The same overlap, a different bond

An overlap integral is not monotonic in the separation once a radial node is involved. A 1s with a 2s overlaps by 0.016 at contact, by 0.282 near four bohr and less again beyond; two 2p orbitals head-on change sign at 5.06 bohr. The corresponding statement about the interaction rather than the overlap is rarely made, though non-monotonic overlaps ought to produce non-monotonic stabilisations — a curve that is seldom drawn.

This essay draws it. The curve is there, and the expected conclusion is not the one the arithmetic supports — like an expected argument its own arithmetic refused and improved by the refusal.

The chain, and its one fitted number #

Turning an overlap into a stabilisation takes three steps and only the middle one is a choice.

Integrate. The overlap at each separation, by numerical quadrature.

Couple. Turn the overlap into a resonance integral by Wolfsberg–Helmholz, $H_{AB} = K S (H_{AA} + H_{BB})/2$ with $K = 1.75$. That is the oldest bridge between an overlap and a matrix element and it is fitted; no resonance integral here is computed from first principles, and that is worth saying rather than hiding inside a curve. It is the same rule a contact distance was computed with, where its being dimensionless was what made the answer a length nobody had put in.

Solve. The two-by-two secular problem with the overlap kept in it, which is where the antibonding level rises by more than the bonding falls.

The one-electron energies come from the wavefunctions themselves by the virial route rather than being quoted, so $K$ is the only fitted number in the whole chain and it is dimensionless.

The expectation the arithmetic refused #

The essay was planned around a specific claim: that the stabilisation would turn over somewhere else than the overlap does, because the secular determinant divides by $(1 \pm S)$ and so cannot be a faithful copy of $S$.

It does not. The two maxima coincide to five decimal places of a bohr — 4.16818 against 4.16817, and 7.58281 against 7.58281 — and refining them off the grid by fitting a parabola through three points, rather than reading off the largest sample, is what makes the comparison sharp enough to say so.

The reason is one line and it is more useful than the expectation was. At fixed atomic energies the stabilisation is a function of the overlap alone — every appearance of the separation in the calculation is through $S$ — and it is a strictly increasing function of it over this range. A strictly increasing function of a curve peaks exactly where the curve does.

So the denominators do move the value, and they cannot move the position, because they are functions of the same variable as everything else. That would have been true of any coupling rule proportional to the overlap, and it would not have been true of a rule that depended on the separation some other way.

The closed form for a matched pair #

For two orbitals at the same energy the whole thing collapses to an expression. Setting $\alpha_A = \alpha_B = \alpha$ and $\beta = KS\alpha$, the bonding root is $\alpha(1 + KS)/(1 + S)$ and the stabilisation of a filled pair is

$$2(1-K)\,\alpha\,\frac{S}{1+S},$$

which the computation reproduces to $10^{-12}$ at every separation tested.

Two things follow immediately. It is monotonic in $S$, which is the previous section’s claim in closed form. And it saturates: the factor $S/(1+S)$ is nearly $S$ when $S$ is small and flattens when $S$ is not. Doubling the overlap from 0.05 to 0.1 buys 1.909 times the bonding; doubling it from 0.4 to 0.8 buys 1.556.

That saturation is worth having as a number because the received account of a bond does not contain it. The picture in which a bond is the overlap of two orbitals suggests a quantity proportional to the overlap, and at chemically ordinary overlaps of 0.2 to 0.3 the departure from proportionality is already a fifth. A double bond is not two single bonds for a related reason: the two components are not additive either.

What actually separates one pair from another #

The maxima coincide, so the interesting comparison is not where but how much, and here the pairs behave completely differently.

At a fixed overlap of 0.25, two 1s orbitals are stabilised by 0.150 hartree and two 2s orbitals by 0.0375 — a ratio of exactly four, which is the ratio of their atomic energies. That is the closed form again: the stabilisation is proportional to $|\alpha|$, so a deep pair is bonded four times as strongly as a shallow pair at the same overlap.

A 1s with a 2s, whose energies differ by that same factor of four, is stabilised by 0.375 at an overlap of 0.10 and by 0.377 at 0.40.

Four hundredths of a per cent, for a fourfold change in the overlap. Over the range where the 1s–2s overlap actually turns over — two to seven bohr, where $|S|$ runs from 0.169 to 0.282, a change of 67 per cent — the stabilisation runs from 0.375334 through 0.375913 and back to 0.375426, a total variation of 0.154 per cent.

So the non-monotonicity of the overlap is there in the bonding, exactly where it is in the overlap, and it is invisible. A curve that varies by a part in six hundred over a range in which its argument varies by two thirds is a flat line with a bump drawn on it at high magnification.

Why a mismatched pair does not notice #

The mechanism is elementary second-order perturbation theory and it is worth writing out, because it is the same argument that governs why overlap is not interaction.

Two levels a distance $\Delta$ apart, coupled by $\beta$, repel by roughly $\beta^2/\Delta$ when $\beta \ll \Delta$ and by roughly $\beta$ when $\beta \gg \Delta$. The 1s–2s pair has $\Delta = 0.375$ hartree and $\beta$ around 0.09 at its largest, so it is deep in the first regime — and in that regime the whole stabilisation is dominated by the gap. A homonuclear pair has $\Delta = 0$ and is in the second regime always, where the stabilisation is the coupling and the coupling is the overlap.

That is the general statement about overlap and bonding: whether an overlap curve’s shape shows up in anything depends on which regime the pair is in, and the regime is set by the energies rather than by the geometry. That is why an electronegativity difference does not fix how much charge moves: the same two-level problem, with the transfer rather than the stabilisation read off it.

What the non-monotonicity would take to see #

Since the effect is real and invisible, it is worth asking what would make it visible, because the answer identifies the case where a non-monotonic overlap matters.

The bonding tracks the overlap when the two orbitals are close in energy and ignores it when they are far apart. So a non-monotonic overlap shows up in a stabilisation exactly when the pair is matched — and a matched pair with a radial node between them is two 2s orbitals, or two 2p orbitals head-on.

Two 2s orbitals do not turn over inside the range where the model is sensible: their overlap falls from 0.975 at 0.8 bohr to 0.261 at twelve, monotonically. Two 2p orbitals head-on do something better than turning over — they change sign, at 5.06 bohr — and a coupling proportional to a quantity that changes sign is a coupling that changes sign.

That is the case where a non-monotonic overlap has a consequence for an energy, and it is not computed here, because the two-level model reports a number and the interesting thing there is a picture.

What this cannot say #

K is fitted and everything scales with it. Wolfsberg–Helmholz’s constant sets the size of every stabilisation here. It does not set any position, because it multiplies the coupling by a constant and the maximum of an increasing function of $S$ does not care; and it does not set any ratio between pairs at a fixed overlap.

The energies are hydrogenic. These are one-electron functions at $Z = 1$, so the 1s–2s gap of 0.375 hartree is hydrogen’s. Real orbitals on real atoms have gaps set by screening, and the regime a pair is in would have to be decided case by case.

No kinetic energy term. A real bonding interaction has a kinetic contribution that the secular determinant with a fitted coupling does not represent separately, and it is the term responsible for the fact that the antibonding level rises more. This model has that effect through the overlap denominators and has it as a consequence rather than as a mechanism.

The two electrons are not correlated. A filled bonding orbital in this model has both electrons in the same spatial function with no repulsion between them, so what is called a stabilisation here is a one-electron sum. What that omits is largest exactly where the two orbitals are closest in energy, which is the regime where the overlap matters most.

And there are only two levels. A real diatomic mixes many orbitals, and a pair whose stabilisation is dominated by its gap would in practice be more affected by a third level nearer to it than by anything in this calculation.

Where this leaves more overlap means more bonding #

Three separate results take apart the sentence more overlap means more bonding, and it is worth putting them in one place because they fail it in three different ways.

Overlap is not interaction: a coupling half again as large with a gap four times worse buys a third less stabilisation. The overlap is one of two inputs.

Closer is not more overlap: with a radial node in the picture, the overlap itself is not monotonic in the distance, so even the input does not behave.

And this one: the stabilisation follows the overlap faithfully — exactly, in the sense that it peaks in the same place — and how much it follows it by depends on the gap, which for a mismatched pair means it barely follows it at all.

There is a fourth failure, and the closed form above makes it visible: the stabilisation saturates in the overlap, so the sentence is quantitatively wrong even in the case where it is qualitatively right. At the overlaps a real bond has, the last increment of overlap buys about four fifths of what the first did.

So the sentence is not wrong in the way a false statement is wrong. It is a statement about one of two variables, which is exactly right when the other is zero and nearly meaningless when the other is large, and the two cases look identical from a picture of two overlapping orbitals.

Why the two maxima had to coincide #

The arithmetic refusing to move the maximum reads as a surprise, and it is worth showing that it could not have gone otherwise — because the reason is structural and it says something about what the whole family of such calculations can produce.

Fix the two diagonal energies. The stabilisation of the lower level is then a function of one variable, because the coupling is taken proportional to the overlap and the secular determinant contains nothing else that varies. Whatever complicated expression that function is, it is a function of $S$ alone.

A function of one variable inherits the extrema of that variable. If $S$ is largest at some separation, then anything monotone in $S$ is largest there too, and it does not matter how the denominators depend on $S$ so long as the dependence never turns round.

So the coincidence to five decimal places is not agreement between two calculations. It is the same extremum, reported twice, in a quantity and in a monotone function of that quantity — and finding it anywhere else would have meant the stabilisation was not monotone in the overlap, which would be the interesting result.

That also says what the expectation got wrong. The denominators do vary along the curve and they do change the stabilisation’s value at every separation; what they cannot do is create a maximum where the overlap has none or move the one it has, because they enter through the same variable.

The general form is worth carrying past this pair. A one-parameter model cannot separate the position of a maximum from the position of its parameter’s maximum. Distinguishing where the strongest bonding occurs from where the largest overlap occurs requires a second variable that moves independently — a change in the diagonal energies, or a coupling that is not proportional to the overlap — and neither is present here.

Which is why the essay’s real finding is the other one. What separates one pair from another is not where the maximum sits, since that is settled by the overlap alone, but how much stabilisation there is at it — a quantity the denominators do control, and the only one in the calculation that they can move.

It is worth saying which of the two a chemist actually needs. Nobody chooses a bond length by looking for the maximum of an overlap — the length is settled by a balance against nuclear repulsion and the rest of the molecule, and it lands wherever it lands. What a comparison between two pairs is for is deciding which of them binds more strongly at the lengths they actually adopt, and that is a comparison of magnitudes rather than of positions. So the quantity the model cannot separate is the one nothing was going to ask it for, and the quantity it can supply is the one that matters.

What the argument requires #

The control, which is that a pair with no radial node falls away strictly monotonically — from 0.356 hartree at 0.8 bohr to $2.8 \times 10^{-4}$ at twelve.

Both nodal pairs turning over inside the range, and both maxima refined off the grid rather than read off it.

The two maxima agreeing to $10^{-4}$ bohr, which is the test the expected conclusion would have failed.

The closed form to $10^{-12}$ at every separation for a matched pair, and its saturation stated as two ratios.

The factor of four between a deep pair and a shallow one at one overlap, to $10^{-3}$, which is the ratio of the atomic energies and is what makes the comparison a comparison rather than a coincidence.

And the sharp one, in both directions: over the measured range the mismatched pair’s overlap changes by more than half and its bonding by under one per cent. Either half alone would be a weaker claim — a flat curve over a range where nothing else moves is not a finding.

Still open: the sign, and a third level #

The obvious open question is the sign. Two 2p orbitals head-on change sign at 5.06 bohr, and a coupling proportional to the overlap changes sign with it — so beyond that separation the bonding combination is the one that would be drawn as antibonding, and a filled pair is stabilised by an orbital whose picture has a node between the nuclei. That is a picture worth drawing and it has not been drawn here, because the two-level model reports an energy and not a shape.

The nearer question is what a third level does. Every pair here is bonded by an amount set by its gap, and a third orbital at an intermediate energy changes that gap for both of the others. The calculation is a three-by-three secular problem with overlaps, a short step from the two-level one, and the question it would answer is whether the regime a pair is in is a property of the pair at all. A heteroatomic ring is where a third level is unavoidable, and the same arithmetic applies there with more terms.