σ-holes

Overview

σ -HolesTimothy Clark1,2∗

σ -Holes are regions of positive molecular electrostatic potential collinear with andopposite to covalent bonds to atoms of Groups IV–VII. They are responsible formany noncovalent bonding interactions, such as halogen bonding. σ -Holes make‘negatively charged’ atoms act as if they were ‘positively charged’. The existenceof σ -hole bonding emphasizes what has been called ‘the fallacy of net atomiccharges’, which means that many covalently bonded atoms cannot be representedadequately by a single charge because they look negative from some directionsand positive from others. Hydrogen bonding can also be regarded as a special caseof σ -hole bonding, although in this case the origin of the σ -hole is rationalizeddifferently than in the heavier elements. Phenomena such as the directionality ofhydrogen bonds and ‘blue-shifted’ hydrogen bonds can be explained very simplyusing the σ -hole concept. C© 2012 John Wiley & Sons, Ltd.

How to cite this article:WIREs Comput Mol Sci 2012. doi: 10.1002/wcms.1113

INTRODUCTION

C hemists work with models that have developedover the years and have proved to be useful

in rationalizing and systematizing the vast body ofknowledge (structures, stabilities, and reactions) thatmakes up the subject. We sometimes lose track ofthe fact that many common concepts (e.g., hybridiza-tion, molecular orbitals, and resonance structures) areindeed simply models designed to rationalize whatwe observe. This generally works well; our survivingmodels have relatively few serious deficiencies andcan serve as useful surrogates for the (as far as weknow) real but more complicated situations. In fact,this article will use concepts such as the linear combi-nation of atomic orbitals (LCAO) approximation andhybridization to explain the σ -hole phenomenon.

Most models have their limitations but thereis one in particular that leads to false conclusionsand cannot rationalize important noncovalent inter-actions: net atomic charges. Although this conceptmight not be familiar to you under that name, we areall acquainted with the practice of writing ‘δ+’ or ‘δ−’next to atoms to indicate their polarity. This implies

∗Correspondence to: [email protected] der Friedrich-Alexander UniversitatErlangen-Nurnberg, Erlangen, Germany2Center for Molecular Design, University of Portsmouth,Portsmouth, United Kingdom

DOI: 10.1002/wcms.1113

that atoms can be assigned a charge, the net atomiccharge. This is not true physically; net atomic chargescannot be measured or even defined uniquely. Politzeret al.1 have pointed out what they call ‘a fallacy ofatomic charges’. This simply means that if we thinkof an atom as having a single charge (monopole) at itscenter (and ignore the influence of other atoms), thenit should have approximately the same molecular elec-trostatic potential (MEP, the interaction energy thata single positive electron charge would have with theatom) all over its surface. This would mean that ‘neg-ative’ atoms look negative from every direction. Thisis very often not the case. Consider CF3Br. Its MEPon an isodensity surface2 that roughly correspondsto the van der Waals radii of the atoms is shown inFigure 1.

The bromine (Br) atom exhibits a prominentpositive (red) area of MEP collinear with and oppo-site to the C Br bond and a large equatorial belt ofnegative MEP. This means that it looks positive fromopposite the carbon but negative from the sides (Fig-ures 1b and 1c, respectively). Is the Br positive ornegative? There is no answer because the net atomiccharge concept cannot reproduce these features. Wemust abandon any hope of net atomic charges describ-ing the electrostatics of CF3Br adequately. In contrast,the MEP is a measurable quantity and is uniquely de-fined (and also fairly easy to calculate). The rest ofthis article will be concerned with the red spot: theσ -hole.3

Volume 00, January /February 2012 1c© 2012 John Wi ley & Sons , L td .

Overview wires.wiley.com/wcms

FIGURE 1 | Three views of the molecular electrostatic potentialprojected onto an isodensity (contour level 0.07 e− Bohr−3) for CF3Br.(a) View showing all the important features of the MEP; (b) view forma point collinear with and opposite to the C Br bond showing theσ -hole; (c) view from a point perpendicular to the C Br bond,showing the negative ‘equatorial’ belt. Data calculated at thePW91/PNP DFT level.

AN EXPLANATION OF THE σ -HOLE

To understand how the Br in CF3Br comes to havea σ -hole, consider the valence state electronic struc-ture of the Br atom as it interacts with the carbon. Interms of the orbital model, it can be represented as4s24px

24py24pz

1 (Figure 2). The half-filled 4pz inter-acts with a carbon orbital. As the single 4pz electronwill be largely localized in the bond region, the elec-tronic density in the outer (noninvolved) lobe of the4pz orbital will be depleted. This ‘empty’ outer lobeis the σ -hole.3 If the depletion of electronic charge issufficient, the σ -hole will have a positive electrostaticpotential and can interact attractively with negativesites. The more electron attracting is the remainder ofthe molecule, and the more polarizable the halogen(I > Br > Cl > F), the more positive will be the σ -hole. Meanwhile, the doubly occupied 4px and 4py

orbitals of the Br create a negative MEP belt aroundits lateral sides, through which it can interact withpositive sites.

The halogens are not the only elements to ex-hibit σ -holes. Atoms of Groups IV–VI also can havepositive σ -holes opposite covalent bonds, and formnoncovalent bonds to negative sites.4–6 These GroupIV–VI σ -holes can be explained in analogous fashionto Group VII (the halogens). The electronic densitydeficiencies that result in σ -holes have been observed,

BOX 1: CHEMICAL MODELS

It is easy to forget that hybridization is neither real nor nec-essary to account, for instance, for the tetrahedral struc-ture of methane. It is useful, although, for explaining itto students without having to go through molecular or-bital arguments. Models such as the LCAO approximationare so deeply ingrained in our thinking that huge contro-versies have arisen as to whether ‘sulfur uses d-orbitals’for bonding. The question has no meaning outside theLCAO approximation. Similarly, molecular orbitals are fic-titious one-electron wave functions—but where would webe without the concepts of highest occupied molecularorbital (HOMO) and lowest unoccupied molecular orbital(LUMO)? The trick is that the HOMO and the LUMO arepretty good representations of the differences between thewave function of the neutral molecule and its one-electronoxidized and reduced forms, respectively, and are a usefulshorthand in, for instance, molecular electronics. An in-credibly pedantic view of molecules is that they consist of acloud of indistinguishable electrons moving in the field of acollection of static (according to the Born–Oppenheimer ap-proximation) positively charged nuclei. There are no atomsor bonds in this picture. As far as we know, it is correct,but it does not help us to understand chemistry, so we usemodels.

as anisotropies in the charge distributions of cova-lently bonded atoms.7–9

The first-row atoms (C, N, O, and F) requirespecial mention. Unlike the heavier elements, the va-lence electronic configurations of the first row can beexpected to have some degree of spn hybridization.10

Thus, the valence state of fluorine, for example, islikely to have some (2s2pz)22px

22py2(2s2pz)1 char-

acter (Figure 3). This means that there will be some2s electronic density in the outer lobe of its bondingorbital, which will to some extent counteract the de-pletion that is the σ -hole. As a result, fluorine andother first-row atoms are less likely to have positiveσ -holes. However, this does not mean (as was oncethought) that fluorine cannot form halogen bonds.If it is linked to a sufficiently electron-withdrawinggroup (e.g., as in F2, FCN, FOF, F3COF, etc.), thenfluorine can indeed have a positive σ -hole and interactthrough it,11–13 as can other first-row atoms.4–6

THE NATURE OF σ -HOLE BONDING

Chemists’ extensive use of models leads to an almostobsessive desire to partition observed phenomena intocontributions defined by the models. That is not the

2 Volume 00, January /February 2012c© 2012 John Wi ley & Sons , L td .

WIREs Computational Molecular Science σ -Holes

FIGURE 2 | Schematic view of the fluorine atomic orbitals in CF3F.

FIGURE 3 | Schematic view of the bromine atomic orbitals in CF3Br. This scheme ignores the fact that the four fluorines are equivalent to makeit consistent with Figure 2.

intention of this article. The problem with detailedanalyses of bonding interactions is that they try todivide total interaction energies (which are physi-cal observables) into contributions that are neitherobservable nor uniquely defined and also cannot bestrictly separated from each other. Nonetheless, manyschemes for analyzing bonding have been proposed.These include complete analyses of intermolecularinteractions such as symmetry-adapted perturbationtheory,14 partitioning of the electron density (e.g.,

atoms in molecules),15 and calculating local proper-ties indicative of bonding (e.g., the electron localiza-tion function).16

The purpose of this section is not to assign thedegrees of importance of contributions such as elec-trostatics (Coulomb interactions and polarization),van der Waals forces (dispersion), repulsion, etc. It ismerely to point out that an adequate rationalizationof most (if not all) observed properties of halogenbonds can be obtained by viewing the interactions



FIGURE 4 | Schematic view of the interactions possible for anorganic halide R X. The color of the contour around the halogenindicates the sign of the MEP at the surface (red is positive, blue isnegative).

as essentially electrostatic (with a dispersion com-ponent). There are obvious exceptions, such as thesymmetrical X3

− anions (X = Cl, Br, I),17 in whichother factors (in this case, complete delocalizationto a three-center, four-electron bond) are important,but our conclusions should be acceptable for weak(≤ 10 kcal mol−1) interactions.

A minimalistic explanation of halogen bondingis electrostatic. Figure 4 shows the interactions ex-pected for an organic halide R X. The electrostaticpotential at the surface of the halogen is color coded(red for positive, blue for negative) schematically.

If the X-bond acceptor is negative (e.g., a lonepair), it will be attracted by the positive σ -hole butrepelled by the ‘belt’ of negative MEP around the

halogen. This leads to a pronounced preference forX-bond acceptors to prefer a linear interaction withhalogens in halogen bonding. Exactly this trend wasfound by analyzing X-ray crystal structures from theCambridge Structural Data Base18 and from the Pro-tein Data Bank.19 Halogen bonds prefer linear orclose-to-linear C X···A interactions, where A repre-sents the X-bond acceptor. However, the halogen canalso interact favorably with positive sites or with H-bond donors at angles around 90◦.20 The pattern ofthe MEP at the surface of halogen atoms explainsboth of these angle preferences and therefore sufficesin this respect. More importantly, it has been shownthat the interaction energy for halogen bonds corre-lates well with the most positive value of the MEP inthe halogen σ -hole.21–23 This is exactly the behaviorthat would be expected of an electrostatic interaction.Thus, in the spirit of the models used in chemistry, wecan conclude that for weak halogen-bonding interac-tions, a simple Coulomb picture (plus dispersion) isusually compatible with the experimentally knownfacts.

POLARIZATION ANDHYDROGEN/HALOGEN σ -HOLES

It is remarkable that halogen and hydrogen bondshave quite similar characteristics, especially theirangle preference. Figure 5 shows that this similarityis no accident.

The (admittedly not very pronounced) morepositive area of MEP found for water collinear with

FIGURE 5 | View along one H O bond in water, showing the σ -hole (Newman projection on the right). Data calculated at the PW91/PNPdensity functional theory level.



FIGURE 6 | Schematic view of the principal polarizationmechanism in bonds to hydrogen. The disk represents the electrondensity that we might assign to the hydrogen.

and opposite to each O H bond resembles a σ -hole.The difference is that the hydrogen atom generallyexhibits positive values of MEP over all of its surface;it is simply more positive opposite the O H bond.The reason for the ‘σ -hole’ found for H-bond donorhydrogen atoms is different to that outlined abovefor halogens and other heavy atoms and has beenknown for more than 30 years,24,25 although in a dif-ferent context. Measurements of bond lengths to hy-drogen by microwave spectroscopy (which essentiallymeasures the positions of the nuclei) are consistentlyapproximately 10% longer than those found indiffraction experiments, which depend on the elec-tronic density. This is a consequence of the singlenuclear charge of hydrogen. It cannot hold ‘its’ elec-tron tightly, so that the region of electronic densitythat we might assign to hydrogen shifts toward itsbonding partner, as shown in Figure 6.

This shift in electronic density causes the dis-crepancy in measured bond lengths and also leavesthe hydrogen with a considerably anisotropic chargedistribution. This results in the hydrogen σ -hole.26

However, as shown in Figure 5, the hydrogen σ -hole is not very prominent in unperturbed water,for instance. This is where a second, far more re-cent discovery comes into play. Fielder et al.27 foundthat the polarizabilities of organic molecules can bereproduced well in ab initio or density functionaltheory if the hydrogen basis set contains diffuse p-orbitals. This is remarkable because it has long beenbelieved that the use of extremely large basis setswith higher (d-, f-, . . .) polarization functions on thenon-hydrogen atoms is necessary to obtain accuratemolecular polarizabilities.28 The physical effect be-hind Fielder et al.27 observation is that it is easiest toshift hydrogen electronic densities along the bonds tohydrogen, as indicated in Figure 6.

What does this have to do with hydrogen bondsand halogen bonds? The H/X-bond acceptor itselfusually looks negative to the donor hydrogen or halo-gen; it is perfectly situated to push the hydrogen elec-tronic density toward R in Figure 6 and therefore toincrease the magnitude of the hydrogen or halogenσ -hole.26 Thus, H/X-bond acceptors that look nega-tive to the donor can induce a σ -hole or strengthenan existing one, thus increasing their own abilities tointeract with the donor. This means that:

• The strengths of hydrogen and halogen bondsdepend strongly on the applied electric field(which may be due to the acceptor). This isthe well known cooperative effect.29–31

• Because electron density can be shifted intothe R H or R X bond on hydrogen or halo-gen bonding, the stretching frequency of thisbond can actually increase. This is the originof blue-shifted hydrogen32 and halogen33,34

bonds. Blue shifting has been reproduced us-ing purely electrostatic perturbations (appliedfields or point charges).33–36

• A σ -hole need not necessarily be evident inthe unperturbed donor, but can appear onpolarization by the acceptor. The fact thatCH3Cl has been found to form a complexwith O CH2,37 CH3Cl····O CH2, despitethe chlorine in CH3Cl not having a positiveσ -hole appears to be due to such polarization,in conjunction with dispersion.38

CONSEQUENCES

Remarkably, although individual examples of halo-gen bonds have been known since 186339 and insome detail since 1954,40,41 their importance hasnot been recognized until relatively recently. Halo-gen bonds were first used systematically in crystalengineering42 and only recognized fairly recently asbeing important in biological systems.19 Before this,halogen-containing groups such as chlorophenyl werevery often introduced into drug candidates, but usu-ally because they make compounds more lipophilicthan the phenyl groups that they replace and thus in-crease binding constants to receptors and enzymes.Even this increase in lipophilicity is remarkable. Whyshould a relatively polar substituent such as chlo-rine make an aromatic compound more lipophilic?The answer is the σ -hole. More recent work hasinvestigated the effect of halogen bonding in protein–ligand interactions in depth.43



Consequences for computational techniques aresevere, as was pointed out by Auffinger et al.19 andPolitzer et al.1 because traditional force fields treathalogens using a single negative partial charge cen-tered at the halogen atom. This representation cannotpossibly reproduce halogen bonding, but will actuallylead to Coulomb repulsions in place of the halogenbonds. Thus, several modified force fields with ad-ditional off-center positive charges on halogens havebeen introduced recently.44–46 Quantum mechanicaltechniques do not have this problem (as long as theycan reproduce the electrostatics of the donor atomscorrectly), so that halogen bonding sites can be rec-ognized by a recent procedure in which local proper-ties calculated using semiempirical molecular orbitaltheory are used to recognize binding sites on ligandsusing an artificial neural net.47

CONCLUSION

The σ -hole can exist for both halogens and, sur-prisingly, for hydrogen as a hydrogen-bond donor.This means that the directionality of both halogenand hydrogen bonds has at least a significant elec-trostatic component and that it is not necessary toinvoke effects such as donor–acceptor interactions toexplain this directionality. Simple electrostatic modelsin which point charges are used to represent acceptors

are able to reproduce the angle dependence of halo-gen and hydrogen bonds.48 Note that such studies,although they may appear primitive, are definitive inthe sense that the point charge can only interact withthe donor molecule Coulombically or by polarization.It cannot donate electrons or enter into dispersioninteractions. In contrast, more complicated analysistechniques are always to some extent arbitrary to theextent that they attempt to divide an observable quan-tity (the interaction energy) into a series of nonorthog-onal components that may not be uniquely definedand are strongly interdependent. Exactly which in-teractions constitute hydrogen and halogen bonds isactually not important. As long as the simple elec-trostatic picture described above explains the exper-imental and accurate theoretical observations (struc-tures, energies, vibrational frequencies, etc.), there isno pressing need to replace or extend it. That appearsto be the current situation.

There are now very many experimental inves-tigations of σ -hole bonding and the field is enjoyingconsiderable interest. These studies have only beenmentioned sparingly above because this article con-centrates on the σ -hole concept and its relevance forhydrogen and halogen bonding. It is fascinating toobserve just how important halogen bonds are turn-ing out to be and to wonder why their contributionremained unrecognized for so long. Are there others?

ACKNOWLEDGMENTS

I especially thank Jane Murray and Peter Politzer for starting the σ -hole odyssey and providingthought and inspiration along the way.

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FURTHER READING

Politzer P, Murray JS, Clark T. Halogen bonding: an electrostatically-driven highly directional noncovalent interaction.Phys Chem Chem Phys 2010, 12:7748–7757.

Politzer P, Lane P, Concha Monica C, Ma Y, Murray JS. An overview of halogen bonding. J Mol Model 2007, 13:305–311.

Parisini E, Metrangolo P, Pilati T, Resnati G, Terraneo G. Halogen bonding in halocarbon–protein mlexes: a structuralsurvey. Chem Soc Rev 2011, 40:2267–2278.

Metrangolo P, Neukirch H, Pilati T, Resnati G. Halogen bonding based recognition processes: a world parallel to hydrogenbonding. Acc Chem Res 2005, 38:386–395.

Metrangolo P, Resnati G, Eds. Halogen Bonding: Fundamentals and Applications, (Structure and Bonding). Berlin: Springer;2010, 126.


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