A d shell in an octahedral field

One of the figures on a d shell in a field: What a set of ligands does to five degenerate orbitals — computed twice, from an integrated point-charge potential and from an angular overlap matrix, which agree on every ratio.

Ten essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

The splitting is a symmetry statement

The five d levels in an octahedral field, computed twice — once by integrating the potential of six point charges over the angular density of each orbital, once by diagonalising an angular overlap matrix. Three levels together, two levels together, and the dashed line is the barycentre the five of them sum to.

The same d shell in four fields at one set of parameters. What symmetry supplies is the pattern in each column — which orbitals are degenerate with which — and it supplies it from the character of a complete l shell under each operation, a quantity fixed by how far the operation turns and by nothing about the metal.

The octahedral case from the point-charge model alone. The pattern it gives is identical to the overlap model’s, because the pattern is the group’s and neither model can move it — and the two models agree about that and about nothing else, which is the essay’s claim in the smallest available form.

Two models, one ratio

Four arrangements of ligands round one d shell, drawn on a common scale in units of the octahedral splitting. An octahedron gives 1, a cube 0.889, a tetrahedron 0.444 and inverted, and a square plane its own four-level pattern. Every number is an eigenvalue of a matrix built from the ligand directions.

The octahedral splitting from both models side by side, each scaled to itself. The patterns are identical: three below, two above, and a barycentre neither model can move. What is not identical is the size, and that is the subject of the last section here.

Eight point charges at the corners of a cube, integrated. The splitting is 0.018605 against the octahedron’s 0.020931 — eight ninths — and it is inverted, as a tetrahedron’s is, because the cube’s directions avoid the axes in the same way.

Where a d–d band falls

The transition being talked about: an electron moves from the lower set of three to the upper set of two, and the energy it needs is the gap. In an octahedral field with σ-only ligands that gap is 3eσ, and both standard models agree about it.

A square-planar field has four levels rather than two, so there are three possible one-electron excitations rather than one, and a square-planar complex accordingly shows a more complicated spectrum than an octahedral one of the same metal. Counting the levels is the easy half; the states built on them are what a spectrometer sees.

The same octahedral field with a π donor rather than a π acceptor. The splitting shrinks and the ordering is unchanged, so the position of the band moves and its assignment does not — which is why the spectrochemical series orders the ligands by a parameter whose sign is what decides the size of the gap.

The pairing energy decides the moment

The two fillings of a d⁶ ion against the splitting, in units of the pairing energy. The high-spin arrangement has four unpaired electrons and the low-spin one has none, so the crossing between these two lines is the difference between a moment of 4.90 and a moment of zero. It is at Δ = P exactly.

d³, which has no choice at all: three electrons, three orbitals in the lower set, and nothing to decide. One line, one unpaired count, one moment — whatever the ligand. Chromium(III) is d³, and its complexes are all magnetically alike.

d⁴, the first configuration with a choice. High spin has four unpaired electrons and a moment of 4.90; low spin has two and a moment of 2.83. Manganese(III) and chromium(II) are d⁴, and both forms are known.

An orbital carries no angular momentum

The splitting the two sets come from: five d functions in an octahedral field, three below and two above, with the degeneracies fixed by the reduction the splitting is a symmetry statement computes rather than by any model of the field. Which set an electron is in is what decides whether it has angular momentum available to it.

The spectrochemical series is not electrostatics

An octahedral field with π-donor ligands, eπ = 0.3eσ. The lower set has been pushed up to 1.2 and the splitting has fallen from 3 to 1.8 — a forty per cent reduction with the σ interaction untouched. This is a halide.

The same complex with π-acceptor ligands, eπ = −0.3eσ. The lower set has been pulled down to −1.2 and the splitting has risen to 4.2. Neither figure changed eσ at all: the whole difference, a factor of 2.3 between them, is the sign of one parameter.

VSEPR does not reach a transition metal

What a square plane is worth over a tetrahedron for every d^n, each geometry in its own ground state and each with the same four ligands. The bar is exactly zero at d⁰ and at d¹⁰, and largest at d⁸ — and d⁸ is where four-coordinate complexes are square planar.

The square-planar level pattern: three levels close together and dx²−y² far above them. Eight electrons fill the lower three and one more, all of them below the gap, and the eight-electron configuration is the one this arrangement was made for.

The tetrahedral pattern for the same four ligands: two levels below three, with a gap four ninths the size of the octahedral one. Eight electrons here leave two of them unpaired in the upper set, and the arrangement gains far less.

The g-value is the orbital coming back

The octahedral splitting, computed two ways — from point charges and from an angular overlap model — with the barycentre preserved in both. The gaps in this diagram are the energy denominators the g-shift sum divides by, so a stronger field gives a smaller shift, and the same ion in two ligand environments has two different g-values.

The electrons repel less inside the complex

The octahedral splitting from two routes — a point-charge model and an angular overlap model — which agree on the pattern and not on why. The nephelauxetic effect is invisible to the first and natural to the second, since only one of them has any covalency in it at all.

Six ligands about a metal, and the d orbitals in the field they make. Every complex analysed here is this arrangement with a different ligand, and the two numbers the spectrum gives — the splitting between the two sets, and the repulsion within them — are properties of different parts of the picture.

The moment a fit invents

The spin-crossover system, for comparison. There the moment varies with temperature because two spin states of one ion are both populated; here it varies because a singlet and a triplet of a coupled pair are. The two are different mechanisms and produce curves of similar shape, which is a further reason a single fitted number is a poor summary of either.