The g-value is the orbital coming back

An orbital carries no angular momentum — not in the sense that the electron is standing still, but in the sense that a real d orbital is an eigenfunction of no component of $\hat{L}$, and the expectation value of every component in it is exactly zero. The ligand field is what does this: it splits the five d orbitals into sets that a rotation about any axis cannot mix, and once the ground state is a single real orbital the moment is quenched.

Which leaves a puzzle, because the quenching is never total. Measured magnetic moments of first-row ions run above the spin-only value by amounts of a few tenths of a Bohr magneton, and electron paramagnetic resonance reports $g$-values that are not the free-electron 2.0023 — 2.39 for copper(II) along one axis, 1.94 for titanium(III).

The orbital moment comes back, and the mechanism is spin-orbit coupling.

The mechanism, and where the numbers come from #

Spin-orbit coupling is a term $\lambda\, \hat{\mathbf{L}} \cdot \hat{\mathbf{S}}$ in the Hamiltonian, and it does not commute with the ligand field. So it mixes the ground state with excited d states, and to second order the mixing changes the response to a magnetic field:

$$g_i = g_e - 2\lambda \sum_{n \neq 0} \frac{|\langle n | \hat{L}_i | 0 \rangle|^2}{E_n - E_0}.$$

Three ingredients, and they have different statuses.

$\lambda$ is quoted. It is a free-ion spin-orbit constant, measured on the gaseous ion, and nothing here computes one.

The energy denominators are measured. They are the ligand-field splittings the optical spectrum gives — the same numbers a d–d band reports.

The matrix elements are computed, and this is the part worth doing rather than looking up. The angular momentum operators are written out in the real d functions the ligand-field problem uses, as $i$ times real antisymmetric matrices, and the elements are read off.

The operators, checked before they are used #

Writing down $\hat{L}_x$, $\hat{L}_y$ and $\hat{L}_z$ in the real d basis is exactly the sort of task where an error is plausible and invisible: the matrices are sparse, the signs are conventional, and a wrong entry produces a g-value that looks reasonable.

Three checks make them safe, and all three are relations rather than values.

Hermiticity. Each is $i$ times a real antisymmetric matrix, so the operator is Hermitian, and every element must satisfy $M_{ij} = -M_{ji}$.

The commutator. $[\hat{L}_x, \hat{L}_y] = i \hat{L}_z$ — which, with $\hat{L} = i M$, reads $M_x M_y - M_y M_x = M_z$. This is what makes the three matrices one operator rather than three unrelated tables, and it is the check with real teeth: it involves all three at once and a single wrong sign anywhere breaks it.

$\hat{L}^2 = 6$. For every d function, since $l(l+1) = 6$ when $l = 2$. A missing element shows up immediately.

The commutator catches a genuine class of error, and it is worth recording what kind. An $\hat{L}_z$ matrix with one of its two blocks given the wrong sign is invisible to a quenching calculation, which reads the eigenvalues of the restricted matrix, and flipping the sign of a two-by-two block sends $\{+2, -2\}$ to $\{-2, +2\}$, which is the same set. Nothing that uses only the eigenvalues can see it. It is wrong all the same — with that sign the three components do not satisfy the commutator — and only a calculation needing all three components exposes it.

Copper, and the four-to-one #

Copper(II) is d⁹, which is best thought of as a single hole in a filled shell. In a tetragonally elongated octahedron the hole sits in $d_{x^2-y^2}$, and the sum above has one term per component.

$\hat{L}_z$ connects $d_{x^2-y^2}$ to $d_{xy}$ with a matrix element of exactly 2. $\hat{L}_x$ connects it to $d_{yz}$ with a matrix element of exactly 1. So

$$g_\parallel = g_e - \frac{8\lambda}{\Delta_{xy}}, \qquad g_\perp = g_e - \frac{2\lambda}{\Delta_{yz}},$$

and the famous eight and two are two squared matrix elements rather than fitted coefficients. The factor of four between them is why the g-tensor of a copper(II) complex is as anisotropic as it is, and it is a property of the shape of the d orbitals.

For copper $\lambda$ is negative — $-830$ cm⁻¹ — because the shell is more than half full, and a negative $\lambda$ with a minus sign in front gives a shift upward. Computed: $g_\parallel = 2.417$ against a measured 2.39.

Why the denominators are not the same #

The two components divide by different splittings, and the reason is worth following because it is where the geometry enters.

$\hat{L}_z$ rotates about the tetragonal axis, and the pair of orbitals it connects — $d_{x^2-y^2}$ and $d_{xy}$ — both lie in the plane perpendicular to that axis. Rotating one into the other is a rotation within the plane, and the energy separating them is the in-plane splitting.

$\hat{L}_x$ rotates about an axis in the plane, and connects $d_{x^2-y^2}$ to $d_{yz}$, which sticks out of the plane. The energy separating those two is a different splitting, set by how much the ligands along the tetragonal axis have been pulled away.

So the anisotropy of the g-tensor has two sources multiplying together: the four-to-one ratio in the squared matrix elements, and the ratio of the two splittings. For a strongly elongated complex the second can be large, and the two act in the same direction — the axial ligands are further away, so the $yz$ orbital is lower, so the denominator is larger and $g_\perp$ is smaller still.

Reading the two g-values therefore gives, in principle, both splittings — and this is how EPR is used on copper proteins, where the optical spectrum is obscured by everything else in the sample and the resonance is not.

The sign counts the electrons #

Titanium(III) and vanadium(IV) are both d¹, with $\lambda$ positive — $+155$ and $+250$ cm⁻¹ — and both shift downward. Computed 1.940 for titanium against a measured 1.94; 1.902 for vanadium against 1.93.

The sign of $\lambda$ is fixed by Hund’s rules: positive for a shell less than half full, negative for one more than half full. So the direction a resonance line moves from the free-electron value is a statement about the electron count, and it is a statement that can be read off a spectrum without knowing anything else about the compound.

That is a genuinely useful piece of chemistry, and it is why EPR is a routine tool for oxidation-state assignment. A $g$ above 2.0023 says the ion is more than half full; below says less. It is one bit of information, obtained from the direction of a shift rather than its size, and it does not depend on any of the quantities that are hard to get right.

What is left over is a covalency #

The computed shifts are consistently larger than the measured ones, and the ratio is the interesting quantity. Dividing measured shift by computed shift gives 0.93 for copper along the axis, 1.00 for titanium, 0.72 for vanadium.

That ratio is the orbital reduction factor, usually written $k^2$, and its meaning is direct. The spin-orbit coupling constant used is the free ion’s, so the calculation assumes the unpaired electron spends all its time on the metal. It does not: the ligand field is not a field of point charges but a set of covalent interactions, and an electron partly delocalised onto a ligand is partly out of reach of the metal’s own spin-orbit coupling — which is much larger than a first-row ligand atom’s.

So a value of $k^2$ below one is a measurement of covalency, obtained from a magnetic resonance. The vanadyl ion’s 0.72 says that something like a quarter of the unpaired electron is not on the vanadium, which is a strong statement about a bond made without any calculation of one.

There is a second, independent route to the same conclusion in this collection: the reduction of the electron-electron repulsion inside a complex, where a spectroscopic fit gives ratios between 0.53 and 0.98. Two entirely different measurements — one magnetic and one optical — reporting the same thing, and agreeing in order of magnitude. Neither is a calculation of a wavefunction, and both are constraints on one.

The half-filled shell, which shows nothing #

The clean control case is a d⁵ high-spin ion — manganese(II), iron(III). Every d orbital singly occupied, total orbital angular momentum zero by construction, and $\lambda$ effectively zero for a half-filled shell.

The prediction is a g-value of exactly 2.0023, and the measurement is 2.00 to two or three decimals. Manganese(II) is the ion whose EPR spectrum is a textbook example precisely because there is nothing anisotropic about it: no shift, no orientation dependence, six sharp hyperfine lines and nothing else.

That is the case an account of the g-shift has to get right for the right reason, and the reason is that there is no excited state of the same spin to mix with. Everything above depends on there being one.

What a distortion does to the answer #

There is a circularity in the copper case that has to be broken, and breaking it needs a separate result.

The perturbation treatment assumes a non-degenerate ground state: the sum has $E_n - E_0$ in the denominator and a degenerate ground state makes some of those zero. A d⁹ ion in a regular octahedron has a doubly degenerate ground state, so the calculation as written does not apply to it.

It never has to. A d⁹ ion in a regular octahedron is unstable against distortion — copper is never quite octahedral — and the distortion is precisely what removes the degeneracy and creates the splittings the sum divides by. The same effect that makes the perturbation legitimate also supplies its denominators.

That is a satisfying arrangement and it is not a coincidence. Both are consequences of the ground state being orbitally degenerate in the undistorted geometry: the degeneracy makes the distortion favourable, and the distortion converts the degeneracy into an energy gap.

The size matters as well as the existence. A copper(II) complex with a small tetragonal distortion has small denominators and therefore a large g-shift, and a strongly distorted one has a smaller shift. So the g-tensor is a probe of the distortion, and comparing g-values across a series of copper complexes is comparing how far each has moved from octahedral.

What this cannot say #

One electron or one hole. The perturbation sum written above is a one-electron expression. For d² or d⁷ the ground state is a many-electron term with its own orbital degeneracy, the treatment needs the full term symbols, and the shift can be far larger — cobalt(II) measures nearly a full Bohr magneton above spin-only for exactly this reason.

Second order only. The expansion parameter is $\lambda/\Delta$, which for copper is about a twentieth and for a second- or third-row metal is not small at all. Fourth-row spin-orbit constants run to several thousand wavenumbers and the perturbation series stops being useful.

A quoted $\lambda$, in the same way the spectrochemical series is a table of measurements rather than a computed ordering. The whole calculation rests on it, and its provenance is a free-ion atomic spectrum. That is the same status as the pairing energies used for spin states: a quantity taken from measurement and used as an input, with the consequences computed.

The two quantities a magnet reports #

It is worth setting this beside the other common magnetic measurement, because the two are usually taught together and they report different things.

A magnetic susceptibility gives an effective moment, which for a first-row ion is compared against the spin-only value $\sqrt{n(n+2)}$ and found to exceed it by an amount that is always positive. That is a scalar: it says how much orbital contribution there is and cannot say which way.

A g-value is a tensor and it has a sign. It says which way, it says how the answer depends on direction, and — through the reduction factor — it says how much of the electron is on the metal.

The two are not independent: the same mixing produces both, and the excess moment can be computed from the same sum. But the susceptibility averages over orientations and over the whole thermal population, so the individual pieces are lost in it. That is a general feature of the two kinds of experiment, and it is the reason the resonance measurement was worth developing at all.

How far from two, and the element that decides it #

The shift’s direction is a count and its size is a ratio, and it is worth putting numbers on the ratio, because the range it spans across the periodic table is four orders of magnitude and it decides which compounds the measurement is useful for at all.

The deviation goes as the spin–orbit coupling constant divided by the splitting to the state being mixed in. The denominator is a ligand field and varies over a factor of about three. The numerator is an atomic property and varies enormously, because spin–orbit coupling scales steeply with nuclear charge — roughly as its fourth power for a given shell.

Divide by a splitting of order ten to twenty thousand wavenumbers and the consequences follow immediately.

An organic radical has essentially no shift. With a coupling of thirty wavenumbers and an excitation energy of tens of thousands, the deviation is in the fourth decimal place: organic radicals have g values of 2.0023 to about 2.008, and the whole of organic radical chemistry lives in that narrow band. A radical centred on sulfur rather than carbon moves to about 2.02, which is a large shift by those standards and comes entirely from the element.

A first-row metal ion has a shift of tenths. Copper(II), with the largest coupling constant of the row, gives deviations of 0.2 to 0.4 along its unique axis, and the g values near 2.2 that are the signature of a copper site in a protein or a mineral.

A heavy metal has no useful reference at all. With a coupling of thousands the perturbative estimate stops being small, and measured g values run from near zero to above six.

That range is what makes the measurement worth different things in different fields. For an organic radical, the g value is nearly useless as a fingerprint and the hyperfine structure carries all the information — which is why organic EPR spectra are read as coupling patterns rather than as positions. For a metal ion it is the reverse: the g value alone identifies the metal, the oxidation state and often the geometry, before any hyperfine is resolved.

So the same experiment is a different instrument on the two sides of the periodic table, and which one it is is decided by a single atomic constant that varies by a factor of a hundred between them.

The same constant sets the boundary of the arithmetic used here. Everything above is a first-order perturbation — one matrix element over one energy denominator — and that is a good approximation while the coupling is small beside the splitting. For a first-row ion the ratio is under a tenth and the expansion is sound. For a third-row ion it approaches one, at which point the states being mixed are not a ground state with a correction but a genuine mixture, and the g value stops being a shift from two and becomes a property of a state with no spin-only ancestor to be shifted from.

Still open: hyperfine structure, and what the reduction factor tracks #

Two directions, and the first is uncomfortable.

The hyperfine structure — the interaction between the unpaired electron and the metal nucleus — carries more information than the g-tensor and is harder to compute, because it depends on the electron density at the nucleus, which is a property of the s character mixed into the ground state. Estimating it needs a wavefunction rather than a set of matrix elements, and an honest account of what a Gaussian basis does to a wavefunction near a nucleus does not encourage optimism.

The second is the systematic one. The reduction factor is measured, per compound, per axis, and it varies. Whether it correlates with anything else measurable — with the nephelauxetic ratio, with a bond length, with a position in the spectrochemical series — is a question about a table rather than a calculation, and it is the sort of question that turns a fitted parameter into a quantity.