The integral that cannot count electrons

Compute one integral per complex — the overlap between chromium’s σ-facing d orbital and each ligand’s donor orbital, at the measured bond length with Slater-screened exponents — and ask how much of the spectrochemical series it accounts for. Quite a lot, and not all: a residual that ordered itself in a way one overlap could not explain.

It closed by naming the term it had left out. “The obvious continuation is the π channel, which is the thing the residual is made of.” An angular overlap treatment has an $e_\pi$ as well as an $e_\sigma$, and the same construction supplies it: the metal’s t₂g orbitals face the bond sideways, the ligand has a p function perpendicular to it, and the overlap between them is a second integral over the same exponents at the same distance.

Computed, it is worth a fifth of the error. And a single word per ligand — donor or acceptor, which is not an integral at all — is worth more than twice as much.

That is a result about what kind of thing a ligand field parameter is, and it is a sort worth collecting. Two models agree on one ratio because the geometry fixes it; the splitting is a symmetry statement because the point group fixes the pattern. What neither symmetry nor geometry fixes is the size, and the size turns out to depend on a count of electrons that no integral over shapes can reach.

The second integral #

The metal’s five d orbitals divide by how they face a ligand: one points at it, and two present a lobe to each side. The first overlaps the ligand’s donor orbital head-on and gives $e_\sigma$; the second pair overlaps a ligand p function perpendicular to the bond and gives $e_\pi$. The octahedral splitting is then $\Delta = 3e_\sigma - 4e_\pi$, with the minus sign because a filled ligand π orbital pushes the t₂g set up and closes the gap.

Both are computed here from the same orbitals: chromium’s 3d at its Slater effective charge, the ligand’s p at its own, and the measured bond length. Nothing about the two calculations differs except which d orbital faces the bond.

That last is worth pausing on. The π overlap runs from 0.55 of the σ one for fluoride to 0.97 for chloride, so anybody’s intuition that a sideways overlap is a minor thing is wrong for a diffuse donor. Whether the interaction is correspondingly large is a different question, because $e_\pi$ is proportional to the square of an overlap divided by an energy denominator that the model does not compute.

Three fits #

Five measured splittings and three models, all least squares.

The second row is the single-overlap question answered arithmetically: adding the integral takes 19.7 per cent off the error.

The third row is the answer that matters. It uses the same two integrals and the same two parameters, and differs only in that cyanide’s π term is subtracted with the opposite sign, because cyanide is a π acceptor. That takes 45.0 per cent off — more than twice what the integral bought.

Why the integral cannot do it alone #

An overlap integral is a number about two functions. It says how much they coincide in space.

What decides whether a π interaction raises the t₂g set or lowers it is not that. It is whether the ligand’s π orbital is occupied. A filled halide p orbital and a filled metal t₂g set repel each other and the metal’s level goes up; an empty cyanide π* orbital takes charge from the metal and the metal’s level goes down. Same geometry, same overlap, opposite effect.

So the model has a term whose magnitude is computable and whose sign is not, and the sign is worth more than the magnitude. That is a description of what a spectrochemical series actually encodes, and it is not the description the electrostatic picture or the overlap picture gives.

The general shape of it is worth naming because it keeps recurring. A one-electron model can compute how strongly two orbitals interact and not how many electrons are in them. An orbital carries no angular momentum until something says how it is occupied; a count is invariant where a charge is not; and here an overlap is invariant where an occupation decides the sign. Occupation is the piece of information a shape calculation does not have.

It also explains why the series is not electrostatics in the specific way that essay finds. A charge model has no room for an occupation, and neither does an overlap model; the ordering that puts cyanide above ammonia above water above the halides is an ordering by π character first and by overlap second.

The halides, which is where to look #

The halides are the pair a second parameter would have to earn its keep on, and it does — with a complication.

Chloride’s measured splitting is 13,600 cm⁻¹ and fluoride’s is 15,200: the smaller, harder ligand splits more. The σ-only model predicts 16,441 and 14,516 — the wrong order, which is the residual the single-overlap model leaves.

Adding the π overlap fixes it: 14,428 and 15,185, in the right order and both within a thousand. Chloride’s near-unit π overlap is doing exactly what it should, closing the gap that its large σ overlap opened.

And the model with the better total gets it wrong again. Once the sign is supplied and cyanide is fitted properly, the halides come back as 14,480 and 14,188 — the wrong order once more.

So no one of the three models gets both ends of the series. The free fit describes the halides and mis-describes cyanide; the signed fit describes cyanide and mis-describes the halides. Two parameters are not enough to carry both, and the natural third — a separate $e_\pi$ scale for donors and for acceptors — would be a parameter fitted to two data points, which is not a model.

What a spectroscopist should take from it #

A single fitted e_π is doing two jobs. For a series containing both donors and acceptors it is standing in for a quantity whose sign changes, and a fit that lets it float will land somewhere between — which is what the free two-parameter fit above does, and why it describes neither end well.

The halides are the diagnostic pair. They are both π donors, they differ mainly in how diffuse the donor orbital is, and their measured ordering is the one an overlap model gets backwards without a π term. Any account of the series should be checked against them first.

And the ordering of the series is not the ordering of the overlaps. Cyanide has the largest σ overlap of the five and the largest splitting, which looks like a confirmation and is not: it also has the largest π overlap, and if that acted as a donor it would fall below ammonia. The agreement at the top of the series is a cancellation of two errors.

What is quoted, and what is computed #

Five measured splittings and five measured bond lengths are quoted, and so is one word per ligand — its π character. That word is the essay’s subject: it is the piece of chemistry an integral cannot supply, and its worth is measured rather than assumed.

It is worth being precise about what kind of thing that word is, because a reader might reasonably ask why it is not computed too. Whether a ligand’s π orbital is filled is a fact about the free ligand — a halide’s p shell is full and cyanide’s π* is empty — so it is not a property of the complex and needs no calculation on the complex to establish. What it would need is a calculation on the ligand, which is a separate job and is not joined up to this one. So the word is quoted for the same reason a bond length is: it is a known fact about a separate object.

Everything else is computed: the Slater effective charges from the rules, both overlap integrals from the orbitals, and all three fits in closed form.

The two channels have to be independent for the fit to mean anything, and they are: the ratio of the π overlap to the σ one is not constant across the series, running from 0.55 to 0.97. A set in which it were constant would give a singular fit, and the fit refuses one rather than splitting an arbitrary way between two proportional columns.

An acceptor pulls the lower set down and widens the splitting where a donor pushes it up and narrows it — the same diagram with one sign changed. Getting that sign from an integral rather than from a fit is the whole of what this essay is asking the integrals to do.

What this cannot say #

Slater orbitals and a fixed metal. The exponents come from Slater’s rules and the metal is chromium throughout, so this compares ligands and not metals. The splitting against something structural names the comparison down a group as the other open question and it is still not done — and it is the harder of the two, because the bond length and the exponent move in opposite directions there where here they moved together.

No energy denominator. The angular overlap model’s $e$ parameters are proportional to the square of an overlap divided by the energy gap between the two orbitals, and nothing here computes that gap. Folding it into the fitted constant is what makes $a$ and $b$ single numbers for the whole series, and it is the largest approximation in the essay.

One geometry. Octahedral, six identical ligands, no distortion. The whole $\Delta = 3e_\sigma - 4e_\pi$ arithmetic is a statement about that geometry, and two models agree on one ratio only because the geometry fixes it.

And five points. Two parameters against five measurements is enough to refuse a model and not enough to establish one. What the essay claims is a comparison between three models on the same five points, which is a much weaker and safer thing than a fitted value — and the same caution a fitted Hückel model attracts two fields over, for the same reason: a fit with more data than parameters can be refused, and a fit with fewer cannot be trusted.

Nor is the ligand’s π character a single number. Water is a weak donor and ammonia is neither, and both are given the same sign here because the model has only two signs to give. A model with a graded π character would be a model with a parameter per ligand, which is not a model.

Every number here is about six ligands on the axes, with three d orbitals pointing between them and two pointing at them. Which orbital faces the bond is the whole difference between the two sets, and it is a geometric fact rather than a computed one.

The splitting matters because it decides how the electrons arrange themselves, and a few thousand wavenumbers is the difference between a high-spin complex and a low-spin one. The errors above are of that size, which is why they are worth measuring rather than dismissing.

What was checked #

The sideways overlap is a real fraction of the head-on one, between 0.3 and 1.5, for every ligand — which is what makes the second channel worth computing rather than assuming away.

The series contains both a π acceptor and a π donor, or the sign would have nothing to decide.

Two parameters fit five measurements better than one, which is arithmetic rather than chemistry and is a check on the regression.

And telling the model which ligands are acceptors beats letting it fit the sign itself, with the same two integrals.

So the useful part of the second channel is a fact about occupation rather than an integral — the improvement from the sign is more than twice the improvement from the overlap. That is the finding, and it is checked as a ratio so that it cannot be satisfied by a model where both are small.

The σ-only model puts the two halides in the wrong order, which is the residual this essay set out to explain.

The second channel puts them in the right one.

And the model with the better total gets them wrong again, so no one fit does both — checked rather than mentioned, because it is the thing a reader would otherwise assume away.

The halide pair is where the second channel earns its keep, at less than half the error, as the single-overlap result suggested it would.

And the refusal is two channels that are proportional across the series, which would make the fit singular and is refused rather than split arbitrarily between them. That is the failure a two-parameter fit has when its two columns carry the same information: the total is fine, the individual parameters are meaningless, and nothing in the output says so.

What a ligand field decides in the end is how many unpaired electrons a complex has, and every number in this essay is upstream of that. The two models disagree by enough to change it, which is the sense in which an integral that cannot count electrons is a real failure rather than a small one.

A sign is not a magnitude, and the model needs one of each #

The result — that one word per ligand beats a computed integral — looks like an embarrassment for the calculation and is better read as a statement about what kind of quantity is missing.

An overlap integral is a magnitude. It is a positive number that says how much two functions coincide in space, and it takes the same value whether the ligand orbital it reaches is full or empty. Nothing continuous can encode a fact that has only two values.

Whether the ligand’s π orbital is occupied is exactly such a fact. A halide’s p lone pairs are full, so mixing them with a filled metal t₂g set pushes that set up; a carbonyl’s π* is empty, so mixing pushes it down. Same geometry, same kind of integral, opposite sign — and the sign is worth more than any refinement of the magnitude, which is what the forty-five per cent measures.

So the model’s failure is structural rather than parametric. It is not that the overlap is computed badly; it is that a one-electron overlap has no place to put an occupancy, and the occupancy is half the answer.

The constructive half is that the missing input need not be quoted. Whether a ligand’s π orbital is occupied at the metal’s energy is a count — the same kind of quantity that is safest everywhere else. Take the ligand, fill its own orbitals with its own electrons, and look at what sits near the metal’s d level: a halide’s np lone pairs are there and are full, a carbonyl’s π* is there and is empty, and an ammonia has nothing there at all, which is why it has no π term in either direction.

That is a determination rather than a fit, it costs no parameters, and it converts one word per ligand, quoted into one count per ligand, computed — which is the difference between an input and an assumption.

What no count supplies is the size, and the size is what the integral is for. So the honest shape of the model is two quantities of different kinds: a magnitude from an overlap and a sign from an electron count, neither obtainable from the other. Fitting a single continuous parameter to carry both is what makes a two-parameter fit look singular when its two columns turn out to be proportional — the fit is being asked to discover a discrete fact by regression, and regression is not how a discrete fact is found.

There is one last way to see that the missing input is not a larger integral, and it is the sharpest, because it asks the integral to supply the answer on its own and reads off what it would have to be.

Still open: more ligands, and the energy denominator #

The obvious open question is the third parameter this essay refused to fit. A separate π scale for donors and for acceptors would carry both ends of the series, and with five points it would be a parameter fitted to one complex — so the honest way to get it is more data rather than more freedom. The chromium series can be extended to bromide, iodide, thiocyanate and carbonyl without leaving the metal or the geometry, and eight or ten points would make a three-parameter fit a measurement rather than an interpolation.

The nearer question is the energy denominator the 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* — they differ by a large factor and in sign of the relevant ordering. Computing those gaps, even crudely from the same Slater orbitals, would turn the fitted constants into predictions and would say whether the sign of the π term is the only thing the occupation is carrying, or whether the denominator is doing some of the work the sign is currently getting credit for.