle reagent shows that it binds with itself to form a loose dimer. We are now in a position to analyze exactly how this binding occurs. To do this, we are going to invoke a technique known as Atoms-in-molecules or AIM. This effectively looks at the curvature of the electron density in the dimer, and from the characteristics of this curvature, identifies a series of so called critical points, or regions where the first derivative of the electron density (referred to as ρ(r) ) with respect to the geometry is zero. These critical points come in four varieties only;

  1. A nuclear critical point, which almost exactly corresponds to where the nuclei are
  2. A bond critical point, which is the key to understanding not only where actual bonds are in the molecule, but also a range of weaker interactions which are conventionally not graced with the term bond, but which nevertheless can be essential in understanding how to molecules interact weakly with each other.
  3. The remaining two types of critical point relate to rings and cages, and we will not be concerned further with them here.

The electron density required for this analysis could in principle come from the X-ray measurements themselves, but it is not easy to acquire this to the required accuracy (although it can be done). In this case, it is easier (and probably no less accurate) to calculate the density from a DFT-based quantum mechanical calculation. The result of this is shown below.

Pirkle dimer. Click on image to obtain model

Pirkle dimer. Click for 3D.


The light blue spheres show the position of selected bond critical points or BCPs in the AIM analysis. So what do they tell us about how two molecules of Pirkle molecule interact with each other? Three different points labelled 1-3 are highlighted for discussion.

  1. Points 1 connect the hydrogen of the OH group with the carbons of the π-face of the anthracene ring (the left ring of the molecule as shown above). This is an unusual type of interaction known as a π-facial hydrogen bond, and it has only been recognized as such in the last 30 years. Note that this interaction is not restricted to occur just between a pair of atoms, but can involve more (in this case almost a whole benzene ring). By finding the value of the electron density ρ(r) at this BCP, one can estimate the energy of interaction resulting from its formation. In this case, ρ(r) ~ 0.014 au, and comparison with other types of hydrogen bond suggests that this value corresponds to an interaction energy of around 2.5 kcal/mol. This is a little weaker than a conventional OH…O hydrogen bond, but is still quite significant. Two of these interactions occur in this Pirkle dimer.
  2. Points 2 are equally unexpected. They connect the oxygen of the same OH group involved in the previous interaction, and one of the ring C-H groups. Again, that C-H…O groups can interact has only been recognized relatively recently. The value of ρ(r) of ~ 0.018 indicates a hydrogen bond strength of ~3 kcal/mol, again hardly insignificant.
  3. There are four specific interactions of the final type 3. These occur in the region of overlap of the two anthracene rings, and these are referred to as π-π stacking interactions. Again, the ρ(r) of ~ 0.005, calibrated against known systems, suggests that each is individually worth around 1 kcal/mol.

So adding up all eight interactions indicates that the two molecules of the Pirkle reagent have an interaction energy of around 15 kcal/mol resulting just from these weak bonds (there are other types of interactions between two molecules known as dispersion forces, which also contribute), and which together provide more than enough free energy to overcome the entropy required to bring the two molecules together.

Armed with tools such as AIM, one can now be more confident in analyzing the various terms that contribute to two molecules interacting with each other, and in the case of chiral molecules, how these interactions may result in chiral recognitions.

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