What couples two spins

Put two magnetic ions a few ångströms apart, with something between them, and their moments stop being independent. In most such compounds the two prefer to be antiparallel, and the preference is worth tens to hundreds of wavenumbers — a substantial energy, comparable with a vibrational quantum.

The obvious explanation is that each ion’s magnetic field is felt by the other. It can be ruled out with an estimate.

Two moments of about two Bohr magnetons, three ångströms apart, interact magnetically with an energy of order $\mu_0\mu_1\mu_2/4\pi r^3$. Putting the numbers in gives about $10^{-24}$ joules, which is 0.06 wavenumbers. The measured couplings are three to four orders of magnitude larger.

So whatever couples two spins is not magnetism — an inference of the same shape as the one what an absence proves draws from a missing band, where a quantity that is far too small settles a question that looked open. It is electrostatics and the Pauli principle, and the smallest system that shows it can be solved exactly.

Two sites, two electrons #

Take two orbitals, one on each ion, with a hopping amplitude $t$ between them and an energy $U$ for putting two electrons on the same site. Put two electrons in. That is the whole model, and its complete solution is four states in the sector with equal numbers of up and down spins, plus a state with both spins parallel.

The state with parallel spins cannot hop at all. Both electrons have the same spin, so moving one onto the other’s site would put two identical fermions in one orbital, which the Pauli principle forbids. Its energy is therefore exactly the energy of two isolated sites: zero, at every value of U, to twelve decimal places.

The state with antiparallel spins can hop, and pays U for the privilege. It does not do so freely — a large U makes double occupancy expensive — but quantum mechanically it can do it virtually, spending a short time in the doubly occupied arrangement, and second-order perturbation theory says that lowers its energy by an amount proportional to $t^2/U$.

The exact answer for the two-site model is

$$E_{\text{singlet}} = \frac{U - \sqrt{U^2 + 16t^2}}{2}$$

and the triplet is at zero, so the singlet–triplet splitting is that same expression. Expanded for large U it is $-4t^2/U$.

The convergence, computed #

The perturbative result is a limit, and a limit is worth checking rather than quoting.

The product $U \times J$ approaches $-4$ from below and is within a tenth of a per cent by $U = 64$. That is the second-order argument confirmed by an exact calculation rather than assumed, and a proper check tests both the limit and the monotonicity of the approach — a single point could be a coincidence, and a sequence converging the wrong way could not.

There is a second check worth naming, and it is about small U. At $U = 2$ the expansion has no business being used, and the coupling is still negative — strongly so, at $-1.24$. The two electrons at small U simply both occupy the bonding orbital, which is a singlet, so the antiferromagnetic preference survives outside the regime the standard argument covers. A check that only tested the limit would pass on a model whose sign was wrong everywhere else.

Why the answer is negative #

The sign is the whole chemistry and it comes from an asymmetry in what is permitted.

An antiparallel pair has two ways to make a virtual excursion: the left electron can hop right, or the right electron can hop left. Each costs U and gains $t^2/U$, and there are two spins involved, giving the factor of four.

A parallel pair has none. Not “fewer” and not “less favourable”: none, exactly, because every hop is blocked by the exclusion principle.

So the arrangement with more available motion is lower, and the arrangement with more available motion is the antiparallel one. That is the mechanism, and it explains why the default in chemistry is antiferromagnetic coupling: almost any pair of magnetic ions with an orbital pathway between them will prefer opposite spins.

The estimate that rules magnetism out #

The order-of-magnitude argument at the top deserves to be done properly, because it is the step that makes the rest of the essay necessary.

Two magnetic dipoles $\mu_1$ and $\mu_2$ separated by $r$ interact with an energy of order $\mu_0\mu_1\mu_2/4\pi r^3$. For two moments of two Bohr magnetons — about $1.9\times10^{-23}$ joules per tesla — at three ångströms, that is

$$\frac{(10^{-7})(1.9\times10^{-23})^2}{(3\times10^{-10})^3} \approx 1.3\times10^{-24}\ \text{J}$$

and a wavenumber is $1.99\times10^{-23}$ joules, so the interaction is about 0.06 cm⁻¹. In temperature units that is under a tenth of a kelvin.

Measured couplings in bridged dimers are 10 to 500 cm⁻¹, and antiferromagnetic ordering temperatures in transition-metal oxides are hundreds of kelvin. The discrepancy is not a factor of two to be argued about; it is three to four orders of magnitude, and it settles the question before any alternative is proposed.

The estimate also predicts where dipolar coupling does matter: at large separations, where the electronic pathway has vanished exponentially and the $r^{-3}$ tail has not. That is the regime of dilute magnetic materials and of the line widths in magnetic resonance, and it is a different subject from the chemistry here.

When it is ferromagnetic instead #

Not every coupling is negative, and the exception has a clean structural condition.

If the two magnetic orbitals are orthogonal — arranged so that the hopping amplitude between them vanishes by symmetry — the mechanism above gives nothing, because there is no hop to be virtual about. What is left is the ordinary exchange interaction, which favours parallel spins for the same reason Hund’s rule does: two electrons in orthogonal orbitals with parallel spins avoid each other and repel less.

So the sign of the coupling is decided by whether the two magnetic orbitals overlap through the bridge, and that is a question of geometry.

The classic instance is the bridge angle in an oxo- or hydroxo-bridged pair. A metal–oxygen–metal angle near 180° gives strong overlap through the same oxygen p orbital and a large antiferromagnetic coupling; an angle near 90° puts the two metals on different oxygen p orbitals, which are orthogonal, and the coupling turns ferromagnetic. That correlation — the sign changing near 95 to 100 degrees — is one of the better-established structure–property relations in inorganic chemistry, and it follows from whether a hopping amplitude vanishes.

What the model is and is not #

Two clarifications, because the words in this area are used loosely.

There is no magnetic term in the Hamiltonian. The model has a hopping amplitude and an on-site repulsion. Spin enters only through the requirement that two electrons in one orbital have opposite spins, which is antisymmetry rather than magnetism — the same principle that gives Hückel theory its filling rules and a moment counts electrons, not orbitals its integers. A coupling between two spins has been produced from a model that never mentions a magnetic field, and that is the result.

The coupling constant is a fitted quantity in practice. Experimenters extract J by fitting a measured susceptibility to a model of two coupled spins, so a quoted J is a parameter of an assumed Hamiltonian rather than a directly observed energy. The relation between what is fitted and what is computed here is that both describe the same singlet–triplet splitting, and the fitting model is the effective spin Hamiltonian that this microscopic model reduces to at large U.

That reduction is worth stating plainly: at large U the Hubbard model’s low-lying states are exactly the states of two spins with a coupling $-4t^2/U$, and everything above them is separated by roughly U — the gap the smallest many-electron calculation follows from the other side. So the spin Hamiltonian people fit is not a phenomenological guess but the low-energy limit of a model with electrons in it — which is why fitted J values can be compared with structures at all.

The same arithmetic makes an insulator #

The system in this essay is two sites. Four sites of the same kind produce a result the solids field could state and not compute.

Both results come from the same term. A large U suppresses double occupancy, and suppressing double occupancy does two things at once: it stops electrons moving, which makes the material an insulator, and it leaves each site with one electron whose spin can still couple to its neighbours’ by virtual hopping. That is why Mott insulators are usually antiferromagnets — the two properties are consequences of one term, and the insulator band theory cannot see computes the first of them.

Why this belongs to chemistry rather than to physics #

The mechanism is a piece of many-body physics and its consequences are chemical, and the reason is the same one that makes the dimers a better test than the oxides.

A chemist can build the system. Choosing the bridging ligand fixes the pathway; choosing the metal fixes how many unpaired electrons each site has; choosing the co-ligands fixes the geometry and therefore the bridge angle. A series of compounds differing in one of those and nothing else is a controlled experiment on a parameter of the Hamiltonian, and series of that kind are what established the structure–property correlations.

The bridge-angle correlation is the clearest. Hydroxo-bridged copper(II) dimers have been made across a range of Cu–O–Cu angles from about 95° to about 105°, and the fitted coupling runs from ferromagnetic at the low end to strongly antiferromagnetic at the high end, crossing zero near 97°. That is a straight line through a dozen compounds, and what it plots is a hopping amplitude against a geometry.

Nothing in the two-site model predicts 97°, because the model has no geometry in it — $t$ is a number. What the model supplies is the reason there should be a crossing at all: the coupling is proportional to $t^2$ with a negative sign, so wherever $t$ passes through zero the antiferromagnetic term vanishes and whatever is left over takes charge.

What is measured #

The compounds where this is cleanest are dimers: two metal ions held at a fixed distance by a bridging ligand, magnetically isolated from everything else.

Copper(II) acetate is the standard case, and it is a d⁹ compound of the kind copper is never quite octahedral is about. Two copper ions, each d⁹ with one unpaired electron, bridged by four acetate groups, with a singlet–triplet splitting near −300 cm⁻¹. At low temperature the compound is diamagnetic, because only the singlet is populated; as it warms the triplet fills and a moment appears. The temperature dependence of the susceptibility is the measurement, and the shape of it — a maximum, then a fall — is the signature of an antiferromagnetically coupled pair.

That behaviour is exactly what the two-site model predicts: two states, separated by J, populated according to temperature. Fitting the measured curve returns J, and the fitted J is what gets compared with structures.

Copper(II) measures 1.90 Bohr magnetons for one unpaired electron in an isolated complex, which is what a moment counts. In copper acetate the two coppers’ spins couple and the measured moment falls far below that — which is the observation this essay is an explanation of, and it needs no magnetic term to produce.

What this model cannot reach #

One orbital per site. A real magnetic ion has up to five, and couplings through different orbital pairs can have different signs and add. That is why the Goodenough–Kanamori rules are a list of cases rather than one formula.

No bridge. The model connects the two sites directly with an amplitude $t$. In a real compound the pathway runs through a bridging ligand, and $t$ is an effective amplitude summarising that route. Where the route runs is the whole of the structural chemistry of the subject, and this model contains it as a number.

Two sites. Extended magnetic order — a Néel state, a ferromagnet, a transition temperature — is a property of many coupled spins, and nothing here says at what temperature a lattice of them orders. The site’s stated boundary applies: a chain of two hundred is a large molecule and can be diagonalised, and the exact solver here is limited to six sites precisely because exactness is the claim, which is the boundary a band with no structure in it drew for the solids field and the same one applies here.

No spin–orbit coupling. Anisotropic couplings, which decide the direction the moments prefer, come from spin–orbit terms this model has not got.

The bridge the model summarises, and the angle where the coupling changes sign #

In a real compound the pathway runs through a bridging ligand, and $t$ is an effective amplitude summarising that route is the model’s largest omission, and it is the one with the most measured structure behind it — because the effective amplitude depends on the geometry of the bridge, and it passes through zero.

The mechanism is visible in the two limiting arrangements.

At a metal–bridge–metal angle of 180°, both metals interact with the same p orbital of the bridging atom, pointing along the line joining them. The hopping route is direct, $t$ is large, and the essay’s $-4t^2/U$ makes the coupling strongly antiferromagnetic.

At 90°, the two metals interact with two different p orbitals on the bridge, and those two are orthogonal to each other. The route from one metal to the other through the bridge closes: $t$ goes to zero, the antiferromagnetic term with it, and what is left is a small ferromagnetic term from the exchange between the two bridge orbitals.

So the coupling should be strongly negative near 180°, pass through zero somewhere between, and be weakly positive near 90°. That is a prediction about a structural angle, and it has been tested on a family designed for it.

The family is the hydroxide-bridged copper(II) dimers, which are planar, have two bridges, and can be made across a range of bridge angles by changing the rest of the ligand. Their couplings, measured magnetically and their angles measured by diffraction, fall on a straight line: the coupling falls by about 75 wavenumbers for every degree the bridge angle opens, and it crosses zero at 97.5 degrees.

Below that angle the compounds are ferromagnetic and their moments rise as they are cooled; above it they are antiferromagnetic and their moments fall. One family, one structural variable, a sign change at a stated angle, and a slope steep enough that two degrees is the difference between a measurable coupling and none.

That is what the model’s single parameter $t$ is standing in for, and the correlation says two things about the summary.

The summary is not a constant of the pair. Two copper ions at the same distance with the same bridging atom have couplings of opposite sign depending on an angle, so nothing about the metals or their separation determines $t$.

And the model’s form survives. The dependence on $t^2$ and the inverse dependence on $U$ are unchanged; what the bridge supplies is the value of $t$, and the value happens to run through zero as a geometry is varied. A model whose parameter is an effective quantity is not thereby a model without content — the content is what the parameter is squared and divided by, and that part is exactly what the family confirms.

Who found it, and when #

Hendrik Kramers proposed the superexchange mechanism in 1934, and Philip Anderson gave it the form used here in 1950 and 1959: a second-order process in which virtual hopping through an intermediate state lowers the antiparallel arrangement. John Goodenough and Junjiro Kanamori worked out the dependence on orbital geometry through the same decade, producing the rules that predict the sign.

The chronology has a feature worth noting. The mechanism was proposed for magnetic oxides — extended solids — a quarter of a century before the molecular chemists began making bridged dimers to test it one pair at a time. The dimers turned out to be the better test, because a pair has two states and a lattice has a thermodynamic limit, and fitting two states to a measured susceptibility is a far less ambiguous business than inferring a coupling from an ordering temperature.