A third kind of correlation

Two kinds of correlation, and only one is small is the essay this one answers. It found an exact identity in the two-site model: the local power of the correlation energy in the repulsion — the slope of one logarithm against another — equals the occupation of the bonding natural orbital, to nine figures, at every repulsion strength. And it closed by asking whether anything like that survives with more orbitals, guessing that the generalisation would be a weighted average of the occupations.

The answer is no, twice over, and the second failure is more interesting than the identity.

It is worth saying at the outset that the earlier essay’s identity is not being overturned. It is exactly true of the system it was measured on, at every repulsion, and it is reproduced here by a more general calculation as the first thing that calculation is asked to do. What is overturned is the hope that it was a special case of something.

What the calculation needed #

Everything in the earlier essay was available from a two-by-two matrix. The dimer’s natural occupations can be had from the algebra of a two-level problem without ever writing down a density matrix, which is what made the identity findable and what makes it useless for anything larger.

What generalises is the one-particle density matrix — the object whose entries are ⟨c†ᵢcⱼ⟩ and whose eigenvalues are the natural occupations. It is defined for any system here, and building it from an exact ground state is a matter of bookkeeping with fermion signs.

The signs are the part to watch. A sign error there produces a matrix that is still symmetric, still traces to the right number of electrons, and has the wrong eigenvectors — so it is checked against two cases with known answers rather than against itself: the dimer, where the occupations must reproduce the two-level algebra at four repulsion strengths, and the non-interacting limit, where every occupied orbital must hold exactly two electrons.

What an exponent is, and why it is the right thing to measure #

A correlation energy is a number with units and its size depends on everything: the system, the basis, the units, the strength of the interaction. Comparing correlation energies between systems is comparing apples.

An exponent has none of those dependences. The local slope of log|E_corr| against log U is dimensionless, is unchanged by any rescaling of either quantity, and answers a question about shape rather than size: how does the error respond when the interaction is turned up? Two means quadratically, one means linearly, and the distinction between them is a statement about what kind of expansion the exact answer admits.

That is why the earlier essay measured it and why this one does too. It is the one quantity that can be put beside the natural occupations of a different system and be meaningfully compared.

The identity, reproduced #

The two-centre row is the check, and it passes: 1.97014 against 1.97014 at U = t, and the same at every other repulsion tried. So the calculation reproduces the result it is being used to test.

Four centres in a chain: not an average of anything #

The guess the earlier essay made was a weighted average of the occupations, and it can be ruled out without knowing the weights.

At U = t the four-centre chain’s exponent is 1.98540 and its occupations are 1.98378, 1.96031, 0.03969, 0.01622. The exponent is larger than every one of them. No weighted average of a set of numbers exceeds the largest of them, so no weighting whatever reproduces the exponent.

That would be enough on its own, and the numbers are more decisive still. At U = 4t the same chain’s exponent is 1.78369 and its largest occupation is 1.78878 — now the exponent is below the largest. So the two quantities cross as the repulsion is raised, which rules out even a one-sided relation between them.

The identity is a two-site accident. It is exact in the model where it was found, and it is exact because a two-dimensional occupied space has one parameter’s worth of structure and both quantities are functions of that one parameter.

Four centres in a ring: an exponent of one #

This is the finding rather than the refutation.

A correlation energy is defined as a subtraction: the exact energy minus the energy of the best single determinant. For a system whose lowest configuration is a non-degenerate closed shell, a weak perturbation shifts that configuration at second order, so the correlation energy goes as the square of the repulsion and the exponent tends to two. Every closed-shell system here does that — the two-centre one from below, the four-centre chain from below, and larger closed-shell rings from above.

A half-filled ring of four does not have a non-degenerate lowest configuration. Its Hückel levels are one lowest, a degenerate pair, and one highest, and four electrons fill the lowest and put two into the degenerate pair — an open shell. A degenerate reference is shifted at first order by any perturbation that connects its members, and the exponent is one.

Not one in the limit of strong repulsion, where every model does surprising things. One at a repulsion of 0.02 of the hopping, which is as weak as an interaction can usefully be made.

The occupations say it too, and more starkly #

The ring’s natural occupations are the other half of the picture and they are worth quoting exactly.

At every repulsion tried, from 0.02 upwards, its four occupations are 1.99994, 1.000000, 1.000000, 0.00006 — and the two middle ones are one to nine decimal places, at every repulsion. The degenerate pair holds exactly one electron each, pinned there by symmetry rather than by the strength of the interaction.

So at a repulsion of one fiftieth of the hopping the ring’s correlation energy is 0.005 β — negligible by any measure — and its description is already maximally multireference: two orbitals half occupied, which is as far from a single determinant as a four-orbital system can get.

That is the separation two kinds of correlation is about, taken to its limit. That essay’s point was that the energy error and the description error do not track each other. Here the description error is total while the energy error is nothing at all.

Two more checks the calculation had to pass #

The density matrix is new here and every result above rests on it, so it is worth setting out what it was made to prove before it was used.

It traces to the electron count, which catches a dropped sign in the bookkeeping immediately: a matrix whose diagonal does not sum to the number of electrons is wrong before any eigenvalue is taken.

It reproduces the one-electron answer when the repulsion is switched off. With U = 0 the exact ground state is a single determinant and every occupied orbital must hold exactly two electrons and every empty one none. It does, to 10⁻⁸.

The second check has a subtlety worth recording, because getting it wrong would have looked like a result. For the half-filled ring of four the non-interacting ground state is degenerate, so the solver returns some member of a manifold rather than a particular state, and the split of two electrons between the degenerate pair is not a property of anything. What is a property is the sum, which is two, and that is what is checked — checking the split would have been checking an artefact of the eigensolver.

That distinction is the same one the whole essay turns on, arriving first as a question about how to write a test.

The molecule this is about #

Cyclobutadiene, which has already appeared three times.

Aromaticity as a computed shell closure fills every ring from three to ten and asks whether the highest occupied shell came out full; cyclobutadiene’s does not. Delocalisation is stabilising, and other things that are false in general computes its delocalisation energy at exactly zero. The vibration that lowers the symmetry is about the distortion that removes the degeneracy.

This is the fourth appearance and it is the one that says why no single-determinant method is even a starting point for the molecule. The other three describe consequences of the degeneracy; this one measures what the degeneracy does to the methods that would be used to improve on the one-electron picture.

The practical form of it: a method built as a correction to a single determinant assumes the correction is small, and “small” means second order in something. For this molecule the correction is first order from the outset, so the method is not slightly inaccurate — it is expanding about the wrong point.

The density matrix built for this essay reads more than one quantity, and it is worth saying so because it is the same object each time. Its eigenvalues are the natural occupations above; its diagonal is how much charge an energy difference actually moves once the electrons repel. One matrix, two readings, and neither of them available from the energy alone.

Three kinds, then #

The earlier essay named two and this adds a third, and the three are distinguished by what happens as the interaction is switched off.

Dynamic. The reference is a good single determinant; the correction is second order; the exponent tends to two; the occupations tend to two and zero. Every closed-shell system here at weak repulsion.

Static. The reference is a poor single determinant because a second configuration has come close in energy; the exponent falls below two; the occupations move away from integers. Every closed-shell system here at strong repulsion, and the regime the earlier essay’s identity lives in.

Degenerate. The reference is not a single determinant at any interaction strength, because the lowest configuration is an open shell that symmetry keeps open. The exponent is one at arbitrarily weak interaction and the occupations are pinned at one, independent of the strength.

The third is not the strong-interaction end of the second. It is a different regime at the weak end, and the two-site model cannot exhibit it at all, because two sites cannot produce a degenerate half-filled shell.

Where the third kind appears in chemistry #

The degenerate case is not a curiosity of a four-site model. It is the situation of every open-shell system whose openness is enforced by symmetry, and there are several important classes of them.

A molecule in a degenerate electronic state is one — which is the starting point of the vibration that lowers the symmetry, where the molecule escapes the degeneracy by distorting. Escaping it is exactly what makes the third kind of correlation a transient condition for a real molecule: the distortion removes the degeneracy and the reference becomes a single determinant again.

A transition metal with a partly filled d shell is another, and there the escape is not always available — copper is never quite octahedral is a case where it is, and a high-spin d⁵ ion is a case where the shell is half filled with parallel spins and the state is a perfectly good single determinant after all.

The general rule the three kinds give is: look at the lowest configuration before looking at the interaction. If it is unique, the correlation is a correction and the usual methods apply. If it is not, no strength of interaction makes it one.

What this says about small models #

There is a general moral and it is one worth stating against these essays themselves.

A model small enough to solve exactly is small enough to have accidents. The two-site identity was exact, was checked to nine figures, and was a property of the number two rather than of correlation. Nothing about the check was wrong; the check verified a true statement about a small system, and the true statement did not generalise.

What makes the situation recoverable is that the failure is informative. Testing the identity required building a density matrix that works for any system, and that calculation immediately produced a result — the degenerate case — that the original model could not have suggested.

It is worth putting beside the other exact statement about how an error scales. An energy is second order in a wavefunction’s error where every property is first — the same shape of claim, about a power rather than a size, and with the same fine print: it holds where the thing being expanded about is a single state, and the ring of four is exactly the case where nothing is.

The generalisation failed and the attempt to generalise found something, which is the usual return on trying to break an exact relation in a small model.

The kinds are regions of a curve, not boxes a system sits in #

One property of the instrument used here deserves separating from the results it produced, because it changes what the taxonomy is a taxonomy of.

The exponent measured throughout is a local slope — one logarithm against another, at a stated repulsion — so it is a function of the coupling rather than a label. The dimer’s runs from two down towards one as the repulsion rises. The ring’s is one at repulsions fifty times weaker than the hopping and does not stay there for ever either.

So dynamic, static and this third degenerate kind are not three boxes that molecules fall into. They are three regions of one curve, and a system moves between them as the ratio of its repulsion to its hopping changes.

That ratio is not fixed for a real molecule. Stretching a bond weakens the hopping while leaving the on-site repulsion alone, so a molecule slides along the curve as it dissociates — which is the familiar observation that a bond near equilibrium is a single-reference problem and the same bond half broken is not, restated as a position on an axis rather than as a change of category.

The consequence is the one that makes reaction chemistry hard. A path from reactants to products passes through geometries at different points on that curve, so the kind of correlation changes along the coordinate. A method chosen because it is right for the reactants is being applied, unchanged, to a transition state that may sit a long way along the axis from them — and the error it makes is not a constant offset but a quantity whose scaling with the interaction has changed underneath it.

Which is why the diagnostics are run at every geometry rather than once. A label attached to a molecule would only ever have been right somewhere.

What would settle the ring’s exponent exactly #

The obvious loose end is whether the ring’s exponent is exactly one, and it is worth saying what would settle it, because the numerical answer here is 1.008 rather than 1.000.

Degenerate perturbation theory gives the first-order shift as an eigenvalue of the perturbation restricted to the degenerate space, which for this system is a two-by-two matrix of on-site repulsion terms. If that matrix has a non-zero eigenvalue at all, the shift is linear in U and the exponent is exactly one in the limit — and the measured 1.008 at U = 0.02, falling towards 1.000 as the repulsion falls further, is what a linear leading term with a quadratic correction looks like.

Writing that matrix down is a short calculation and it is not done here, because doing it would replace a measurement with a derivation and the measurement comes first. What the measurement establishes on its own is that the exponent is nowhere near two, which is the whole of the claim.

What is left #

The systems here have two and four centres, and the largest that fits comfortably in an exact diagonalisation is six. A six-centre ring at half filling — benzene’s π system, in this model — is a closed shell and shows an exponent slightly above two at weak repulsion, which is a fourth behaviour and is not accounted for above. Whether it is a finite-size effect or something about the ring geometry is not settled here.

The model has repulsion only when two electrons are on the same site, and nothing when they are on neighbouring ones. That is the crudest possible representation of a repulsion, and it is the reason everything here is exactly solvable. Whether the degenerate case’s exponent of one survives a longer-ranged interaction is a question this model cannot ask.

And the exponents are local slopes measured by finite differences on the computed energies, so they are numerical rather than analytic — the same caution a method that is not additive records about its own measured departures. The values quoted are stable to five figures under changes of the step size, which is enough to distinguish one from two and not enough to say whether the ring’s exponent is exactly one or merely very close to it as the repulsion vanishes.