The splitting against something structural

Every chromium(III) spectrum gives two numbers instead of one: the splitting Δ and the Racah parameter B, solved together in closed form from two measured bands. What would make the pair mean something is a series in which both are compared against something structural — a bond length, an overlap integral, a ligand ionisation energy — and two numbers per complex is a small dataset per compound and a large one across a series. It is already established that the series is not electrostatic — charge orders it worse than a coin — so what orders it is still open.

This essay does the comparison for one of the two. The angular overlap model makes a specific claim about what Δ is proportional to, and the quantity it names is one that can be computed.

What the model actually claims #

The angular overlap model’s justification is perturbation theory. A metal d orbital that can mix with a ligand donor orbital is pushed up by the square of their interaction divided by the gap between them; the interaction is taken proportional to the overlap; so the σ parameter goes as the square of the overlap, and for an octahedron $\Delta = 3e_\sigma$.

That is a claim with an exponent in it. Not the splitting increases with the overlap — the square, and no other power. Which makes it testable in a way a monotone claim is not.

Five complexes, one metal, one integral #

The series is five chromium(III) complexes — chloride, fluoride, water, ammonia and cyanide — chosen to span the spectrochemical series and to be the same metal in the same oxidation state, so that Δ and B and this overlap are three measurements of one series rather than three series. Chromium(III) is also the ion whose d–d bands solve in closed form, and the one whose splitting pattern is fixed by symmetry before any parameter is chosen.

For each, the σ overlap is computed between a chromium 3d(z²) and a donor orbital pointing at it, at the measured metal–ligand distance, with Slater-type radial functions at effective charges from Slater’s rules. Nothing about the ligand field enters: this is an ordinary overlap integral, done by the same quadrature as any other.

The ratio Δ/S² runs 4.41, 4.91, 5.51, 5.58 and 5.76 in units of $10^5$ wavenumbers, a spread of 1.31 across a series whose splittings span a factor of 1.96. Fitting the exponent instead of imposing it returns 1.22 rather than 1, which is close enough to say the model has the right power and far enough to say it is not exact.

The pair it gets wrong #

Nine of the ten pairs come out in the right order. The one that does not is fluoride against chloride: chloride’s computed overlap is larger — 0.1757 against 0.1651 — and its measured splitting is smaller, 13,600 against 15,200.

That is not a random failure. Fluoride and chloride are the two ligands at the π-donor end of the series, and a π donor lowers Δ by pushing the t₂g set up, which is an effect a σ-only calculation contains nothing about. The same channel run the other way is what puts carbon monoxide at the far end.

So the arithmetic is doing what the physics says it should: one term accounts for most of the series and the residual is concentrated where the missing term is largest. That is a better outcome than a fit that worked everywhere, because a fit that worked everywhere would have to be absorbing the π contribution into the σ one.

The control that nearly wins #

There is a comparison here that has to be reported rather than buried, because it is the honest form of the result.

Two other quantities are available for each complex — the bond length and the donor atom’s effective charge — and each can be given a fitted power law and asked to do the same job. The bond length, even with its exponent free, manages a residual spread of 1.87 against the overlap’s 1.31, and orders only four of the ten pairs correctly.

The donor’s effective charge does better: 1.21, at an exponent of −1.38, and it orders nine of ten and misses the same halide pair.

The argument for the overlap is therefore not that it fits best. It is that it fits nearly as well without being asked to, at an exponent a model wrote down before the data were looked at. A power of −1.38 on an effective charge is not predicted by anything and could have been any number; a power of 2 on an overlap is what the perturbation argument says and could not have been.

That distinction is the whole content of the comparison, and it is worth stating in general terms because it is easy to lose. A model with a free exponent that fits well has demonstrated that its variable is correlated with the answer. A model with a fixed exponent that fits well has demonstrated something about the mechanism. The two are not comparable by residual.

Why the effective charge does so well #

The two are not independent, which is most of the answer and is worth making explicit rather than leaving as a coincidence.

The overlap integral depends on two things: the separation, and how diffuse the donor orbital is. Across this series the separations vary by twenty per cent and the effective charges by sixty, so the overlap is mostly tracking the exponent. A donor with a low effective charge has a diffuse orbital that reaches the metal, and the ligands with low effective charges here — carbon in cyanide at 3.25, nitrogen in ammonia at 3.90 — are the ones at the strong end of the series.

So the effective charge is not an independent predictor that happens to beat the overlap. It is the larger of the overlap’s two inputs, used on its own, with a fitted exponent standing in for the integral that would have combined it with the other one.

The bond length is the smaller input, and used on its own it is nearly useless — which is the third row of the figure above and is the check that the decomposition is right. That a bond length alone says little is itself worth knowing, since a structure is what a crystallographer supplies and it is tempting to reach for. Neither input alone is the answer; one of them dominates; and the integral that combines them correctly is what the model asked for.

What an octahedral field does to the five d orbitals is the diagram whose gap this whole essay is about. The pattern is symmetry’s and exact; everything argued here concerns the size.

What the numbers look like laid out #

It is worth having the five rows in one place, because the argument has three columns in it and they do not move together.

The sign of the overlap alternates with the donor’s shell and is a sign convention rather than a fact — chloride’s 3p has a radial node the others’ 2p functions do not, so the integral comes back with the opposite sign for the same geometrical arrangement. Only the square enters the model, which is fortunate, because the sign here carries no physical content at all.

What does carry content is that the two inputs are not ordered the same way. The bond lengths run 2.34, 1.93, 1.96, 2.07, 2.08 — chloride longest, fluoride shortest — while the effective charges run 4.90, 5.20, 4.55, 3.90, 3.25, falling steadily down the column apart from fluoride. Only the integral combines them, and it is the combination that orders the series.

Every point in the table above is one value of an overlap against separation, at a distance a structure determination supplied and with an exponent a screening rule supplied. Those two inputs are where the quoted numbers come from and where the uncertainty in them lives.

The two sets whose separation is Δ are fixed by symmetry and exact; the size is the parameter this essay computes an overlap for. Reading the two apart is the discipline all of ligand field theory runs on.

What this does not settle #

The essay’s opening question was whether covalency measured optically agrees with covalency measured any other way, and this essay answers half of it. Δ, which is one of the two optical numbers, tracks a computed overlap. The other optical number — the nephelauxetic ratio β, which measures how much the d electrons’ mutual repulsion has been reduced — has not been compared against anything here.

That is deliberate rather than an omission. β is a repulsion parameter and no repulsion integrals between metal and ligand functions are computed here; comparing it against a σ overlap would be comparing it against the nearest available quantity rather than against the right one. The two optical numbers are independent — the ligands that split most are not the ligands that reduce the repulsion most — so there is no reason to expect one structural quantity to account for both. The nephelauxetic ratio’s own caution — that it is partly an artefact of a truncated model — applies to any such comparison before it starts.

Where the model stops #

Slater’s rules for the exponents. Every radial function here comes from an arithmetic recipe on a configuration, applied by hand for the ions because the recipe is stated for neutral atoms. Elsewhere in this collection the same recipe was found inadequate where the answer is a length; here the answer is a dimensionless ratio and the recipe’s errors partly divide out, but they are present.

One orbital per ligand. A real donor is a molecular orbital of the ligand — the lone pair of ammonia, the σ orbital of cyanide — and is modelled here by a single p function on the donor atom. That is the crudest possible representation of it and it is what makes cyanide, whose donor orbital is a carbon-centred hybrid with substantial s character, the least trustworthy row.

The splittings are quoted. All five come from published spectra, and the two that are also in this collection’s earlier table agree with it. Nothing about Δ is computed here; the essay computes the other side of the comparison.

And the metal’s own exponent is Slater’s too. Chromium(III)'s 3d effective charge of 5.30 comes from the same recipe applied to an ion, and it enters every one of the five overlaps identically — so it cannot affect the ordering and it does affect the constant of proportionality, which is why the slope of the line in the first figure is not a number worth quoting.

And five is a small series. Ten pairs, of which one is discordant, is not a statistic. What it is, is a specific prediction — that the discordant pair should be a π pair — which was made before the ordering was looked at and which held. That is the same kind of evidence a refusal that names its own failure case provides, and it is worth more than a tighter fit would have been.

One more thing the table settles, and it is small. The two σ overlaps that are largest belong to the two ligands whose donor atoms are lightest and least screened, and both of those ligands are at the strong end of the series. A reader could take from that a rule of thumb — a diffuse donor is a strong-field ligand — and the rule of thumb is right about four of the five and wrong about fluoride, which has the most contracted donor orbital in the set and is not the weakest field. Chloride, whose donor is diffuse, is. The rule of thumb and the calculation part company on the same pair, for the same reason, and neither has the π channel in it.

The π channel this essay leaves out is computed elsewhere for the acceptor end of the series, and its absence is what puts the two halides in the wrong order here. That is the obvious next integral to compute, and it is named rather than glossed.

The comparison that separates the two variables #

The decomposition here could not tell the bond length from the orbital exponent, because across a series of ligands the two move together. Going down a group instead of across a series separates them, and one step of that descent separates them completely.

The measured splittings for the hexaammine complexes of one group are

— a rise of 49 per cent from the first row to the second and 21 per cent from the second to the third.

Now put the bond lengths beside them. The metal–nitrogen distance grows from about 1.97 ångström at cobalt to about 2.07 at rhodium, and then stops: iridium’s is the same as rhodium’s to within a couple of picometres, because the contraction that follows the filling of the 4f shell cancels the increase that going down a period would otherwise give.

That makes the second step a controlled experiment. Between rhodium and iridium the distance is fixed, the ligand is fixed, the charge is fixed and the geometry is fixed — and the splitting still rises by a fifth. Nothing about the distance can account for it.

What is left is the radial extent of the metal orbitals. A 5d function is more diffuse than a 4d one at the same distance from the nucleus, so it reaches further into the ligand’s orbital and the overlap integral is larger. The model’s prediction — that the splitting goes as the square of that overlap — then requires the overlap to have grown by about ten per cent between the two, at a fixed separation, which is a computable statement about two radial functions and not about a molecule.

The first step is the messier one and is worth reading against the second. From cobalt to rhodium both variables move, and they move in opposite directions: the bond lengthens by five per cent, which reduces the overlap, and the orbital expands, which increases it. The splitting rises by half, so the expansion wins comfortably — but the size of the win cannot be attributed without doing both integrals.

So the descent supplies exactly what the series across ligands could not: one comparison in which the distance is held fixed by an accident of the periodic table, and the splitting moves anyway. The received statement that Δ increases down a group is usually explained by the larger orbitals, and this is the pair for which that explanation has nothing else it could be.

The same pair also settles something about the effective-charge control that beat the overlap here. An effective nuclear charge computed by a screening rule rises steadily down a group, and it rises between rhodium and iridium by an amount that has nothing to do with the splitting — the two metals differ by thirty-two protons and a filled f shell. A control that tracks a quantity across ten ligands at one metal is not thereby a control that tracks it across three metals at one ligand, and this is the comparison on which the two candidates would have to be separated properly.

What the comparison requires #

Every overlap an ordinary number, between 0.05 and 0.5, which is the check that would catch a radial function or a unit going wrong.

The proportionality within forty per cent with nothing fitted, across a series whose splittings span more than a factor of 1.9 — both halves, because a tight ratio across data that barely varies is no result at all.

The fitted exponent within 0.3 of the model’s, which is the check that the power is the model’s rather than the data’s.

Nine of ten pairs, and the tenth being the halides, and both of them π donors.

And both controls, including the one that wins: the bond length ordering worse and unable to reproduce the size even with a fitted exponent; the effective charge ordering exactly as well, missing the same pair, and beating the overlap on residual. The last is stated rather than omitted, because a result that survives only when its best rival is left out is not a result.

Still open: the π channel, and the metal #

The obvious open question is the π channel, which is the thing the residual is made of. An angular overlap treatment has $e_\pi$ as well as $e_\sigma$, and the same construction gives it: the overlap between a metal t₂g orbital and a ligand p function perpendicular to the bond, at the same distance and with the same exponents. Two computed integrals instead of one, and the halide pair is where the second would have to earn its keep.

The nearer question is about the metal. Everything here holds the metal fixed and varies the ligand, which is the way the spectrochemical series is stated. The same integral computed down a group — chromium, molybdenum, tungsten, with their much larger d orbitals — would say whether the well-known increase in Δ down a transition series is the overlap growing, and that is a comparison in which the bond length and the exponent move in opposite directions rather than together — which is exactly the case the decomposition above could not separate, because here they moved together.