The gap that would have to be smaller

The integral that cannot count electrons fitted a π scale to a chromium series and refused to fit a second one, on the ground that five points cannot support three parameters. It ended by naming the quantity the whole model folds away:

An $e_\pi$ is an overlap squared over an energy gap, and the gap between a metal d orbital and a halide p orbital is not the gap between it and a cyanide π*. Computing those gaps, even crudely from the same Slater orbitals, would turn the fitted constants into predictions.

Half of that is available and half is not, and the half that is not is instructive.

The half that is measured #

The ligand side of the gap is the energy of the orbital doing the donating, and for a halide that is a lone pair on the atom itself. Its energy is a measurement: minus the first ionisation energy, which is measured for every halide.

The metal side is not measured and is not computable here, so it is swept rather than assumed. That turns out to be enough, because the quantity wanted is a ratio of two $e_\pi$ values, and a ratio is bounded whatever the metal level is.

What the gap alone predicts #

With the metal orbital at energy $-x$, the gap for ligand $i$ is $\mathrm{IE}_i - x$, and $e_\pi \propto 1/(\mathrm{IE}_i - x)$. The ratio of iodide’s to fluoride’s is then

$$\frac{17.42 - x}{10.45 - x},$$

which is 1.667 at $x = 0$ and grows without limit as $x$ approaches 10.45, past which iodide would stop being a donor at all.

The fitted ratio is 1.429 — smaller than anything the expression can produce.

So no metal level reproduces the fitted trend from the denominator alone, and the failure is not marginal: the weakest the denominator effect can be is already sixteen per cent too strong, and every physically admissible metal level makes it stronger.

The same holds ligand by ligand rather than only at the extremes:

It is worth being clear about what kind of failure this is, because the model over-predicts can mean two very different things. It is not that the denominator is a poor approximation to something — it is exact arithmetic on four measured numbers, and the only uncertainty in it is where the metal level sits, which the sweep removes. What over-predicts is a one-term account: the claim that the trend in $e_\pi$ down the halide group is the trend in the gap. That claim is refuted by a bound, and a bound does not care how good the approximations elsewhere are.

The spectrochemical series is not electrostatics makes the same kind of argument about the σ half of the series: an account that predicts an ordering can be refuted by the ordering without any of its parameters being fitted.

What that requires of the overlap #

An $e_\pi$ is an overlap squared over a gap, so what the fit requires of the overlap is the fitted ratio divided by the gap’s:

The overlap has to be smaller for every heavier halide, by about fifteen per cent, and roughly the same amount for all three — 0.851, 0.872, 0.857 against fluoride’s 1. That flatness is itself informative: if the residual had run steadily down the group it would look like a second trend the model is missing, and a residual that is a constant offset for three ligands looks more like a single systematic difference between fluoride and everything else. Fluoride is the ligand whose lone pair is by far the most tightly bound, and it is the only one of the four whose valence shell has no inner p shell beneath it. That is the opposite of the usual expectation — a larger, more diffuse orbital is normally taken to overlap more — and it is a statement about an overlap arrived at without computing one.

There is a reason it might be right, and it is the reason closer is not more overlap gives: an overlap is a competition between how far an orbital reaches and how far away the other atom is, and a heavier halide brings both a larger orbital and a longer bond. The bond length grows faster than the orbital here, and the fifteen per cent is what that costs.

What the arithmetic cannot do is check it, because computing a metal–ligand π overlap needs a metal 3d orbital and a bond length, and neither is used here.

What a fifteen per cent residual is worth #

Before the crude route, one caution about the residual itself, because a number obtained by dividing two ratios inherits both their errors.

The fitted π scales are quoted to two decimal places — 0.14, 0.16, 0.18, 0.20 — which is a resolution of about six per cent on the smallest of them. The residuals are 0.851, 0.872 and 0.857, which differ from each other by about two per cent. So the flatness of the residual is inside the resolution of the numbers it is made from, and the right reading is a constant offset of about fifteen per cent rather than three offsets that happen to agree.

What survives the resolution comfortably is the offset itself. Fifteen per cent against a quoting precision of six is a real discrepancy, and the bound that produced it — 1.667 against 1.429 — is a sixteen per cent gap between a fitted number and the smallest value a whole family of predictions can take.

The crude route, priced #

The natural suggestion is to compute the gaps “even crudely from the same Slater orbitals”, meaning the hydrogenic estimate $-Z_{\text{eff}}^2/2n^2$. It is worth doing once, because the answer is definite.

Three to five times too deep — and, worse, not by a consistent factor. If it were wrong by five everywhere it would still give the right ratios, and the ratios are what the comparison needs. It is wrong by a factor that itself varies by 1.75 across the four, so it cannot supply the ratios either.

That is a definite answer to even crudely: the crude route is not available, and the reason is that Slater’s rules were built to reproduce orbital sizes rather than energies. Using them for an energy is using a quantity outside what it was fitted for, which is what a size a confound cannot supply warns about.

What the denominator does account for #

The sign, exactly.

A donor’s occupied lone pair lies below the metal orbital, the gap is positive, and the interaction pushes the metal level up — a positive $e_\pi$. An acceptor’s empty π* lies above it, the gap is negative, and the metal level goes down. That is the whole content of the sign, and it is not a small thing: it is why back-bonding is two interactions rather than one, and why the spectrochemical series has donors at one end and acceptors at the other.

So the answer to the question is precise. The sign of the π term is not the only thing the occupation is carrying — the denominator carries more than the whole trend, and the overlap gives some back.

That is a better answer than either of the two one would anticipate — either the denominator explains the trend, which would have turned two fitted constants into one, or the denominator explains only the sign, which would have left the fitted constants where they were. What comes out is a third thing: the denominator over-explains, so the two terms of the model are pulling against each other, and neither can be read off the fitted parameter alone.

Why the metal level could be swept away #

The design of this calculation is worth extracting, because it is the reason a comparison with one unmeasured quantity in it can still be decisive.

The unmeasured quantity — where the metal orbital sits — enters every gap the same way, as a common subtraction. It does not cancel from a ratio, because the gaps are differences rather than products, but it moves the ratio monotonically: the closer the metal level comes to the ligands, the more the small gaps shrink relative to the large ones and the wider the predicted ratio grows.

A monotone dependence with a bounded range is as good as no dependence at all, provided the answer is outside the range. Here the range is [1.667, ∞) and the fitted value is 1.429, which is below the whole of it. The metal level is unknown and cannot help, and establishing that took a bound rather than a value.

That is a more useful pattern than it looks. Several arguments are blocked on one quantity nobody has measured — a partial charge, a bend force constant, a metal level — and in each the question worth asking first is not what is it but does the conclusion depend on it monotonically. If it does, one bound settles the matter.

What is quoted, and what is computed #

Four measurements are quoted — the first ionisation energies of fluorine, chlorine, bromine and iodine — and five fitted parameters, the π scales fitted to a chromium series. Nothing else.

The spectrochemical series is not electrostatics supplies the σ half of the same table, and neither half is recomputed here.

Everything else is arithmetic on those. The gaps, the ratios, the sweep over the metal level, the bound on it, and the residual left for the overlap. There is no diagonalisation here and no orbital.

The metal level is the one quantity that is neither measured nor computed, and the whole design of the calculation is to avoid needing it: every statement is about a ratio of two π scales, every ratio is monotone in the metal level, and the bound comes from the definition of a donor rather than from a guess.

What this cannot say #

The fitted π scales are fitted, and to one metal. They come from a chromium(III) series, and a fitted $e_\pi$ has a reliable sign and a less reliable size. Everything above is a comparison of ratios of quantities whose absolute values are uncertain, which is the right way to use them and does not make them exact.

An ionisation energy is not an orbital energy. Using minus the first ionisation energy as the level is Koopmans’ approximation, and Koopmans’ theorem is exact for nothing shows how badly that can fail. For a halide lone pair it is one of the better cases, and it is still an approximation with an error of an electronvolt or so — which is a per cent or two of the gaps here and does not move the conclusion, since the discrepancy is sixteen per cent.

One ratio is doing most of the work. The bound is stated on iodide over fluoride, which is the widest pair, and the intermediate ligands are reported rather than used. They agree — chloride and bromide are over-predicted by the same sixteen per cent — but a reader should know that the refutation rests on the two ends and the middle is a consistency check rather than independent evidence.

And the halide is not the ligand. A coordinated halide is not a free halide ion, and its lone pair is not at the free atom’s ionisation energy. What the comparison uses is the ordering and spacing of the four, which survives a common shift and does not survive a differential one — and nothing here bounds a differential one.

What was checked #

The fitted π scales span a factor of 1.43 from fluoride to iodide, which is the number the whole comparison is against and is read from the collection’s own table rather than retyped.

The denominator over-predicts it at every admissible metal level, checked over the whole sweep rather than at its ends.

And the over-prediction grows as the metal level falls, which is the monotonicity the bound depends on — a quantity that turned round would make the sweep’s minimum an interior point and the argument would need the minimum rather than the endpoint.

The Slater estimate is several times the measurement and not by a consistent factor, checked as two separate claims, because the second is the one that kills the crude route and the first alone would not.

And the refusal is the bound. A metal level below the least tightly bound donor gives a negative gap, which is an acceptor — so the sweep stops there, and a routine that ran past it would be reporting donors as acceptors and finding whatever ratio it liked.

Why a shrinking overlap is less surprising than it sounds #

The overlap has to shrink down the group is called the opposite of the usual expectation, and the usual expectation is only half of the calculation. Putting the other half in makes the required shrinkage look modest rather than perverse — and makes it the small residue of two large competing terms.

The expectation that overlap grows down a group is about the ligand’s orbital: a heavier halide’s np function is more diffuse, so at a fixed separation it reaches further into the metal’s d orbital and the integral is larger. That is right and it is not the whole comparison, because the separation is not fixed.

The bond lengths grow, and they grow a great deal. A chromium–fluoride distance is near 1.90 ångström and a chromium–iodide near 2.72 — more than eight tenths of an ångström further, across the group.

An overlap between compact atomic functions falls off exponentially with separation, with a decay length of a few tenths of an ångström. Over eight tenths of an ångström that is a fall of a factor of three or four, taken alone.

The measurement requires a fall of about fifteen per cent per step — roughly forty per cent over the whole group, which is far less than the distance alone would produce.

So the two effects are both large and they oppose each other, and the residual the fit reports is what is left after a cancellation. The diffuseness is doing exactly what the usual expectation says, at close to the size it would be expected to do it, and the distance is doing more.

That reframes the finding. It is not that the naive expectation is wrong; it is that the naive expectation names one term of two and the term it names is the smaller. A comparison down a group varies the ligand’s radial extent and its distance together, and the two enter the same integral with opposite signs — which is the same entanglement of variables that could not be separated across a series of ligands, met again in a different pair.

Still open: the overlaps themselves, and the acceptors #

The obvious open question is the overlap the residual has now specified. Fifteen per cent smaller for every heavier halide is a definite prediction about a quantity that is routinely computed: a metal 3d against a halide np at the crystallographic bond length. What it needs is two things not used here — a metal orbital at an effective charge, and four bond lengths — and both are a table lookup rather than a calculation. If the computed overlaps come out fifteen per cent down, the model is closed; if they come out up, the fitted π scales are carrying something neither the gap nor the overlap explains.

The nearer question is the acceptors, deliberately not touched here. A π* is a molecular orbital of a diatomic, not an atomic level, so its energy cannot be had from an ionisation energy — but a cyanide’s and a carbonyl’s π* energies are both measurable, by electron transmission spectroscopy, and both are quoted in the literature. Putting those two numbers in 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.