Half of it is given back at one bond

Where the electrons are, without subtracting anything computed where the electrons of a correlated ring actually are, without subtracting anything from anything: the opposite-spin pair distribution, which is a property of the wavefunction and needs no reference state.

It found more than a hole, and it found it without the subtraction the reference decides the correlation warns about. The distribution is deep at zero separation, enhanced at one, depleted at two and enhanced at three — an antiferromagnetic arrangement in the charge — and the two features move at different rates as the repulsion is turned up. It ended by asking which of them a correlation energy is mostly made of.

The question has an arithmetic answer, and the answer is that the Hubbard model cannot see the second feature at all.

The hole, as a count of pairs #

The pair distribution is a ratio and is awkward to price. What an interaction multiplies is a number of pairs, so the useful form of the hole is the change in how many electron pairs sit at each separation, against the same ring with no repulsion at all.

For a ring of six at a repulsion of eight:

The correlation removes 1.2653 pairs from zero separation and puts 1.1126 of them one site away. Eighty-eight per cent of what it took goes exactly one bond, and everything further out moves a few hundredths of a pair.

That is a much tidier statement than the distribution makes on its own, and it is still a property of the wavefunction alone: no interaction has been applied to it. The weighting comes next and is a separate decision.

Priced by an on-site interaction: a hundred per cent, as an identity #

A Hubbard interaction is $U$ on one site and exactly zero everywhere else. So the energy of the hole is

$$E = U \times (-1.2653) = -10.12, \qquad \text{and } 0 \times 1.1126 = 0.$$

A hundred per cent of the correlation energy is at separation zero, and it is not a measurement — it is an identity. Nothing at any larger separation can contribute anything, however large it is, because it is being multiplied by zero.

That is the same shape of failure a predictor’s ties expose: a quantity that cannot respond to something is a quantity that says nothing about it, and the not-responding is invisible in the output.

So a Hubbard correlation energy is not a measurement of how strongly electrons avoid each other. It is a measurement of the on-site hole, and the model has no opinion about anything else. That distinction is invisible in the model’s own output, because the model’s own output is one number.

Priced by anything that reaches a neighbour #

Give the interaction a nearest-neighbour term of $U/2$, and the second row of the table stops being free.

Forty-four per cent of the on-site saving is given straight back at one bond, and forty-seven per cent with the Coulomb tail. The correlation energy is nearly halved by an interaction change that leaves the wavefunction untouched.

The sign is the part worth dwelling on, and it is the same reversal a pair inside a trio meets: an arrangement that is favourable under one accounting is a bill under another, with the wavefunction unchanged between them. The enhancement at separation one is a cost, not a saving: the correlation has pushed electrons onto neighbouring sites, and if neighbouring sites repel each other then that is a bill. The model gets the saving and skips the bill.

And everything beyond one neighbour is negligible under every weighting — −0.031 and +0.328 against −10.122, which is three per cent between them. The alternation in the pair distribution is real and is worth almost nothing, except at its first term, where it is worth nearly half of everything.

It is not a weak-repulsion artefact #

A cancellation of forty-four per cent invites the suspicion that it is a coincidence of the parameters, so the repulsion was swept over a factor of thirty-two.

It settles. From a repulsion of four upwards the give-back is 44 to 47 per cent and does not move, which is the behaviour of a ratio between two quantities that are both saturating — the on-site depletion approaches its floor and the enhancement at one approaches its ceiling, and the pairs removed have nowhere else to go.

So the factor of about two is a property of this system rather than of a chosen repulsion — the kind of stability across a parameter that a fitted exponent conspicuously lacks, and a correlation energy computed with an on-site model and applied to a system whose electrons repel at a distance is wrong by that factor, stably.

Why the two features move at different rates #

The two features of the distribution saturate at different repulsions — the enhancement at one bond settles by a repulsion of eight while the on-site hole is still deepening at sixteen — and that looks like a decomposition the energy cannot see. It is now possible to say what it is.

The two are not independent. Pairs removed from zero separation have to go somewhere, and the only nearby somewhere is one bond away — so the enhancement at one is very nearly the depletion at zero, redistributed. What limits it is that a site can only hold so much: the on-site depletion can keep deepening towards a state with no double occupancy at all, while the enhancement at one is bounded by how much charge a neighbouring site can accept without doubly occupying itself.

That is why the give-back ratio settles. Both quantities approach ceilings, their ratio approaches a constant, and the constant is a property of the ring’s geometry rather than of the repulsion.

And it is why the ratio is a better thing to quote than either number. The on-site depletion and the enhancement at one both depend on the repulsion, strongly; their ratio does not, above a repulsion of about four. A quantity that stops depending on a parameter is a quantity that has found something about the system, and this one has found how much of the correlation energy an on-site model over-counts.

What this says about a correlation energy #

What a correlation energy is a property of has several answers already, and this adds the sharpest one.

A correlation energy is a contraction of the hole against the interaction. It is one number produced from a function of separation and a function of separation, and two systems with very different holes give the same number if their contractions agree. The reference decides the correlation gave the first answer; this adds a second thing it is a property of, and the answer is the range of the interaction.

The practical consequence: the hole is the transferable object and the energy is not. A hole computed on one system can be weighted by a different interaction; an energy computed on one system cannot be reweighted at all, because the weighting has already been performed and the separations have been summed over.

That is an argument for reporting holes. It is also the reason computing the distribution rather than an energy difference was the right decision, and this is what it was right for.

The one number a reader should take away #

Strip the argument to its shortest form.

A correlation energy is a sum over separations of how many pairs moved, times what the interaction costs where they moved to. The first factor is the wavefunction’s business and the second is the Hamiltonian’s, and a model that sets the second to zero beyond one site has not made an approximation to the sum — it has computed a different sum.

For this ring the two sums differ by a factor of 1.8. The wavefunction is identical in both; only the weighting differs; and the weighting is a modelling decision made before any of the computation, usually without comment.

That is worth stating as a rule of thumb because there is one available: an on-site correlation energy roughly doubles the correlation energy of a system whose electrons repel at a distance. The factor is 1.8 here and would be smaller under screening, and no version of it is one.

It is worth saying what the rule is not. It is not a claim that the on-site model is wrong: a model of a system whose interaction really is short-ranged is right, and the give-back computed above is then zero because the weighting has no neighbour term in it. The rule is about transfer — taking a correlation energy computed in one interaction and applying it in another — which is what happens whenever a lattice model’s parameters are fitted to one property and its correlation energy quoted for another.

The same arithmetic explains why a bad hole can give a good energy #

The operation performed here — take a hole, weight it by an interaction, and see what each separation contributes — is not a device invented for a lattice. It is the operation that accounts for one of the most-remarked-on facts in electronic structure: that the crudest density functional works far better than its assumptions deserve.

The reasoning runs the way this essay runs. A correlation energy is an integral of the exchange-correlation hole against the interaction, and the interaction depends only on the distance between two electrons. So the energy cannot see the hole’s shape. It sees the hole’s spherical average, weighted by one over the separation, and nothing else.

That has two consequences and the essay’s numbers illustrate both.

A hole can be badly wrong in shape and right in energy. The local approximation replaces the hole of a real molecule with the hole of a uniform electron gas of the same density — an object that is spherical by construction, when a real molecule’s is not, and centred on the wrong place. Judged as a hole it is a poor description. Judged through the weighting, which is the only way an energy judges it, it satisfies the same sum rule and has roughly the right weighted average, and that is enough.

And the weighting decides everything. The lattice version of that statement is what is measured here: the identical hole is worth one hundred per cent of its depletion under an on-site interaction and 56 per cent under one that reaches a neighbour. Nothing about the electrons changed between those two numbers. What changed was the function the hole was integrated against.

So the honest reading of a correlation energy is that it is not a measurement of a hole. It is a measurement of one particular weighted moment of a hole, and a great deal of the hole is invisible to it — which is the same statement, in another setting, as the observation that a second moment fixes a band’s width and not its shape.

It also says what a functional has to get right and what it may get wrong. The sum rule — that the hole integrates to exactly one electron for exchange and zero for correlation — is a constraint on the weighted average at long range, and imposing it is worth more than getting the shape right at short range. Modern functionals are constructed by imposing such constraints for exactly this reason, and the reason is the arithmetic in this essay’s third row: an interaction is a weighting, and a weighting can forgive a great deal.

What is quoted, and what is computed #

Nothing is quoted. The ring, its filling, the repulsion and the three interaction shapes are all stated parameters. Every pair distribution comes from the exact ground state of the whole configuration space, and the uncorrelated reference is the same ring at zero repulsion rather than a mean-field approximation to it.

The three interactions are declarations rather than fits: on-site only, on-site plus a half at one neighbour, and $U/(1+r)$. No conclusion depends on which of the three is right, because the finding is the spread between them.

The pair counts use $n(n-1)$ on a site rather than $n^2$, which matters more than it looks — without it a site holding one electron reports a pair with itself and the exclusion principle appears to be violated by the bookkeeping.

What this cannot say #

Six sites is a ring, not a solid, and the exact solver is deliberately limited to sizes a whole configuration space fits into. The separations available are 0 to 3 and the alternation has nowhere to develop; a longer ring would have more separations and the tail could carry more than the three per cent it does here. That would strengthen the finding rather than weaken it, since every extra separation is another place the on-site model sees nothing.

A real interaction is not any of these three. In a solid the Coulomb interaction is screened, and how quickly it is screened is exactly what decides the give-back. The forty-four per cent is a number for an unscreened neighbour term; heavy screening would take it towards the on-site answer, which is presumably why the on-site model is used at all.

Nothing here is about a real material. A ring of six sites with one orbital each is a model of a question, not of a substance, and the forty-four per cent should be read as the give-back is of the order of the saving rather than as a number to apply to anything. A third kind of correlation is where the kinds are separated; this is about how any of them is priced.

And the wavefunction was computed with the on-site interaction. Weighting its hole by a longer-ranged interaction is a first-order estimate: a system whose electrons really repelled at a distance would have a different ground state, with less enhancement at one bond. The give-back is therefore an upper bound on the correction, and the true correlation energy of the extended system lies between the two columns.

What was checked #

The uncorrelated ring’s opposite-spin pair distribution is one at every separation — the control, and the statement that makes every difference here a hole rather than a distribution.

A Hubbard correlation energy is entirely on-site, checked as a share of exactly one and as exactly nothing at every other separation, because the claim is an identity and a tolerance would understate it.

An interaction reaching one neighbour gives back between a third and three fifths of the on-site saving, checked as a band because the number moves with the repulsion and the finding is that it is large rather than that it is 44.

Everything beyond one neighbour is worth under a twentieth of the on-site term, which is the other half of the answer to the question about alternation.

And the hole is a depletion on one site and an enhancement on the next — checked as two signs, because a hole that were a depletion everywhere would make the whole argument vacuous.

Still open: the extended model solved outright, and a hole with a real width #

The obvious open question is the self-consistent version. The give-back above is computed by weighting a wavefunction optimised for an on-site interaction with a longer-ranged one, which is a variational bound; solving the extended model outright — an on-site $U$ and a nearest-neighbour $V$ in the Hamiltonian rather than in the weighting — would give the true correlation energy and the true hole, and the difference between the two is the second-order term the bound is missing. Six sites is small enough to do it exactly, and the interesting output is whether the enhancement at one bond survives being charged for.

The nearer question is the width of the hole, which is now concrete. A lattice hole is one bond wide by construction, so the distinction between short-range correlation and the near-degenerate kind cannot be drawn spatially here — and the give-back shows why that matters: the whole give-back lives at exactly one bond, which is the only length the model has. A continuum model with two electrons in a box would have a hole with a real width, and the same weighting would then say how much of the correlation energy sits inside it.