The spectrochemical series is not electrostatics

Ligands can be put in order. Take one metal ion, make a series of complexes with different ligands, measure where the first d–d absorption falls in each, and the ligands line up:

I⁻ < Br⁻ < Cl⁻ < F⁻ < H₂O < NH₃ < en < CN⁻ < CO

The order is called the spectrochemical series, it has been known since 1938, and its most striking property is how well it travels. Change the metal, change its oxidation state, change the other ligands, and the ordering survives — the magnitudes move around considerably, but iodide stays at the bottom and carbon monoxide stays at the top. That reproducibility is what makes the series worth explaining rather than tabulating.

The explanation offered in most first courses is electrostatic, and it follows from the crystal field model directly: a ligand that repels d electrons harder pushes the orbitals pointing at it higher, so a more negative ligand should split more. It is a clean argument from a model that is right about a great deal, and the series refutes it in the plainest possible way.

The tally #

The claim is testable and the test is a count, so this site counts it rather than gesturing at counter-examples.

Of the thirty-six pairs that can be made from the nine ligands, twenty involve ligands of different charge, and those are the pairs where an electrostatic prediction says anything at all. Three of the twenty are in the predicted order. Seventeen are not.

Three from twenty is not a model with exceptions. A rule that assigned the order at random would be right about ten, so the electrostatic prediction performs substantially worse than chance — which is the signature of a rule that has the correct variable and the wrong sign, or the wrong variable entirely.

A caution about the counting, since a tally is only as good as the set it runs over. Nine ligands is a small sample and it was not chosen at random: it is the list that appears in textbooks, which is to say the list chosen to display the series. What that biases is the range — a randomly chosen nine would probably span less — and it does not bias the direction, because the charges of these nine are distributed across the range rather than concentrated at one end. Four carry a full negative charge and sit at positions one to four; three are neutral and sit at five, six and seven; and the two at the top are one of each. Any subset large enough to test the electrostatic prediction refutes it.

The counting is done exactly as electronegativity is not one quantity counts the disagreements between four electronegativity scales: concordant and discordant pairs, tallied. The same arithmetic applied to the π parameter puts thirty-five of thirty-five comparable pairs in order.

There is a harder version of the same test, and it is worth running because a count of ordered pairs is a weak instrument: it asks only for a direction, and a quantity can get every direction right while being wrong about every magnitude. The stronger question is whether a candidate predictor reproduces the size of the splitting across a series, and the honest way to ask it is to give each candidate the same freedom — fit an exponent to each, and compare how tightly the residuals cluster.

The third bar is the interesting one and it is not a win for the model. The donor atom’s effective nuclear charge, fitted freely, does slightly better than the overlap at 1.21 — at an exponent of −1.377 that no theory predicted and that nothing explains. Five complexes cannot separate a spread of 1.21 from one of 1.31, so the correct reading is that this sample does not distinguish them, and the reason to prefer the overlap is that it arrived with its exponent already fixed while the charge had to be told what power to take. One over the bond length, given the same freedom, spreads by 1.87 and is simply worse.

The single fact that does most of the damage is the one at the top of the list. Carbon monoxide is electrically neutral and splits a d shell harder than anything else in common use. Ammonia and ethylenediamine, also neutral, sit above every halide. Iodide carries a full negative charge and is last. A model in which splitting is caused by ligand charge has no room to move here.

What does order it #

The angular overlap model has a second parameter, and the second parameter is the whole answer.

Each ligand interacts with the d shell through channels classified by their behaviour about the metal–ligand axis: σ, with no nodal plane containing the axis, and π, with one. The σ interaction touches the orbitals pointing at the ligands — the upper set in an octahedron — and the π interaction touches the ones pointing between, the lower set. The splitting is the difference:

$$\Delta = 3e_\sigma - 4e_\pi$$

Both terms are computed here rather than assumed: the coefficients three and four fall out of the eigenvalues of the angular overlap matrix for six ligands, and the sum rule that the trace is $6e_\sigma$ holds throughout.

The sign convention is where the chemistry is. A π donor has filled orbitals of π symmetry — a halide’s lone pairs — below the metal d level, and a filled orbital below pushes up. So the lower set rises and the splitting shrinks. A π acceptor has empty orbitals of π symmetry above the metal d level — carbon monoxide’s π*, cyanide’s — and an empty orbital above pulls down. The lower set falls and the splitting grows.

Two figures, one parameter changed in sign, and a factor of 2.3 in the splitting. That is the size of the effect the electrostatic model has no representation of, because a point charge has no orbitals and therefore has neither lone pairs to donate nor empty levels to accept.

Why the π parameter runs the way it does #

The ordering of ligands by $e_\pi$ is not arbitrary and does not have to be quoted either.

The halides have filled p orbitals perpendicular to the bond, and they are high in energy and diffuse — iodide’s most of all, because it is the largest and least electronegative. A filled donor orbital raises the metal level it interacts with, and it raises it more the closer in energy the two are. So iodide is the strongest π donor of the four, then bromide, then chloride, then fluoride, whose lone pairs are held far too tightly to donate much. That is exactly the order of the halides in the series, bottom to top, and it is the reverse of their order by electronegativity and the reverse of what any argument from charge density would give.

Water has two lone pairs and donates weakly. Ammonia has one, and it is used for the σ bond, so ammonia has essentially nothing left for π — which puts it at $e_\pi \approx 0$, above every halide, without needing to be a better σ donor than they are.

Cyanide and carbon monoxide have empty π* orbitals, and those orbitals are low enough to accept. The metal’s lower set is stabilised by donating into them, which both increases the splitting and does something else measurable: it weakens the C–O bond, by an amount visible in the stretching frequency. Back-bonding is two interactions computes that and compares it with six measured frequencies, and it is the independent evidence that the π-acceptor half of this story is not a device invented to save the series.

The halide reversal deserves its own sentence because it is the sharpest available test. Fluoride is the most electronegative element there is and carries the same charge as iodide; on any argument from charge density it should split hardest of the four halides, and it splits hardest of the four — but at the top of the halide group, next to water, while iodide is at the bottom of the entire series. The direction is right and the reason is inverted: fluoride wins among the halides by being the worst π donor, not the best electrostatic repeller. The two explanations agree about the order within the halides and disagree about everything else, which is what makes the halides a poor place to test the question and the neutral ligands a good one.

The arithmetic of that comparison is worth stating with the numbers in it. A hexafluoridocobaltate ion carries six fluoride ligands, a full negative charge on each, and splits by about 10,200 cm⁻¹ — less than a third of what six electrically neutral carbon monoxide molecules produce round a metal in the same geometry. Six units of negative charge against none, and the six units lose by a factor of three.

That leaves one thing to check, because so far the π parameter has only been shown to order the series and the model claims more than an ordering. The angular overlap model says the interaction is proportional to the square of an overlap integral, which is a quantity computable from the metal and donor orbitals at the measured bond length without reference to any spectrum.

Thirty-one per cent is not agreement to spectroscopic precision and is not claimed as any. What it is, is a computed quantity with a fixed exponent tracking a measured one across a doubling, which is more than an ordering and is the reason the σ and π account is worth preferring to a relabelling of the series.

What the two models can and cannot each do #

It is worth being precise about the division, because “crystal field theory is wrong” is too coarse and is not what the computation shows.

What the point-charge model gets exactly right is every ratio between geometries: four ninths for a tetrahedron against an octahedron, eight ninths for a cube, the inverted ordering in both — computed in two models, one ratio to eight decimal places, agreeing with a model that shares none of its arithmetic. It also conserves the barycentre exactly, which is a real theorem and not a convenience.

What it cannot do at all is order two ligands in the same geometry. Every complex in the series is octahedral; the geometry factor is identical for all nine; so the entire content of the series lies in a quantity the point-charge model represents by a single number, the charge, which is the number that fails.

What the angular overlap model does is put the difference where the chemistry is: in two parameters describing an interaction between orbitals. It orders the series because it has a parameter that can change sign, and a change of sign is what separates a lone pair from an empty antibonding orbital.

What neither does is compute $e_\sigma$ or $e_\pi$ from anything. Both are fitted to spectra. This site takes them as quoted numbers and says so, exactly as it takes Hückel’s heteroatom parameters as quoted in Hückel with a heteroatom — the model supplies the structure of the answer and somebody else’s spectrometer supplies its scale.

What the series is not #

Three careful denials, because each is a way of over-reading a genuine regularity.

It is not a bond-strength ordering. Δ is a separation between two sets of metal orbitals; a metal–ligand bond energy is a different quantity, and the two need not run together. A ligand can raise Δ substantially through $-4e_\pi$ while contributing no more σ bonding than a weak-field ligand does. Cyanide splits harder than water by a factor of two and the argument that it therefore binds twice as strongly does not follow from anything here.

It is not a single number per ligand. The magnitudes depend strongly on the metal and its charge: a given ligand splits a trivalent ion more than a divalent one, and a second-row metal more than a first-row one. What survives across those changes is the order, which is why the series is stated as an order.

It is not the only order ligands can be put in. Rank the same ligands by how much they reduce the interelectronic repulsion in the metal — the nephelauxetic series — and the order is different, with iodide near the top instead of the bottom. Two properties, two orderings, both reproducible, and the pattern is a familiar one: electronegativity is not one quantity found four scales that disagree about the direction of the polarity of ordinary bonds, C–H among them. A word that names a quantity is not evidence that there is one quantity.

The nephelauxetic column also settles what the series does not say about colour. Reading the same nine splittings as wavelengths, the order spans a factor of nearly five in energy — iodide at 1,429 nm, carbon monoxide at 294 nm — and only three of the nine fall inside the visible range at all. A ligand’s position decides where a complex absorbs; whether that absorption is something anyone can see is a separate question with a different answer.

What the order decides #

The series would be a curiosity if it only predicted colours. It predicts the spin state, which is a structural and magnetic property, and the mechanism is arithmetic.

A complex takes the low-spin arrangement when the splitting exceeds the energy needed to pair two electrons in one orbital, and the crossover is at exactly $\Delta = P$ — computed, for all four fillings that have a choice, in the pairing energy decides the moment. Pairing energies for the first-row ions are around 15,000 to 25,000 cm⁻¹ and do not depend on the ligand. So the series is a list of which ligands can clear that bar.

Plotting a d⁶ ion’s two possible fillings against the splitting makes the crossing a single point: the two total energies are straight lines in Δ, they meet at Δ = P exactly, and which of them is lower on either side is decided by nothing but the sign of Δ − P. Six fluorides put an iron(III) ion to the left of that crossing and six cyanides put it to the right.

Iron(III) with six fluorides has a splitting near 10,000 cm⁻¹ and a pairing energy near 30,000: high spin, five unpaired electrons, and a large magnetic moment. The same ion with six cyanides has a splitting near 35,000: low spin, one unpaired electron. Two compounds of one ion, differing only in the ligand, with magnetic moments differing by a factor of four — and the property that decides between them is the one an electrostatic argument gets backwards, since cyanide and fluoride carry the same charge.

That is what makes this a wrong-field essay rather than a footnote about a model’s accuracy. The mistaken explanation is not merely inelegant; applied to a real question with two ligands of equal charge, it has nothing to say at all.

The halides are the case where that goes furthest. On all four electronegativity scales they run fluorine first, then chlorine, bromine, iodine; in the spectrochemical series they run in exactly that order too — and for the opposite reason, fluoride leading because it is the worst π donor rather than the best repeller. Two orderings of one set of ligands that agree completely about the four halides and disagree about everything else is why the halides are the wrong place to test the question.

One ligand in two places in the series #

There is a test that separates the two accounts more sharply than any comparison between different ligands, because it holds the ligand fixed and changes only which end of it faces the metal.

Some ligands can bind through either of two atoms. Thiocyanate binds through its sulfur or through its nitrogen; nitrite binds through its nitrogen or through an oxygen. In each case the ligand is the same species, with the same charge, the same number of electrons and very nearly the same size — and it appears in two well-separated places in the spectrochemical series depending on which atom is doing the binding.

Sulfur-bound thiocyanate sits near the weak-field end, among the chlorides. Nitrogen-bound thiocyanate sits in the middle, above water. Nitrogen-bound nitrite sits near the strong end, above ammonia and the diimines.

An electrostatic account has nothing to say about that. The charge is identical, the distance is similar, and the point-charge model’s only inputs are those two. It predicts one position and the ligand has two.

The σ and π account predicts exactly two, because which atom binds decides both the σ donor orbital and what π orbitals are pointed at the metal, and those differ completely between a sulfur end and a nitrogen end of the same ion.

The consequence is visible without a spectrometer. Pentaamminecobalt(III) complexes of nitrite exist in both forms and they are different colours — the nitrogen-bound one yellow, the oxygen-bound one red — with the same formula, the same charge and the same metal. One of the two converts slowly into the other on standing, so a single sample changes colour while its composition does not.

Two compounds identical in every quantity an electrostatic model contains, differing in a splitting large enough to change what colour they are, is as clean a refutation as this subject offers. It also explains why the series has to be a list of ligands as bound rather than of substances, which is how it is always printed and rarely explained.

Who found it, and when #

Ryutaro Tsuchida published the series in 1938, from absorption spectra of cobalt and chromium complexes, and stated it as an empirical ordering with no explanation attached. The explanation took another quarter of a century: the π-bonding account belongs to the ligand field theory of the 1950s and to the angular overlap model of the 1960s, by which time the failure of the electrostatic reading was thoroughly established among the people doing the work.

The gap between 1938 and the 1960s is the interesting part of the history. The series was useful immediately — it predicts spin states, colours and reactivity well enough to design experiments with — and being useful is not the same as being understood. A rule that works can be taught with a wrong reason attached for a long time, because the rule keeps working and the reason is never the thing being tested.

Why the wrong explanation survives #

It survives because it is right about everything except the one thing it is asked about.

The electrostatic model is right about how many levels there are, right about which orbitals go up, right about every ratio between geometries, and right about the barycentre. Every picture it produces is the correct picture. It is wrong only when two ligands in the same geometry are compared — and comparing two ligands is nearly the whole of coordination chemistry.

It also survives because the alternative costs something. Explaining the series needs two parameters instead of one, a sign convention, and the idea that an empty orbital on a ligand can lower the energy of a filled orbital on the metal — which is back-donation, and is a harder thing to draw than a negative charge. A first course that has ten minutes for coordination chemistry will spend them on the picture that is right about the geometry.

There is a lesson here that recurs across chemistry and is worth naming. A model can reproduce a body of results for a reason unrelated to the mechanism it describes, and the way to find out is to ask it a question whose answer depends on the mechanism rather than on the structure. The geometry ratios do not depend on why the field splits the shell; the spectrochemical series does. That is the same test hybridisation does not explain applies to hybrid orbitals and the same one a spectrum counts environments, not atoms applies to a spectrum: find the question whose answer separates the accounts, and ask that one.