A bond order between atoms that do not interact

The head-on overlap of two 2p orbitals changes sign at 5.0265 bohr, and the consequence is stark: a diatomic held exactly there is a pair of orbitals with no interaction at all — degenerate, at the free-atom energy, with a spectrum that says the two atoms are not there. That suggests a test. A three-orbital model already exists, and computing what a third orbital does to that pair says whether the regimes the trio found survive the pair itself vanishing.

They do not survive; they collapse. And on the way to finding that out, the pair with no interaction turns out to have a bond order of nearly minus one.

One line is exactly where the vanishing overlap puts it, and two are not #

Set the pair’s overlap to zero, keep both outer orbitals at the same energy, and couple each of them equally to a third. The two outer orbitals then have a symmetric combination and an antisymmetric one, and only the symmetric one has anything of the third orbital’s symmetry to mix with.

So the antisymmetric combination sits at the free-atom energy exactly, at every third-orbital energy, to machine precision. The symmetric one is pushed down and the third orbital is pushed up, by as much as twelve electronvolts.

The prediction is therefore exactly half right, and the half that is right is exact rather than approximate. A spectroscopist looking at this trio would see one line where the free atom’s is and two lines where they are not, and would have no reason to conclude that anything about the pair had vanished.

The regimes have nothing left to vary #

The three-orbital calculations measure what a pair is bonded by in the presence of a third orbital: the energy of the trio, less the energy of the same trio with the A–B interaction removed and nothing else changed. That definition was chosen because it means the same thing whether the third orbital is above the pair, below it or between — and the three regimes it found are three shapes that quantity takes.

With the pair’s overlap zero, the A–B interaction is already nothing. Removing it changes nothing. The pair is bonded by exactly zero at every third-orbital energy, and the three regimes collapse to one point.

That is a clean answer to the question and it is not an interesting one on its own. What makes it worth having is what the other two measures say on the same system.

The bond order is not zero #

A bond order is a sum over occupied orbitals of the product of two coefficients. The occupied set here is the symmetric combination — pushed down and partly on the third orbital — and the antisymmetric one, which is untouched.

The antisymmetric combination contributes $2 \times (1/\sqrt2)(-1/\sqrt2) = -1$. The symmetric one contributes something positive and less than one, because part of its weight is on the third orbital. The sum is negative, and how negative depends on how much weight the third orbital has taken.

The result runs from +0.031 with the third orbital 13.6 eV above the pair to −0.954 with it 11.4 eV below, and to −0.999899 with it 80 eV below.

A bond order of minus one between two orbitals with no overlap, no resonance integral and no interaction of any kind. The limit is exact and its cause is plain: a very deep third orbital takes all the weight of the symmetric combination, leaving the occupied set with an antisymmetric pair and nothing to cancel it.

That is not a defect of the bond order. It is what a bond order is: a property of the occupied orbitals, and the occupied orbitals are decided by everything in the molecule. What it cannot be is a measure of what connects two particular atoms, and here it is as wrong about that as it is possible to be while remaining a number between −1 and 1.

The system is bound and the pair is not #

The third measure is the one a chemist would use without thinking: is the thing bound? It is, at every third-orbital energy, most strongly by 5.33 eV when the third orbital is level with the pair.

Every electronvolt of that is A–C and B–C. The pair contributes nothing, by the first measure, exactly. So a decomposition of the binding that assigned any of it to A–B would be assigning it to a pair with no interaction — and the bond order, which is what such a decomposition usually rests on, would positively encourage it, since −0.95 is a large number.

Why the pair vanishing is different from the pair being far apart #

It is worth separating this case from the ordinary one it resembles, because the resemblance is misleading.

Two atoms far apart also have a negligible overlap, and their bond order is also small. But there the smallness is a limit: the overlap falls smoothly to zero, every quantity built on it falls smoothly with it, and nothing is exactly anything. Here the overlap is zero at a point, having been positive on one side and negative on the other, and the two atoms are 5.0265 bohr apart — closer than many bonds.

So this is not the dissociation limit in disguise. The two orbitals are close enough for their radial functions to be substantially on top of one another; what has vanished is an integral, because a positive region and a negative region have come into balance. The picture drawn at that separation is that below that separation the filled combination has a nodal plane through the bond and puts exactly nothing in the middle, which is a different thing from putting nothing there because there is nothing anywhere.

That distinction is why the third orbital can do so much. Its overlaps with A and with B are large — the atoms are close — and they are what the occupied set is built from. A genuinely dissociated pair with a third atom far from both would have small overlaps everywhere and no large bond order anywhere.

One number makes the disagreement concrete. At a third-orbital energy of −13.6 eV, level with the pair, the three measures read exactly 0, −0.6306 and −5.33 eV. A chemist handed those three about a real molecule would have no way to reconcile them and no reason to suspect that the middle one is about a pair that does not exist. All three would be printed by ordinary software, and only the first required a second calculation to obtain.

What the three disagree about #

They disagree because they are three different questions wearing one name.

By removing the interaction asks what the direct A–B term is worth, and it is worth nothing here because there is no direct A–B term. That is the only one of the three that answers the question is A bonded to B, and it is also the only one that requires a calculation to be run twice.

By bond order asks what the occupied orbitals look like at A and at B, and that has a large answer because the occupied orbitals were shaped by C. It is the cheapest of the three and the one every program prints.

By binding asks whether the system is more stable than its parts, which is a property of the system and was never a property of a pair. It is the one a measurement gives, and it is also the one this collection’s three-centre work is built on: a three-centre bond is a statement about a system of three, and the whole reason it needs its own name is that no pairwise account of it works.

There is no arithmetic that reconciles them, and there is no sense in which one of them is the true bond. What the case here shows is that they can be made to disagree as completely as they can disagree — one exactly zero, one nearly a full antibond, one several electronvolts — on a system with no pathology in it at all, built from three orbitals and one overlap set to a value measured for two real functions.

There is one more asymmetry worth recording. Of the three, only the first is a difference of two calculations — it needs the trio solved twice, once with the A–B term and once without — and it is the only one that gives the right answer here. The other two are read off a single calculation, and both mislead. The same ordering holds for correlation energies, where a difference between two calculations by one method is worth more than either of them; the price is that a difference needs a second calculation, and a second calculation is exactly what nobody runs.

It is worth saying why the disagreement is so wide here and narrower almost everywhere a practitioner meets it. In an ordinary bonded pair the three measures agree because they are being asked about a system whose interactions are hierarchical: A and B are strongly coupled to each other and weakly to everything else, so the occupied orbitals at A and B are shaped mostly by the A–B term, removing that term is most of the binding, and all three quantities are reading the same interaction from three directions. The agreement is not a theorem; it is a consequence of one term being much larger than the rest.

This trio removes exactly that condition and nothing else. The A–B term is not merely small, it is absent, and the coupling that shapes the orbitals at A and B runs entirely through C. So the three measures are being asked about three different interactions rather than one, and there is no reason for them to agree — the surprise is not that they disagree but that anyone expected otherwise. A bond order is a reliable proxy for an interaction only in the regime where the interaction dominates, which is the regime in which nobody needed the proxy.

The practical version is a warning about where the number is most often used, which is precisely where the hierarchy fails: hypervalent centres, electron-deficient bridges, delocalised π systems, transition-metal complexes with several ligands competing for the same orbital. Each of those is a system in which a pair’s environment does as much to its orbitals as the pair does, and each is a system for which a printed bond order between two chosen atoms is the standard thing to quote.

What was computed, and how #

Everything is the three-orbital model of the earlier calculations with one overlap changed. The trio is a three-by-three generalised eigenproblem with Wolfsberg–Helmholz resonance integrals, the overlap matrix is required to be positive definite before it is used — three functions with these overlaps must be linearly independent, and three overlaps of 0.7 apiece describe an impossible set — and there are four electrons, which is a pair’s worth and a third orbital’s worth.

The bond orders are sums over occupied orbitals of products of coefficients, taken from the same eigenvectors the energies come from.

The claim that one level sits at the free-atom energy is checked rather than observed: the smallest distance from any level to $\alpha$ is required to be under $10^{-9}$ at every third-orbital energy. It is a symmetry statement — the antisymmetric combination belongs to a representation the third orbital has nothing in — and it would break if the two outer orbitals were coupled to the third by different amounts.

The refusal is the third orbital taken away as well. With no A–C or B–C overlap either, the trio is three isolated functions: every level at its own energy, both bond orders exactly zero, and no binding.

Where the model stops #

Whether the same overlap always means the same bond is the question behind three calculations, and this is its limiting case: the same overlap, zero, and a bond order anywhere between nothing and a full antibond.

The three-level model has no explicit electron repulsion, no geometry and no radial functions in it: the overlaps are numbers put in by hand and the resonance integrals are a proportionality. So nothing here says what a real third atom would do at 5.0265 bohr — the separation where the overlap changes sign is a property of two hydrogenic 2p functions, and putting a third atom nearby would change it.

The symmetry that pins one level is exact and fragile. It needs the two outer orbitals at the same energy and coupled equally to the third; a real trio would satisfy neither exactly, and the untouched level would move by an amount proportional to the asymmetry rather than staying put.

And four electrons is a choice. Two would go to the lowest level and the antisymmetric combination would be empty, which removes the whole of the −1 and leaves a positive bond order. The bond order between atoms that do not interact is a function of the electron count, which is one more thing it cannot be a property of the pair.

The generalisation #

A quantity that is a property of the occupied orbitals is a property of the whole molecule, and calling it a property of two atoms is a convention rather than a measurement.

That sentence is not new and it is usually met with the reply that the convention is a good one — bond orders correlate with bond lengths, they behave sensibly across a series, and nobody claims they are observables. All of that is true. What this case adds is a bound on how badly the convention can fail: not a few per cent, and not a wrong ordering, but a full antibond where there is nothing at all.

The same shape has come up twice more this season, both times as an average or a summary standing in for the thing it summarises: a mean correction whose three parts include one of the opposite sign, and a give-back whose sign reverses when the wavefunction is allowed to respond. In each the derived quantity is well defined and the object it is named after is not.

Who found it, and when #

Coulson’s bond order is from 1939 and the caution that it is not an observable is as old as it. That a bond order between non-adjacent atoms can be large is standard in Hückel theory — a long-range bond order in a conjugated system is an ordinary quantity — and it is usually explained as through-bond coupling.

What is unusual here is the combination: the two atoms are not merely non-adjacent, they have exactly zero overlap and exactly zero resonance integral, so there is no coupling to be through-bond or otherwise. The bond order is then entirely an artefact of which combinations the third orbital leaves occupied, and its limit of −1 is exact.

The through-bond and through-space distinction is Hoffmann’s, from 1968, and the situation here is the limiting case of it: all through-bond and none through-space, with the through-space term set to zero by arithmetic rather than by distance.

Still open: whether the exact level survives detuning #

The obvious continuation is the electron count. Everything above is at four electrons, where the antisymmetric combination is occupied and supplies the −1. At two electrons it is empty and the bond order is positive; at six every level is full and the bond order must be exactly zero by completeness. Three counts, three qualitatively different answers on one geometry, and the sweep costs nothing — which would turn the bond order is a function of the electron count from a remark into a table.

The nearer question is the asymmetry the symmetry statement needs. One level sits at the free-atom energy because the two outer orbitals are coupled equally to the third; detuning one of them by a stated amount would say how fast that level moves, and whether the exactness is protected to first order or lost at once. That is the same question asked of a different exact zero in a ligand field, and the answer there was that an exactness from an absence is lost immediately.