The smallest many-electron calculation

A Hückel energy comes from a one-electron model. A Hückel matrix has one number per atom and one per bond; the levels are filled two at a time; and the total energy is a sum over the occupied orbitals. That structure carries a long way and covers a great deal of chemistry.

It has one feature that is easy to state and hard to remember while using it, and it is the caution Hückel theory and delocalisation both carry. Nothing in it can say that two electrons cost more when they are close together. The matrix has no term describing what one electron does to another, so the electrons in it are independent — the energy of one does not depend on where the others are.

Add exactly one term to fix that: an energy $U$ paid whenever a site holds two electrons at once. The result is the Hubbard model, and it is no longer a matrix of size $n$.

Why one term changes the size of the problem #

In a one-electron model the question is which orbitals exist, and the answer is the eigenvectors of an $n \times n$ matrix. Filling them is bookkeeping done afterwards.

With a repulsion term the energy depends on which sites are occupied at once, so the states of the system are no longer orbitals but configurations: which sites hold an up electron, which hold a down one. Two sites with two electrons give four such configurations in the sector with one of each spin; four sites at half filling give thirty-six; six sites give four hundred.

The Hamiltonian is a matrix over those configurations, and diagonalising it is the calculation. Nothing is variational, nothing is self-consistent, nothing is fitted: the whole space is written down and solved, so the answer is the answer.

That exactness is why the systems here are tiny. The number of configurations grows combinatorially, and an exact treatment has to stop at about six sites rather than approximate — because exactness is the claim, and a calculation that quietly truncated would lose the one property that makes it worth doing.

The check that ties it to everything computed before it #

The first thing to ask of a new machine is whether it reproduces the answers of the machine it extends.

At $U = 0$ the Hubbard model is the one-electron model, so the exact many-body ground state must equal the sum over filled orbitals computed the old way. It does, for three systems:

Machine precision, from two calculations that share no code path: one diagonalises an adjacency matrix and fills levels, the other diagonalises a matrix over thirty-six configurations.

The second half of the check is that the agreement must stop. With $U = 4$ the four-site ring’s exact ground state is −2.103 against the one-electron −4.000, and the one-electron answer is too low — it has been counting a stability the electrons cannot actually have, because it allowed them to occupy the same site for free.

Both halves matter. Agreement alone would pass on a model with no U in it, and disagreement alone would pass on a model that was simply wrong.

How the matrix is built, and the sign that is easy to lose #

The construction is short enough to describe and has one step that is easy to get wrong invisibly.

The up-spin electrons and the down-spin electrons do not hop into each other, so the kinetic part of the Hamiltonian is a sum of two independent pieces — one hopping matrix for each spin species, acting on its own occupations. Everything that couples the two is the diagonal repulsion term. That structure is exact rather than an approximation, and it is what makes the matrix cheap to assemble.

The part that has to be right is the fermion sign. Moving an electron past another one of the same spin changes the sign of the amplitude, because the state is antisymmetric under exchange, and a program that ignored it would produce a smooth, plausible, wrong spectrum. Nothing about the resulting numbers would look odd.

Two checks catch it. A matrix with a dropped sign is generally not symmetric, and a symmetric eigensolver cannot be trusted on one that is not — so a matrix that passes the symmetry test is partly checked already. The stronger one is the trace: the sum of all the eigenvalues must be U times the number of configurations with a double occupancy, counted independently, and that can be checked for every system before its levels are used.

The habit goes back to Hückel theory: an eigenvalue problem is checked against a quantity computable without solving it. A trace is the cheapest such quantity there is.

What correlation is, as a number #

The word “correlation” is used in quantum chemistry for whatever a one-electron model leaves out, which makes it sound like a residual. It has a direct measurement.

Ask how often two electrons are found on the same site. In an independent-electron ground state at half filling the answer is exactly a quarter per site — an up electron is somewhere, a down electron is somewhere, and the two facts are unrelated, so they coincide a quarter of the time.

The exact answer starts at the quarter and falls: 0.199 at $U = 1$, 0.085 at 4, 0.0023 at 32. The electrons have arranged themselves to avoid each other, which is what “correlated” means — their positions are no longer statistically independent.

The starting value is the check worth having. A many-body ground state coming back at exactly 0.25 when the repulsion is switched off is evidence that the state really is the one-electron product state there, and it is not something a broken calculation would produce.

The repulsion actually paid #

A second measurement makes the same point more sharply, and its shape is a surprise.

Multiply the double occupancy by U and the result is what the ground state actually pays in repulsion. An uncorrelated state would pay $U/4$ per site, rising for ever. The exact state pays 0.448 at $U = 0.5$, 1.217 at 2, 1.358 at 4, 0.998 at 8 and 0.565 at 16.

It peaks and then falls. At small U the electrons do not bother avoiding each other, because avoiding costs kinetic energy and the repulsion is cheap; at large U they have almost entirely stopped meeting, so there is almost nothing to pay. The worst case is in the middle, where the two costs are comparable — which is exactly the regime real materials occupy, and the reason the middle of this range is where the interesting physics is.

The difference between what an uncorrelated state would pay and what the exact one does pay is the correlation energy, and it is not a small correction to an energy: at $U = 16$ it is 15.4 out of 16, or ninety-six per cent of the naive estimate.

Cyclobutadiene, and an earlier phase corrected #

Hückel theory treats square cyclobutadiene cleanly. Four π electrons, a degenerate non-bonding pair holding two of them, Hund’s rule applied — and a predicted triplet ground state, which is wrong in the specific way that makes the Jahn–Teller distortion discussable — the distortion delocalisation is not always stabilising computes, and the molecular twin of the one in a chain cannot stay even.

The exact calculation says what was missing.

At U = 0 the singlet and the triplet are exactly degenerate, to twelve decimal places. That is the one-electron model’s answer stated honestly: it has nothing to choose between them, and Hund’s rule is a statement about electron repulsion, which it does not contain. The rule was imported.

At every U above zero the singlet is lower. The gap is 0.0037 t at $U = 0.25$, 0.048 at 1, 0.296 at 4 and 0.332 at 8. So repulsion, added to the model, produces a singlet — the opposite of what the rule imported to stand in for repulsion predicted.

The direction is not an accident of these parameters. Lieb’s theorem states that the half-filled Hubbard model on a bipartite lattice with sublattices of equal size has a ground state of total spin $|A - B|/2$ — zero here. The check verifies that the four-ring is bipartite with two sites in each sublattice before making the claim, and confirms that a three-site chain, whose sublattices are unequal, is the case the theorem treats differently.

So the Hückel result stands as a statement about the model it was computed in, and the exact calculation says what the model was missing. That is the right relation between the two: the Hückel answer was not a mistake, it was the answer to a different question, and both are worth having.

What “many-electron” means, and what it costs #

The last figure’s caption carries the deepest of the differences, and it is worth stating directly.

In a one-electron model a state is a list: this orbital occupied, that one empty. In the exact calculation the ground state is a superposition of configurations, with amplitudes on all thirty-six, and no assignment of electrons to orbitals describes it. That is why the energy is not a sum over orbital energies and why a correction to the orbital energies cannot repair it.

The cost of exactness is size. Thirty-six configurations for four sites, four hundred for six, and roughly a hundred million for sixteen — which is why exact diagonalisation is a tool for small clusters and everything larger uses an approximation of some kind. This site takes the small clusters and says so.

What is bought is a class of statement no approximation supplies: these are the levels, this is the ground state, and no assumption entered. Given that the whole point is to say what a one-electron model gets wrong, using an approximate many-electron method instead would have replaced one set of unstated assumptions with another.

One thing the exact solution is good for is the singlet–triplet splitting of two sites, which approaches −4t²/U and is the microscopic origin of magnetic coupling. A quantity second order in the hopping is invisible to a one-electron model, and this is the smallest system that has one.

Two electrons, and what a bond is #

The dimer at half filling is the hydrogen molecule’s problem in its simplest form, and the two limits it interpolates between are the two pictures of a bond set side by side in molecular orbital and valence bond.

At $U = 0$ the ground state is the molecular orbital answer: both electrons in the bonding combination, which means both are equally likely to be found on either site — including a quarter of the time on the same site, twice over. That configuration, an ionic one with both electrons on one atom, is a quarter of the wavefunction and costs nothing in a model with no repulsion.

At large $U$ the ground state is the valence bond answer: one electron on each site, spins opposite, with only a small admixture of the doubly occupied configurations. The double occupancy at $U = 32$ is 0.002, which is the ionic contribution measured.

So the exact solution is a continuous interpolation between two descriptions that are usually presented as rivals, and the parameter that moves between them is the repulsion. The molecular orbital picture over-weights the ionic configurations; the valence bond picture under-weights them; the exact answer weights them by what they cost.

That is the sharpest form of the argument molecular orbital and valence bond makes qualitatively, and it is available here because two sites and two electrons is small enough that “the exact answer” is a thing one can hold in one hand.

The one-electron picture of the same dimer is two orbitals with the pair in the lower one, and that is the U = 0 end of the exact calculation — where every Hückel energy lives.

The one-electron treatment this extends is rings filled and asked whether their highest occupied shell came out complete. The four-ring is the one whose shell does not close, which is why it is the ring taken here to an exact solution.

Where this leads #

Three further results depend on the exact solution.

What couples two spins takes the singlet–triplet splitting as the origin of magnetic exchange, and shows that it is three to four orders of magnitude larger than any magnetic interaction between the same two moments.

The insulator band theory cannot see computes the charge gap of the four-site ring, which is exactly zero at $U = 0$ — the ring’s half-filled shell is degenerate, so band theory calls it a metal — and grows without limit thereafter.

The pairing energy decides the moment takes the pairing energy as a quoted parameter, and the relation is exact: U is that model’s pairing energy, and the exact solution shows why a measured pairing energy is a property of a whole distribution rather than of one integral.

All three are cases where a one-electron model states something it cannot compute, which is the position a half-filled band is not always a metal leaves in writing: band theory cannot see electron repulsion.

What the model is not #

It is not a molecule. One orbital per site, repulsion only when two electrons are on the same site, and no repulsion at all between neighbouring sites. A real molecule has all of those and more, and the difference between a model of a medium and a model of a substance is the distinction a basis is not a thing keeps making, and the Hubbard model is a caricature chosen because it contains the one term whose consequences are wanted.

U is a parameter. Nothing here computes it. Estimates for a carbon 2p orbital are around 10 electronvolts against a hopping of 2 to 3, so the physically interesting range is $U/t$ of three to five, which is the middle of every plot in this essay.

The systems are too small for a thermodynamic limit. A four-site ring has a charge gap and a coupling; it does not have a phase transition, an ordering temperature or a band structure. Where a claim needs size, an exact calculation this small cannot supply it, and the honest course is to say so rather than extrapolate.

Why the smallest calculation is also nearly the largest #

Four sites give thirty-six configurations, which is small enough to solve exactly is the sentence this essay turns on, and it is worth asking what happens to that number, because the answer is why exact results of this kind are computed on two sites, four or six.

At half filling the count is the number of ways of placing the up electrons times the number of ways of placing the down ones:

Each pair of sites added multiplies the problem by roughly four, and diagonalising a dense matrix costs the cube of its size. Twelve sites is already a matrix with $7\times10^{11}$ entries, which cannot be stored, let alone diagonalised — so the honest limit for a complete spectrum, which is what several arguments about repulsion need, is eight or ten sites.

Methods that ask only for the ground state reach further, because the matrix is sparse and never has to be written down; twenty sites is within reach that way, and the record for such calculations is in the low twenties. Twenty is not a step towards $10^{23}$. It is not even a step towards a molecule of interesting size.

So the wall is exponential, and it arrives immediately. That is not a limitation to be apologised for on the way to something bigger; it is the reason the entire apparatus of approximate quantum chemistry exists, and the reason a four-site answer is worth computing carefully — it is one of very few places where an exact answer is available at all, and everything else in the subject is measured against answers of exactly this kind.

Who found it, and when #

John Hubbard wrote the model down in 1963, and Martin Gutzwiller and Junjiro Kanamori arrived at it independently in the same year — the problem in view being why some materials with partly filled bands are insulators, which Nevill Mott had raised in 1937 and which band theory had no room for.

Elliott Lieb and Fa-Yueh Wu solved the one-dimensional case exactly in 1968, by Bethe ansatz, and found that the chain is an insulator for every $U > 0$ — no transition, no threshold. Lieb’s theorem about the ground-state spin came in 1989.

The model’s later career is worth a sentence, because it explains why a two-site calculation is worth doing carefully. In two dimensions the Hubbard model is unsolved, it is the standard model of high-temperature superconductivity, and enormous computational effort has gone into it for four decades. What can be said exactly is said about small clusters, and the small clusters are where the mechanisms are visible.