The overlap the model is not proportional to

A denominator that fails both ways tested the angular overlap model’s own justification for its π parameter. That justification is perturbation theory: a metal d orbital mixing with a ligand orbital is pushed by the square of their interaction over the gap between them, so $e_\pi \propto S_\pi^2/\Delta E$. It swept the metal level over every admissible value and found the denominator alone over-predicting the halide trend at every one of them.

From that it drew a conclusion about a quantity it could not compute. If the denominator over-predicts, the overlap must run the other way — shrink down the group — to cancel part of it. It closed by naming that as the thing to check, and overlaps between Slater functions at stated separations can be computed directly.

It does, and the check is one integral away. The σ half was already here: a metal 3d(z²) against a donor’s p(z) at the measured bond length, with the radial functions from Slater’s rules and nothing fitted anywhere. The π half is the same integral with both orbitals turned — 3d(xz) against p(x) — which is what a π interaction is.

The requirement is refused #

Fluoride’s π overlap squared is $8.317\times10^{-3}$. Chloride’s is $2.919\times10^{-2}$ — 3.5 times larger.

So the overlap does not shrink down the group. It grows, and it grows in the same direction the denominator already over-predicts in. The two factors compound rather than cancel: the denominator gives at least 1.67 across that pair and the overlap gives 3.5, for a product near six, against a fitted ratio of $0.16/0.14 = 1.14$.

The model’s derivation, evaluated with the model’s own kind of inputs, over-predicts the one trend it is most often quoted for by about a factor of five.

That is a cleaner failure than the denominator argument could produce, and the reason is worth naming. Its argument had a free quantity in it — the metal level — and a free quantity lets a conclusion be shaped: whatever the denominator did, the overlap could be asked to compensate. With the overlap computed there is nothing left free, and the two halves can be wrong in the same direction, which is what they do.

There is a way of putting the failure that makes it useful rather than only negative. The model has two factors and the fit has one number, so a fit can always be achieved: whatever the overlap does, the denominator absorbs the rest. What the computation removes is that freedom on one side, and what is left is a statement about the other. Fluoride needs a denominator ratio of 2.2 relative to a reference, chloride needs 5.9 — so the model is not merely mis-scaled, it needs the two ligands’ denominators to differ by a factor of 2.7 in a particular direction.

Whether that is plausible is a question not settled here, though it can be, because both overlaps are now numbers rather than arguments. Two models, one ratio first noticed two accounts of the same quantity and no way to separate them; this is the separation, arriving one factor at a time.

The two columns, side by side #

The fitted ep in this collection’s series is already a ratio to $e_\sigma$, so it compares with the computed ratio directly and no scale has to be chosen.

They disagree by 2.2 for fluoride, 5.9 for chloride, 3.5 for water and 5.0 for cyanide. And they disagree in shape as well as size: the computed ratios go 0.305, 0.946, 0.348, 0.355, 0.501 across fluoride, chloride, water, ammonia and cyanide, while the fitted ones go 0.14, 0.16, 0.10, 0.00, −0.10. The computed column has chloride as a huge outlier; the fitted column has it as an unremarkable neighbour of fluoride.

The chloride outlier has a cause and it is not π bonding. A 3p radial function has a node, and a 3d–3p overlap at 2.34 Å catches the outer lobe with much less cancellation in the π geometry than in the σ one. The computed ratio is picking up a property of the radial node; the fitted parameter is not.

The column that cannot be an overlap #

The sharpest of it is not a discrepancy in size at all.

The fitted $e_\pi/e_\sigma$ runs +0.14 for fluoride, +0.16 for chloride, +0.10 for water, exactly 0 for ammonia, and −0.10 for cyanide. The sign is the donor–acceptor distinction, and it is the whole content of that end of the spectrochemical series.

A ratio of two squared overlaps is positive by construction. So no computation of the kind the model’s derivation calls for can produce that column — not a better basis, not a better bond length, not a better metal charge. The sign lives entirely in the energy denominator: a donor’s lone pair is below the metal level and an acceptor’s π* is above it, so $\Delta E$ changes sign while $S^2$ cannot.

That is the denominator argument arriving as an arithmetic impossibility rather than as a preference. And it means the parameter being fitted is a product of two things, only one of which is an overlap — which is exactly what the model says it is, and exactly what is forgotten when $e_\pi$ is described as measuring π overlap.

Ammonia is the single clearest case. Its fitted $e_\pi$ is exactly zero — not small, zero, by decision — and its computed $S_\pi^2/S_\sigma^2$ is 0.3546, larger than fluoride’s. The zero records something true that the computation cannot see: nitrogen in ammonia has no p orbital to spare, because its other two p functions are bonded to hydrogens. Putting a bare p orbital on the donor atom, which is what the computation does, gives an overlap for a function that is not available.

So the gap between the two columns is not only calibration. Part of it is that the fitted parameter knows about the rest of the ligand and the computed overlap knows only about the donor atom.

What the ratio is actually a function of #

There is a confound running through every comparison above, and it is worth separating because it accounts for a good part of the chloride outlier.

At its own 2.34 Å, chloride’s $S_\pi^2/S_\sigma^2$ is 0.9456. At a common 2.05 Å it is 3.9152 — four times as large. Fluoride’s moves the other way, from 0.3052 to 0.2380.

The chloride is both more diffuse than the fluoride and further from the metal, and at the measured geometry those two nearly cancel. So a series compared at its own bond lengths is comparing two variables at once, and the model’s parameters are fitted to complexes in which both move together.

The separation dependence is steep and it is different for the two shells. Chloride’s π overlap falls far more slowly than fluoride’s, so their ratio grows from 5.5 at 1.8 Å to 40.8 at 2.6 Å. Their measured bond lengths differ by 0.41 Å, which is most of that range.

So the number that answers “how much more π overlap does a chloride have than a fluoride” is anywhere between 3.5 and 40 depending on whether the comparison is made at the real geometries or at a common distance — and both are defensible questions. The model asks the first; anyone thinking about “which ligand is the better π donor” usually means the second.

The geometry factor — the own-length ratio over the common-length one — runs from 0.24 for chloride to 1.28 for fluoride across five ligands. It is not a small correction and it is not the same for any two of them.

The last two columns are the summary. The computed ratios span a factor of three; the fitted ones span a factor of 1.6 among the donors and then cross zero. The fitted parameters are smoother than any computed column, which is what a parameter fitted to a smooth experimental series looks like, and not what an overlap looks like.

What survives, and it is most of the model #

It is worth being explicit about what is not being claimed, because a five-fold discrepancy in a named quantity reads worse than it is.

The angular overlap model works. Its five parameters reproduce the d–d spectra of the complexes they were fitted to, they transfer between complexes of the same ligand, and every calculation that used them got answers that agreed with measurements. Nothing above touches any of that.

What is refuted is a sentence about where the parameters come from — that $e_\pi$ is $S_\pi^2/\Delta E$ with the overlap computed from the orbitals involved. That sentence is what licenses two common habits. It licenses reading a large $e_\pi$ as a large π overlap, which the chloride row shows is not safe; and it licenses predicting an $e_\pi$ for a ligand nobody has fitted, which is exactly what the denominator argument tried for dinitrogen and what cannot be done from the overlap alone.

So the parameter is empirical, and the value of computing the overlap is to have found out by how much — a factor of five, in a known direction, with the sign supplied entirely by something else. A parameter known to be empirical is used differently from one believed to be derived, and that difference is the whole return on the integral.

What was computed, and how #

The radial functions are Slater’s: a 3d on chromium(III) at an effective charge of 5.30 from the [Ar]3d³ configuration, and each donor’s p at the effective charge its own configuration gives. Nothing there is fitted and nothing is taken from the ligand-field literature.

The separations are measured bond lengths, quoted, for five chromium(III) complexes whose octahedral splittings are also measured. The overlap is a three-dimensional quadrature, with the two orbitals displaced along the axis between them, so a σ overlap is 3d(z²) against p(z) and a π overlap is 3d(xz) against p(x) — the same routine, the same convergence, different orientations.

The fitted $e_\pi$ values are the fitted spectrochemical series and are ratios to $e_\sigma$, so no scale is introduced when they are compared with a ratio of squares.

One convention has to be stated because it looks like a result. The sign of an individual overlap flips between the 2p donors and the 3p one, because a 3p radial function has a node and the phase of a Slater function is a convention. Only squares are used anywhere above, and the sign is not read.

Five things are checked. Chloride’s π overlap must exceed three times fluoride’s, which is the refusal of the denominator argument’s requirement. The fitted values must be within a third of each other over the same pair. Every computed ratio must be positive and at least one fitted ratio must be negative, which together are the impossibility. Ammonia’s fitted value must be exactly zero while its computed ratio exceeds 0.2. And the common-separation ratio for chloride must exceed three times its own-length one, which is the confound.

The check that would catch a broken quadrature is the σ channel, which the splitting against something structural already tested against the measured splittings — so a fault in the integral would show there first, on numbers that have already passed.

Where the model stops #

A Slater function is a one-parameter guess at a radial shape and the overlaps here inherit that. What survives it is the comparison rather than any single number: both donors are treated identically, so a systematic error in the radial form is common to them and the ratio is more reliable than either overlap.

The donor is a bare atom. Every ligand here except the halides is a molecule, and using a p orbital on the coordinating atom ignores the rest of it — which for ammonia is the whole of why the fitted value is zero, and for cyanide is why the acceptor orbital is a π* of the C–N bond rather than a p function on carbon at all — the distinction back-bonding is two interactions is built on. So the cyanide row is the least meaningful of the five, and it is included because its sign is the point rather than its size.

The metal is a bare ion at a Slater effective charge. A chromium(III) in a complex is not that; its 3d functions contract or expand with the ligand field, and that feedback is what a self-consistent calculation supplies and this does not.

And none of the above touches whether $e_\pi$ is a good parameter. It plainly is: the series it produces reproduces spectra, and it has been used throughout, from where a d–d band falls onward. What is refuted is the claim that it is an overlap squared over a gap, which is the story told about where it comes from.

The generalisation #

A parameter’s derivation and a parameter’s value are two different things, and the derivation can be checked. $e_\pi$ is derived as $S^2/\Delta E$ and fitted to spectra, and almost nobody evaluates the derivation because the fit works. When it is evaluated, it over-predicts by a factor of five and cannot reproduce the sign. That does not make the parameter wrong; it makes the story about it wrong, and the story is what gets used when the parameter is extrapolated to a ligand nobody has fitted.

The same shape appears in four electronegativity scales that disagree about a direction and in a bond order that is not a property of the pair it names: the quantity is useful, its name is a claim, and the claim is checkable separately.

And a sign is a stronger constraint than a magnitude. Every discrepancy above could have been argued away with a better basis or a better effective charge — a factor of five is within what a crude radial function might cost. The sign cannot: no choice of anything makes a ratio of squares negative. When a model’s parameters take both signs and its derivation produces only one, the derivation is missing a factor rather than being inaccurate, and looking for which factor is a more useful question than improving the arithmetic.

Who found it, and when #

The angular overlap model is Schäffer and Jørgensen’s, from the mid-nineteen-sixties, and the perturbative reading of $e_\lambda$ as $S_\lambda^2/\Delta E$ is theirs too. That $e_\pi$ is negative for acceptors is the standard way the model handles back-bonding and is not a discovery here — it is stated in every account. What is not usually said in the same breath is that the negative value cannot come from the $S^2$ factor, so the model is using a signed denominator whose value is never quoted.

Computing the overlaps with the same Slater functions the effective-charge rules give is a deliberately crude test, and it is the test the model’s own derivation invites. Its outcome — right kind of quantity, wrong by five, wrong sign at one end — is the sort of result a fitted parameter usually gets when it is asked to justify its name.

Still open: the metal level solved for, and the metal’s contraction #

The obvious open question is the metal level, which is now the only free quantity left. With both overlaps computed, $e_\pi/e_\sigma = (S_\pi^2/S_\sigma^2)(\Delta E_\sigma/\Delta E_\pi)$ can be solved for the denominator ratio each ligand would need — one number per ligand, with nothing fitted. If those numbers are sensible and smooth, the model is right in structure and its overlaps are simply bad; if they are wild, the structure is wrong. Either answer is worth more than the five-fold discrepancy on its own, and every input is already computed.

The nearer question is the metal’s own contraction. Every overlap above uses 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.