The splitting is a symmetry statement

A free transition-metal ion has five d orbitals and nothing to distinguish them. They are five solutions of the same equation with the same energy, and which five functions are used to span the space is a choice — an argument set out at length in real and complex harmonics, where the same subspace is written two ways and neither is more real than the other.

Put six ligands round that ion and the degeneracy breaks. It does not break into five; it breaks into two, a set of three and a set of two, and the gap between them decides the compound’s colour, its magnetic moment, and often its shape. Almost every property a coordination compound is used for hangs off that one number.

The usual account of why is electrostatic and is worth stating carefully, because it is not wrong so much as beside the point. Ligands are negative, or at least have their negative ends pointing inwards; a d electron in an orbital whose lobes point straight at a ligand is closer to that negative charge than one whose lobes point between; so the orbitals pointing at the ligands go up.

That story predicts the right pattern. It is also unnecessary, and this essay is about why.

The pattern is a reduction #

The five d functions on the metal span a representation of the molecule’s point group. Reducing that representation says how many levels there can be and how degenerate each is, and it says so without any reference to what the ligands are, what charge they carry, or how far away they sit.

The answer is $E_g \oplus T_{2g}$: one two-dimensional piece and one three-dimensional piece. That is the whole of the pattern. Two levels, one doubly degenerate and one triply degenerate, in an octahedral field, for any metal and any ligand whatever.

The method is the one character tables and reduction applies to methane’s hydrogen orbitals and the one degeneracy is a group theorem uses to argue that degeneracies are dimensions of irreducible representations rather than coincidences. The only new part is the basis: the d functions sit on the central atom, which no operation of a point group moves, so the character is not a count of unmoved atoms but the l = 2 shell’s own — $\sin(5\theta/2)/\sin(\theta/2)$ for a rotation through $\theta$, and the same at $\theta + \pi$ with a sign for an improper operation.

That formula is worth watching work, because it needs nothing but the operation’s matrix. The trace of a proper rotation is $1 + 2\cos\theta$ and of an improper one $-1 + 2\cos\theta$, and the determinant says which is which — so the angle comes out of the matrix and the character follows. The identity leaves everything alone and contributes five. A threefold rotation gives $\sin(300°)/\sin(60°) = -1$. A fourfold gives $-1$ as well, a twofold $+1$, the inversion $+5$ because a d function is even, and the mirrors and improper rotations $+1$ and $-1$ in their turn. The reducible character across the ten classes of the octahedral group is therefore 5, −1, −1, 1, 1, 5, −1, −1, 1, 1 — and $E_g \oplus T_{2g}$ reproduces every one of those ten numbers, which is the check the reduction formula performs before it reports anything.

None of that consulted a table of which orbitals are which. It consulted the group.

Three of the five functions being reduced, drawn at a contour enclosing ninety per cent of their density, look nothing like one another — and what the reduction says about them is a statement about how they transform rather than about how they look.

What the reduction does not settle #

It says two levels. It does not say which is higher.

That is not a gap in the argument; it is the boundary of what a symmetry argument can do, and it is the same boundary symmetry forbids a dipole runs into from the other side. A group tells what can happen. Deciding what does happen needs a model with energies in it, and the model is where the electrostatics — or something better — comes back.

Any model at all will do for the ordering, which is the reason the electrostatic account survives being wrong about the mechanism. Both models used here agree, and so does every other one anybody has tried: in an octahedron the two orbitals pointing at the ligands go up, and the three pointing between them go down.

One d orbital among the six ligand directions is the picture the point-charge account reasons from: an orbital pointing at the ligands is raised and one pointing between them is lowered. That is a correct account of the sign and it is not where the pattern comes from.

The sum is exact, and it is the one thing the model states without hedging #

Every one of these figures draws a dashed line for the barycentre, and it is there because the sum of the five energies is not approximately the same for every geometry, it is exactly the same.

The reason is Unsöld’s theorem: summed over a complete l shell, $\sum_i |Y_i|^2$ is a constant on the sphere. A complete shell has no shape. So whatever potential the ligands make, the total energy of five electrons distributed one to each d orbital is the energy they would have in a spherically averaged version of the same potential — and a splitting can only redistribute levels about a centre of gravity it has no power to move.

The site checks this by integration rather than by citation. The point-charge model computes each of the five energies as a separate quadrature over the sphere, and their sum comes back at five times the spherical average to ten decimal places, for octahedral, tetrahedral, cubic and square-planar arrangements alike. The tripwire is the same sum over four of the five orbitals, which misses by a great deal — so the check is measuring the completeness of the shell rather than a coincidence of the quadrature.

The angular overlap model reaches the same statement from a different direction. Its σ-only matrix has trace $N e_\sigma$ for any arrangement of $N$ ligands whatever, because the ligand’s σ function is a unit vector in the five-dimensional space and the sum of its squared components is one. Two models, two arguments, one conserved quantity.

That is worth insisting on because of what it rules out. A ligand field diagram is often read as though it showed the metal–ligand bonding, with the lower set representing stabilisation. It does not. It shows a redistribution whose total is fixed, and the bonding that actually holds the complex together is elsewhere — in the ligand σ orbitals that this site counts in eighteen is a count.

The tetrahedron: the same two pieces, the other way up #

Four ligands at the tetrahedral angle give a group with no inversion in it, and the reduction there returns $E \oplus T_2$ — again two levels, again of two and three, with the subscript g gone because there is no centre for a function to be even about.

So the count is the same and the labels are nearly the same, and a reader who stopped there would conclude that a tetrahedral field does what an octahedral one does. Both models say otherwise, and they agree with each other: in a tetrahedron the pair goes below the trio. The orbitals pointing between the ligands are now the set of three, because a tetrahedron’s four directions avoid the axes rather than lying along them, and the picture inverts.

That inversion is the half of the result a symmetry argument cannot supply and a ratio cannot carry either. It therefore has to be checked separately: in an octahedron dx²−y² must sit above dxy, and in a tetrahedron below it, both by more than arithmetic noise. A check comparing only the sizes of the two splittings would pass on a model that had lost the sign — and the sign is what decides what a tetrahedral complex looks like.

The square plane needs four levels, and the simplest model gives three #

The most informative case is not the octahedron. Take the same reduction in a square-planar complex and it gives $A_{1g} \oplus B_{1g} \oplus B_{2g} \oplus E_g$ — four pieces, of dimensions one, one, one and two. Four levels, one of them doubly degenerate.

Now compute the splitting with only σ interactions switched on and three of the levels come out at exactly the same energy — dxz, dyz and dxy all at zero, because none of them has any amplitude along a ligand direction lying in the plane. The model has produced a degeneracy the point group does not require.

That is not a bug and it is not harmless. It is the model having more symmetry than the molecule, which is a specific and recognisable failure: a σ-only description cannot distinguish an orbital sticking out of the plane from one lying in it, and the molecule certainly can. Turn on the π interactions and the accidental degeneracy lifts, giving the four sets the group demanded from the start.

A complete check covers both halves: that the computed degeneracies match the irreducible dimensions in all three geometries, and that exactly one of the three has an accidental degeneracy when π is switched off. A check that only tested agreement would pass while quietly hiding the interesting case.

Where the octahedron came from #

One thing here has not been assumed and is worth pointing at, because it is computed independently of all of this.

The six ligand directions are not a convention. Six points repelling each other on a sphere settle into an octahedron, and VSEPR, computed minimised that energy rather than quoting the arrangement — finding two distinct angles, 90° and 180°, from a numerical minimisation started from random points.

The octahedral arrangement six regions produce is what a repulsion minimisation returns, with the two distinct angles measured off the minimised coordinates. The field the d shell sits in is that arrangement, and nothing about the splitting pattern depends on how the arrangement was arrived at.

So the chain from nothing to a splitting diagram runs: minimise repulsion, find an octahedron, recover its point group from the coordinates, reduce the d shell in that group, and read off two levels of three and two. Every step is a computation and none of them needed a table.

The one link in that chain worth doubting is the first. Repulsion between electron domains is a good account of main-group geometry and a poor one for transition metals, for reasons taken up in VSEPR does not reach a transition metal: the d shell contributes its own preference, and it is strong enough to overturn the repulsion answer for some fillings. The octahedron survives because six ligands want an octahedron on almost any account. Four do not.

The complex the reduction is done on has its group recovered from the coordinates by the same search that recovers methane’s, and the character table used is generated by closing those operations rather than typed in. Every symmetry statement in this essay is downstream of that.

What this decides downstream #

The splitting is one number and an enormous amount hangs from it.

Colour. A transition between the lower set and the upper is an energy, and an energy is a wavelength. Where a d–d band falls works out where the band lands and, more usefully, where it does not.

Magnetism. Whether the electrons pair up in the lower set or spread across both decides how many are unpaired, which is a measured quantity. A moment counts electrons, not orbitals does that count, and the pairing energy decides the moment is about the quantity that competes with the splitting.

Shape. For a d⁸ metal the square plane’s lowest four levels can hold all eight electrons below a large gap, and that is worth more than the tetrahedron’s better repulsion arithmetic.

Distortion. An unevenly occupied degenerate set is unstable against a distortion that splits it — the Jahn–Teller argument that applies to a square four-ring, and makes again for a copper complex in copper is never quite octahedral.

The last of those is worth a moment, because it turns the present essay’s conclusion into a warning. A degeneracy that symmetry requires is exactly the situation in which a molecule looks for a way out. The reduction says an octahedral d shell has a triply and a doubly degenerate level; put nine electrons in and the upper pair is occupied unevenly, and the complex responds by ceasing to be octahedral. So a symmetry argument can be perfectly correct about a structure that the compound then declines to adopt, and the way to find that out is to compute the energy of the distortion rather than to admire the diagram.

What the picture cannot show #

Three things, and they are the standing cautions of the whole field.

There are no electrons in this argument. The reduction is about functions, not about occupancies. Nothing here says how many d electrons the metal has, and nothing here can: putting two of them in one orbital costs an energy that no one-electron model contains, which is the subject of the pairing energy decides the moment and, computed exactly on a small system, of the smallest many-electron calculation.

The size of the splitting is not in the symmetry. Two levels, yes; how far apart, no. The gap ranges over a factor of nearly five across ordinary ligands, and what orders that range is not charge — the spectrochemical series is not electrostatics is the essay on it.

The lower set is not non-bonding. Drawing three levels at the barycentre and labelling them t₂g invites the reading that those orbitals do nothing. In a σ-only model they are indeed untouched, which is why the model gives them exactly zero — but the moment a ligand has π orbitals, they move, and for the strongest-field ligands they move a long way. The apparent inertness of the lower set is an artefact of leaving something out.

The number of bands is not the number of gaps #

There are no electrons in this argument is the sharpest of the three cautions, and it has a consequence that is easy to check and easy to be caught by: the spectrum of an octahedral complex usually has more bands than this argument has gaps.

The reduction gives two levels and therefore one separation. A naive reading predicts one absorption band per complex.

A chromium(III) complex shows three. Hexaaquachromium absorbs near 17,400, 24,600 and 37,800 wavenumbers, and the same three-band pattern appears in every octahedral d³ complex with the positions shifted. Hexaaquanickel, a d⁸ ion, shows three as well — near 8,500, 13,800 and 25,300.

Neither is a failure of the symmetry argument. It is the argument being applied to the wrong object.

What a spectrum connects are states of the whole d shell, not one-electron orbitals. A d³ ion has many ways of arranging three electrons in five orbitals, those arrangements combine into many-electron terms, and it is the terms that the point group classifies and the light connects. Doing the reduction on the terms rather than on the orbitals gives a ground state and three excited states of the right spin — and three allowed transitions, which is what is measured.

So the same group theory produces one answer for orbitals and another for states, and the second is the observable one. The gap between them is exactly the electron–electron repulsion the one-electron picture omits: with no repulsion, all three of those transitions would coincide at the single splitting energy, and what pulls them apart is the different amount of repulsion in each arrangement.

That is worth carrying because it decides what a measured band position means. A single absorption maximum is the ligand field splitting only for a d¹ or a d⁹ ion, where there is one electron or one hole and no repulsion between d electrons to speak of. For everything in between, extracting a splitting from a spectrum requires disentangling it from the repulsion — which is a two-parameter problem, and belongs to the analysis of spectra rather than to symmetry.

What symmetry settles before any model #

What has been established is small and hard. The number of levels and their degeneracies come from the group, exactly, and match the computed splitting in three geometries; the barycentre is conserved, exactly, in both models; and one of the three geometries has an accidental degeneracy that says precisely which interaction the simplest model is missing.

Everything else in ligand field theory depends on there being a gap and on its size. Two models, one ratio takes the next step: computing that size two independent ways and finding that the two agree on a number neither was told.