The hole that is not repulsion

Two electrons in a bond stay out of each other’s way. The reason offered is always the same — they are both negative and they repel — and the reason is at best half of the answer.

The way to find out is to switch the repulsion off and look. There is a model in which that is a well-defined thing to do: the Hubbard system of the smallest many-electron calculation, with electrons hopping between sites and paying an energy $U$ whenever two of them share one. Setting $U = 0$ leaves independent electrons, exactly, and the ground state is then a product of two determinants.

Ask that state how the spins on neighbouring sites are arranged. The quantity is $\langle S^z_i S^z_j\rangle$, and if the two electrons really were arranged independently it would be zero — the average of a product of independent things is the product of their averages, and each average is zero by symmetry.

It comes out at $-0.125000000$.

Where the number comes from without any repulsion #

A single Slater determinant is antisymmetric, and antisymmetry has a consequence for same-spin electrons that has nothing to do with the interaction: two electrons of the same spin cannot be in the same place. In a lattice model that is exact and immediate — two up-spin electrons cannot occupy one site, and the wavefunction has no amplitude for it.

So a determinant already correlates. The up electrons avoid each other and the down electrons avoid each other, entirely because of the antisymmetry, and the effect has a name: the Fermi hole, or the exchange hole.

The size of it is computable from the one-electron problem alone. For a closed-shell determinant with one-particle density matrix $\rho_{ij} = \sum_{k\,\mathrm{occ}} c_{ki}c_{kj}$ per spin,

$$\langle S^z_i S^z_j\rangle = -\tfrac12 \rho_{ij}^2 \qquad (i \ne j)$$

with no interaction anywhere in the derivation. For the two-site system the bonding orbital is $(1,1)/\sqrt2$, so $\rho_{12} = \tfrac12$ and the correlation is $-\tfrac12 \cdot \tfrac14 = -0.125$.

That is the number the exact many-body calculation returns.

Two routes with nothing in common #

The agreement is worth setting out as the check it is, because the two calculations share nothing but the lattice.

One route builds the full many-body Hamiltonian over every configuration of up and down electrons, keeps the fermion signs, diagonalises a matrix of dimension up to nine hundred, takes the ground-state vector and sums a weighted product of site spins over every configuration in it.

The other route diagonalises a matrix of the size of the system — two by two for the dimer — takes the occupied eigenvector, multiplies two of its coefficients together and squares the result.

They agree to nine decimal places for the two-site system, for a four-site chain, and for a six-site chain: $-0.125000000$, $-0.075000000$ and $-0.065027726$. Nothing in the many-body code knows the formula, and nothing in the formula knows about configurations.

The check has a refusal attached. A half-filled four-ring has two levels exactly at the Fermi energy, so there is no single closed-shell determinant to compare against, and the exchange formula refuses it rather than averaging over a multiplet nobody chose. That open shell is the same one that makes the four-ring a metal in the one-electron picture and the same one the smallest many-electron calculation had to handle carefully when it computed the double occupancy.

What repulsion then adds #

Having established that the hole exists at zero repulsion, the question becomes how much the repulsion deepens it. The answer is: substantially, and less than a factor of two.

For the two-site system the neighbouring spin correlation runs $-0.1250$, $-0.1405$, $-0.1553$, $-0.1809$, $-0.2134$, $-0.2368$ and $-0.2463$ at $U = 0$, $0.5$, $1$, $2$, $4$, $8$ and $16$. The limit is $-0.25$, which is what two localised spins pointing opposite ways would give. So at a repulsion four times the hopping — a reasonable figure for a real bond — the exchange hole accounts for $59\%$ of the total and the repulsion for the rest.

Two other quantities move at the same time and they are worth having side by side, because the three are usually conflated.

Double occupancy falls from $0.250$ to nearly nothing. That is the quantity “electrons avoiding each other” most directly names, and it starts at exactly a quarter, which is what independence means for a half-filled system.

The local moment $\langle (S^z_i)^2 \rangle$ rises, from $0.125$ to $0.246$. A site that is empty or doubly occupied carries no spin, so squeezing out double occupancy leaves more sites carrying one — and the moment is a measure of how much of a spin there is to correlate rather than of how well it is correlated.

Two quantities moving in opposite directions out of one wavefunction is the sort of thing that catches a sign error, and the check requires both directions.

How the exchange hole depends on the system rather than on the pair #

The exchange value is $-\tfrac12\rho_{ij}^2$, and the density matrix element $\rho_{ij}$ is a property of the whole occupied set rather than of the two sites. That has a consequence worth drawing out, because it is where the lattice model says something a two-electron picture cannot.

For the two-site system $\rho_{12} = 0.5$ and the correlation is $-0.125$. For a four-site chain the two occupied orbitals give $\rho_{12} = \rho_{34} = 0.447$ and $\rho_{23} = 0.224$, so the end bonds have a correlation of $-0.100$ and the middle bond has $-0.025$ — a factor of four between two bonds in one molecule, with the same two atoms at each end and the same interaction.

The middle bond is the one whose atoms are shared with two neighbours each, and its density matrix element is smaller because the occupied orbitals have less amplitude concentrated there. So the exchange hole between two atoms depends on what else the molecule is attached to, which is not something a picture of a bond as a pair of electrons between two nuclei can express.

That is the same lesson bond order from the eigenvectors draws about bond orders, and it comes from the same object: $\rho_{ij}$ is the bond order, and the exchange correlation between two sites is minus half of its square. The two quantities differ only by an arithmetic operation, which is a connection the vocabulary of “correlation” and “bond order” thoroughly hides.

The four-ring, and why an open shell has no determinant #

The refusal above is worth one section rather than one clause, because it is a case where the honest answer is that the comparison does not exist.

A half-filled ring of four has levels at $2, 0, 0, -2$ in the one-electron problem. Four electrons fill the lowest and then meet two degenerate levels with two electrons to place. There is no unique way to do it: the state can be a triplet with one electron in each, or one of several singlets, and at $U = 0$ they are all at the same energy.

So “the determinant” is not a well-defined object for it, and any exchange value computed from one would be a value for whichever member of the multiplet the diagonaliser happened to construct. The formula therefore refuses, checking the gap at the Fermi level and failing when it is smaller than $10^{-8}$.

What the exact calculation returns for that system is nonetheless meaningful: $-0.0594$ at $U = 0$, which is a genuine expectation value of a genuine degenerate ground state chosen by the diagonaliser. It is simply not comparable with a single determinant, and reporting it beside the closed-shell numbers would invite exactly that comparison.

The four-ring is the same system whose singlet–triplet degeneracy at $U = 0$ corrects the usual Hückel result about cyclobutadiene. A model that has no preference to express is a recurring object, and each time it appears the right response is to say so rather than to pick one of the states it cannot choose between.

The Coulomb hole, and what this model is not #

The part that repulsion adds is the Coulomb hole, and the vocabulary is worth keeping straight because the two holes behave differently in one important respect.

The Fermi hole applies only between electrons of the same spin. Two electrons of opposite spin are not forbidden from meeting by antisymmetry at all, and in a determinant they meet as often as independence allows. The Coulomb hole applies to every pair, and its main effect in a closed-shell system is therefore on the opposite-spin pairs, which is exactly where the determinant has nothing.

That is why the double-occupancy curve is the useful one here. A doubly occupied site holds one up and one down electron, so its population is entirely an opposite-spin quantity: at $U = 0$ it takes the independent value of a quarter, and everything below that is the Coulomb hole.

What the model is not is a statement about real space. A Hubbard site is not a point and the repulsion is not $1/r_{12}$; the whole interaction has been compressed into one number that applies when two electrons are on the same site and vanishes otherwise. A real Coulomb hole is a statement about the pair density as a function of the separation $|\mathbf{r}_1 - \mathbf{r}_2|$, it has a cusp at zero separation, and neither is computed here.

The claim being made is therefore narrow and is worth stating in its narrow form: in a model where “the same place” is well defined and the interaction is switched off exactly, the electrons still avoid one another by an amount computable from a determinant. That the same is true in real space is standard and is quoted.

What this does to the word “correlation” #

The standard definition of correlation energy is the difference between the exact energy and the Hartree–Fock energy — everything a single determinant misses.

That is a perfectly serviceable definition of a number, and it is a very poor definition of a phenomenon, for the reason this essay has been about. A determinant contains the entire exchange hole. Defining correlation as what the determinant misses therefore defines away the majority of the electron avoidance in a typical closed-shell system, and leaves a residue that is called “the correlation” because of where the subtraction was drawn.

The consequence is a familiar confusion. A calculation is described as having “no correlation” when it is a single determinant, and a reader takes that to mean the electrons are moving independently, which they are not. They are avoiding one another by $-0.125$ out of a possible $-0.25$ in the two-site system at $U = 4$, and the determinant knows all of it.

The same shape of problem turns up in a different currency. Delocalisation is not always stabilising is about a quantity whose value depends on a reference state that is chosen rather than measured, and the correlation energy is another: it is a difference from a stated approximation, so it says as much about the approximation as about the system.

Where the two holes matter differently #

Two cases separate the exchange hole from the Coulomb hole cleanly, and both are computed already.

A triplet state has only the exchange hole between its two electrons, since they have the same spin and cannot be on one site at all. Its double occupancy is exactly zero at every $U$, which is why the smallest many-electron calculation finds the triplet’s energy independent of $U$ — the parallel-spin state pays no repulsion because it never has two electrons in one place to pay it for.

A dissociating bond has neither hole doing enough. Where molecular orbital theory dissociates shows the restricted determinant putting half its weight on ionic configurations at every separation, which is precisely a failure to make a large enough Coulomb hole. Exchange cannot help, because the two electrons have opposite spins.

Between them those two cases say what each hole is for. Antisymmetry handles same-spin pairs completely and opposite-spin pairs not at all; the interaction handles all pairs and is the only thing that handles the second kind.

A note on which spin component is being measured #

Everything above uses $\langle S^z_i S^z_j\rangle$, the correlation of the $z$ components, and a reader who knows that spin has three components may reasonably ask why.

The answer is convenience with a known factor. The full spin correlation $\langle \mathbf{S}_i \cdot \mathbf{S}_j\rangle$ is three times the $z$ part for a state with no preferred direction, which every ground state here has. So the numbers above are exactly a third of the quantity a magnetic measurement would relate to, and the factor is the same at every $U$, so every ratio and every trend in this essay is unaffected.

The $z$ part is the one computed because it is diagonal in the configuration basis. Each configuration has a definite up or down occupation at every site, so $S^z_i S^z_j$ is a number rather than an operator that moves amplitude around, and the expectation value is a weighted sum over configurations with no matrix elements to evaluate. The transverse parts would require the off-diagonal elements, which is more work for a result that differs by a known constant.

That is the sort of choice worth stating rather than leaving implicit, since a reader comparing these numbers with a published $\langle \mathbf{S}\cdot\mathbf{S}\rangle$ would otherwise find them a factor of three too small and have no way to know why.

The two holes are normalised differently, which settles the matter #

There is an exact statement separating the two contributions that needs no model and no parameter, and it is sharper than any comparison of magnitudes.

Sit on one electron and ask how the density of the others is depleted around it. The depletion caused by exchange integrates, over all space, to exactly one electron. Not approximately one and not one in some limit: the Pauli principle removes precisely one electron’s worth of same-spin density from the neighbourhood of any electron, in any system, whatever the interaction is doing. It is a counting statement about antisymmetry, and it survives switching the repulsion off — which is why a determinant with no interaction in it reproduces the number above.

The depletion caused by repulsion integrates to exactly zero. Electrons pushed out of each other’s way have to go somewhere, and the total number of them is fixed, so whatever is removed at short range reappears at longer range. Repulsion redistributes and does not deplete.

So the two are not two sizes of the same thing. One carries a whole electron and the other carries none, and no amount of turning the interaction up changes either total.

That also explains why a Coulomb hole must be paid for somewhere. A depletion that sums to zero has a corresponding enhancement, and where that enhancement lands — and what the interaction charges for it there — is a question with a number attached rather than a detail.

What switching the repulsion off establishes #

Exact solution of a small Hubbard system is usually shown for its own sake. Here it answers a question it was not built for, and the answer contradicts the reason usually given.

Three things are now on the record. Exchange correlates electrons without any interaction, by an amount that agrees between an exact many-body calculation and a one-electron formula to nine decimal places. Repulsion adds to it, roughly doubling the correlation at a realistic interaction strength. And the two apply to different pairs, which is why one of them is useless at a dissociating bond and the other is not.

The open question is the real-space version: a pair density as a function of separation, with a cusp at zero and a hole around it. That needs two-electron integrals over continuous functions, which is not attempted here, for a reason worth stating — a wrong two-electron calculation produces plausible numbers, and plausible wrong numbers are the failure the whole method here is arranged against. What can be said in the meantime is what a lattice can say, and a lattice can say this much exactly.