What is a bond?

My piece on the chemical bond is now published in Nature. I hope it attracts more comment – already I’m pleased to see remarks from the IUPAC team who are redefining the hydrogen bond (I had no room to talk about this in any detail, or to supply the link), and also some comment on Bader’s perspective, to which again I could only allude in the briefest of terms – it deserves more space.

... ah, Julie's post about the inaccessibility behind Nature's firewall makes me feel bad, so here's the whole piece after all, before final editing so with a few more refs and details included:

******************************

Not so long ago the chemistry student’s standard text on the theory of chemical bonding was Charles Coulson’s Valence (1952). Absent from it was Coulson’s real view of the sticks that generations of students have drawn to link atoms into molecules. ‘A chemical bond is not a real thing: it does not exist: no one has ever seen it, no one ever can. It is a figment of imagination which we have invented,’ he wrote [1].

There is a good reason for postponing this awkward truth. The bond is literally the glue that makes the entire discipline cohere, and so to consider it an objective reality is necessary for any kind of chemical discourse. Chemistry is in fact riddled with such convenient (but contested [2]) fictions, such as electronegativity, oxidation state, tautomerism and acidity.

Disputes about the correct description of bonding have ruffled chemists’ feathers since the concept of molecular structure first emerged in the mid-nineteenth century. Now they are proliferating, as new theoretical and experimental techniques present new ways to probe and quantify chemical bonds [3]. Traditional measures such as crystallographic atomic distances and dissociation energies have been supplemented by spectroscopic techniques for determining vibrational frequencies, shifts in the electronic environment of the atom, magnetic interactions between atoms, measurements of force constants, and a host of quantum-chemical tools for calculating such aspects as electron distributions, electron localization and orbital overlap.

The nature of the chemical bond is now further complicated by the introduction of the dynamical dimension. Molecules have traditionally been regarded, if not as static, then as having platonic architectural frameworks which are merely shaken and rotated by thermal motions. The bonds get stretched and bent, but they still have an equilibrium length and strength that seems to justify their depiction as lines and stalks. Now, thanks to ultrafast spectroscopies, we are no longer restricted to these time-average values to characterize either structure or reactivity. What you ‘measure’ in a bond depends also on when you measure it.

Some chemists argue that in consequence the existence (or not) of a bond depends on how the problem is probed; others are committed to absolute criteria [4]. This difference of opinion goes to the heart of what chemistry is about: can all be reduced to quantum physics or are fuzzy heuristics essential? More pressingly, the issue of how best to describe a chemical bonding pattern has tangible implications for a wide range of problems in chemistry, from molecules in which atoms are coerced out of their usual bonding geometry [5] to the symmetric hydrogen bond (where the hydrogen is shared equally between two atoms) [6,7] and new variations on old themes such as aromaticity (special patterns of ‘smeared-out’ bonding like that in benzene) [8].

Just about every area of chemistry harbours its own bonding conundrums, almost any of which illustrate that we have a far from exhaustive understanding of the ways in which quantum rules will permit atoms to unite – and that in consequence our chemical inventiveness suffers from a limited view of the possibilities.

Carving up electrons

We can all agree on one thing: chemical bonding has something to do with electrons. Two atoms stick together because of the arrangement of electrons around their nuclei. In the nineteenth century it was commonly thought that this attraction was electrostatic: that atoms in molecules are positively or negatively ionized. That left the puzzle of how identical atoms can form diatomic molecules such as H2 and O2. American chemist G. N. Lewis proposed that bonding can instead result from the sharing of electrons to create filled shells of eight, visualized as the corners of a cube [9].

In the 1920s and 30s Linus Pauling showed how this interaction could be formulated in the language of quantum mechanics as the overlap of electron wavefunctions [10]. In essence, if two atomic orbitals each containing a single electron can overlap, a bond is formed. Pauling generalized earlier work on the quantum description of hydrogen to write an approximate equation for the wavefunction created by orbital overlap. This became known as the valence-bond (VB) description.

But an approximation is all it is. At the same time, Robert Mulliken and Friedrich Hund proposed another way to write an approximate wavefunction, which led to an alternative way to formulate bonds: not as overlaps between specific orbitals on separate atoms but as electron orbitals that extend over many atoms, called molecular orbitals (MOs). The relative merits of the VB and MO descriptions were debated furiously for several decades, with no love lost between the protagonists: Mulliken’s much-repeated maxim ‘I believe the chemical bond is not so simple as some people seem to think’ was possibly a jibe at Pauling. By the 1960s, for all Pauling’s salesmanship, it was generally agreed that MO theory was more convenient for most purposes. But the debate is not over [11], and Roald Hoffmann of Cornell University insists that ‘discarding any one of the two theories undermines the intellectual heritage of chemistry’.

Both options are imperfect, because they insist on writing the electronic wavefunction as some combination of one-electron wavefunctions. That’s also the basis of the so-called Hartree-Fock method for calculating the ground-state wavefunction and energy of a molecular system – a method that became practical in the 1950s, when computers made it possible to solve the equations numerically. But separating the wavefunction into one-electron components is a fiction, since the distribution of one electron depends upon the distributions of the others. The difference between the true ground-state energy and that calculated using the Hartree-Fock approach is called the correlation energy. More recent computational methods can capture most of the correlation energy – but none can give an exact solution. As a result, describing the quantum chemical bond remains a matter of taste: all descriptions are, in effect, approximate ways of carving up the electron distribution.

If that were the limit of the bond’s ambiguity, there would be little to argue about. It is not. There is, for example, the matter of when to regard two atoms as being bonded at all. Pauling’s somewhat tautological definition gave the game away: ‘there is a chemical bond between two atoms or groups of atoms in case that the forces acting between them are sufficient to lead to the formation of an aggregate with sufficient stability to make it convenient for the chemist to consider it as an independent molecular species’ [1]. Pauling himself admitted that although his definition will in general exclude the weak van der Waals (‘induced dipole’) attraction between entities, occasionally – as in the association of two oxygen molecules into the O4 cluster – even this force can be strong enough to be regarded as a chemical bond.

It’s no use either suggesting (as Coulson did) that a bond exists whenever the combined energy of the objects is lower than that when they are separated by an infinite distance. This is essentially always the case, at least for electrically neutral species. Even two helium atoms experience mutual van der Waals attraction, which is after all why helium is a liquid at very low temperature, but they are not generally thought to be chemically bonded as a result.

Besides, the ‘bonded or not’ question becomes context-dependent once atoms are embedded in a molecule, where they may be brought into proximity merely by geometric factors, and where there is inevitably some arbitrariness in assigning them an individual electronic configuration. The resulting ambiguities were illustrated recently when three experts on inorganic compounds failed to agree about whether two sulphur atoms in an organometallic compound are linked by a bond [12]. The argument involved different interpretations of quantum-chemistry calculations, tussles over the best criteria for identifying a bond, and evidence of precedent from comparable compounds.

All this is merely a reminder that the molecule is ultimately a set of nuclei embedded in a continuous electron cloud that stabilizes a particular configuration, which balls and sticks can sometimes idealize and sometimes not. This doesn’t mean that disputes about the nature of the chemical bond are simply semantic. It matters, for example, whether we regard a very strong multiple bond as quintuple or sextuple, even if this is a categorization that only textbooks, and not nature, recognize.

Besides, how we choose to talk about bonds can determine our ability to rationalize real chemical behaviour. For example, the different descriptions of the bonds in what are now called non-classical ions of hydrocarbons – whose relative merits were furiously debated in the 1950s and 60s – have direct implications for the way these species react. Whether to consider the bonding non-classical, in the sense that it involved electrons spread over more than two atomic nuclei, or tautomeric, involving rapid fluctuations between conventional two-atom bonds, was not just a question of convention. It had immediate consequences for organic chemistry [13]

Perhaps one might seek a distinction between bonded and not-bonded in terms of how the force between two atoms varies with their separation? Yes, there is an exponential fall-off for a covalent bond like that in H2, and a power-law decay for van der Waals attraction. But the lack of any clear distinction between these two extremes has been emphasized in the past two decades by the phenomenon of aurophilicity [14,15]. Organometallic compounds containing gold with only a few chemical groups attached tend to aggregate, forming dimers or linear chains. In aurophilic bonds, the basic interaction has the same origin as the van der Waals force: the electron clouds ‘feel’ each other’s movements, so that random fluctuations of one induce mirror-image fluctuations of the other. But that interaction is modified here by relativistic effects: the changes in electron energies resulting from their high speeds in orbitals close to gold’s highly charged, massive nuclei [15,16]. Aurophilic bonds have therefore been described as a ‘super van der Waals’ interaction. Does that make them true bonds? It’s chemically meaningful to treat them that way (they’ll even serve for cementing new ‘designer’ molecular crystals [17]), but perhaps at the cost of relinquishing potential distinctions.

In uniting ‘closed-shell’ atoms, aurophilicity has sometimes been compared to hydrogen bonding, which is of comparable strength. Hydrogen bonds have traditionally been rationalized in electrostatic terms: positively polarized hydrogen atoms drawn towards regions of high electron density, due for example to ‘lone pairs’. But the bond has some covalent, electron-sharing character too, as is clear from its directional nature (it tends to have a 180o bond angle). Quantifying that is not at all straightforward, however, and has only very recently been done experimentally [18], prompting a task group of the International Union of Pure and Applied Chemistry to propose a new definition of the hydrogen bond (open for comment until this March) to replace the older electrostatic picture [19]. It’s an indication of how new methodology can restructure thinking about apparently familiar – and vitally important – modes of bonding. Even then, the IUPAC report warns that ‘there will be borderline cases for which the interpretation of the evidence might be subjective’: an explicit admission that categorizing bonds must remain an art, informed but not wholly determined by scientific criteria.

Moving target

How dynamics colours the notion of a chemical bond is an increasingly subtle matter. Atomic motions make even a ‘simple’ molecule complex; any movement of one nucleus demands that the entire electron cloud adjusts. So a jiggle of one group of nuclei can make it easier to cleave off another.

This complication never used to matter much in chemistry. The movements were too rapid to be observable, much less exploitable. But ultrashort pulsed lasers have moved the goal posts. For example, we can pump energy into a vibrational mode to weaken a specific bond, enabling selective molecular surgery [20]. We can ask about the chemical behaviour of a molecule at a particular moment in its dynamical evolution: even a strong bond is weakened when a vibration stretches it beyond its average, equilibrium length, so in ultrafast chemistry it may no longer be meaningful to characterize bonds simply as strong or weak. As Fleming Crim of the University of Wisconsin-Madison puts it, ‘a bond is an entity described by quantum mechanics but not a fixed ‘entity’ in that it will behave differently depending on how we perturb and interrogate it.’ The trajectory of a chemical reaction must then be considered not as a simple making and breaking of bonds but as an evolution of atoms on a potential-energy surface. This was always implicit in classical drawings of transition states as molecular groupings containing dashed lines, a kind of ‘almost bond’ in the process of breaking or forming. Now that is explicitly revealed as a mere caricature of a complicated dynamical process in space and time.

Underlying most these discussions is an unspoken assumption that it is meaningful to speak, if not of a ‘bond’ as an unchanging entity, then at least of an instantaneous bound state for a particular configuration of nuclei. This assumes that the electrons can adjust more or less instantly to any change in the nuclear positions: the so-called Born-Oppenheimer approximation. Because electrons are so much lighter than nucleons, this assumption is usually justified. But some clear breakdowns of the approximation are now well documented [21]. They are best known in solid-state systems [22], and in fact superconductivity is one of the consequences, resulting from a coupling of electron and nuclear motions. Such things may also happen in molecules, particularly in the photochemistry of polyatomic molecules, which have a large number of electronic states close in energy [23]; they have also been observed for simple diatomic molecules in strong electric fields [24]. As a result, the molecular degrees of freedom may become interdependent in strange ways: rotation of the molecule, for example, can excite vibration. In such situations, the very notion of an electronic state begins to crumble [21].

Embrace the fuzziness

These advances in dynamical control of quantum states amount to nothing less than a new vision of chemistry. The static picture of molecules with specific shapes and bond strengths is replaced by one of a bag of atoms in motion, which can be moulded and coaxed into behaviours quite different from those of the equilibrium species. It does not demand that we abandon old ideas about chemical bonds, nor does it truly challenge the ability of quantum theory to describe atoms and their unions. But it recommends that we view these bonds as degrees of attraction that wax and wane – or as cartoon representations of a molecule’s perpetual tour of its free-energy landscape. At a meeting in 1970, Coulson asserted that the simple notion of a chemical bond had already become lost, and that it seemed ‘something bigger’ was needed to replace it. ‘Whether that ‘something bigger’… will come to us or not is a subject, not for this Symposium, but for another one to be held in another 50 years time’, he said [25]. That moment is almost upon us.

But we needn’t fret that the ‘rules’ of bonding are up for grabs — quite the converse. While there may be some parts of science fortunate enough to be exhaustively explained by a single, comprehensive theory, this isn’t likely to be a general attribute. We are typically faced with several theories, some overlapping, some conflicting, some just different expressions of the same thing. Our choice of theoretical framework might be determined not so much by the traditional criterion of consistency with experiment but for more subjective reasons. According to Roald Hoffmann of Cornell University, these preferences often have an aesthetic component: depending on factors such as simplicity, utility for ‘telling a story’ about chemical behaviour, the social needs of the community, and the question of whether a description is productive.

As Hoffmann says, ‘any rigorous definition of a chemical bond is bound to be impoverishing’. So his advice to ‘have fun with the fuzzy richness of the idea’ seems well worth heeding.


References

1. C.A. Coulson, The Spirit of Applied Mathematics, 20-21 (Clarendon Press, Oxford, 1953).
2. Jansen, M. & Wedig, U. Angew. Chem. Int. Ed. 47, 10026-10029 (2008).
3. J. Comput. Chem. special issue, 28, 1-466 (2007).
4. Cortés-Guzmán, F. & Bader, R. F. W. Coord. Chem. Rev. 249, 633 (2005).
5. Merico, G., Médnez-Rojas, M. A., Vela, A. & Heine, T. J. Comput. Chem. 28, 362-372 (2007).
6. Jensen, S. J. K. & Csizmadia, I. G. Chem. Phys. Lett. 319, 220-222 (2000).
7. Benoit, M., Marx, D. & Parrinello, M. Nature 392, 258-261 (1998).
8. Abersfelder, K., White, A. J. P., Rzepa, H. S. & Scheschkewitz, D. Science 327, 564-566 (2010).
9. Lewis, G. N. J. Am. Chem. Soc. 38, 762 (1916).
10. Pauling, L. The Nature of the Chemical Bond (Cornell University Press, Ithaca, 1939).
11. Hoffmann, R., Shaik, S. & Hiberty, P. C. Acc. Chem. Res. 36, 750-756 (2003).
12. Alvarez, S., Hoffmann, R. & Mealli, C. Chem. Eur. J. 15, 8358-8373 (2009).
13. Brown, H. C. The Nonclassical Ion Problem (Springer, Berlin, 1977).
14. Schmidbaur, H. Gold Bull. 13, 3-10 (2000).
15. Pyykkö, P. Chem. Soc. Rev. 37, 1967-1997 (2008).
17. Schmidbaur, H., Cronje, S., Djordjevic, B. & Schuster, O. Chem. Phys. 311, 151-161 (2005).
17. Katz, M. J., Sakai, K. & Leznoff, D. B. Chem. Soc. Rev. 37, 1884-1895 (2008).
18. Isaacs, E. D. et al., Phys. Rev. Lett. 82, 600-603 (1999).
19. Arunan, E. et al., ‘Definition of the hydrogen bond’, recommendation submitted by IUPAC task group 2004-026-2-100, October 2010. See http://media.iupac.org/reports/provisional/abstract11/arunan_310311.html
20. Crim, F. F. Science 249, 1387 (1990).
21. Sukumar, N. Found. Chem. 11, 7-20 (2009).
22. Pisana, S. et al., Nature Mater. 6, 198-201 (2007).
23. Worth, G. A. & Cederbaum, L. S. Ann Rev. Phys. Chem. 55, 127-158 (2004).
24. Sindelka, M., Moiseyev, N. & Cederbaum, L. S., Preprint http://www.arxiv.org/abs/1008.0741.
25. Coulson, C. A. Pure Appl. Chem. 24, 257-287 (1970).