The force with no force in it

Two electrons avoid one another more than their charge accounts for, and two photons crowd together for no reason that any interaction between them supplies. Both are usually attributed to an exchange force, and the name is a considerable obstacle to understanding what is happening.

There is no force. There is a restriction on which states exist, and the restriction has consequences that look exactly like a force, are of the size of the largest forces in chemistry, and cannot be written as one.

The calculation with nothing in it #

Take two free particles in one dimension, in a box, with no interaction between them at all. Build the two-particle state three ways: symmetric under exchanging the labels, antisymmetric, and neither. Then ask, given that one particle is found at the origin, how likely the other is to be found a distance away.

For an ideal gas of identical fermions at zero temperature the answer is exact and has no free parameters:

$$g(r) = 1 - \left(\frac{\sin k_{\text{F}}r}{k_{\text{F}}r}\right)^2$$

with the sign reversed for bosons and the correction absent for distinguishable particles. Those are the three curves in the opening figure.

The correlation function this essay is about is built out of the occupied states of a free-fermion gas and nothing else — no interaction, no potential, no force of any kind. That is why the Fermi wavevector sets its scale: the only length in the problem comes from how densely the states are filled, and the “force” is a consequence of counting rather than of anything acting.

The fermion curve reaches exactly zero at zero separation. That is the exclusion principle stated as a probability: two identical fermions in the same spin state are never found at the same place. The boson curve reaches exactly two, which is the same statement with the opposite sign — identical bosons are twice as likely to be found together as independent particles would be.

And the distinguishable curve is flat at one, which is what no interaction ought to give. All three come from the same Hamiltonian.

The three cases differ in one thing only: whether the two-particle state was built by adding the two single-particle products, subtracting them, or taking one of them. That is a choice about the state, not about the dynamics, and it is not a free choice — nature makes it, and which way it goes is decided by the particles’ spin. Everything below follows from a choice made once and for all about which states exist.

The hole holds one particle #

The property that turns this from a curiosity into the dominant energy scale in chemistry is a sum rule.

Integrate the density times the departure of $g$ from one over all separations, and the fermion case gives exactly minus one particle. The hole around each electron is not a tendency or a preference: it is one missing particle, and the count is the same at every density, because a denser gas digs a proportionally narrower hole.

Around every electron in an atom is a region from which the others are excluded by the same counting, and the size of that region is what shifts one level below another as the nuclear charge rises. The 4s and 3d crossing is that effect made visible in the periodic table — a reordering produced by exclusion rather than by any change in the Coulomb attraction.

That one particle is worth a great deal of energy. An electron in a metal is surrounded by a region a Fermi wavelength across from which one unit of like-spin negative charge has been removed, so the Coulomb repulsion it feels is smaller than a random arrangement would give — by an amount comparable with the Fermi energy itself, which is electronvolts.

It is worth pausing on why the count is exactly one and not something that depends on the density, because that exactness is the whole reason the effect is so robust. The correlation hole’s width is set by the Fermi wavelength, which shrinks as the density rises; its depth at contact is fixed at zero by antisymmetry, whatever the density. The product of a shrinking width and a fixed depth, integrated against a rising density, is a constant — and the constant is one, because what has been removed is precisely the particle at the origin, which cannot also be somewhere else. The sum rule is a statement of normalisation rather than of dynamics, which is why no approximation that respects normalisation can get it wrong.

The consequence is that the electron gas is far more stable than a naive calculation predicts, and the difference is called the exchange energy. It is not a correction. In a simple metal it is a substantial fraction of the total electronic energy, and it is why density functional theory — whose whole business is approximating that energy as a functional of the density — works at all.

What the spins have to do with it #

The best-known instance of exchange is the splitting between singlet and triplet states, and it is where the misleading name does most damage.

The total state of two electrons must be antisymmetric. The state factorises into a spatial part and a spin part, so a symmetric spin state requires an antisymmetric spatial one and vice versa. The three triplet states are symmetric in spin, so their spatial part is antisymmetric and vanishes when the electrons coincide; the singlet is antisymmetric in spin, so its spatial part is symmetric and does not.

Since the electrons repel, the state that keeps them apart has the lower Coulomb energy. The triplet is therefore lower — for the spatial reason, with the spin serving only as a label for which spatial symmetry is available.

The size of the splitting is the exchange integral, which depends on how much the two orbitals overlap. That figure measures it: the exchange term follows the square of the overlap, so it dies exponentially as the orbitals separate while the direct Coulomb term dies only as one over the distance. Exchange is a contact effect with a Coulomb-sized magnitude, and that combination is what makes it so hard to describe in classical language.

The magnetism that is not magnetic #

In helium the singlet–triplet splitting of the lowest excited configuration is 0.8 electronvolts. That is a temperature of about 9,000 kelvin.

The ordering temperature of a magnet is set by the exchange coupling, and for iron it is a thousand kelvin. That is the number which shows the effect is not magnetic: the actual magnetic interaction between two neighbouring moments corresponds to about one kelvin, three orders of magnitude too small. What holds a ferromagnet together is electrostatics with a counting constraint on it, and the magnetism is the symptom.

The magnetic dipole interaction between two neighbouring atoms is about $10^{-4}$ electronvolts, which is a tenth of a kelvin. So if iron’s magnetism were magnetic — dipoles aligning with each other’s fields — it would disorder above about a tenth of a kelvin and no permanent magnet would exist at room temperature.

Iron orders at 1,043 kelvin. What aligns the spins is exchange: the Coulomb repulsion between electrons, sorted by which spatial states each spin arrangement permits. The whole of ferromagnetism above liquid-helium temperature is an electrostatic effect wearing a magnetic name, and the actual magnetic interaction is four orders of magnitude too small to appear in the answer.

The size of the exchange coupling also explains a fact about magnetic materials that is otherwise arbitrary: why so few elements are ferromagnetic. The coupling depends on the overlap between neighbouring orbitals and changes sign as the ratio of the atomic separation to the orbital radius varies, so ferromagnetism requires that ratio to fall in a narrow band. Iron, cobalt and nickel are in it and manganese, whose atoms are slightly closer together relative to their d-orbitals, is not — manganese orders antiferromagnetically instead. Alloying manganese to push the atoms apart makes it ferromagnetic, which is a direct test of the account and was performed long before anybody could compute the integral.

That is worth restating because it inverts the ordinary account. A permanent magnet is not held together by magnetism. It is held together by the same principle that gives matter its volume, and the magnetic field it produces is a by-product rather than the cause.

The bond that is the same argument #

The other place exchange sets the energy scale is chemistry, and there the sign runs the other way.

Bring two wells together and each level splits into a lower symmetric state and a higher antisymmetric one — and the lower one is the bond. The splitting is the same exchange arithmetic in its simplest setting, which is why a covalent bond and a ferromagnet are the same phenomenon at different signs and different scales.

Bring two hydrogen atoms together. The symmetric combination of the two atomic orbitals puts electron density between the nuclei and is lower in energy; the antisymmetric combination has a node there and is higher. Two electrons in the symmetric orbital must have antisymmetric spin, which is a singlet — so the covalent bond is a singlet, and the triplet state of the same pair is unbound.

That is the reverse of helium’s ordering, and the reversal is not a contradiction. In helium the two electrons are in different spatial orbitals, and exchange lowers the state that keeps them apart; in a bond they occupy the same spatial orbital, and what decides is whether that orbital is bonding. Same principle, opposite conclusion, because the spatial states available are different.

Every stable molecule with an even number of electrons is a singlet for this reason, and the exceptions are informative: molecular oxygen is a triplet in its ground state because its two highest electrons occupy different degenerate orbitals, which restores the helium situation. That is why oxygen is paramagnetic, why liquid oxygen sticks to a magnet, and why oxygen’s reactions with singlet molecules are spin-forbidden and therefore slow — which is a substantial part of why organic matter does not spontaneously burn.

The bosons doing the opposite #

Everything above has an inverted counterpart for particles with symmetric states, and the inversion has consequences of its own.

Bosons do the opposite, and the contrast makes the mechanism plain. Where fermions keep apart because the wavefunction must change sign under exchange, bosons pile up because it must not — and the pile-up at zero separation in a boson correlation function is the same tendency that produces a condensate at low temperature. One rule about symmetry, two opposite consequences, and no force in either.

The boson curve’s value of two at zero separation is the same statistics that make a thermal light source bunch and that make stimulated emission possible: a photon is more likely to be emitted into a mode that already contains photons, in proportion to how many are there, and that proportionality is what a laser amplifies.

At low temperature the same preference produces a condensate, where a macroscopic fraction of the particles occupies one state — which for fermions is forbidden outright and for bosons is favoured. Two families of behaviour, one sign, and no interaction responsible for either.

Two photons at a beam splitter #

The pair correlation drawn at the top of this essay is a statistical statement about a gas. There is a two-particle version of it that can be done with exactly two particles, and it is one of the cleanest experiments in the subject.

Send two identical photons into the two input ports of a half-silvered mirror so that they arrive at the same instant. Ask how often one emerges from each output.

Counting classically there are four possibilities: both reflect, both transmit, or one of each does one of each. Two of those four put one photon in each output, so a quarter of the time each detector should fire — and coincidences between the two detectors should be common.

They never happen. The two ways of getting one photon into each output are two amplitudes for the same final state, and the beam splitter’s phase relation makes them equal and opposite, so they cancel exactly. The photons always leave together, by the same port, chosen at random.

Nothing interacts. The photons do not see each other, there is no nonlinearity in the glass, and the mirror is the same mirror it would be for one photon. What cancels is a pair of two-particle amplitudes, and the cancellation is the boson half of this essay’s opening figure evaluated at one point in an apparatus rather than averaged over a gas.

Hong, Ou and Mandel demonstrated it in 1987, and the measurement is now routine. Sweeping the relative arrival time traces out a dip in the coincidence rate — zero when the two photons are simultaneous and indistinguishable, rising to the classical quarter when they are separated by more than their coherence time. The depth of that dip is the standard measure of how identical two photons are, and it is the enabling effect for two-photon logic gates in photonic quantum computing.

The fermionic counterpart has been done as well, with electrons in a two-dimensional gas playing the part of the photons and a quantum point contact playing the beam splitter. There the result is exactly inverted: two identical electrons arriving together at a splitter never leave by the same port, and the coincidence rate rises to one where the photons’ fell to zero.

Two experiments, one sign, and in neither is anything exerting a force.

The rule that fills a shell #

Helium’s triplet-below-singlet ordering is the two-electron case. Applied to a shell it becomes a rule that determines the ground state of most of the periodic table.

Hund’s first rule states that among the terms arising from a given configuration, the one of largest total spin lies lowest. The traditional argument is the essay’s: a large total spin means a symmetric spin state, therefore an antisymmetric spatial state, therefore electrons kept apart, therefore less repulsion.

The consequence is a strong preference for parallel spins wherever there are degenerate orbitals to put them in, and the sharpest signature is a half-filled shell. Five d orbitals can hold five parallel-spin electrons with no orbital doubly occupied at all, so a half-filled d shell is unusually stable, and the same is true of a half-filled f shell with seven.

That stability is visible in the periodic table as a set of anomalies in the ground-state configurations. Chromium is not $3d^4 4s^2$ as the filling order would suggest; it is $3d^5 4s^1$, promoting an s electron to complete a half-filled d shell. Copper is $3d^{10}4s^1$ for the corresponding reason at the full shell. Molybdenum, silver and gold do the same. Half a dozen elements refuse the tidy filling sequence and every one of them refuses it in the direction Hund’s rule prefers.

The same rule gives the magnetic moments of the transition-metal and rare-earth ions, which are measured routinely and which come out at the values maximising the spin — the reason gadolinium, with its half-filled f shell and seven parallel spins, has the largest magnetic moment of any element.

There is a correction to the traditional explanation that is worth having, because it is a good example of a rule surviving its own justification. Detailed calculations on atoms show that the high-spin state’s electron–electron repulsion is actually a little larger than the low-spin state’s, not smaller. What makes it lower overall is the electron–nucleus energy: the exchange-induced avoidance lets the electrons screen one another less effectively, so their orbitals contract toward the nucleus and the attraction to it strengthens by more than the repulsion costs.

The rule is right, the ordering is right, exchange is still what produces it — and the intermediate step everybody quotes points the wrong way.

Why the name survives anyway #

If there is no force, it is fair to ask why every textbook calls it one, and the answer is not simply carelessness.

An effective potential is exactly what an exchange effect can be summarised as, provided everybody remembers that the potential is a bookkeeping device rather than an interaction. That is why the name survives: it is enormously convenient to write the consequence as a force, and enormously misleading to then ask what carries it.

The effect can be summarised as a force, and doing so is often the only practical route. In a many-body calculation the exact antisymmetric wavefunction is unmanageable, so the usual move is to work with a simpler wavefunction and add an effective potential that reproduces what antisymmetry would have done. That effective potential is repulsive between like fermions and attractive between bosons, has the right magnitude, and can be handed to a classical simulation. Nuclear physics does this constantly, and so does every semi-empirical method in chemistry.

The trouble comes when the summary is read back as a mechanism. An effective potential of this kind has none of the properties a real interaction has. It does not fall off as a power law — it dies with the overlap, exponentially. It does not act between particles that are far apart, however strongly it acts when they are close. It has no carrier and no propagation speed. It changes when a third particle is added, because the antisymmetrisation is over all of them at once, so it is not pairwise even approximately. And it depends on the spin state without involving any magnetic interaction whatever.

A quantity with those properties is not a force in any useful sense of the word. It is the energetic consequence of a restriction on the state space, and the safest habit is to keep the word “exchange” and drop the word after it.

What the pictures cannot show #

The correlation drawn is for spinless particles. Real electrons come in two spin states, and the exchange hole is dug only around electrons of the same spin — so the correlation between opposite spins is flat in this approximation, and the full hole around an electron is half as deep as the figure suggests until Coulomb correlation is added.

Nothing here has any Coulomb interaction. Real electrons repel, and that produces a further hole — the correlation hole — on top of the exchange one. Separating the two is the central difficulty of electronic structure theory, and the fact that exchange is exactly calculable while correlation is not is the reason the split is made at all.

The gas is one-dimensional and ideal. In three dimensions the sinc-squared becomes a different function with the same properties at both ends, and the sum rule is unchanged; the shape between is not.

Temperature is absent. The correlation drawn is the zero-temperature one; heating a fermion gas fills the hole in, and above the Fermi temperature the three curves converge, because the states are then sparsely enough occupied that two particles rarely compete for one. That is why exchange matters enormously for electrons in a metal, whose Fermi temperature is tens of thousands of kelvin, and hardly at all for the atoms of an ordinary gas.

And the exchange integral is computed for two Gaussian orbitals with a softened kernel, which is a model rather than a molecule. What it demonstrates is the scaling with overlap, which is the claim being made; the numerical value belongs to the model.

The ladder from here #

Later rungs on this anchor: the Hartree–Fock approximation, where exchange is treated exactly and correlation not at all, and what that costs; the Heisenberg model, which packages the exchange integral into a spin Hamiltonian and is the starting point of all magnetism; superexchange and double exchange, which are how the coupling survives between ions that do not overlap; and the exchange hole’s role in density functional theory, where an approximate functional for it is what makes calculation on real materials possible.

The neighbouring ladders are no two in the same state, which is the principle stated as a counting rule, four states, and one of them is odd, where the singlet and triplet are constructed, and the pressure that is not a temperature, where the same exclusion holds up a star.