The pairing energy decides the moment

An iron(III) ion has five d electrons. Six fluoride ligands round it give a compound with five unpaired electrons and a magnetic moment near 5.9 Bohr magnetons; six cyanide ligands give one unpaired electron and a moment near 2.3. Same metal, same oxidation state, same number of ligands, same geometry.

The difference is a competition between two energies, and both of them have to be known before the answer can be predicted.

The competition #

Filling a split d shell is a sequence of decisions, and each decision has the same form.

The first three electrons go into the lower set with parallel spins: nothing has to be paid, because there are three orbitals and no reason to double up. The fourth electron has a choice. It can go into the upper set, which costs the splitting Δ, or it can pair with one of the three already there, which costs the pairing energy P.

Whichever is smaller wins, and the comparison repeats for the fifth, sixth and seventh electrons. From the eighth on there is no choice again: the lower set is full whatever happens, and the remaining electrons go into the upper one.

That is the whole model, and running it over every filling gives an answer that is usually presented as a list to memorise.

Four configurations, counted #

Exactly d⁴, d⁵, d⁶ and d⁷ have a spin choice in an octahedral field. The other seven fillings give the same unpaired count either way.

The enumeration is worth watching rather than believing. d¹, d² and d³ never have to pair, so the two rules agree. d⁸, d⁹ and d¹⁰ have a full lower set under both rules, so they agree too. d⁰ is trivial. What is left is four.

That the count is four rather than some other number is the sort of fact that is normally quoted from a textbook. Here it is an output of two filling rules and a loop, which means it can be interrogated — and it was, with a result the next section records.

Every crossover is at Δ = P #

The four configurations do not merely have a choice; they change their minds at the same place.

Computing where the two fillings cross for each of the four gives $\Delta/P = 1$ in every case, to twelve decimal places. That is not obvious in advance: the four differ in how many electrons are in each set, in how many pairs each arrangement makes, and in how much ligand field stabilisation each collects, so there was no reason for four different competitions to balance at the same ratio.

The reason they do is a cancellation. Going from high spin to low spin moves one electron from the upper set to the lower — a gain of Δ — and creates one additional pair, at a cost of P. That trade is the same trade for all four, however many other electrons are present, because the other electrons are in the same places in both arrangements.

So the rule “low spin when Δ > P” is exact within this model, and it is exact for a reason rather than by approximation. What it needs is a value of P, and that is where the model stops.

The pairing energy is not computed here #

Δ is a one-electron quantity: it is a difference between orbital energies, and the whole of ligand field theory computes it.

P is not. Pairing two electrons in one orbital costs energy because the two electrons repel each other, and repulsion between electrons is exactly what a one-electron model does not contain. No amount of angular overlap arithmetic produces a pairing energy, and this site quotes them:

Two features of that table carry chemistry. Pairing energies rise with the charge on the ion: manganese(II) at 25,500 against manganese(III) at 28,000, iron(II) at 17,600 against iron(III) at 30,000. A more highly charged ion is smaller and its d orbitals are more contracted, which puts two electrons in one orbital closer together, and closer costs more.

And they are all of the same order as the splittings in the spectrochemical series, which run from 7,000 to 34,000 cm⁻¹. The competition is real: neither quantity dominates the other across the range of ordinary chemistry, which is why both spin states occur and why the choice is a useful diagnostic.

The tetrahedral four, and a claim the enumeration refuses #

The tempting claim is that a tetrahedral field offers no spin choice at all, on the strength of low-spin tetrahedral complexes being essentially unknown.

The enumeration refuses it. With two orbitals below three rather than three below two, four configurations come back with a choice: d³, d⁴, d⁵ and d⁶. They are the octahedral four reflected, as they have to be — swapping a set of three below a set of two for a set of two below a set of three is the particle–hole map, and it takes d⁷ to d³.

So the rarity is not combinatorial and the enumeration cannot explain it. What explains it is a magnitude, and the magnitude is the one two models, one ratio computes: a tetrahedral splitting is four ninths of an octahedral one for the same ligands.

Four ninths of the largest splitting in the table — carbon monoxide’s 34,000 cm⁻¹ — is about 15,100. The smallest pairing energy in the table is 17,600. So even the strongest field known, in the geometry that halves it, does not reach the weakest pairing energy in ordinary chemistry, and the low-spin tetrahedral complex has nowhere to come from.

That is a better answer than the one first written down, and the difference matters. “No choice exists” would have been a statement about arithmetic; “the choice exists and the field is never strong enough” is a statement about magnitudes that predicts what would have to change — a much larger splitting, or a much smaller pairing energy — for the exception to appear.

What the enumeration is doing #

Two filling rules, eleven configurations and a comparison is not much of a program, and it is worth saying why running it beats reading the answer off a table.

It can be asked new questions. Changing one line — two orbitals below three instead of three below two — turns the octahedral enumeration into the tetrahedral one, and the tetrahedral answer is not in the textbooks because the compounds are not. Getting it required no new thought, and it produced the result that overturned the sentence this essay’s code was written around.

It can be wrong in a visible way. A filling rule that never paired would report ten choices; one that always paired would report none. The check covers both lists and the relation between them — that the two are related by the particle–hole map, so that d⁴ pairs with d⁶ and d⁵ with d⁵ — which no single error reproduces.

It makes the exceptions computable rather than memorable. “Only d⁴ to d⁷” is a fact to be recalled; “the fillings for which the two rules give different unpaired counts” is a definition, and the four follow from it. The same view applies to the 4n+2 rule in aromaticity as a shell closure and of the eighteen-electron rule in eighteen is a count, and it is the habit rather than any one result that is the point.

What a pairing energy is made of #

The quoted numbers can be unpacked a little, even though they are not computed here, because the two contributions behave differently.

Putting two electrons in the same orbital costs Coulomb repulsion: they are in the same region of space and repel. That term is large and positive.

Separating two electrons into different orbitals with parallel spins gains exchange stabilisation: electrons with parallel spins avoid each other for reasons of antisymmetry rather than of charge, so they repel less. Pairing therefore also loses whatever exchange stabilisation the parallel arrangement had.

P is the sum of the two, and it is why the pairing energy depends on how many other parallel spins are present — which is why the tabulated values differ between configurations of the same ion.

Both terms are two-electron quantities, and both are outside every one-electron model.

The smallest system in which a pairing energy exists at all is two sites and two electrons with a cost U for double occupancy. At U = 0 the singlet is the one-electron answer; as U rises it climbs, and the triplet — which never puts two electrons on one site — does not move at all. That is a pairing energy with no d shell anywhere near it, and the competition in this essay is the same arithmetic with a splitting on the other side of it.

That system is the subject of the smallest many-electron calculation, and the connection is exact: the on-site repulsion U in that model is the model’s pairing energy. What the exact solution adds is that the energy actually paid is less than U, because the electrons rearrange to avoid paying it — which is why a measured pairing energy is a property of a whole electron distribution rather than a number that can be read off one integral.

Where the compounds sit #

The distribution of real compounds across the crossover is uneven in an informative way.

First-row metals have splittings of 10,000 to 25,000 cm⁻¹ and pairing energies of 17,000 to 30,000, so the two are comparable and both spin states are common. Almost all of the interesting magnetic chemistry of the first row lives here.

Second and third-row metals have splittings roughly 30 to 50 per cent larger for the same ligand, and slightly smaller pairing energies because their d orbitals are more diffuse. The result is that almost every second- and third-row complex is low spin, which is a strong regularity and follows from a shift in two numbers rather than from any change in the argument.

Spin-crossover compounds are the ones sitting almost exactly at Δ = P. They convert between the two states with temperature, pressure or light, and their existence is direct evidence that the crossing computed here is a real balance rather than a bookkeeping device.

What the model leaves out #

P is quoted, so half the prediction is borrowed. Everything computed here about spin states is the Δ side and the structure of the competition. The number that decides real cases comes from somebody else’s measurement, and that is said plainly rather than presenting a rule with one measured half hidden inside it.

Temperature is not in the argument. The comparison is between two energies at zero temperature. A high-spin state has more ways of being arranged than a low-spin one, so it is favoured entropically, which is why spin-crossover compounds convert towards high spin as they warm. Nothing here computes that, and the same omission applies to the coupled pairs of what couples two spins, whose measured susceptibilities are temperature curves rather than single energies.

The filling rules are rules. Deciding which orbital the next electron enters by “lowest first, singly before pairing” is Hund’s rule plus an ordering, and both are statements about repulsion imported into a model that has none. The place where that shows most sharply is a classic case: square cyclobutadiene, where the rules predict a triplet and the molecule is a singlet.

The other side of the comparison moves too #

The competition is between two quantities and the essay’s arithmetic varies one of them. It is worth saying that the pairing energy is not a constant of the ion either, because when both sides move the outcome can be decided before any ligand is chosen.

The pairing energy has two parts and only one is the obvious one. Putting two electrons in one orbital costs a Coulomb term, because they are now in the same region of space. It also costs an exchange term, because two electrons of parallel spin in different orbitals were enjoying a stabilisation that pairing them destroys. The second is why the pairing energy depends on how many parallel spins there were to lose, and therefore varies across the series rather than being one number per element.

Both parts scale the same way with the size of the d orbitals: electrons in larger, more diffuse orbitals repel less and exchange less. So a metal whose d orbitals are more extended has a smaller pairing energy.

That is the same property that makes the splitting larger. A 4d or 5d metal has more extended d orbitals than a 3d one, so it overlaps its ligands better — the splitting rises by about half from the first row to the second — and its own electrons are further apart, so the pairing energy falls.

Both sides of the comparison therefore move towards low spin, and neither moves back.

The consequence is categorical rather than statistical. Complexes of the second and third transition series are essentially always low spin, whatever ligand they are given: the weak-field ligands that keep a first-row iron high spin do not keep a ruthenium or an osmium high spin, because Δ has grown and P has shrunk and the crossing has been left far behind. High-spin second- and third-row complexes are rare enough to be remarked on individually.

That is a prediction with no free parameters in it, and it is a stronger statement than anything the first-row comparison supports. Within the first row the outcome depends on the ligand, so the model earns its keep by getting the boundary right one compound at a time. Below the first row the model says the boundary has been passed altogether, and the chemistry agrees by not producing the compounds that would test it.

It also explains why the spin-state question is a first-row question. A textbook’s high-spin and low-spin examples are all 3d, and the reason is not that anyone chose them for familiarity — it is that the first row is the only place where the two energies are close enough for a ligand to decide between them.

The measurement that closes it #

A prediction about spin states is only worth having because it is checkable, and the check is a magnetic measurement.

A moment counts electrons, not orbitals establishes that a magnetic measurement returns the number of unpaired electrons, well separated between adjacent counts. For each of the four configurations with a choice, the two states differ by two or four unpaired electrons — d⁶ by four, from 4.90 to zero — and no measurement confuses those.

So the competition of this essay is directly observable, one compound at a time, and the observations line up with the comparison. Iron(II) with six water molecules is high spin; with six cyanides it is diamagnetic. Cobalt(III) with six fluorides is high spin and with six ammonias is low spin — and the ligand that flips it in each case is the one further up the spectrochemical series, exactly as the spectrochemical series is not electrostatics orders them.

That chain — a series ordered by a π parameter, a splitting compared with a pairing energy, a crossover at Δ = P, an unpaired count, a moment — is four computations and two quoted tables long, and every link in it is checkable separately. It is the kind of chain chemistry can actually use.

Who found it, and when #

The distinction between what were then called ionic and covalent complexes — now high spin and low spin — is Linus Pauling’s, from the 1930s, and was framed in terms of which orbitals were used for bonding. Ligand field theory in the 1950s replaced that with the comparison of this essay, and the replacement is a clear improvement: Pauling’s version predicts the same two classes and gives no way to say which a given compound will fall into, while the comparison of two energies does, provided both are known.

The pairing energies themselves were extracted through the 1960s from spectra, by fitting the many-electron term structure — the same Racah parameters that make a d–d band energy differ from a splitting, as where a d–d band falls records. So the number this essay quotes and the number that essay could not compute are the same number, arrived at from the same measurements, and both are outside what a one-electron model can reach.