A denominator that fails both ways

The quantity an angular overlap parameter hides — an $e_\pi$ is an overlap squared over an energy gap, and the gap is never written down — can be computed for the π donors, whose level is an ionisation energy and is tabulated for every atom. The denominator over-predicts the halide trend at every metal level a donor allows: the fitted π scales span 1.4286 from fluoride to iodide and the denominator gives at least 1.667.

It stopped at the sign change, and said why. A π* is a molecular orbital of a diatomic, not an atomic level, so its energy cannot be had from an ionisation energy — but it is measurable, by electron transmission spectroscopy, and the numbers are quoted in the literature. Putting those two numbers into the same arithmetic would extend the comparison across the sign change, which is the one place a denominator argument is on its strongest ground and has never been tested.

It is not on strong ground there. It fails on that side too, in the opposite direction, and the one free parameter the argument has makes both sides worse at once.

An acceptor’s level is a measurement #

Electron transmission spectroscopy fires slow electrons through a gas and looks for the energy at which they are briefly captured. For a small π system that capture is into the π* orbital, and the energy at which it happens — the vertical attachment energy — is the π* level above the vacuum.

Carbon monoxide’s is 1.50 eV and dinitrogen’s is 2.30 eV, both measured on the molecule itself. Cyanide’s cannot be measured that way: an electron cannot be attached to an anion and read as a resonance. Hydrogen cyanide’s π* is used in its place, at 2.26 eV, which is the same orbital with a proton on the nitrogen instead of a metal.

That substitution is the weakest link in everything below, and it is named rather than buried. What it costs is unknown; what can be said is that the finding does not depend on it delicately — cyanide’s π* would have to lie below carbon monoxide’s, rather than 0.76 eV above it, for the direction of the result to change.

The same arithmetic, the other way up #

For a donor the gap is the metal orbital minus the ligand’s lone pair, which lies below it. For an acceptor the roles swap: the gap is the π* minus the metal orbital, and the π* lies above. Nothing else changes, and the sign of $e_\pi$ follows.

The metal level is the one number that is not measured, so it is swept, exactly as it was on the donor side. The range is the same: a donor’s lone pair must lie below it and an acceptor’s π* above it, so it sits between iodine’s ionisation energy at 10.45 eV below the vacuum and the vacuum itself.

It under-predicts where it over-predicted #

The fitted π scales for the two acceptors with values are −0.10 for cyanide and −0.18 for carbon monoxide, a ratio of 1.8000. The denominator’s ratio is the reciprocal of the two gaps, which at the vacuum level is $2.26/1.50 = 1.5067$ and falls towards one as the metal level goes down.

So the acceptor side is under-predicted at every metal level: 0.837 of the fitted spread at the vacuum, 0.592 at the bound. The donor side is over-predicted at every one: 1.167 at the vacuum, 12.9 at the bound.

The one lever moves both the wrong way #

The argument has exactly one free number, and it does not help.

Lowering the metal orbital widens the donors’ gaps by different fractions — fluorine’s 17.42 eV gap grows proportionally less than iodine’s 10.45 eV one — so the ratio between them grows and the over-prediction gets worse. Lowering it also widens both acceptors’ gaps, and there the ratio $\,(2.26 + m)/(1.50 + m)$ falls towards one, so the under-prediction gets worse too.

The two sides need the metal level moved in opposite directions and it is already at the end of its range for both. At the vacuum the donor side is 17 per cent over and the acceptor side 16 per cent under — nearly symmetric errors, in opposite directions, at the one point where both are least bad.

That symmetry is worth noticing and not worth over-reading. What it means for the overlap is definite: an $e_\pi$ is an overlap squared over a gap, so the overlap must shrink down the halides to cancel part of the over-prediction and grow from cyanide to carbon monoxide to make up the under-prediction. Two statements about an overlap, obtained without computing one, and they do not have the same sign.

There is one more thing the sweep says that the donors alone could not. Both sides are least bad at the same point — the vacuum level — and that point is the one physically indefensible value in the range, since a metal d orbital in a complex is bound by several electronvolts at least. So the model’s best behaviour is at the end of its own range that it has no right to be at, and every physically sensible metal level makes both sides worse. Where a d–d band actually falls is the measurement that would fix the metal level, and it is available: it is nowhere near the vacuum.

The one number nobody fitted #

Dinitrogen has a measured π* resonance and no π scale in this collection’s series, so the denominator can be asked for a prediction with nothing to fit.

The gap ratio is $2.30/1.50 = 1.5333$, and carbon monoxide’s fitted −0.18 divided by it gives $e_\pi = -0.1174$ for dinitrogen. That places it between cyanide and carbon monoxide, nearer carbon monoxide, which is where the spectrochemical series puts it.

It is one prediction and it is a weak test — the ordering of three ligands, got right by a model already known to get the sizes wrong by a sixth. What it does show is that the denominator has the right sign and the right rough scale on the acceptor side, which is what makes its failure to reproduce the ratio a failure rather than an irrelevance.

Why the two sides differ, and what that does not explain #

There is an obvious candidate explanation for the two failures having opposite signs, and it is worth setting out and then setting aside.

The donors’ gaps are large — 10.45 to 17.42 eV before the metal level is subtracted — and their ratio is therefore close to the ratio of the two ionisation energies, which is 1.67. The acceptors’ gaps are small — 1.50 to 2.30 eV — so their ratio is 1.51 and collapses towards one the moment the metal level is included. The two sides sit at opposite ends of the same sensitivity: on the donor side the gap ratio is nearly the level ratio, and on the acceptor side it is nearly one.

That explains the shapes and it does not explain the failures, because the failures are against the fitted values and the fitted values do not care where the gaps sit. Fluoride to iodide is fitted at 1.4286, which is less spread than the donors’ levels give; cyanide to carbon monoxide is fitted at 1.8000, which is more than the acceptors’ levels give. So the residue the overlap has to carry is a compression on one side and an expansion on the other, and nothing about the sizes of the gaps says why.

The argument that the spectrochemical series is not electrostatics finds the same shape from another direction: the ordering of the series follows a quantity the electrostatic picture does not contain, and reproducing the ordering is much easier than reproducing the spacings. This is the spacings, on both sides at once.

Two smaller readings follow from the same figure and are worth taking, because they are the practical residue of an argument that is otherwise about a refutation.

The first is that a π parameter’s sign is a much more important piece of metadata than it is usually treated as. A table of $e_\pi$ values runs from negative to positive with no break, and the numbers on either side of zero are used identically: fitted on one complex, carried to another, added into a splitting. But the sensitivity to the metal level is 9.5 per cent per 2 eV on one side and 19.3 on the other, and the two run opposite ways — so a set of parameters transferred across metals is not merely uncertain, it is sheared, with the donors and the acceptors moving apart. A fit that then adjusts a single global scale to compensate makes the donors worse in exactly the proportion it makes the acceptors better.

The second is about what a good agreement on the ordering is worth. Both sides reproduce the order of the series correctly at every metal level in the sweep — fluoride below chloride below bromide below iodide, cyanide above carbon monoxide — and neither reproduces the spacings. Ordering is cheap here because it is decided by the numerator, which is a set of measured levels and is not in dispute; the spacings are the denominator’s, and the denominator is the thing being tested. A model checked against an ordering has been checked against the part of it that was quoted rather than computed.

What was computed, and how #

Every ligand level is quoted: the donors’ from tabulated first ionisation energies of the donating atom, the acceptors’ from published vertical attachment energies. The fitted π scales are the spectrochemical series’ own, as used in the argument that the series is not electrostatics, unchanged, so the two sets of numbers are the same numbers.

What is computed is the comparison. The metal level is swept over the whole range in which every gap is positive, and the positivity is checked at three points across it rather than assumed — a sweep that produced a negative gap would be reporting an acceptor as a donor and would go unnoticed in a ratio.

The refusal is the direction of each interaction. A donor’s level must be below the metal orbital and an acceptor’s π* above it, at every metal level in the sweep, and the two conditions between them are what bound the window.

One consequence for the practice is worth stating. An $e_\pi$ fitted to one metal is routinely carried to another, and the denominator is the reason it is expected to transfer badly — a different metal has a different orbital energy, so every gap changes. The sweep above says how badly, and the answer is that it depends entirely on which side of zero the ligand is. Moving the metal level by 2 eV changes the donors’ predicted ratio by 9.5 per cent and the acceptors’ by 19.3, in opposite directions. A set of π parameters fitted on one metal and used on another is therefore distorted in a way that depends on the sign of each parameter, which is not how anyone treats them.

Where the model stops #

One more limit belongs on the list because it bounds how far the sweep can be pushed. The metal orbital energy is swept over the window in which every gap keeps its sign, and that window is narrower than the range of real metals: a first-row metal and a third-row one differ by more than the sweep allows, because somewhere between them a ligand that is a donor becomes an acceptor. The sensitivities quoted are therefore local, measured where all eight ligands keep the character the fit assigned them, and the shear they describe gets worse rather than better outside it.

The angular overlap model puts one $e_\pi$ on a ligand for both perpendicular directions, which is right for a halide with two equivalent lone pairs and wrong for anything one-sided. Carbon monoxide has two equivalent π* orbitals so it is fair there too; a ligand with one π system would need two parameters and the comparison would not be the same comparison.

Nothing here computes an overlap. The whole argument is that $e_\pi$ is a product of two things and one of them can be estimated, so the other is whatever is left over — which is a statement about a ratio and not a calculation of either factor. The Slater estimate cannot supply the levels either, being wrong by a factor between 3.0 and 5.3, so there is no cheap route to the second factor from this side.

Whether the two channels are even separable is a question already answered, and the answer bears on this one: a back-bonding ligand acts through a σ channel as well, and the σ and π contributions to a splitting are not measured separately. So the fitted eπ values carry whatever the fit could not put elsewhere.

And the fitted π scales are fitted. They come from a series that was itself assembled from measured splittings on a range of metals, so a discrepancy of a sixth is not obviously outside what they are worth. What the result rests on is the direction being opposite on the two sides, which no uncertainty in either fitted value can reverse.

The generalisation #

A model with one free parameter and two constraints is not underdetermined; it is refuted, and the way to find out is to write both constraints down. With one constraint the most that can be reported is that the parameter is in the wrong place. With two the parameter has no place at all, because the two constraints move it in opposite directions.

That is a cheaper test than it sounds, and it is available whenever a model is being used on both sides of something — a sign change, a threshold, a change of state. The same move works for a correction transferred along a second axis, with the same shape of answer: the mechanism generalised and the calibration did not.

What survives here is the mechanism. A π parameter really is an overlap over a gap, the gaps really are measurable on both sides, and the sign of $e_\pi$ really does follow from which side of the metal level the ligand orbital sits on. What does not survive is the idea that the gap is the part doing the work.

Who found it, and when #

The angular overlap model is Schäffer and Jørgensen’s from 1965, and its parameters have always been understood as second-order perturbation quantities of the form $|H_{\text{ML}}|^2/\Delta E$ — the denominator argument is the model’s own, not something imposed on it.

Electron transmission spectroscopy is Schulz’s, from the nineteen-sixties, and the compilation of vertical attachment energies for small molecules is largely Burrow and Jordan’s. That those energies are the natural π* levels for a back-bonding argument is not a new observation; using them to test the denominator’s ratio against a fitted series does not appear to be a standard exercise.

Still open: whether the residue is an overlap #

The obvious open question is the overlap. The argument has now produced two requirements on it — shrink across the halides, grow across the acceptors — and neither has been checked against a computed overlap. Overlaps between Slater functions at stated separations are already computed here, and a metal d orbital against a ligand p or π* is the same integral with one more centre in it. Two numbers would settle whether the residue the denominator leaves is really an overlap or is something the model has no name for.

The nearer question is a third acceptor with a fitted scale. Everything above rests on two fitted numbers, −0.10 and −0.18, and a ratio of two numbers has no redundancy in it at all. Nitrosyl and isocyanide are both in the spectrochemical series, both have measured π* resonances, and adding either would turn a ratio into a trend — which is what the comparison of the two channels needed and could not have — at which point the under-prediction could be a slope rather than a single comparison, and the direction of the failure could be established rather than inferred from one pair.