An integer nobody measured

The electron count of a complex is the same number under two conventions that assign the metal two different oxidation states. That leaves what the arithmetic cannot answer: what the metal’s charge actually is.

That is computable rather than conventional, and it is computed here. The answer has the shape it was expected to have — the number depends on how the density is divided up, and there is no privileged division — and it has one feature that was not expected.

The oxidation state is not in the range at all. Not a rough member of the family, not the extreme member: outside it, at the positive end, by a whole electron on the most ligand-favouring partition available.

The model, which is nine orbitals and six donors #

An octahedral complex, treated with molecular orbitals rather than with a bookkeeping rule. The metal brings one s, three p and five d orbitals; each of six ligands brings one σ donor orbital pointing at the metal. Fifteen orbitals, and $d^n + 12$ electrons.

The couplings are the Wolfsberg–Helmholz form — proportional to the overlap and to the mean of the two diagonal energies — with the angular factors the orbitals themselves supply: one for the s orbital, the direction cosine for each p, and the σ component of each d orbital along the bond, which is the same decomposition the angular overlap model uses. The overlap is kept rather than set to zero, which is the whole reason there is anything to argue about.

Solving $Hc = \varepsilon Sc$ gives nine occupied orbitals for a $d^6$ metal: six metal–ligand bonding orbitals and the three non-bonding t₂g. Eighteen electrons, which is the count the rule is about.

The model is deliberately the smallest one that has the question in it. It has no repulsion, no geometry and no π channel to begin with, and every one of those absences is a term whose effect on the count is nothing and whose effect on the charge is large — which is the essay in one sentence, and is the reason the model does not need to be better than this to make the point.

Why the charge is not one number #

A population is unambiguous only where two basis functions do not overlap. Where they do, the term $2c_ic_jS_{ij}$ belongs to neither orbital, and every population analysis in use is a rule for splitting it.

Mulliken’s rule is half each. That is a choice, not a derivation: there is no argument that says the electrons in an overlap between a chromium d orbital and a carbon lone pair are half chromium’s. Give the ligand a fraction $\lambda$ of every shared pair and the metal the rest, and the whole family is available:

The range is 2.06 electrons wide, which is larger than most of the charges chemists quote. And it is linear in λ, because the overlap population enters linearly — so there is no minimum, no plateau, and nothing in the arithmetic that picks a value out.

Löwdin’s rule is not on that line. It orthogonalises the basis first, so that there is no overlap left to divide, and then counts. That gives −2.369, which is outside the interval spanned by λ between 0 and 1 at the negative end. Two reasonable-sounding rules, three-tenths of an electron apart, and a family that contains neither of them at any distinguished point.

This is the same complaint a valence bond weight attracts in the other tradition, and it is the same complaint for the same reason: an overlap is a quantity that belongs to two things at once, and dividing it is a convention.

Where the oxidation state sits #

An oxidation state is computed by giving each ligand the whole of every bonding pair it is involved in — not a share of the overlap but both electrons — and then asking what the metal is left with. For chromium hexacarbonyl that gives $d^6$ on a neutral chromium, so the oxidation state is 0.

The most ligand-favouring member of the population family gives the metal −0.979. So the bookkeeping answer is a whole electron more positive than the most extreme population analysis anyone would write down, and more than two electrons more positive than Mulliken’s.

That is not a small discrepancy to be noted and moved past. It says the oxidation state is a different kind of object: it is not an approximation to a charge, because the family of approximations to the charge does not contain it.

What it is is a count in disguise. Assigning both electrons of every bond to the ligand is the ionic convention, and the number it leaves on the metal is the d count — so an oxidation state is the group number minus a count of orbitals, which is why it is always an integer and why it survives being computed two ways. Reading it as a charge is reading a count as a measurement, and the two are separated here by two electrons.

The term that moves the charge and not the count #

A σ-only model of a carbonyl gives the metal a charge of −2, and that is physically absurd — a chromium atom does not sit in a molecule carrying two extra electrons, and Pauling’s electroneutrality argument says so on general grounds. The missing term is well known and is computed here too: back-donation, where the filled metal d shell gives charge back into the ligands’ empty π* orbitals.

Adding it — two empty π* orbitals per ligand, coupled to the t₂g set at the same angular factors the σ channel uses — changes the answer completely.

The Mulliken charge moves from −2.009 to −0.001, and between 1.40 and 2.62 electrons end up in the ligands’ empty orbitals depending on the partition. Electroneutrality, arrived at from a calculation rather than assumed.

That coincidence is a coincidence of the parameters and is not offered as a result — a different $H_{\pi}$ would give a different number. What is a result is the contrast: adding a whole new interaction to the Hamiltonian moved the charge by more than two electrons and left the electron count at eighteen.

It could not have done anything else. The count is the number of electrons in filled orbitals, and no term in a Hamiltonian changes how many orbitals are filled — it changes where they are and what they are made of. Every partition, every parameter and every interaction leaves it alone.

Why the rule is about the count #

Put those two together and the eighteen-electron rule stops looking like a piece of accountancy and starts looking like the only statement of its kind that survives.

The charge is a coordinate. It moves with the partition, it moves with the overlap, it moves with whether the model has a π channel in it, and none of those is a measurement.

The count is a number. It is invariant under the partition — checked here at five values of λ — and invariant under adding a whole interaction, and it agrees with what the bookkeeping rule computes without any wavefunction at all, which is the cross-check that makes it worth having.

That is why the rule that worked was the one about counting. The count is invariant under a change of convention; the charge is the thing it was invariant instead of, and the thing is much larger than the difference between the two conventions.

What is quoted, and what is computed #

Nothing is quoted except the complexes. The formulas, their charges and their ligand counts are chemistry; the diagonal energies, the overlap and the Wolfsberg–Helmholz constant are model parameters, stated rather than fitted.

Everything else is computed: the fifteen-by-fifteen generalised eigenvalue problem, the occupancies, every population at every λ, the Löwdin populations through a symmetric orthogonalisation of the same overlap matrix, and the electron counts. The count from the diagonalisation is compared against the count from the bookkeeping rule for every complex, and they must agree — two routes to one number, which is what makes it the number worth having.

What this cannot say #

One set of parameters for every metal. The diagonal energies do not change from chromium to cobalt in this model, so the gap between the oxidation state and the computed charge comes out the same for every d⁶ complex in the census. A real calculation would put the metals in different places, and the constancy here is a property of the model rather than of chemistry.

No electron repulsion. Every charge here is a one-electron quantity, and the thing that most strongly opposes piling charge onto a metal is the repulsion between the electrons already there. The smallest honest version of that is a different model and is not used here.

The ligands are one orbital each. A real donor has a shape and a spread, and how much charge it will part with is a property of the bond rather than of the free molecule — which is the same finding this essay reaches, one field over and by a different calculation.

Only σ and π, and no geometry. The bond length appears nowhere; the overlap is a parameter. So this cannot say how the charge changes as a ligand is brought closer, which is the quantity a real spectroscopic argument would want.

And a partial charge is not an observable at all. That is the deepest limitation and it is not a limitation of this model: there is no experiment that measures the charge on an atom in a molecule, because there is no operator for it. Every number in this essay is a number about a division of a density, and the density is what exists.

That last point is worth separating from the others, because it is the one that does not go away with a better calculation. A complete-basis, fully correlated, relativistic treatment of chromium hexacarbonyl would produce a density that every method agreed on to many decimal places — and the question how much of it is chromium’s would still have as many answers as there are ways of drawing a boundary. The spread computed above is therefore not an error bar on a quantity; it is a measurement of how much freedom the question leaves, which is what a convention is.

What the observable quantity actually tracks #

The measurement nearest to a charge exists and is made routinely: the binding energy of a metal’s core electrons, which shifts with the metal’s environment and is the basis of an entire analytical technique. It is worth saying what that shift is found to track, because the answer is neither of the candidates and the reason is instructive.

The shift is real and it is small. Across the whole accessible range of oxidation states of a first-row transition metal, a core level moves by a few electronvolts on a binding energy of several hundred — roughly an electronvolt per unit of formal oxidation state, as a rough trend rather than a law. So the correlation with oxidation state exists and is used, and a practitioner reading a spectrum will assign a state from it.

It also fails, and there is a standard case where it fails completely. Copper metal and copper(I) oxide have essentially the same copper core binding energy — the two differ by about a tenth of an electronvolt, which is less than the width of the line. A shift that is supposed to report an oxidation state cannot distinguish zero from plus one in the element it is most often applied to. What distinguishes them in practice is not the binding energy at all but the presence of satellite structure a few electronvolts away, which copper(II) has and the other two do not, and which is a statement about the final state rather than about any charge.

That points at the reason. A binding energy is the energy required to remove an electron, so it contains two things: where the level sat before, which is the initial-state charge the essay’s partitions are trying to compute, and how much the remaining electrons relax afterwards, which is a property of how polarisable the environment is. The two are of comparable size and they run in opposite directions — a metal with more electron density around it has a lower initial-state binding energy and relaxes more strongly, and a measured shift is the residue.

So the honest answer to the essay’s question is that the observable is a difference of two quantities, only one of which is the charge, and neither is the oxidation state. It does not vindicate any member of the family of partitions above and it does not vindicate the integer either.

What it does supply is the thing the whole essay has been missing: a quantity that is not a convention. A core-level shift is measured, it depends on the environment, and it is a number two laboratories can disagree about and settle. That is more than can be said for a Mulliken charge, which two programs can compute exactly and report differently, and more than can be said for an oxidation state, which is an integer arrived at by a rule and compared with nothing.

What was checked #

The complex’s own charge is the same at every λ, exactly, because a partition moves electrons about and creates none. A scheme that failed this would be an arithmetic error rather than a convention.

And so is the electron count, because it is a count of occupancies and has no overlap in it.

The metal’s charge is not: it moves by more than half an electron across the family, and it rises monotonically with the share the ligand is given, so the family is ordered and there is no interior member for anything to prefer.

The oxidation state is more positive than every member of the family, which is the finding.

The orthogonalised population is a different number again, by more than a thousandth of an electron — which is the check that Löwdin’s rule is genuinely a fourth answer rather than a relabelling of Mulliken’s.

Adding the π channel does not move the count and moves the charge by more than an electron, with more than an electron of population appearing in the ligands’ empty orbitals — all three checked, because the first alone would pass on a model where the π channel did nothing.

Every complex’s filled orbitals hold what the ionic count says, which is the cross-check between the diagonalisation and the bookkeeping, and the census contains both a closed count and an open one so that the check is not being made on eighteen alone. A twenty-electron aqua complex is as legitimate an answer as an eighteen-electron carbonyl: which count closes a shell depends on the ligands, and the check is about agreement rather than about the number.

And the refusal is an overlap large enough to make the basis linearly dependent. At that point there is no population analysis to do, and the solver stops rather than returning a charge.

Still open: a charge that can be measured #

The obvious open question is the quantity that is observable and is nearest to a charge. A metal’s core-level binding energy shifts with its environment, it is measured routinely, and it is a genuine expectation value rather than a partition — so a calculation that produced a shift could be compared against something. The interesting question is whether the shift tracks the oxidation state, the Mulliken charge, or neither, and it is the only way to turn a convention into a measurement.

The nearer question is about the π channel’s own count. Back-donation puts electrons into orbitals that are not the metal’s and not the σ donors’, and it does so without changing how many orbitals are filled — but it does change which combinations are filled, and the eighteen-electron rule’s derivation assumes the nine metal orbitals are the ones being counted. Whether a strong enough π channel can make one of the ligand π* combinations dip below a metal orbital, and what the rule would say then, is a computation this model can do by turning one parameter.