What orders the spectrochemical series, and what does not

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.

13 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.

Where a d–d band falls

Nine measured splittings converted to wavelengths, on a logarithmic scale, with the visible range shaded and each band drawn at a typical width. Three of the nine land inside it. The halides absorb in the near infrared, from 980 to 1,429 nanometres; carbon monoxide absorbs at 294, in the ultraviolet.

The same nine ligands ordered by their π parameter. A ligand at the acceptor end is one with low-lying empty orbitals — exactly the ligands whose complexes have accessible metal-to-ligand charge transfer bands, which is a second route to intense colour that has nothing to do with the splitting either.

The pairing energy decides the moment

The splittings the pairing energies compete with, as wavelengths. Seven of the nine ligands give a splitting below 22,000 cm⁻¹ — that is, below most of the pairing energies in the table — which is why high spin is the common case and low spin belongs to the strong-field end of the series.

The spectrochemical series is not electrostatics

Nine ligands’ measured octahedral splittings, plotted twice: against the ligand’s charge, which is what an electrostatic model would order them by, and against a π parameter whose sign says whether the ligand donates π density to the metal or accepts it. The first plot is a scatter and the second is a line.

The charge panel alone, which is the whole of what an electrostatic argument has to work with. The nine splittings are stacked against two values of one variable — four ligands at a full negative charge, three at neutral, and the pair at the top one of each — so the plot is two vertical columns rather than a trend, and the columns overlap over most of their range. Carbon monoxide and ammonia sit in the neutral column above every member of the charged one.

Three candidate predictors across five chromium(III) complexes, scored by how far the residual ratio spreads — one would be a perfect account of the magnitudes. The square of the computed σ overlap is shown twice: at the exponent the angular overlap model requires, with nothing fitted but a constant, where it spreads by 1.31; and at the best exponent a fit can find, 1.222, which buys almost nothing. That is the point of the pair — a quantity fitted at its predicted exponent and a quantity fitted freely landing in the same place is what it looks like when the exponent was not doing the work of fitting.

The double hump and what removes it

Where the optical splittings come from: ligands ordered by how hard they split a d shell, with water part way along. The numbers in the previous paragraph are for the aqua ions, and the range they span across the series is itself a fact — the splitting varies by nearly a factor of two between manganese(II) and chromium(II).

The electrons repel less inside the complex

Seven chromium(III) complexes, with Δ from the first band, B solved from the second, and β = B/918 against the free ion. Every β is below one. The two orderings at the foot are the same seven ligands sorted two ways, and they are not the same ordering.

The splitting against the square of one computed overlap, which is where the ordering the whole essay leans on comes from. It is not a ranking by charge and it is not a ranking by size: what sorts the ligands is how much their donor orbital overlaps the metal’s, squared, and the nephelauxetic ordering follows the same axis for the same reason.

What orders the spectrochemical series and what does not: charge on the ligand explains part of the ordering and π character explains more, and neither explains it fully. The nephelauxetic ordering has a different explanation again, and the two are only loosely related.

The splitting against something structural

The series ordered by charge, which is the account this essay is against. Each ligand acts on the d shell through channels of stated strength and the geometry enters only as a rotation — so what varies between ligands is the channel strength, and charge does not order it.

The measured splitting of five chromium(III) complexes against the square of the computed metal–ligand σ overlap, with the proportionality the model requires drawn through the origin. Across a series in which the splitting nearly doubles, the ratio varies by thirty-one per cent — and its only free parameter is the slope of the line.

The series by charge and by the π parameter: charge fails to order it, the π parameter orders every pair. The overlap computed here is a σ quantity, so the pair it misses being a π-donor pair is the prediction rather than the excuse — and the claim is framed so that it would fail if the discordant pair were any other.

The count that cannot be broken by strength

Where the ordering that decides everything comes from: the ligands ranked by what their σ and π interactions do, with the donors at one end and the acceptors at the other. The crossing located here is the boundary between the two halves of that series.

The gap that would have to be smaller

The four donor levels and the band of metal levels a π donor is compatible with. A donor’s lone pair has to lie below the metal orbital, which bounds the sweep from below.

The predicted ratio against the metal level, with the fitted value drawn across. The curve starts above the line and goes up.

Each halide’s π scale as a multiple of fluoride’s, fitted and from the gap alone. The gap over-predicts at every one.

A denominator that fails both ways

The measured π* levels of the acceptors, and the π scales the series carries for them. One of the three rows is a stand-in and one has no fitted value at all.

Every ligand’s fitted π scale against the energy of the orbital it uses — a lone pair below the vacuum for the donors, a π* above it for the acceptors.

Every energy denominator in the argument, across that window. All of them are positive throughout, which is the condition that a donor donates and an acceptor accepts.

The overlap the model is not proportional to

Both overlaps, squared, for five chromium(III) donors at their measured bond lengths.

The computed Sπ²/Sσ² beside the fitted eπ/eσ, for the five donors.

The two overlaps, the computed ratio and the fitted one, with each ligand’s π character.

A contraction that cannot reach three of them

The metal effective charge each ligand would need for the model’s ratio to equal the fitted one, against the range Slater’s rules allow chromium.

Chromium’s 3d effective charge in each oxidation state, by Slater’s rules: eighteen core electrons screening completely and each companion 3d electron by 0.35.

The model’s computed π/σ ratio for each ligand as the metal’s 3d effective charge is swept across the whole Slater window, with the fitted parameters drawn as dashed lines.

The correction that moves three of them backwards

For each complex, the whole range of π/σ ratios the model can produce as its donor atom is taken through every oxidation state it has, with the fitted parameter marked beneath.

How far each ligand’s computed ratio travels across a whole window, for the metal’s contraction and for the donor’s.

Each ligand’s computed ratio at the neutral donor used earlier and at the charge its donor actually carries, with the fitted parameter marked.

The channel that points at the metal

Each ligand’s filled π and empty π* against a metal d level, on one energy scale with the vacuum at zero.

The coefficient each ligand’s filled π and empty π* place on the donor atom the metal touches and on the far atom beyond it.

Each ligand’s donor and acceptor term — a squared overlap over a measured gap — drawn to one scale.