A regime that belongs to the neighbours

There is a division that runs through every question about overlap and bonding. Two orbitals at the same energy are stabilised in proportion to their overlap, so the bonding tracks the integral; two far apart in energy are stabilised by an amount that barely notices the overlap at all — a 1s with a 2s is bonded by 0.376 at an overlap of 0.10 and by 0.377 at 0.40, a change of 0.4 per cent for a fourfold change.

That leaves whether the regime a pair is in is a property of the pair. A third orbital at an intermediate energy changes the gap each of the others sees, and answering it — a three-by-three secular problem with the overlaps kept — is a short step from a two-by-two.

It is not a property of the pair, and the way it fails is worse than shifted. Overlap is not interaction is an old finding; the same integral now turns out not even to fix the sign.

Defining what a pair is bonded by, in a trio #

The two-level answer is a stabilisation: the lowest level’s drop below the mean of the two atomic energies. That definition stops meaning anything the moment a third orbital is present, because when the third one is the lowest thing in the problem the lowest level is the third orbital and not the pair.

So the quantity here is different and it means the same thing in every case:

the energy of the trio, less the energy of the same trio with the A–B interaction removed and nothing else changed.

Take out the A–B overlap and the A–B coupling, leave everything else, solve again, and subtract. That is what the pair is bonded by in the presence of the third orbital, and it reduces to the ordinary answer when the third orbital is decoupled — a control that is checked below and comes back equal to fourteen decimal places.

Four electrons rather than two, which needs saying. Two electrons go to whichever orbital is lowest, so a third orbital below the pair would take them both and the pair would be unoccupied; four fills the two lowest levels, which is a pair’s worth and a third orbital’s worth, and is the configuration in which the question means anything at all.

The couplings are Wolfsberg–Helmholz throughout — $\beta = K S (\alpha + \alpha')/2$ — so a coupling is never a free parameter beside its overlap. That matters here more than usual: a third orbital cannot be handed a large coupling and a small overlap, which is the manoeuvre that would make any of this come out any way at all.

The bond goes negative #

With the third orbital decoupled, the pair gains 0.539 eV — an ordinary two-centre bond.

With the third orbital coupled to both, the same A–B interaction costs between 1.14 and 3.10 eV, at every placement from twenty-four electronvolts below the pair to two above.

The worst is at −8 eV, where the third orbital sits exactly on one of the pair.

This is the familiar four-electron repulsion, arriving in a place it is not usually looked for. Four electrons in three orbitals is a closed shell, and a closed shell’s interactions are net repulsive: the system is already bonded through the third centre, and adding a direct A–B term fills an antibonding combination as well as a bonding one.

So the sign of what a pair is bonded by is decided by its neighbours, not by the pair. It is the arithmetic behind a fact computed from the other end: a three-centre bond puts nothing on its middle atom in one of its orbitals, which is the same statement about where the interactions actually live.

The regime inverts #

The sign is striking and the sensitivity is the finding, because the sensitivity is what the two regimes are made of.

Quadrupling the A–B overlap, from 0.1 to 0.4, and asking what happens to what the pair is bonded by:

Alone, the pair is squarely in the same energy regime: quadrupling the overlap multiplies the bonding almost twelvefold.

With a third orbital present the response collapses by a factor of eight or more, and for two of the three placements it goes below one — a fourfold increase in the overlap between the two atoms leaves them less strongly interacting than before.

That is not the other regime. The second regime is a flat response, a ratio near one; a ratio of 0.74 is a response with the wrong sign, and neither of the two regimes contains it. A third orbital does not move a pair from one regime to the other. It puts the pair somewhere neither regime describes.

Why more overlap can buy less #

The mechanism is worth writing down because it is not a numerical accident.

A larger A–B overlap does two things. It increases the A–B coupling, which is the effect the two-level picture is about. It also makes the overlap matrix more nearly singular — the three functions become more nearly linearly dependent — and the generalised eigenproblem’s response to that is to push the antibonding level up faster than it pushes the bonding level down.

In a closed shell both are occupied. So the larger overlap raises the filled antibonding combination by more than it lowers the filled bonding one, and the net is a loss that grows with the overlap. It is the same arithmetic that makes the antibonding level go up more than the bonding one goes down, applied to a system where both are filled.

The overlap is doing its usual job. What has changed is which orbitals are occupied, and that is decided by the third centre.

The linear-dependence refusal #

A three-function basis has a failure mode a two-function one does not, and it is worth meeting properly because it is easy to write down an impossible molecule.

Suppose A overlaps C by 0.99, and B overlaps C by 0.99. Then A and B are each nearly the same function as C — and they cannot be orthogonal to each other. An A–B overlap of zero describes no set of functions at all.

The arithmetic notices: the overlap matrix has a negative eigenvalue, so it has no inverse square root, and the generalised eigenproblem cannot be formed. The overlap matrix is checked before solving and the set is refused rather than returned as numbers, alongside a check that a possible set is still solved.

That refusal has no analogue in the two-level problem. Any $|S| < 1$ describes a possible pair; three overlaps have to be mutually consistent, and most triples of plausible-looking numbers are not.

What this does to the overlap integral as a measure #

The relation between an overlap integral and a bond has been taken apart in nine essays, which found, in order: that closer is not more overlap; that overlap is not interaction; that a symmetry-forbidden overlap is exactly zero; that the same overlap gives different bonds; and now that the same pair at the same overlap gives a bond whose sign depends on what is nearby.

The through-line is that the overlap integral is an input to an arithmetic problem and not a measure of anything on its own. Each essay has found a different way for it not to be a measure, and this is the strongest: not that the relation is weak, but that it inverts.

The practical form is a caution about a common move. Comparing two bonds by comparing two overlaps — a bigger overlap here than there, so a stronger bond here than there — is valid only if the two are in the same environment and have the same occupation. Neither condition is usually stated and neither is usually true. A hypervalent molecule does not stop at three centres either, which is the same caution applied to a whole molecule rather than to a pair.

What survives, and it is not nothing #

It would be easy to read all of this as saying that an overlap integral is useless, and that is not what it says.

Three statements survive intact. A symmetry-forbidden overlap is exactly zero and nothing about neighbours changes that, because a zero from an odd integrand stays zero in any environment. An overlap of zero means no direct coupling, whatever else is present. And the ordering of two overlaps between the same pair of functions at two separations is a fact about the functions and is not an environmental quantity at all.

What does not survive is the step from an overlap to a bond energy, and the reason is that the step passes through an occupation and an environment. Those are two extra inputs, both of them usually supplied silently.

The honest form of the two-regime finding is therefore narrower than it looked and more useful for being narrow: at a stated occupation, in a stated environment, what a pair is bonded by is a function of its overlap and its gap. Change either of the two stated things and the function changes, including its sign.

That is not a retreat. A stabilisation is measured from somewhere is the standing rule about references, and this adds a second clause to it: a stabilisation is also measured in somewhere.

What is quoted, and what is computed #

Nothing is quoted except the model’s own constant. The atomic energies are stated parameters in electronvolts, chosen to be of ordinary size; the Wolfsberg–Helmholz $K$ of 1.75 is fitted rather than derived, is quoted, and no conclusion here depends on its value.

Every level comes from a generalised eigenproblem solved by symmetric orthogonalisation — the same procedure used for Hückel problems with overlap. Its overlap matrix is checked for positive definiteness before use.

The pair’s bond is a difference of two solved trios rather than a formula, which is why it means the same thing at every placement.

What this cannot say #

There is no electron repulsion. The four-electron destabilisation here is the one-electron kind — a filled antibonding combination — and a real closed-shell repulsion has an exchange contribution this model does not contain. The sign is right for both reasons, which makes it robust and makes the size untrustworthy.

The Wolfsberg–Helmholz convention is doing work at the edges. The coupling scales with the mean of the two diagonal energies, so a third orbital placed far below the pair acquires a large coupling rather than a negligible one. That is why the effect does not die away at the left of the sweep, and it is a property of the convention rather than of chemistry — a real deep orbital is compact and overlaps little.

Nothing here is a molecule. Three energies, three overlaps and an electron count are a model of an arrangement, not a compound, and no conclusion above should be attached to a named substance. What it is a model of is the question — whether a pair’s bond is the pair’s — and that question has an answer at this level of description.

And three is not many. A real neighbour set is four or six orbitals, and whether the inversion strengthens or washes out as more are added is a question answered by growing one matrix.

What else is a property of a pair, and is not #

The finding — that a regime belongs to the neighbours rather than to the pair — has a reach beyond this model, because a great deal of chemical vocabulary consists of labels attached to pairs of atoms.

Ionic and covalent is the standing example. A bond is described as one or the other on the strength of the two elements at its ends, from a table indexed by element pairs, and the arithmetic here says that a third orbital nearby can change how the pair responds to its own coupling by more than an order of magnitude. A pair’s character is not a property of the pair when the pair has neighbours, and every bond in every molecule larger than a diatomic has neighbours.

Bond order is exposed in the same way, and it can be measured: three ways of asking how strongly two atoms are bonded, on one system, disagreeing completely.

And transferability is what the exposure costs. A bond energy taken from one molecule and used in another assumes that what the bond is worth is a property of its two atoms — which is the assumption additivity rests on, and which works because in most molecules the neighbourhoods are similar rather than because the assumption is sound.

So the negative result is a general caution with a stated mechanism. A label attached to a pair is safe when its neighbours are the same, and is a property of the neighbourhood otherwise — and the way to find out which case a molecule is in is to change a neighbour and see whether the label moves.

What was checked #

The pair on its own is stabilised, and quadrupling its overlap multiplies that severalfold — the two-regime finding, reproduced in the same currency so that the comparison is between like quantities.

With a third orbital coupled to both, the same A–B term is destabilising everywhere, checked across the whole sweep rather than at a point.

The pair’s response to its own overlap collapses by more than a factor of four, and for some placements inverts — the two halves of the finding, checked separately because the second is the surprising one and the first would be a much weaker claim on its own.

A third orbital present but decoupled changes nothing, to fourteen decimal places, wherever it sits — the control that makes the effect the coupling’s rather than the presence’s.

And an impossible set is refused. Two functions each nearly identical to a third and orthogonal to one another are not three functions, and the calculation says so rather than returning eigenvalues; a possible set is still solved, so the refusal is about the overlaps and not about the procedure.

Still open: the pair term against the electron count #

The obvious open question is the occupation. Everything above is a closed shell, and the sign of the pair term is a consequence of both combinations being filled — so the same trio with two electrons rather than four should give a pair term that is positive, and the crossover is somewhere between. Computing what the A–B interaction is worth against the electron count, from zero to six, would turn the sign depends on the neighbours into the sign depends on the neighbours and the filling, with a boundary that is an integer.

The nearer question is the sign change that is named and not drawn. Two 2p orbitals head-on change the sign of their overlap at 5.06 bohr, so beyond that separation the combination that is bonding is the one whose picture has a node between the nuclei. The same arithmetic reaches it directly: put a negative $S_{ab}$ into the trio and the arithmetic is unchanged, so the question is only what the resulting orbitals look like — and drawing a filled bonding orbital with a node in the middle of the bond is a picture long overdue.

A third open question is the one every essay on overlap circles. Each has found a way for the overlap integral not to be a measure of bonding, and the integral itself has never once misbehaved — a symmetry-forbidden overlap comes out at arithmetic noise, the curves turn over where the argument says, and the closed forms hold. Writing down what the integral is a measure of, positively rather than by a list of exclusions, is an essay worth writing — and two kinds of correlation is the nearest thing to a model of how to write it.