Where two-centre bonding stops

Diborane has twelve valence electrons and eight bonds to draw. Aluminium has three valence electrons and twelve nearest neighbours. Neither fact can be accommodated by a picture in which a bond is two electrons shared between two atoms, and neither is exotic.

The two-centre bond is a special case that happens to cover most of organic chemistry, and treating it as the general case is why the rest of the periodic table looks like a collection of exceptions.

The counting problem, stated fairly #

A Lewis structure allocates two electrons to each line drawn between two atoms. That works when the number of electrons is at least twice the number of bonds required, and a great deal of chemistry satisfies that condition.

Boron does not. Diborane, B₂H₆, has two boron atoms and six hydrogens; the structure has two bridging hydrogens, each apparently bonded to both borons, and there are twelve valence electrons for what looks like eight bonds. Two electrons short by four.

The resolution is that two of the “bonds” are not two-centre bonds. Each bridge is a three-centre two-electron bond: one orbital spread over boron, hydrogen and boron, holding two electrons and doing the work of what would otherwise be two separate bonds.

That is the same construction as the three-centre four-electron bond in hypervalent molecules, with two electrons rather than four, and the two cases together are worth setting side by side because they are usually taught separately as unrelated oddities.

Electron-deficient systems — boranes, and organolithium and organoaluminium compounds — have too few electrons for two-centre bonds and use three-centre two-electron ones.

Hypervalent systems — sulfur hexafluoride, the interhalogens — have too many for an octet and use three-centre four-electron ones.

Neither needs a new principle. Both need the abandonment of one: that a bond involves exactly two centres.

The clusters, where counting becomes a science #

Boron’s carboranes and the transition-metal cluster compounds push the arithmetic much further, and the fact that any counting rule works at all is remarkable.

A closo-borane $\mathrm{B}_n\mathrm{H}_n^{2-}$ has $n$ vertices and $n+1$ skeletal electron pairs. That relation — Wade’s rule — predicts the shape of the cluster: $n+1$ pairs gives a closed deltahedron, $n+2$ gives one with a vertex missing, $n+3$ two missing.

The rule works because the cluster’s skeletal orbitals form a pattern with a single low-lying totally symmetric orbital and then a set of degenerate ones, which is exactly the pattern a ring’s orbitals have and for the same reason: high symmetry produces degeneracies, and degeneracies produce a shell structure.

So Wade’s rules are a three-dimensional version of the $4n+2$ count, and the phrase “three-dimensional aromaticity” that is sometimes used for boranes is more literal than it sounds.

The three-dimensional case does not have to be left as an analogy, and it should not be, because the pattern the rule depends on is a statement about a graph and a cage is a graph. Each vertex of a closo deltahedron carries one orbital pointing inwards at the centre — the radial set — and the energies of those orbitals are the eigenvalues of the cage’s own adjacency matrix, computed exactly as a ring’s are. The two smallest closo cages are the octahedron and the icosahedron, and the pattern is visible in both.

Adding six more vertices does not change the shape of that answer, which is the point. The gap below the nodeless orbital stays large, the levels above it stay degenerate in sets, and the shell structure survives the change of cage exactly as a ring’s survives a change of size.

That is the analogy made computable, and it is worth being exact about how far it goes. What is computed is the radial set, which supplies one of the $n+1$ skeletal pairs; the other $n$ come from the tangential orbitals, which need a treatment with two orbitals per vertex that is not here. So the pattern — one low totally symmetric orbital, then degenerate sets, then a gap — is verified in three dimensions, and the count $n+1$ is still quoted.

The molecule that shows both at once #

Benzene is the case where the two descriptions can be laid side by side, which makes it the best place to see that the localised one is optional rather than obligatory.

Its lowest π orbital has the same coefficient, $1/\sqrt{6}$, on all six carbons and no node anywhere: an orbital that belongs to the ring rather than to any bond in it, and the one that carries the ring current. What the six of them produce is six identical bonds, each with a computed π order of exactly two thirds — not one and a half, which is what averaging two Kekulé structures gives, and not one or two, which is what either structure alone gives. Both numbers are computed in Hückel theory and both are drawn there.

A reader can insist on drawing three localised pi bonds in benzene, and the insistence is legitimate — a transformation exists that produces them, and the density is unchanged. What the transformation cannot do is make them equivalent to one another and to the ring’s symmetry at the same time. Three localised pi bonds pick out three of the six bonds, and no operation of D₆ₕ preserves that choice.

So benzene sits exactly on the boundary this essay is about: two-centre bonding is available and it is unnatural, and the diagnostic is that the localised description has less symmetry than the molecule.

One sign change is enough to put a ring on the other side of that boundary, and it is the cleanest available demonstration that the shell structure belongs to the topology of the delocalised set rather than to any arrangement of bonds. Close a ring with a half turn in the ribbon of p orbitals and exactly one resonance integral reverses in sign.

There is no set of two-centre bonds that produces that, and no way to see it coming from a picture made of them. The whole of the difference lives in a property of the ring as a ring.

From a molecule to a metal #

The step from a cluster to a solid is not a step at all, and the arithmetic makes that plain.

A ring of $n$ carbons has levels $2\beta\cos(2\pi k/n)$. As $n$ grows, the levels fill the interval from $-2\beta$ to $+2\beta$ more and more densely, and the width of that interval does not change. Adding atoms adds levels without widening the range they occupy.

The narrowing is not left to be interpolated either, since the gap between the highest occupied and lowest empty level of a chain has a closed form.

Take $n$ large and the levels are a continuum four β wide. That is a band, and the calculation that produced it is the one that produced benzene’s six levels — the tight-binding model of a solid is Hückel theory with more atoms.

Two consequences follow immediately and are worth stating because they are usually presented as separate facts about solids.

Whether a solid conducts is a shell-closure question. A band that is partly filled has empty levels immediately above the occupied ones, so an electron can be moved by an arbitrarily small field: a metal. A band that is exactly full has a gap: an insulator or a semiconductor. That is the $4n+2$ question asked of a very large ring.

Band width goes with overlap. The span is $4\beta$, and β is an interaction between neighbours. Atoms that overlap strongly give wide bands; weakly interacting ones give narrow bands and localised electrons. Overlap decides in a solid for the same reason it decides in a diatomic.

There is no point along the sequence where a molecule stops and a solid starts. What changes is the spacing between levels relative to the thermal energy, and that is a continuous parameter.

The surprise: a bond is the derived concept #

The conclusion this essay reaches by accumulation is worth stating directly, because it inverts the order everything is normally taught in.

Molecular orbitals are the general case. They are spread over whatever atoms are present, in patterns fixed by the symmetry and by the connectivity, and they do not know what a bond is.

A bond is what appears when the delocalised orbitals happen to admit a transformation into localised ones that each sit on two atoms — which requires enough electrons, and a structure without high symmetry, and it is a change of basis rather than a discovery. For most of organic chemistry the transformation exists and is excellent, so the localised picture is used and the general one is forgotten.

For boranes it does not exist, for metals it does not exist, and for benzene it exists but is unnatural. Those are not three anomalies. They are three cases where the special condition that makes two-centre bonding available happens not to hold, and the general arithmetic goes on working.

That reframing costs nothing and repairs a great deal. Electron-deficient compounds stop being electron-deficient — they have exactly as many electrons as their orbitals require — and the word only makes sense relative to a bookkeeping scheme that was never obligatory.

What was computed, and how #

Every level pattern on this page comes from diagonalising an adjacency matrix, checked three ways: against the closed form $2\cos(2\pi k/n)$ for a ring, against the two trace relations which follow from the graph rather than from the solution, and against the pairing theorem, which is tested in both directions so that a non-alternant system whose levels came out paired anyway would be refused.

The shell-closure check is the one that carries the argument. For every ring from three to ten, the closure found by filling the computed levels must agree with the $4n+2$ predicate, and a mismatch would falsify the rule. The twisted ring is checked against its own closed form, $2\cos((2k+1)\pi/n)$, and against the 4n predicate rather than the 4n+2 one — so a sign convention that had quietly not reversed anything would be refused rather than drawn.

The two cages are checked in two ways. Every radial level is an eigenvalue of the cage’s adjacency matrix, and the claim that the top level is nodeless and uniquely so is Perron’s theorem, checked rather than observed. The cage is also required to be a deltahedron: its edge list is built from the shortest inter-vertex distances, and the separation between the edges kept and the next pair not kept is tested, which is what refuses an eight-vertex cage where that ratio is exactly one and the shape is ambiguous.

And the chain gaps are checked against $2\cos(k\pi/(n+1))$ level by level, against the asymptote as a bound approached from below, and against a monotone ratio — three claims, of which the second was written the wrong way round first and refused by the arithmetic.

Nothing on this page computes a whole borane or a metal. The cluster electron counts and the band structures described are quoted; what is computed is the ring case, the radial set of two cages, and the chain gap — the smallest systems in which the shell-closure logic can be exhibited and checked in two dimensions and in three.

That division is deliberate and it is the honest limit of what an adjacency-matrix model reaches. A real borane needs the tangential orbitals as well as the radial ones, with several atom types; a real metal needs a band-structure calculation. Neither is here, and pretending otherwise would be exactly the failure worth avoiding.

What it costs #

The arithmetic costs a small eigenvalue problem per ring.

What the reframing costs is the loss of a very useful picture, and it is worth being clear that the loss is real.

Localised bonds are how chemists think. A reaction mechanism is a story about bonds breaking and forming, arrow-pushing requires bonds to push arrows between, and the whole vocabulary of functional groups presupposes that a molecule decomposes into transferable pieces. None of that survives literally in a delocalised description.

The delocalised picture predicts badly at the level of a reaction. It is right about spectra, about shell closures and about solids, and it says almost nothing useful about why an ester hydrolyses.

So the practical answer is the one this site gives everywhere: use whichever description makes the problem easy, and be clear that the choice is a choice. What changes after this essay is only the direction of the exception — the localised picture is the special case, available under conditions that are usually met and sometimes not.

Where the model stops #

Four limits.

Nothing here is a calculation of a solid. A band from a tight-binding chain has one orbital per site and no self-consistency; a real band structure has many orbitals per site, electron repulsion, and a lattice.

Wade’s rules are quoted, not derived. The pattern they rest on — one nodeless orbital, well separated, then degenerate sets — is now computed in three dimensions for two cages. The count $n+1$ itself is not, because it needs the tangential orbitals, and it is quoted.

Electron counting is a bookkeeping scheme. Skeletal electron pairs, oxidation states and formal charges are all allocations of electrons to places, and electrons do not come allocated. The schemes work because they are consistently applied, not because they describe where anything is.

And the boundary between “molecule” and “solid” is not drawn anywhere here because there is not one. What this essay establishes is that the same arithmetic covers both; where a reader draws the line is a matter of what question is being asked.

The word “bond” in three senses #

Since the essay has spent itself dismantling the concept, it is worth reassembling what remains of it, because three quite different things travel under the name.

A line in a structural formula. A convention for recording connectivity, and an excellent one — it is the input to every Hückel calculation, and Hückel theory uses nothing else. This sense survives everything above intact. It says which atoms are adjacent and claims nothing about electrons.

A localised orbital. Available when a transformation into two-centre orbitals exists, which requires enough electrons and not too much symmetry. This is the sense that fails for boranes and metals, and it is a basis choice throughout.

A region of accumulated density between two nuclei. An observable, measurable by diffraction, and present in diborane’s bridges and in a metal alike. This sense survives too, and it is the one that would let somebody say a borane bridge “is” a bond without contradicting anything above.

Most of the confusion in this area comes from arguments in which one party means the second sense and the other means the first or the third. Separating them dissolves the disagreement without anybody having to concede a point about the chemistry.

One practical consequence closes the argument. A chemist meeting an unfamiliar structure and finding that no Lewis structure fits has learned something specific rather than hitting a wall: the electron count and the orbital count do not match, so the localised transformation is unavailable, and the delocalised description is the one to reach for. That is a diagnosis rather than a failure.

Who found it, and when #

Diborane’s structure was contested for twenty years. The bridged form was proposed in the 1920s, rejected in favour of an ethane-like structure with a boron–boron bond, and settled experimentally only in 1951 by infrared and electron diffraction work — with Longuet-Higgins and Bell having given the three-centre bonding account in 1943, ahead of the evidence.

Lipscomb’s systematic treatment of borane structures won the Nobel Prize in Chemistry in 1976, and Wade’s rules followed in 1971 with Mingos generalising them shortly after.

The band picture arrived from an entirely different direction: Bloch’s theorem in 1928, in solid-state physics, with no reference to chemistry at all. That it is the same calculation as Hückel’s, published three years later for a completely different purpose, was noticed slowly — and Hoffmann’s extended Hückel work from the 1960s onward is largely the project of making the identity explicit and useful in both directions.

Hoffmann’s 1988 Nobel lecture makes the point this essay makes, and makes it better: that chemists and solid-state physicists had been doing the same arithmetic in different notations for fifty years.

Two ways a localised picture fails, and they are not the same failure #

The precondition is given as enough electrons and not too much symmetry, and the two halves are worth separating, because they fail differently and a reader meeting either should be able to tell which.

Too few electrons. Count the pairs available for the skeleton and count the connections a localised description would need. In an ordinary saturated molecule the two match: ethane has seven skeletal connections and seven pairs to put in them. In a closed borane cage they do not — a twelve-vertex cage has thirty edges and nowhere near thirty pairs — so there is no assignment of pairs to edges, and the shortfall is arithmetic rather than aesthetic.

The signature of that failure is an absence. There is no localised picture to draw, no set of two-centre bonds that uses the electrons available, and any drawing that appears to show one has miscounted.

Too much symmetry. Benzene’s π system has three pairs and enough of them: three double bonds fit. The difficulty is that they fit in two ways, related by a symmetry of the molecule, and neither is preferred.

The signature there is a multiplicity. There are localised pictures, there are several, they are equivalent, and the molecule has no more reason to adopt one than the other — which is what a set of resonance structures is, and why the resolution is a single delocalised description rather than a rapid interconversion.

So the two failures produce opposite symptoms. A shortage of electrons produces no picture; an excess of symmetry produces too many. A reader who finds themselves drawing several equivalent structures has met the second, and one who cannot draw any has met the first, and the arithmetic that distinguishes them is a count of pairs against a count of connections done before anything is drawn.

Related arguments, and what is new here #

The three-centre case with four electrons is hypervalency without d orbitals.

The arithmetic throughout is Hückel theory.

The count that decides whether a shell closes is aromaticity as a shell closure.

And the transformation that makes a localised picture available when it is available is the localisation transformation.

The honest summary is that nothing in this essay is new physics. Every case it covers was solved decades ago, by people who mostly did not think of themselves as working on the same problem. What is new is the framing — that boranes, hypervalent molecules, benzene and metals are one situation seen at four sizes — and a framing is worth having when it turns four exceptions into one rule with a condition attached.

It is also worth noticing which direction the generalisation runs. Nothing here says the localised picture is wrong; it says the localised picture is a special case with a precondition, and the precondition is that there be enough electrons and not too much symmetry. Most of chemistry meets it. The parts that do not are not anomalies waiting for a better bonding theory — they are the general case, appearing where the special one runs out.

What the pictures here cannot show. No metal appears in any figure on this page, and no whole borane does either — the two cages are their radial sets, which is one skeletal pair out of $n+1$. What every figure has in common is that it is an eigenvalue problem on a graph, in two dimensions or three, because that is the system an adjacency-matrix model can compute and check. A reader wanting a picture of a band structure is wanting a calculation that is not here, and the last figure above is the nearest this page comes to one: a gap that the model sends to zero, drawn beside the length at which a real polymer stops obeying it.