The same count, two oxidation states

Two methods are taught for counting the valence electrons of a transition-metal complex, they are both correct, and every textbook that gives both says they agree.

They do agree, on the total. This essay computes both for ten complexes to show exactly which quantities are invariant and which are not, and then asks a measurement which of the non-invariant ones is describing the electrons.

The two methods #

The neutral method gives the metal its group number of electrons and every ligand the electrons it would bring as a neutral fragment: two for a carbonyl, one for a chloride radical, five for a cyclopentadienyl radical. Then the complex’s charge is subtracted.

The ionic method assigns the metal an oxidation state, gives it whatever d electrons are left after that, and gives every anionic ligand a full pair: two for chloride, six for cyclopentadienide.

The two differ in where the electrons of a metal–ligand bond are booked, and nowhere else. An anionic ligand either brings one electron and leaves the metal neutral, or brings two and has taken one from the metal — and the sum is the same either way, by construction.

What is invariant #

The total. Eighteen for chromium hexacarbonyl, iron pentacarbonyl, nickel tetracarbonyl, ferrocene, hexaamminecobalt(III) and hexaaquairon(II); sixteen for tetrachloroplatinate; twenty for hexaaquanickel; seventeen for vanadium hexacarbonyl.

That invariance is the whole content of the rule, and it is why the rule is useful. It is a count of electrons in orbitals, and electrons do not know which convention was used to book them.

The three complexes that are not at eighteen are worth a sentence each, because the rule is usually quoted without them. Tetrachloroplatinate is square planar and stops at sixteen, because one metal orbital is pushed too high to use. Hexaaquanickel is at twenty, with two electrons in weakly antibonding orbitals that water is too weak a field to make prohibitive. And vanadium hexacarbonyl is a stable seventeen-electron radical, which is why it is paramagnetic.

What is not #

The oxidation state. Hexaamminecobalt(III) is cobalt(III) ionically and neutral cobalt on the other convention. Ferrocene is iron(II) or neutral iron. The word “state” suggests a property of the compound and it is a bookkeeping choice.

The d count. This is the one that matters, because it is what a ligand-field diagram is labelled with. Hexaaquairon(II) is d⁶ ionically and d⁸ by group number; hexaaquanickel is d⁸ or d¹⁰; hexaamminecobalt(III) is d⁶ or d⁹.

And everything computed from the d count: the spin state, the number of unpaired electrons, the ligand-field stabilisation, the predicted magnetic moment, and whether the complex should distort. All of them — including whether the compound is a Jahn–Teller case at all, which is a statement about a degenerate configuration and therefore about a count.

The measurement that decides #

A d count is not observable and something computed from it is. The number of unpaired electrons is measurable, through the magnetic moment, and the spin-only formula converts one to the other.

The three discriminating cases are worth reading individually.

Hexaamminecobalt(III) is diamagnetic, measured. The ionic count says d⁶ in a strong field, which is six electrons in three low orbitals, all paired, moment zero. The group-number count says d⁹, which has an odd number of electrons and cannot be diamagnetic under any field strength whatever.

Hexaaquairon(II) has a moment of 5.40. Ionic d⁶ in a weak field gives four unpaired electrons and a spin-only 4.90 — the remaining 0.5 is an orbital contribution, which can be computed and is a known feature of that ion. Group-number d⁸ gives two unpaired and 2.83, which is not within reach of any correction.

Hexaaquanickel has a moment of 3.20. Ionic d⁸ gives two unpaired and 2.83; group-number d¹⁰ gives a filled shell and predicts diamagnetism, which is contradicted by the measurement outright.

So the convention that describes the electrons is the ionic one, and the neutral one is a device for getting the total right. That is not a criticism of it — the total is what the rule is for — but a d count read off the neutral method’s “metal contribution” is not a d count.

Why the total has to be invariant #

The invariance is worth proving rather than observing, because a proof says what would break it.

Let the metal’s group number be $g$, the complex’s charge $q$, and let there be $\ell$ neutral two-electron donors and $x$ anionic ligands. The neutral method gives

$$N = g + 2\ell + x - q$$

The ionic method assigns an oxidation state $\mathrm{ox} = x + q$, so the metal contributes $g - x - q$ d electrons, and every ligand contributes a pair:

$$N = (g - x - q) + 2\ell + 2x = g + 2\ell + x - q$$

The same expression. The two methods differ by adding and subtracting $x + q$, which is why nothing about the total can depend on the choice.

What that derivation exposes is where the conventions do differ: they differ in $x$ — in which ligands are classified as anionic. And the total is invariant under moving a ligand between the two categories only if the oxidation state moves with it, which is exactly the nitrosyl case below.

The ligand that is a choice as well #

There is a case where the ionic convention does not settle the oxidation state either, and it is not exotic.

Nitric oxide brings three electrons as a neutral fragment. Counted as NO⁺ it is a two-electron donor, like a carbonyl, and the metal has been reduced by one. Counted as NO⁻ it is a four-electron donor and the metal has been oxidised by one. Both appear in the literature, applied to the same compounds, and the choice is usually justified by the M–N–O angle: linear nitrosyls are counted as cations, bent ones as anions.

The ambiguity is not a defect of nitrosyls specifically. It is the general case of a non-innocent ligand — one whose own oxidation state is not determined by the structure — and the class includes dithiolenes, quinones, nitrosyls and dioxygen complexes. For every one of them the total electron count is well defined and the metal’s oxidation state is not.

That is worth stating as the general form of the finding: a quantity that is a difference between two conventions is not a measurement, and the total is not a difference.

The one place the neutral method is better #

It would be unfair to leave the neutral method as merely a device, because there is a class of compounds where it is the more useful of the two.

Organometallic compounds of low-valent metals — carbonyls, olefin complexes, metal–metal bonded clusters — have bonds that are not remotely ionic, and assigning an oxidation state to the metal in them is an exercise in fiction. Dicobalt octacarbonyl’s cobalts are cobalt(0) by convention and are bonded to each other; ferrocene’s iron is iron(II) and its bonding to the rings is largely covalent.

For those the neutral method’s bookkeeping matches the chemistry: each fragment brings what it has, nothing is transferred, and the count comes out. It is also the method that handles metal–metal bonds naturally, since each bond contributes one electron to each metal.

So the two conventions are suited to two ends of a spectrum — ionic complexes of high-valent metals at one end and covalent organometallics at the other — and the compounds in the middle are the ones where the choice is visible and the arguments happen. That is the ordinary situation for a classification, and the mistake is not choosing badly but forgetting that a choice was made.

What a d count is really doing #

There is a reason the ionic d count wins the comparison above, and it is not that the ionic method is more honest about the bonding.

The d count is the input to a ligand-field diagram: five orbitals split by a field, filled by that many electrons, with the spin decided by the competition between the splitting and the pairing energy. That model was built for ions in fields, its parameters are fitted to ionic compounds, and the count it takes is the count of electrons on the metal after the ligands have taken their pairs.

So the ionic convention wins because it is the convention the model was written in, not because it is a better description of where the electrons are. A calculation of the actual charge on the metal in hexaaquairon gives something well short of +2, because the bonds are not fully ionic — and a partial charge is itself a quantity with a convention in it.

The model the d count feeds is five d orbitals split by an octahedral field, computed two ways that agree about the pattern. The splitting is what the d count is read into, and reading the count off the wrong convention puts the ion in the wrong column of that diagram.

What the count decides is a spin state: for six electrons it is a competition between the splitting and the pairing energy, with the crossover at Δ = P exactly. A d⁶ ion has a choice and a d⁸ one does not, so getting the count wrong can change whether there is a question at all.

The measurement is read through a formula, and the formula has its own failure #

The moment decides between the two d counts, and it decides by a wide margin: 4.90 against 2.83, with 5.40 measured. It is worth noticing that the winning prediction is also wrong, by half a Bohr magneton, and that the direction of that error is predictable from the same d count — which makes the measurement decide twice rather than once.

The spin-only formula counts unpaired spins and nothing else. It assumes the electrons’ orbital angular momentum contributes nothing, and that assumption is not free: it holds when the ground term is orbitally non-degenerate and fails when it is not.

Which is decidable from the configuration. In an octahedral field the ground term of a high-spin d⁶ ion is $^5T_{2g}$ — orbitally triply degenerate, because there is a single hole in the t2g set and it can sit in any of three orbitals. A degenerate set of that kind carries orbital angular momentum, the orbital moment adds to the spin one, and the measured value exceeds spin-only. It duly does: 5.40 against 4.90.

The pattern holds across the row and is worth setting out, because it turns a nuisance into a second prediction:

The A terms sit on the formula and the T terms sit above it, every time, and the excess is largest where the orbital degeneracy is greatest.

So the hexaaquairon measurement confirms the ionic count on two independent grounds. Its magnitude is near 4.90 and nowhere near 2.83. And its deviation from 4.90 is positive and of the size a $T$ ground term produces — where the alternative d count, being d⁸, would predict an $A$ ground term, a moment near 2.83 and essentially no deviation at all.

That second test is the stronger of the two, because it does not depend on the spin-only formula being right. It depends only on the formula failing in a direction the configuration predicts, and a wrong configuration would have to be wrong about the failure as well as about the value.

It also sharpens what the essay is claiming about conventions. The oxidation state is a convention and the d count that follows from it is a convention. But one of the two available conventions makes a prediction that survives being checked twice against the same measurement, and the other does not — which is as close as a convention ever comes to being right.

There is a boundary on that argument worth stating, because it is the place the table above stops working. The orbital contribution is quenched by anything that destroys the orbital degeneracy, and a real complex has several such things: a distortion away from perfect octahedral symmetry, a low-symmetry ligand set, the crystal field of the surrounding lattice. So the measured values for the $T$-term ions are a range rather than a number — cobalt’s runs from 4.3 to 5.2 across compounds — and the range is a measure of how far each compound has been distorted rather than a scatter in the measurement.

Which means the second test is directional rather than quantitative. It establishes that the moment must lie above spin-only and cannot say by how much, because the amount is a property of a geometry this arithmetic never sees. The first test carries the magnitude and the second carries the sign, and a convention that gets both is doing more work than a bookkeeping rule has any right to.

Where the model stops #

The spin-only formula ignores orbital angular momentum, which is why the measured moments here exceed the predictions for the later ions. The comparison survives because the discrepancy between conventions is several times the formula’s own error, and the argument needs exactly that margin.

Only octahedral complexes have moments predicted. A square plane and a trigonal bipyramid split their d orbitals differently, so the five-orbital-in-two-sets model is not the right one and no moment is quoted for them.

A ligand’s classification is an input. Whether a ligand is counted as anionic is decided before the arithmetic begins, and the nitrosyl case is the one where that decision is visible; for the rest of the table it is standard and unambiguous.

Nothing here is a stability argument. A complex obeying the eighteen-electron rule is not thereby stable and one violating it is not thereby unstable — the twenty-electron and seventeen-electron entries in the table are perfectly ordinary compounds, and the count is a shell closure rather than a law.

And nothing here computes a charge. The oxidation state is a bookkeeping quantity throughout, and the fact that it predicts the magnetism is a fact about the ligand-field model rather than evidence that the metal really carries that charge.

A prediction that depends on the d count and not on the total is which geometry a d⁸ complex takes, computed from the ligand-field stabilisation against the pairing energy. That is exactly the kind of claim this essay is a caution about: the invariant total is safe and the d count is not.

Even the invariant total has a condition on it: which count closes an octahedral shell depends on the π character of the ligands. Eighteen for a π acceptor, twelve for a strong donor — so the total is invariant under a change of convention and not under a change of ligand.

What a carbonyl is actually doing is neither convention’s picture: donating from its σ and accepting into its π*, with the two interactions computed separately. A ligand counted as a two-electron donor is doing two things, and the count records one of them.

Three practices that follow #

The arithmetic recommends three habits, and all three are cheap.

Count by whichever method is faster and quote the total. The total is the invariant and is what the rule is for; a formalism argument about how to get there is an argument about nothing.

Quote the d count and the convention together. “d⁶” without “cobalt(III)” beside it is ambiguous by three electrons for a first-row metal, and every ligand-field prediction downstream of it changes.

And treat an oxidation state as a label rather than a measurement, particularly where a non-innocent ligand is involved. Where the compound’s magnetism, its structure or its spectrum can decide, they should — which is the same preference this collection applies to a moment that has been fitted rather than measured and to a structure that has been assumed rather than determined.

None of that is unfamiliar to a coordination chemist. What the table adds is the size of the discrepancy when the second habit is skipped: three Bohr magnetons on a measurement whose typical precision is a tenth.

What is quoted, and what is computed #

Quoted: the classification of each ligand as neutral or anionic; the measured magnetic moments; the complexes’ charges and formulas.

Computed: both totals, both oxidation states, both d counts, the spin states in a stated field regime, the unpaired-electron counts and the spin-only moments; and the comparison between each prediction and the measurement.

What the comparison requires #

Both conventions give the same total in every row. A single disagreement would mean one of the two had been written down wrongly.

With the metal at oxidation state zero the two d counts agree, and where the oxidation state is not zero they do not. Both halves, so the disagreement is traceable to the oxidation state rather than to an arithmetic slip.

Where they differ and a moment is measured, the ionic d count predicts it and the group-number count does not — by more than half a Bohr magneton, which is more than the spin-only formula’s own error.

And the three nitrosyl readings give the same total and a different d count each.

Still open: the metal’s actual charge #

The count is invariant, and which count closes the shell depends on the ligands. Between them they leave one question that the arithmetic cannot answer: what the metal’s charge actually is.

That is computable rather than conventional, and the analogous main-group case is routine — a population from a wavefunction rather than a formal charge from a bookkeeping rule. Doing it for a complex would put a number on the gap between an oxidation state of +2 and whatever charge the metal carries, and the interesting part is that the number would be a different one for every method of dividing the density up. The oxidation state would then be an integer nobody measured, sitting beside a charge that depends on how it was computed, and a report of both would show exactly how much of an oxidation state is convention.