The electrons repel less inside the complex

A d–d spectrum has more bands than a one-electron picture has transitions, and the reason is that the d electrons repel each other. Where a d–d band falls is decided by two quantities: the ligand-field splitting $\Delta$, which is a property of the ligands, and the electron–electron repulsion, which is a property of the metal — or would be, if it did not change when the complex forms.

It does change. The repulsion between the d electrons inside a complex is measurably smaller than the repulsion between the same electrons in the free ion, and by amounts up to a factor of two. That effect has a name — the nephelauxetic effect, from the Greek for cloud-expanding — and it is the second number a d–d spectrum reports.

Two bands, two unknowns, no iteration #

A d³ ion in an octahedral field has three spin-allowed transitions from its ⁴A₂ ground state, and the standard way to analyse them is to lay a ruler across a Tanabe–Sugano diagram. It does not need to be done that way. The two ⁴T₁ states come from a two-by-two matrix, so both are a square root:

$$E(^4T_2) = \Delta, \qquad E(^4T_1) = 7.5B + 1.5\Delta \mp \tfrac{1}{2}\sqrt{225B^2 + \Delta^2 - 18B\Delta},$$

where $B$ is the Racah parameter measuring the repulsion.

Two checks are available on those expressions before any measurement is used, and they cost nothing. With $\Delta = 0$ they must give 0 and $15B$, which is the free ion’s ⁴F–⁴P gap. With $B = 0$ they must give $\Delta$ and $2\Delta$, which are the two strong-field configurations $t_{2g}^2 e_g$ and $t_{2g} e_g^2$. Both hold exactly.

Then the solution. The first band is $\Delta$ outright. Substituting into the second, squaring, and cancelling the $B^2$ terms leaves $B$ linear:

$$B = \frac{\Delta^2 - 4u^2}{18\Delta - 60u}, \qquad u = \nu_2 - 1.5\Delta.$$

Nothing is iterated and nothing is fitted. Two measured numbers go in and two parameters come out.

The third band is a prediction #

Fitting two parameters to two numbers is not, on its own, a test — any two numbers can be reproduced by two parameters. What makes it a test is the third band, which was not used.

For the hexafluoride the prediction is 35,442 cm⁻¹ against a measured 34,400: three per cent. For the aquo complex, 38,529 against 37,800: two per cent.

That is a strong result for a model with two parameters describing a many-electron problem, and it is the reason the Tanabe–Sugano treatment survived. It also says something about what the parameters are: two numbers extracted from the low-energy part of the spectrum predict a band 20,000 wavenumbers away, so they are not curve-fitting constants but quantities with some claim to describe the ion.

What the Racah parameter is #

$B$ deserves a paragraph of its own, because it is quoted constantly and defined rarely.

A free ion with several d electrons has its states split by electron–electron repulsion into terms — ⁴F, ⁴P, ²G and so on for d³ — and the energies of those terms are combinations of a small number of integrals over the d orbitals. Racah’s contribution was to find the combinations that make the term energies come out simple: three parameters $A$, $B$ and $C$, of which $A$ shifts everything together and drops out of any difference.

For the quartets of d³ only $B$ appears, and the ⁴F–⁴P gap is exactly $15B$. So $B$ is measurable on a free ion from an atomic spectrum — 918 cm⁻¹ for Cr(III), 1041 for Ni(II), 862 for V(III) — and it is a property of that ion in isolation.

The whole of this essay is the observation that the same parameter, extracted from a complex, comes out smaller. Not a different parameter measured a different way: the same combination of integrals over the same kind of orbital, and it has changed. Which means the orbitals have changed, which means they are not free-ion d orbitals any more.

That inference is short, and it is why a two-parameter fit is worth taking seriously. $\Delta$ is a new quantity that has no free-ion counterpart and could mean anything; $B$ has a free-ion value to be compared against, so a change in it is a statement rather than a number.

Two series, not one #

The measured $\beta = B/B_0$ runs from 0.976 for the hexafluoride to 0.534 for the hexacyanide. So the fluoride complex has electrons repelling each other almost exactly as they do in the bare ion, and the cyanide complex has lost nearly half of it.

Ordering the ligands by $\Delta$ gives F⁻ < urea < H₂O < oxalate < NH₃ < en < CN⁻, which is the spectrochemical series.

Ordering the same seven by $\beta$ gives F⁻ > H₂O > NH₃ > urea > en > oxalate > CN⁻.

They are different orderings. Urea moves three places, oxalate moves three the other way. The two ends agree — fluoride and cyanide are extreme in both — and the middle does not, which is the shape of two correlated but distinct quantities rather than one quantity measured twice.

What a reduced repulsion means #

The interpretation is direct and it is a statement about covalency.

The repulsion between two electrons scales inversely with how far apart they are, so a smaller $B$ means the d electrons are further apart on average. They are further apart because the orbitals they occupy are not pure metal d orbitals: they are combinations with ligand orbitals, spread over a larger volume, and an electron pair spread over a larger volume repels itself less.

So $\beta$ is a covalency measured optically, and its ordering is the nephelauxetic series, which sorts ligands by how much they delocalise the metal’s electrons onto themselves. Fluoride is small, hard and holds its electrons tightly: little mixing, $\beta$ near one. Cyanide has low-lying empty π* orbitals and mixes strongly: $\beta$ near a half.

The same conclusion is available from a completely different measurement. The g-value of a d⁹ or d¹ ion shifts by less than the free-ion spin–orbit constant predicts, and the shortfall — the orbital reduction factor — is between 0.72 and 0.93 for the ions measured there. One is optical and one is magnetic; both report a fraction of an electron that is not on the metal; and they agree in size.

Two independent measurements constraining the same quantity is worth more than either alone, and neither is a calculation of a wavefunction. What they are is bounds on one.

Why the point-charge model cannot see it #

This is the sharpest thing the effect says, and it is a negative result about a model used constantly.

A crystal-field calculation treats the ligands as point charges. The d orbitals in it are pure metal d orbitals — unchanged, undelocalised, exactly what they were in the free ion — and there is no mechanism whatever by which the repulsion between two of them could change. A pure crystal-field model predicts $\beta = 1$ for every complex, and the measurement is that it is never 1 and is sometimes 0.53.

That is not a small discrepancy in a parameter; it is a quantity the model cannot produce at all. It sits alongside the other things the point-charge picture gets wrong here: the ordering of the spectrochemical series, which the model gets backwards for charged ligands, and the sheer magnitude of the splitting, which point charges at real distances cannot supply.

The model survives because it gets the symmetry right, and symmetry is what fixes which orbitals go up and which go down. The splitting pattern is a symmetry statement and the numbers are not, and the nephelauxetic effect is one more entry in the column of things that are not. Two models can give one ratio and disagree about everything else, which is the same lesson from the other side.

The two ends, and what makes them extreme #

Fluoride at 0.976 and cyanide at 0.534 are the ends of both series, and it is worth saying why the two effects agree at the extremes and not in the middle.

Fluoride is small, highly electronegative and holds its electrons very tightly. It donates weakly through a σ orbital that lies far below the metal d orbitals in energy, so the mixing is slight — and it is a π donor, with filled p orbitals that push the $t_{2g}$ set up, which reduces $\Delta$. Weak mixing gives $\beta$ near one; π donation gives a small $\Delta$. Both extremes, from the same cause.

Cyanide is the opposite in both respects. Its σ donation is from a carbon lone pair lying close in energy to the metal d orbitals, so the mixing is strong. And it is a π acceptor, with empty π* orbitals that pull the $t_{2g}$ set down and increase $\Delta$ substantially — the interaction that back-bonding describes and that a carbonyl shares.

In the middle of the series the two mechanisms decouple. Urea splits weakly and delocalises quite strongly; oxalate splits moderately and delocalises very strongly. There is no reason a ligand’s σ donation strength and its π behaviour should track each other, and in the middle of the series they do not.

What the closed form refuses #

A pair of bands that no $(\Delta, B)$ can produce has to be refused rather than fitted, and the case is easy to construct: a second quartet band lying below the first. The expressions above give a negative $B$ for it, which is not a repulsion, and reporting a negative repulsion as if it were a measurement is how a fitted model launders bad data.

There is a second refusal implicit in the third-band check. A model that reproduced two bands and missed the third by a factor would have been fitted rather than tested, and the two per cent agreement is what makes the parameters worth naming. Had the third band come out badly, the right conclusion would have been that a two-parameter description is inadequate for chromium — not that the parameters needed adjusting.

What is being assumed #

One configuration at a time. The expressions treat the quartet states as arising from $t_{2g}^n e_g^m$ configurations mixed only among themselves. Configuration interaction with higher states is neglected, which is part of why the fitted $B$ absorbs some of the effect and why the values should be read as effective rather than exact.

Octahedral, and undistorted. All seven complexes here are treated as regular octahedra. The tris-chelates — oxalate and ethylenediamine — are not: they have D₃ symmetry, and their bands are split by a small amount that is being averaged over.

Spin-allowed transitions only. The spin-forbidden bands, which are sharp and sit at energies that depend on $B$ almost alone, are a much better probe of the repulsion than the broad allowed ones. They are also weak, and using them would need intensities rather than positions.

A free-ion $B_0$ from an atomic spectrum. 918 cm⁻¹ for Cr(III), quoted. Everything about $\beta$ is a ratio to that.

The two orderings, and the family that reverses between them #

A complex is characterised by two numbers rather than one is the finding, and the sharpest evidence for it is that the two numbers order the ligands differently — with one family running in exactly opposite directions on the two lists.

Ranking ligands by how much they split the d shell gives the spectrochemical series, and among the halides it runs

$$\mathrm{I^-} < \mathrm{Br^-} < \mathrm{Cl^-} < \mathrm{F^-}.$$

Ranking the same ligands by how much they reduce the d–d repulsion gives the nephelauxetic series, and among the halides it runs

$$\mathrm{F^-} < \mathrm{Cl^-} < \mathrm{Br^-} < \mathrm{I^-}.$$

Exactly reversed. Fluoride splits hardest and shares least; iodide splits least and shares most. Two quantities extracted from the same two bands of the same spectra, put in order, and the order is inverted across a whole group of the periodic table.

The chemistry behind that is the hard–soft distinction stated in optical units. A fluoride is small, compact and reluctant to be polarised: it sits close, it perturbs the d orbitals strongly through its charge and its σ donation, and it keeps its electrons. An iodide is large and diffuse: it is a poor σ donor at the distance it has to sit at, so it splits weakly — and its outer electrons mix readily with the metal’s, so the d electrons find themselves spread over a larger volume and repel each other less.

The reversal is not a general anti-correlation, and cyanide is the case that shows so. It is near the top of the spectrochemical series and well up the nephelauxetic one — a strong-field ligand and a strongly covalent one. So the two rankings are not one ranking read backwards; they are two independent properties that happen to run oppositely for the halides and together for cyanide.

Which is the strongest possible form of the argument. If the two numbers measured one underlying thing, no pair of ligands could reverse between them and no pair could agree while another reversed. Both happen, and a complex therefore needs both numbers — one saying how far apart the ligands push the levels, the other saying how much of the metal’s electrons are no longer on the metal.

It also identifies which of the two the older electrostatic picture was ever entitled to. A point-charge model contains charges and distances, so it can produce a splitting; it contains no mechanism whatever for the metal’s electrons to spend time on the ligands, so the free ion’s repulsion is the only repulsion it has and $\beta$ is identically one in it. The nephelauxetic effect is not a quantity the electrostatic model gets wrong — it is a quantity the model cannot have. Measuring it at all is therefore evidence of a kind the spectrochemical series cannot supply: a reduction of forty-seven per cent in a quantity the rival account fixes at zero.

A number that changes when the model does #

There is a caution about $\beta$ that is easy to miss and that has turned up before in another form.

The value of $B$ extracted depends on which states are included in the treatment. Adding configuration interaction with higher quartets lowers the computed energies of the ⁴T₁ states at fixed $B$, so reproducing the same measured band needs a larger $B$ — and $\beta$ moves closer to one. The nephelauxetic effect is therefore partly an artefact of a truncated model, and how much of it is real is a question the two-parameter fit cannot answer from inside itself.

That is the same shape of caution as the ligand-field stabilisation fitted to a set of hydration enthalpies, where a quantity extracted as a residual carries whatever the model left out. The defence is the same too: the effect is large, its ordering is chemically sensible, and it agrees in size with a magnetic measurement that shares none of its assumptions. A truncation artefact would not be expected to reproduce the nephelauxetic series or to match an orbital reduction factor.

What would settle it is a calculation with the ligands in it — an angular overlap treatment that computes the delocalisation rather than absorbing it into a parameter. That is available in principle and it needs overlap integrals between metal d orbitals and ligand orbitals at real bond lengths, which is a step that has been taken for one bond at a time and not for six at once.

Still open: the second repulsion parameter, and covalency across a series #

The immediate open question is the second repulsion parameter. A d–d spectrum in general needs two — $B$ and $C$ — and the treatment above uses only $B$ because the quartet states of d³ happen not to involve $C$. Fitting both needs a spin-forbidden band, which is where the sharp lines of ruby come from, and it is the point at which the analysis stops being closed-form.

The other is the one all of this points towards: a series of complexes in which $\Delta$ and $\beta$ are measured for the same metal across many ligands, and both are compared against something structural — a bond length, an overlap integral, a ligand ionisation energy. Two numbers per complex is a small dataset per compound and a large one across a series, and the question of whether covalency measured optically agrees with covalency measured any other way is exactly the sort of question a table can answer and a single calculation cannot.