An orbital carries no angular momentum

A moment counts electrons, not orbitals starts from the spin-only formula: feed in the number of unpaired electrons and $\sqrt{n(n+2)}$ comes out, and for most first-row complexes it is right to about a tenth of a Bohr magneton — a formula the pairing energy decides the moment then uses to tell high spin from low.

The essay also records where it fails. Cobalt(II) measures $4.80$ against a spin-only $3.87$; iron(II) measures $5.40$ against $4.90$. Every discrepancy is in the same direction — the measurement is always larger — which is the sign that something has been left out rather than that the model is noisy.

What has been left out is the orbital contribution. This essay is about why it is usually absent, why it is sometimes not, and why the group decides which.

The theorem, which is one line #

Write the operator for the $z$ component of orbital angular momentum in the five real d functions everybody draws. It comes out as $i$ times a real antisymmetric matrix: $d_{xz}$ connected to $d_{yz}$ by one unit, $d_{xy}$ connected to $d_{x^2-y^2}$ by two, and $d_{z^2}$ connected to nothing at all.

An antisymmetric matrix has zero expectation value in any real vector:

$$\langle \psi | L_z | \psi\rangle = i\,\psi^{\mathsf T} M \psi = 0 \qquad \text{for } M^{\mathsf T} = -M,\ \psi \text{ real.}$$

So any orbital that can be drawn with real lobes carries exactly no orbital angular momentum along any axis, before any field is applied and whatever the field’s strength.

That is checked here rather than assumed: ten random real combinations of the five functions, and the expectation comes back at $10^{-15}$ or smaller every time.

The usual account — the crystal field quenches the orbital angular momentum — has the causation the wrong way round. The field does not remove the angular momentum from the orbitals; it decides which combinations of them are the states, and a real combination has none to remove. Where the states are complex, no field however strong quenches anything.

That is the same choice real and complex harmonics is about, one shell down: $p_x$ and $p_y$ are real combinations of functions with $m = \pm 1$, and drawing them is choosing a basis that carries no angular momentum. The choice is legitimate exactly when the two are degenerate, and it costs the picture the property the complex functions had.

And a set can carry what none of its members does #

Take the three $t_{2g}$ functions — $d_{xz}$, $d_{yz}$ and $d_{xy}$ — and ask what $L_z$ does inside that subspace.

$d_{xz}$ and $d_{yz}$ are connected to each other, and both are in the set. $d_{xy}$ is connected only to $d_{x^2-y^2}$, which is not. So the restriction of $L_z$ to $t_{2g}$ has eigenvalues $+1$, $0$ and $-1$: an effective unit of angular momentum, exactly as a set of $p$ functions would have.

The two $e_g$ functions behave oppositely. $d_{z^2}$ is connected to nothing and $d_{x^2-y^2}$ only to $d_{xy}$, which is outside the pair. The restriction of $L_z$ to $e_g$ is zero: the set carries none either.

This is the whole mechanism, and it is worth stating in the form that generalises: quenching is a property of a state, and non-quenching is a property of a degenerate set. A single real orbital has no angular momentum; three degenerate orbitals that the operator maps among themselves have a full unit to share out, and which of them an electron is in is not decided until something breaks the degeneracy.

Which configurations are permitted #

Fill the two sets and the criterion falls out. The $t_{2g}$ set carries angular momentum only if it is partly filled: empty, half filled or full, the contributions of its members cancel exactly.

For high-spin octahedral complexes that permits exactly four configurations:

That is the rule every inorganic course states and rarely derives: A and E ground terms are quenched, T terms are not.

Against the measurements #

Nine measured moments, and the criterion sorts them.

The two largest departures from spin-only are cobalt(II) at $+0.93$ and iron(II) at $+0.50$ — d⁷ and d⁶, both permitted. The largest departure among the forbidden configurations is nickel(II) at $+0.37$, which is real and is a smaller effect with a different cause.

And the permission is not a prediction, which is the finding rather than a caveat.

Titanium(III) is d¹ — permitted, a T ground term, and by the argument above it should carry a full unit of orbital angular momentum. It measures $1.73$, which is spin-only to two decimal places.

The reason is that the permission rests on a degeneracy, and a degeneracy is the least robust thing a symmetry argument produces. A distortion of a few hundredths of an ångström removes it, the three $t_{2g}$ orbitals stop being interchangeable, the ground state becomes a single real orbital, and the theorem in the first section applies again. Which is copper is never quite octahedral’s subject arriving in a different field: a degenerate electronic state distorts, and the distortion is what removes the property the degeneracy was supplying.

So the criterion sorts the ions into cannot and may, and the second category is decided by whether anything in the molecule’s real environment has broken the degeneracy — solvent, a lower symmetry, spin–orbit coupling to an excited state, or a Jahn–Teller distortion the complex makes for itself.

The same criterion, from the other direction #

There is a second route to the permission list and it uses no matrix at all, which is worth having because two routes that share nothing are how this site checks anything.

A partly filled $t_{2g}$ set means an orbitally degenerate ground term — there is more than one way to arrange the electrons among the three orbitals at the same energy. A half-filled or full one does not: there is exactly one arrangement. So the criterion “the $t_{2g}$ set is partly filled” is the same statement as “the ground term is orbitally degenerate”, which is the form every textbook gives it in, and the T-versus-A-or-E language is a way of saying which.

The two routes agree on all eleven fillings, and they are genuinely different arguments. One is about whether a matrix restricted to a subspace has non-zero eigenvalues; the other is about how many ways electrons can be arranged. That they give the same list is the sort of agreement degeneracy is a group theorem sets up: a counting statement and an operator statement about the same object.

Where nickel’s excess comes from #

The forbidden ions are not exactly at spin-only either, and nickel(II)'s $+0.37$ is the largest of them. It has a different mechanism and it is worth separating.

An A or E ground term has no orbital angular momentum of its own, and spin–orbit coupling can mix in an excited T term that has some. The admixture is second order — it goes as the coupling divided by the energy gap to that term — and it produces a small, temperature-independent excess that grows for heavier metals, where the coupling is larger.

That is why the excess for the forbidden ions rises across the row and why it is a few tenths rather than a unit. It is also why the spin-only formula works better for the first transition series than for the second and third: the coupling constant roughly triples down a group while the ligand field splittings do not — a size comparison of the same kind as the spectrochemical series is not electrostatics makes for the splitting itself.

What a moment is not measuring #

Two things follow that are worth saying plainly, because a magnetic moment is one of the few chemical measurements that returns something close to an integer and it is therefore over-read.

A moment near spin-only does not prove the ground term is A or E. It proves the orbital contribution is small, which a distorted T term also achieves.

A moment above spin-only does not measure the number of unpaired electrons badly. It measures a different quantity — the total moment — and the count of unpaired electrons is what the spin part of it corresponds to. Reading $4.80$ for cobalt(II) as “somewhere between three and four unpaired electrons” is a category error: the number is three, exactly, and the extra $0.93$ is not electrons.

What the two units in the matrix mean #

One detail of the operator is easy to pass over and carries a fact. The connection between $d_{xz}$ and $d_{yz}$ is one unit; the connection between $d_{xy}$ and $d_{x^2-y^2}$ is two.

That is not a scaling accident. The real d functions are built from complex ones with $m = 0, \pm 1, \pm 2$, and $L_z$ multiplies each by $m$. The pair built from $m = \pm 1$ therefore carries one unit between them and the pair from $m = \pm 2$ carries two, while the $m = 0$ function carries none. Reading the matrix backwards recovers which complex functions each real one was made from — the picture remembers its parentage.

It also says which degeneracies would matter in a lower symmetry. A field that leaves $d_{xy}$ and $d_{x^2-y^2}$ degenerate — which a square-planar or tetragonal field does not, but a linear one does — would leave a set carrying two units rather than one, and the orbital contribution would be correspondingly larger. That is why linear complexes of the later transition metals have moments far above spin-only, and why the criterion above is stated for the octahedral case rather than in general.

The same theorem, one shell up and one shell down #

The argument is about d functions because that is where magnetic moments are usually measured, and it applies unchanged to every shell.

For a p shell. $p_x$ and $p_y$ are real combinations of $m = \pm 1$, so each carries no angular momentum and the pair carries one unit. A degenerate pair of p functions therefore behaves exactly as $t_{2g}$ does — which is the content of the “T–P isomorphism” an inorganic course states as a rule for reading Tanabe–Sugano diagrams, and which is here a statement about which real functions the operator connects.

For an f shell. Seven real functions, connected in pairs carrying one, two and three units, with one function carrying none. The consequences are larger for the lanthanides and for the same reason: their f electrons are barely touched by the ligands, so the degeneracies survive, so nothing quenches, and their moments are far from spin-only.

And for a σ or π orbital of a linear molecule. A π level is doubly degenerate and its two real combinations carry one unit between them, which is why a linear molecule in a π state has orbital angular momentum along its axis and why molecular term symbols carry a Λ at all.

The pattern is the same in all four cases and it is the one the first section states: realness is what quenches, and degeneracy is what prevents it. Which of the two a given molecule has is decided by its symmetry, and by whether anything has disturbed it.

Where the model stops #

Nothing here computes a moment. The criterion says which ions may have an orbital contribution; it does not say how large. Getting the size needs the spin–orbit coupling constant and the energy gaps, neither of which is computed here.

Octahedral, high spin, and one ion. The filling table is for the octahedral case; a tetrahedral field inverts the sets, at the four-ninths ratio two models, one ratio computes, so the criterion inverts with it, and a low-spin complex fills differently again.

No temperature. Real magnetic measurements are made against temperature and the interesting cases are the ones where the moment changes with it. That dependence is where the orbital contribution’s mechanism is usually established, and it is outside what a ground-state argument can reach.

The measured moments are quoted. As everywhere in this field, the comparison is between a computed permission and a measured number, and the measurement is not this site’s.

One asymmetry in the result is worth stating explicitly. The theorem in the first section is exact and unconditional — no real orbital carries angular momentum, whatever the molecule, whatever the field. The criterion in the fourth is a permission and holds only for an idealised octahedron with a definite filling.

That is the usual relation between the two halves of a symmetry argument: what symmetry forbids is exact, and what it permits is a list of candidates the chemistry then sorts. Reading the second half with the confidence appropriate to the first is how a permission gets quoted as a prediction, and titanium(III) is the reminder that they are different kinds of statement.

Degeneracy is necessary and not sufficient #

A single real orbital carries nothing and a set of three carries a unit, and the natural reading is that degeneracy is what restores the angular momentum. It is not quite that: an octahedron has a degenerate pair as well as a degenerate triple, and the pair is quenched too.

The reason is one matrix element at a time. Angular momentum operators move an orbital within a set of functions of the same $l$, and the question is whether the result lands back inside the set being considered.

For the lower set the answer is yes. Acting on $d_{xz}$ with the operator for rotation about $z$ returns $d_{yz}$, which is another member of the same three — so the three of them are closed under it, they behave as a unit of angular momentum among themselves, and an unevenly filled set of them has a moment.

For the upper pair the answer is no, twice over. One of the two, $d_{z^2}$, is annihilated outright: it is the $m = 0$ function and rotating about $z$ does nothing to it. The other, $d_{x^2-y^2}$, is sent to $d_{xy}$ — which is a perfectly good d orbital and is not in the pair. So every matrix element of the operator within the upper set is zero, and a partly filled upper set carries no orbital moment at all.

That refines the criterion in a way that matters for reading the table of ions. It is not is there an unevenly filled degenerate set; it is is there an unevenly filled set of three. An ion with a half-filled lower set and an unevenly filled upper one — high-spin d⁶ has the first, high-spin d⁴ has the second — is permitted a contribution in the first case and forbidden one in the second, although both have a degeneracy in their ground configuration.

It also explains why the exceptions to the spin-only formula among the first-row ions are the ones they are. Manganese(II) and iron(III) are half-filled throughout and quenched; chromium(III) has a filled lower set and is quenched; nickel(II) has a filled lower set and an unevenly filled upper one and is nearly quenched. The ions that miss the formula badly are the ones with an unevenly occupied set of three, which is a much shorter list than the ions with a degeneracy.

Where orbital angular momentum fits among the magnetic arguments #

The other essays on magnetism use the spin-only formula and record where it fails, decide high spin against low spin as a competition between the splitting and the pairing energy, and compute what couples two spins in a bridged pair.

This one accounts for the failure. Quenching is not a consequence of a field being strong; it is a consequence of an orbital being real, which is a one-line theorem about an antisymmetric matrix. The exception is not an orbital either — it is a set of them, and the group says exactly which fillings leave such a set partly occupied. The criterion sorts nine measurements correctly at the top and the bottom, and its failures in the middle are themselves informative: an ion the group permits and the measurement quenches is an ion whose degeneracy something has removed.

Still open is the size rather than the permission, which needs spin–orbit coupling as a quantity rather than as a mechanism — and to the temperature dependence, where the two contributions separate.