A bond with nothing in the middle

What a pair of orbitals is bonded by can invert when a third orbital is nearby, and the same arithmetic points at a picture not yet drawn:

Two 2p orbitals head-on change the sign of their overlap, so beyond that separation the combination that is bonding is the one whose picture has a node between the nuclei. The arithmetic is unchanged, so the question is only what the resulting orbitals look like.

The arithmetic is unchanged and the picture is available. It is also the other way round from the sentence above, and the reason is worth an essay.

The sign change #

Two hydrogenic 2p functions on an axis, both pointing the same way along it — which is the convention every function in this collection uses, since overlap places one at −R/2 and the other at +R/2 and turns neither.

In that convention the lobes that face each other across the gap carry opposite signs, so the region between the nuclei contributes negatively to the integral. The outer lobes carry the same sign as each other and contribute positively. The overlap is a difference between two regions, and which one wins depends on the separation.

At three bohr the outer lobes win and the integral is +0.4825. At seven bohr the gap wins and it is −0.2649. In between, at 5.0265 bohr, they cancel exactly.

Which combination is filled #

Write the two combinations as $p_A + p_B$ and $p_A - p_B$. The first is zero at the midpoint at every separation, and that is a fact about where the nuclei are rather than about any interaction: each function is odd about its own centre, the midpoint is equidistant from both, and the two terms cancel there whatever else is true.

The energies come from the two-level result with $\beta = K S \alpha$ — the relation every extended-Hückel model uses, and the one used here to price a closed-shell contact. It ties the sign of the interaction to the sign of the overlap, and that is what this essay is about.

The two levels cross exactly where the overlap does, which they must: the interaction is proportional to it.

So the picture exists at ordinary separations rather than beyond the crossing. Below 5.03 bohr the filled combination is the one with the node; above it, the filled combination is the ordinary-looking one. The expectation was the opposite because it assumed the standard sign convention, in which the two p orbitals are drawn pointing at each other and the sign of every overlap is flipped. Nothing physical distinguishes the two conventions and one of them is used consistently here, which is exactly why the essay has to say which.

What is between the nuclei #

The picture is worth having, and the number underneath it is worth more.

Two electrons in the filled combination, at the midpoint, as a multiple of what the two free atoms put there:

Below the crossing it is exactly nothing — not small, zero, by the same cancellation that makes the node. And the model calls the pair bonded there by 0.165 hartree at two bohr, which is the largest stabilisation anywhere in the table.

The three numbers above the crossing are worth a second look too. They fall as the atoms move apart — 1.844, 1.581, 1.501 — towards 1.5 rather than towards 1. A pair of separated atoms should put exactly what two atoms put; the excess of a half is normalisation rather than bonding, since dividing by $\sqrt{2-2S}$ with $S$ still at $-0.33$ at nine bohr inflates a combination that has stopped interacting. The accumulation is measuring the normalising constant as much as the physics, and at the separations where the model is doing something interesting it cannot be read as a bond order.

A pair the model calls maximally bonded, with every electron removed from between the nuclei. That is the sharpest form yet of a recurring finding: an overlap integral is not a measure of bonding, and here it is not a measure of density between the nuclei either.

Why the plus combination is always zero there #

The node is worth one paragraph of its own, because it is the least surprising thing in the essay and the reason the rest of it is surprising.

A 2p function is odd about its own centre: $p(-\mathbf{r}) = -p(\mathbf{r})$. The midpoint of the bond is at $+R/2$ from one centre and $-R/2$ from the other, so the two functions take values there that are equal in size and opposite in sign — always, at every separation, at every effective charge, for any orbital with this symmetry. Their sum is zero and their difference is not.

So the node is a fact about geometry rather than about bonding, and it never moves. What moves is which combination the model puts lower, and that is decided by an integral over all space with a sign that has nothing to do with the midpoint. The two facts meet at the midpoint and neither is about the other, which is why the picture is worth drawing rather than merely describing: the model fills an orbital that is zero exactly where the bond is supposed to be, for reasons that never mention the bond.

What the model is actually saying #

None of this means the arithmetic is wrong, and it is worth separating three statements that are easy to run together.

The two-level result is exact for the model it belongs to: given $\alpha$, $\beta$ and $S$, those are the two levels and there is nothing approximate about them.

The relation $\beta = K S \alpha$ is a model — a good one, fitted once on the noble gases and carried everywhere in this collection — and it is what makes the sign of the interaction the sign of the overlap. There is no theorem behind it.

And the overlap’s sign here is set by a cancellation between two regions, one of which is nowhere near the bond. So a model that reads the interaction off the overlap is reading it off a quantity whose sign is decided partly by what is happening outside the molecule.

That is the whole of the difficulty, and it is specific rather than general: for two 1s functions the integrand has one sign everywhere and the overlap is a measure of the region between the nuclei, which is why the same model behaves impeccably there. For two p functions head-on it is not, and no amount of care with $K$ repairs it.

What the overlap integral does and does not measure #

Every essay on overlap has found a way for the integral not to be a measure of bonding, and the integral itself has never once misbehaved. That pattern now has a shape.

The integral is exactly what it is defined to be. A symmetry-forbidden overlap comes out at arithmetic noise, the closed forms hold, the curves turn over where the argument says. None of it has ever been a complaint about the integral.

Every failure has been about a step taken after it — treating it as an energy, as a density, as a ranking, or as a sign. The overlap is a number about two functions, and each case has been somebody turning it into a number about a bond. Two pairs at the same overlap and a different bond is the same observation made about the energy; moving the atoms closer without increasing the overlap is it made about the distance; an interaction that is not the overlap is it made about the interaction.

Which suggests what is still missing, and it is not another failure: a positive statement of what the integral does measure. This sharpens the question rather than answering it — the integral measures the extent to which two functions are not linearly independent, and every use of it as a chemical quantity is a claim that some chemical quantity happens to track that. Sometimes one does.

What is quoted, and what is computed #

Nothing is quoted. There is no molecule here: two hydrogenic 2p functions at a stated effective charge, at separations chosen to span the sign change, and a two-level problem written out.

The overlap is a mapped three-dimensional quadrature and the sign change is found by bisection on it. The midpoint densities are evaluations of a normalised combination and the zeros in them are exact rather than small — the plus combination’s value at the midpoint is checked to be zero to a part in ten to the twelfth at three separations, and the minus combination’s is checked not to be.

What the crossing does to a numerical check #

An accidental zero is not a symmetry zero, and a numerical check can confuse the two.

A natural way to verify an overlap is to recompute it with a second quadrature rule — a different point count, a different map — and the check has two clauses. The two rules must agree to a part in ten thousand, and they must agree about vanishing, because a forbidden overlap coming back as a small number rather than as zero is the failure that matters most.

At the crossing the first rule gives −1.1 × 10⁻¹³ and the second gives 6.4 × 10⁻⁷. Both are nothing; the first happens to fall under a vanishing threshold and the second does not, so a check built that way reports a disagreement where there is none. The two kinds of zero separate cleanly under refinement: change the point count or the map and a symmetry zero stays at arithmetic noise every time, while an accidental zero moves, because it is a genuine value of a function that happens to pass through nothing at that separation.

The right form keeps the claim and drops the accident: the two rules must agree about vanishing unless both are below a millionth, at which point neither says anything. A forbidden overlap wrongly returning a real number still fails that test, twice over, because a real number is not below a millionth either.

A symmetry zero is arithmetic noise under every rule. An accidental zero is a small number under every rule, and can fall below any threshold by luck. That distinction is the whole content of the statement that a forbidden overlap is exactly zero, and it takes a curve that crosses zero to show that a check can lack it.

What this cannot say #

These are hydrogenic 2p functions and there is one electron in the model. A real 2p on a real atom is not this function, and two of them at five bohr are not any molecule. What is established is a property of a model used everywhere in these essays, which is why it is worth establishing.

The two-level model is not the only account available and the others are not consulted. A variational calculation on the same two functions with the full one-electron Hamiltonian would give an interaction that is not proportional to the overlap, and might fill the other combination. This establishes what a model in common use says, and does not establish that the model is right about it — which is a distinction the caution about that constant already draws, in the other direction.

The crossing separation is not transferable. At Z = 1 it is 5.03 bohr; at the effective charge a carbon 2p has it is 3.09, and at a fluorine-like charge 1.55. So the regime this essay is about is reachable at ordinary bond lengths for compact orbitals and not for diffuse ones, and no single number says where it is.

And the node is not an observable. It is a feature of one orbital in a one-electron model, and the density that would be measured is a sum over occupied orbitals. What is being shown is that the model’s own account of bonding disagrees with the density its own filled orbital carries — an internal inconsistency rather than a claim about a measurement.

A picture that flips while nothing else does #

The most useful thing about this result is the mismatch it exposes between what the calculation does and what a drawing of it shows, and the mismatch is worth stating in its own right.

Sweep the separation through 5.03 bohr and every computed quantity passes smoothly. The two levels approach each other, touch and separate again; the energies are continuous; their derivatives are continuous; nothing in the arithmetic has a corner in it. The overlap passes through zero, which is a smooth thing for a function to do.

The picture does not pass smoothly. On one side of that separation the filled combination has density piled between the nuclei; on the other it has a nodal plane through the midpoint and exactly nothing there. Those are two qualitatively different drawings, and a reader shown them in sequence would say that something happened.

Nothing did. The two orbitals swapped which of them was lower, which is what happens whenever two levels cross, and the labels bonding and antibonding followed the ordering rather than the functions.

That says something about the labels rather than about the molecule. A picture that changes discontinuously while every computed quantity changes smoothly is a picture, not a property, and the discontinuity lives in the convention that the filled orbital is the one called bonding.

It also disposes of a criterion that is stated constantly. A bond is an accumulation of electron density between the nuclei is offered as the definition of covalent bonding, and here is a filled orbital, in a bound arrangement, with a nodal plane exactly where the density is supposed to be. Whatever holds such a system together, it is not the density in the middle — and the account of what does, through the kinetic energy and the virial theorem, has no term corresponding to density between the nuclei anywhere in it.

What was checked #

The overlap changes sign once, near five bohr, with the crossing found by bisection rather than read off a plot.

The filled combination has a node below the crossing and none above it, checked separately on each side, because a single check covering both would pass if the model never changed its mind at all.

Where it has a node, two electrons in it put exactly nothing at the midpoint — checked as an equality with zero rather than as a bound, since the whole point is that the cancellation is exact.

And the refusal is the midpoint zero itself, which must hold at every separation and for both signs of the overlap. It is a consequence of the functions being odd about their own centres, so a combination that failed it would mean the orbitals had been placed wrongly rather than that anything physical had changed.

Still open: whether the node survives being filled #

The obvious open question is the occupation, which nothing here has touched. Everything here is two electrons in one combination; four electrons fill both, and the sum of the two densities has no node anywhere. So the whole finding is about a singly occupied pair of levels, and asking what the same arithmetic says at zero, two, four and six electrons would say whether the node survives being filled — with a boundary that is an integer.

The nearer question is the one the sign change makes available for nothing. If the interaction’s sign follows the overlap’s, then a diatomic held at exactly 5.0265 bohr 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. Computing what a third orbital does to that pair, which is the three-orbital model, would be a clean test of whether the regimes it found survive the pair itself vanishing.