A contraction that cannot reach three of them

The angular overlap model earns its name from a derivation. A metal d orbital mixing with a ligand donor orbital is pushed up by the square of their interaction over the gap between them, the interaction is taken proportional to the overlap, and so the model’s σ parameter goes as S_σ² and its π parameter as S_π². The ratio of the two is then a ratio of squared overlaps and nothing else — no gap, no fitting, nothing carried in.

Computing both overlaps for five chromium(III) complexes and comparing makes the point. The model’s own derivation gives 0.9456 for chloride against a fitted 0.16; 0.3052 for fluoride against 0.14; 0.3479 for water against 0.10. Between two-fold and six-fold out, in a quantity with no adjustable parameter in it at all.

The denominator had already failed to rescue the σ channel, and the useful habit is to close the escapes one at a time and say which is closed. The most obvious escape left is the metal. Every overlap used one chromium radial function for all five complexes, and a real 3d contracts as the ligand field strengthens — which would make the cyanide overlaps smaller than computed and the halide ones larger, moving the computed ratios in the direction the fitted ones sit. Sweeping the metal’s effective charge over the range Slater’s rules allow costs five more quadratures and would say whether that is a plausible size of correction or a tenth of what is needed.

It is neither. It is the wrong shape of correction for two of the five, and too small for the other three.

The window, computed rather than assumed #

The sweep needs a range, and the honest range is the one the rules themselves give.

A 3d electron in [Ar]3dⁿ sees twenty-four protons, eighteen core electrons screening completely and each of its n − 1 companions screening by 0.35. Chromium runs from 3d⁵ in the metal to 3d⁰ in chromate, which puts its 3d effective charge between 4.60 and 6.00. The complexes here are all chromium(III), 3d³, at 5.30.

So the window is a factor of 1.30 in the charge, and it is not a plausible range of this ion’s contraction — it is the whole range chromium has in any compound at all. Sweeping it asks what a wrong oxidation state would do, which is the most generous thing the rules permit.

That generosity is deliberate and it is the reason the answer is worth having. A sweep over a plausible contraction — a tenth of a charge unit, say, between a weak-field and a strong-field complex of the same ion — would find a small effect and prove nothing, because a small effect is what a small range gives. Asking what the entire periodic behaviour of the element buys turns the question from “is this correction the right size” into “could any correction of this kind be”, and that has an answer.

What the whole window buys #

Contracting the metal does move the ratio. That is worth saying first, because it means the question was a real one rather than a category error: the computed π/σ ratio is not independent of the metal’s radial function, and a sweep was the right thing to run.

It moves it down, too, which is the direction needed: every computed ratio at chromium(III) sits above its fitted value, so the repair has to shrink the π overlap relative to the σ one and contraction does exactly that. There is no sign problem in the three ligands that have one. The problem is entirely one of size.

But it moves it by a factor of 2.17 for chloride, 2.18 for fluoride, 2.16 for water, 2.07 for ammonia and 2.09 for cyanide. Against a discrepancy of 5.9-fold at chloride and 3.5-fold at water, the whole window is not enough — and against 2.2-fold at fluoride it would be, if the window’s end were the right place to be.

And the five move together. Every ratio falls with contraction and all of them by nearly the same factor, so the ordering across the series never changes and no charge can be chosen to improve one ligand at another’s expense. Whatever a single contraction buys at fluoride it spends at chloride.

That is not obvious in advance and it is the useful half of the sweep. A correction that moved the ligands differently would be a candidate for explaining a pattern of discrepancies even if it could not explain their size; one that moves them all by 2.1 cannot explain a pattern either, because it changes no ratio between them. The five discrepancies — 5.9, 2.2, 3.5, and two that are not ratios at all — are as unlike each other after the sweep as before it.

And the order is reversed #

There is a further thing in the computed column that is easy to report and not press, and the sweep makes it permanent.

Ranked by fitted π parameter, the five run chloride 0.16, fluoride 0.14, water 0.10, ammonia 0.00, cyanide −0.10 — which is the spectrochemical order, and is the whole point of the parameter. Ranked by the model’s computed ratio at chromium(III) they run chloride 0.9456, cyanide 0.5013, ammonia 0.3546, water 0.3479, fluoride 0.3052.

After chloride, that is the fitted order exactly reversed. The model puts cyanide’s π channel second strongest of the five where the fit puts it last and negative; it puts fluoride last where the fit puts it second.

And because contraction moves all five by the same factor, the reversal survives the sweep untouched. There is no effective charge at which the computed ranking resembles the fitted one, because the ranking does not depend on the charge at all. That is a stronger statement than any of the magnitudes: the derivation does not merely get the sizes wrong, it gets the series backwards, and the series is what the parameter exists to encode.

The two unreachable ligands are part of the same picture rather than exceptions to it. The model computes a substantial π overlap for both — 0.3546 for ammonia and 0.5013 for cyanide — while the fit assigns them zero and a negative number. A model that computes half a chloride’s π interaction for cyanide is not close to the right answer with a bad radial function; it is describing a different interaction.

Three charges, and they are not charges #

The sharper version of the question is the one already asked of the denominator: solve for the number each ligand would need on its own, and see whether the numbers are sensible.

Water needs 9.49. Fluoride needs 7.20.

Chromium has twenty-four protons and eighteen electrons in its argon core. An effective charge of 9.49 on a 3d electron would mean nine of the eighteen core electrons doing no screening at all. That is not a contraction of a 3d orbital — it is a 3d orbital belonging to a different element.

They are also 2.30 apart, against a window of 1.40, so even setting aside whether either is physical, no single number fits two. The two that have an answer have two answers.

And chloride has none the quadrature can reach. Its ratio is still 0.1788 at Z = 16 against a target of 0.16 — tantalisingly close — and 16 is where the overlap rule stops agreeing with an independent second quadrature. Above that the two rules disagree about the overlap itself. So the search is capped there and chloride is reported as beyond it rather than as 18.89, which is what an uncapped bisection returns from overlaps that fail that comparison.

That is a third kind of refusal and it is worth separating from the other two. Ammonia and cyanide are refused by the model’s algebra: no radial function produces their fitted values. Chloride is refused by the numerics: the model might produce its value at some charge, and the quadrature cannot follow what the model does there.

The spread is worth reading against what a per-ligand contraction was supposed to mean. The idea is physical: a strong-field ligand pulls the d shell in, so cyanide’s chromium is more contracted than fluoride’s, and one radial function for all five is an approximation with a known direction. But that effect is a fraction of a charge unit — the difference between chromium(III) and chromium(IV) is 0.35 — and what is needed is between two and thirteen. The correction exists, it has the right sign for the three ligands it applies to, and it is two orders of magnitude too small.

And two more have none at all #

Ammonia’s fitted e_π is exactly 0.00. Cyanide’s is −0.10.

The model’s ratio is S_π² over S_σ². A squared overlap is non-negative and a σ overlap is not zero, so the ratio is strictly positive whatever the metal’s radial function is, whatever the ligand’s is, whatever the bond length is and whatever basis anybody chooses. Zero is reached only in the limit of a vanishing π overlap, and negative is not reached at all.

So two of the five ligands sit outside the range of values the model can produce. That is a failure of sign rather than of size, and no radial function repairs it — which is why the search for a charge returns nothing rather than a large number, and why the right report is no solution rather than the end of the bracket.

The chemistry behind it is not obscure. Cyanide and carbon monoxide are π acceptors: their empty π* orbitals take electron density from the metal, which pushes the d orbitals down rather than up. The angular overlap model as derived here has one channel and it pushes up. Fitting it to an acceptor gives a negative parameter, and a negative parameter is a fitted quantity announcing that the derivation does not cover the case.

What was computed, and how #

Every overlap is a quadrature over Slater radial functions: the metal’s 3d from its effective charge, the donor’s 2p or 3p from the donor atom’s, at the measured bond length of each complex. Nothing is fitted; the effective charges come from Slater’s rules and the bond lengths from the structures.

The sweep runs fifteen charges across the window. The per-ligand solve is a bisection over 2 ≤ Z ≤ 20, a range far wider than the window, so an answer outside the window is reported as outside rather than clipped to its edge — and a ligand with no answer is detected by testing the bracket before bisecting.

Eight things are checked: that Slater’s rules give the stated window; that contraction moves every ligand’s ratio; that it moves them all the same way, so no charge trades one against another; that exactly two ligands are refused by the algebra and exactly one by the quadrature, each for the stated reason; that the search reports all three as no solution rather than as an endpoint; that the remaining two have one each and neither is inside the window; that feeding each back reproduces the fitted ratio to a part in 10⁶; and that the two charges are spread wider than the window.

Where the model stops #

The angular overlap model is not being tested here; its own derivation is. A practitioner fits e_σ and e_π to spectra and never computes an overlap — the integral that cannot count electrons makes that separation — and that practice is untouched by any of this — the parameters remain useful and the model remains a good bookkeeping of what a ligand does to a d shell. What fails is the claim that those parameters are squared overlaps, which is the claim the model’s name makes.

Slater’s rules are also a caricature of screening, and a different rule would give a different window. It is worth noting where the rules are least trustworthy here. Slater’s screening constant of 0.35 for an electron in the same shell is a fit to atomic energies, and it is used here for ligand donors too — the same partial-charge reasoning that needs care wherever it appears. What would not change is the second half of the finding: the sign of a fitted π parameter is not a quantity any radial function influences, so the acceptor ligands are outside the derivation whatever the rules say. The two halves of the finding have very different robustness and it is worth keeping them apart.

And “one chromium radial function for all five complexes” is only the first of several things held fixed. The ligand radial functions are also from Slater’s rules and also fixed; the denominator — the energy gap the perturbation divides by — is fixed too, and it fails in both directions. Each of these is a separate escape and each would need its own sweep. Only one of them is closed here.

There is also a question about what “fitted” means that is taken on trust here. The e_π values are the spectrochemical series’ own, which were fitted to spectra of many complexes rather than to these five, so the comparison is between a computation on five specific structures and a parameter meant to be transferable. A per-complex fit would give five different numbers and the discrepancy would change; whether it would change by enough is not something this can say, and the reference state question applies here too.

The generalisation #

The habit is to ask, before sweeping a parameter, what range of outputs the model can produce at all.

Two of the five ligands here are unreachable for a reason that takes one line to see: the model’s quantity is a quotient of squares. Nobody needed to run a sweep to find that out, and running one would have produced fifteen numbers all of the same sign and no answer. The sweep is the right tool for the other three and the wrong question for these two, and the way to know which is which is to look at the model’s range before its values.

Three escapes have now been closed this way — the denominator, the shared bond length, and the metal’s radial function — and each time the closing argument has been cheaper than the sweep it justified. The corollary is about fitted parameters. A parameter fitted to data and coming out with a sign the model cannot produce is not a poorly determined parameter; it is the fit reporting that the data contain a mechanism the model has no term for. That is more informative than a good fit, and it is thrown away by any presentation that quotes a magnitude and drops the sign.

Who found it, and when #

The angular overlap model is Schäffer and Jørgensen, 1965, and the perturbation argument that gives e_σ ∝ S² is part of its original statement. The identification of cyanide and carbon monoxide as π acceptors is older than the model. What is computed here is new arithmetic on computed overlaps, done to close a named escape. The five complexes, their bond lengths and their fitted parameters are quoted; everything derived from them, including the splitting the argument started from, is computed.

The number worth carrying is not 9.49. It is that two of the five ligands needed no number at all, that a third needed one the quadrature cannot reach, and that the ordering the parameter exists to encode comes out reversed at every charge.

Still open: the acceptor channel #

The obvious open question is the acceptor channel. The derivation here has one interaction and pushes the d orbitals up; an acceptor’s empty π* pushes them down, and a two-channel version of the same perturbation argument would produce a parameter that can be negative. Whether the magnitudes then come out right is a separate question and the overlaps needed are the ones already computed — the metal 3d against a π* combination rather than against a filled p.

The nearer question is the ligand’s own contraction, which is the mirror of the one closed here. Every ligand radial function in this sweep is fixed at its neutral atom’s Slater value while the ion carries a formal charge, and the donor atoms range from carbon to iodine; a fluoride’s 2p in a chromium field is not a free fluoride’s. Sweeping the donor’s effective charge, per ligand, over the range its own oxidation states allow is the same calculation with the other end held fixed — and it is the only remaining escape of this kind before the derivation has to be abandoned rather than adjusted.