Two models, one ratio

Symmetry settles how many levels there are and how degenerate each one is. What symmetry cannot settle is how far apart they are, and that needs a model.

Two are in common use, and both are run here.

The first model is the electrostatic one, taken literally rather than as a story. Put a point charge at each ligand position, write down the potential it makes, and integrate that potential over the angular density of each d orbital:

$$E_i = \sum_k \int \frac{|Y_i(\hat{n})|^2}{|a\hat{n} - R\hat{k}|} \, d\Omega$$

with the d electron on a shell of radius $a$ inside a ligand shell of radius $R$. A product Gauss rule in $\cos\theta$ and the midpoint rule in $\phi$ — forty by eighty points, which is generous for an integrand this smooth — and no expansion in Legendre polynomials anywhere. Nothing quoted.

The second model has no charges in it at all. The angular overlap model says each ligand interacts with the d shell through three channels distinguished by how they behave under rotation about the metal–ligand axis: σ, π and δ. Their strengths are named parameters. The geometry enters as the rotation that carries the bond axis onto z, so the interaction matrix for one ligand is a diagonal matrix conjugated into the molecular frame, and the total is the sum over ligands, diagonalised.

The two share the ligand directions and nothing else. One is an integral over a charge distribution; the other is a rotation of a diagonal matrix.

What the two agree on, to eight decimal places #

Set both models on an octahedron and a tetrahedron with the same ligands at the same distance, and take the ratio of the splittings.

The point-charge integral gives $0.020931$ for the octahedron and $0.009303$ for the tetrahedron, in whatever units the charge and the radii imply. Their ratio is $0.44444444$.

The angular overlap model gives $3e_\sigma$ and $\tfrac{4}{3}e_\sigma$. Their ratio is $0.44444444$.

Four ninths. Neither model was told the number; it is the ratio of two quadratures in one case and the ratio of two eigenvalue problems in the other. The cube gives the same kind of agreement at $0.88888889$ — eight ninths, exactly twice the tetrahedron’s, because a cube is two interpenetrating tetrahedra and each contributes its own.

That a cube with eight ligands splits less than an octahedron with six is the fact that kills the obvious explanation. If splitting were about how much ligand there is, more ligands would split more. It is not: the ligand count enters the trace, and the trace is exactly $N e_\sigma$ for any arrangement whatever — six for an octahedron, eight for a cube, four for a tetrahedron. What the geometry decides is how that fixed total is distributed, and eight charges arranged so that none of them lies on an axis distribute it less unevenly than six that all do.

Why the tetrahedron inverts #

The ratio is only half the result. The other half is the sign, and a ratio cannot carry it.

An octahedron’s six directions lie along $\pm x$, $\pm y$ and $\pm z$. A tetrahedron’s four lie along body diagonals, $(\pm1, \pm1, \pm1)/\sqrt{3}$, and every one of them makes the same angle with every axis. So the two orbitals built from the axes — dz² and dx²−y² — point at ligands in the first case and between them in the second, and the picture turns over.

The inversion is checked explicitly, in both models, by comparing dx²−y² with dxy: above in an octahedron, below in a tetrahedron, both by more than arithmetic noise. That is deliberate. A check that compared only the magnitudes of two splittings would pass on a model that had lost the sign entirely, and the sign is the half that decides what a tetrahedral complex looks like.

Whether the integral can be trusted #

A ratio that comes out at eight decimal places invites the question of what the quadrature is doing, since a numerical integral is exactly the sort of thing that produces a plausible wrong answer — a caution that applies equally to say what it encloses, where a contour level is found by integrating and the integral is validated against a closed form before anything is drawn.

Three things make this particular integral easy, and it is worth saying which.

The integrand is smooth. The d electron sits on a shell of radius $a$ strictly inside the ligand shell of radius $R$, so the denominator never approaches zero and the function being integrated has no singularity anywhere on the sphere. The calculation refuses $a \ge R$ rather than integrating through a pole.

The φ direction is periodic, and the midpoint rule on a periodic smooth function converges faster than any power of the step — the same reason a Fourier series of a smooth periodic function converges quickly. Eighty points is far more than needed.

The θ direction is polynomial in disguise. Both $|Y_i|^2$ and the expansion of the inverse distance are polynomials in $\cos\theta$ of low degree, and Gauss–Legendre integrates a polynomial of degree $2n-1$ exactly with $n$ points. Forty points integrate exactly anything of degree seventy-nine.

So the eight decimal places are not luck, and the evidence that they are not is the barycentre: the sum of the five computed energies reproduces five times the spherical average to ten decimal places, and that is a quantity the quadrature had no way to arrange. A rule converging poorly would break the sum long before it broke the ratio.

What is switched off, and why it matters #

The angular overlap model has three parameters per ligand and the figures here use one. That is a choice and it is not innocent.

$e_\delta$ describes an interaction between a d orbital and a ligand function with two nodal planes containing the bond axis. Ordinary ligands have nothing of the kind to offer, so it is set to zero everywhere here, and the only effect of that choice is on the two orbitals that would have received it.

$e_\pi$ is another matter. Switching it off is what produces the accidental degeneracy in the square plane that the previous essay used, and switching it on is the whole content of the spectrochemical series. A σ-only model of an octahedron puts the lower three orbitals at exactly zero — the value they take when nothing touches them — and that exact zero is a statement about σ symmetry rather than about chemistry: those orbitals point between the ligands, so their overlap with a ligand σ function vanishes for the same reason the forbidden integrals in exactly zero vanish, by cancellation in pairs rather than by being small.

Which means the figures in this essay are pictures of the σ skeleton of the problem. Everything that distinguishes fluoride from carbon monoxide lives in the parameter that has been set to zero to draw them.

The exact statements each model makes #

Neither model is exact about anything a chemist would measure. Both are exact about something, and the somethings are worth having because they are what makes each model checkable.

The point-charge model conserves the barycentre. The five energies sum to five times the spherical average, for every arrangement, because a complete l shell has no shape. Computed to ten decimal places, with a tripwire: the same sum over four of the five orbitals misses badly, so the identity is about the completeness of the shell and not about the quadrature.

The angular overlap model conserves the trace. The σ-only matrix has trace $N e_\sigma$ for any arrangement of $N$ ligands. The reason is the same completeness fact seen from the other end: the ligand’s σ function is a unit vector in the five-dimensional space, so its squared components sum to one, and the trace counts ligands.

These are the same statement twice, which is itself a small piece of evidence that the two models are not as different as their descriptions sound. Both are angular; both see only how the ligands are arranged; both are blind to what the ligands are.

One matrix, two constructions #

There is a third agreement worth recording, because it is the one that makes the calculation trustworthy.

The angular overlap σ matrix can be built two ways. The first conjugates $\mathrm{diag}(e_\sigma, e_\pi, e_\pi, e_\delta, e_\delta)$ by the five-dimensional rotation that carries z onto the bond axis — and that rotation is got from ordinary 3×3 arithmetic, because the l = 2 functions are the symmetric traceless matrices and a rotation acts on them as $T \to RTR^{\mathsf{T}}$. No Wigner elements, nothing to mistype.

The second evaluates the five real harmonics at the ligand direction, scales them so their squares sum to one, and forms the outer product.

The two constructions agree to $1.3\times10^{-15}$ for every geometry drawn here. That is a check on the rotations, on the harmonics, and on the normalisation all at once, and it is the sort of agreement that is either exact or absent — there is no way for a rotation matrix to be nearly right.

Where the arrangement came from, again #

Both models take the ligand directions as given, so it matters where those came from.

The tetrahedral arrangement is what four points minimising their repulsion on a sphere actually give, with the single distinct angle 109.47° measured off the minimised coordinates rather than quoted. The ligand-field calculation uses that arrangement rather than a typed-in one, so the two halves of the argument share a geometry.

The octahedron and the tetrahedron are what VSEPR, computed finds by minimising repulsion on a sphere, and the ligand-field directions used here have the angle spectrum the minimiser produces. Two of the five geometries deliberately fail that test — the cube, which is beaten by the square antiprism, as above six coordination established, and the square plane, which is beaten by the tetrahedron. Those two are included precisely because they are not what repulsion would choose, and something else has to explain why they occur.

Eight points minimised on a sphere give a square antiprism and not a cube, so the cube in this essay is a geometry chosen to make a point about ratios rather than one a repulsion argument would produce. That is worth saying plainly: the ratio is exact for the cube as a geometry, and no molecule is obliged to adopt it.

Four ninths, and what it does #

The number is not an ornament. It decides one of the most reliable regularities in coordination chemistry, and the argument takes two lines.

A tetrahedral splitting is four ninths of an octahedral one for the same ligands at the same distance. The energy required to pair two electrons in one orbital does not change when the geometry does. A complex takes the low-spin arrangement when the splitting exceeds the pairing energy — at exactly $\Delta = P$, as the pairing energy decides the moment computes. Four ninths of a splitting that was already comparable with the pairing energy is not comparable with it any more.

So low-spin tetrahedral complexes are almost unknown, and the reason is a factor of $4/9$ rather than any statement about which fillings can have a choice. The enumeration says d³ to d⁶ do have a choice in a tetrahedral field — the same four as the octahedral case, reflected by the particle–hole map — and it is arithmetic about magnitudes, not combinatorics, that makes the choice moot.

What both models get wrong #

The same thing, and it is the magnitude.

Put reasonable numbers into the point-charge integral — a formal ligand charge, a metal–ligand distance of two ångströms, a plausible d-electron radius — and the splitting comes out far from the ten to thirty thousand wavenumbers real complexes show. Choosing the parameters to fit one complex leaves the next one wrong. The model reproduces every ratio here exactly and cannot be trusted with a single absolute number.

The angular overlap model does not even pretend: $e_\sigma$ and $e_\pi$ are fitted to spectra by other people, and what the model supplies is the geometry factor multiplying them.

That is the honest position, and it is the position the spectrochemical series is not electrostatics starts from. The ordering of real ligands by how much they split a d shell is measured, it is highly reproducible, and it is not the ordering of charge — which is a fact about chemistry that no amount of angular geometry will produce, because the geometry is identical in every complex being compared.

What four ninths predicts, and a class of compounds that does not exist #

A ratio computed to eight decimal places by two models is satisfying and does not, on its own, say anything about chemistry. This one does, and what it says is categorical rather than a trend.

The splitting of an octahedral first-row complex runs from about eight thousand wavenumbers for weak-field ligands to about twenty-five thousand for the strongest. Multiply by four ninths and a tetrahedral complex of the same metal with the same ligands has a splitting between roughly three and a half thousand and eleven thousand.

Now put that beside the quantity it has to beat. Whether a complex is low-spin depends on the splitting against the pairing energy — what it costs to put two d electrons in one orbital rather than leave them apart — and for first-row ions that is fifteen to twenty-five thousand wavenumbers.

The largest tetrahedral splitting available is therefore below the smallest pairing energy. Not marginally: by a factor of about two, across the whole range of ligands and the whole first row.

So no tetrahedral complex of a first-row transition metal should be low-spin, and none is. That is not a statistical tendency with awkward cases; it is a class of compound that a ratio computed from four directions says cannot exist, and inorganic chemistry has not produced one.

The prediction is worth contrasting with the other well-known difference between the two geometries, because they have different sources. Tetrahedral complexes are usually much more intensely coloured than octahedral ones — cobalt chloride’s tetrahedral blue against its octahedral pink is the standard demonstration — and that has nothing to do with the ratio at all. It follows from a tetrahedron having no centre of inversion, so the d–d transitions are not forbidden by the parity rule that forbids them in an octahedron.

One difference comes from the angular distribution of four directions against six, and predicts a spin state. The other comes from the presence or absence of one symmetry element, and predicts an intensity. Both are consequences of the geometry, neither is a consequence of the other, and only the first is what this ratio is about.

Who found it, and when #

The point-charge calculation is Hans Bethe’s, from 1929, and it was not written about complexes at all: it is a paper about the splitting of atomic terms in crystals of stated symmetry, and the group theory in it is the reason the field’s vocabulary is group-theoretical to this day. John Van Vleck connected it to magnetic susceptibilities through the 1930s, which is where the model got its reputation, because susceptibilities are what it predicts best.

The angular overlap model is thirty-five years younger — Claus Schäffer and Christian Klixbüll Jørgensen, 1965 — and was written by people who had concluded that the electrostatic account could not be right, for the reasons the next essay measures. Its parameters are overlaps rather than charges, and it treats a ligand field as bonding seen from an unusual angle instead of as an electrostatic perturbation of a free ion.

That the two agree exactly on every ratio computed here is therefore not a coincidence of two attempts at the same theory. It is the discovery that both are, in their angular part, the same theory: a statement about how a set of directions distributes a fixed total among five functions. The disagreement is entirely in what supplies the total, and that is the half chemistry decides.

What is left #

Two models agreeing on a ratio to eight decimal places while disagreeing with reality about the magnitude is a specific and useful situation. It says the angular part of the problem is solved and the radial part is not — that whatever makes one ligand split harder than another is a property of the metal–ligand interaction rather than of where the ligands are.

The rest of this field takes that seriously. The spectrochemical series is not electrostatics measures what does order the series; eighteen is a count counts what the σ interactions do to the bonding as opposed to the splitting; and back-bonding is two interactions computes the π half, which is the part the point-charge model has no representation of at all.