Two sites, two electrons, every level exactly

One of the figures on when repulsion is in the model: Hubbard systems small enough to diagonalise exactly: the singlet a one-electron model cannot find, the coupling between two spins, and a gap where band theory says there is none.

Six essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

What couples two spins

What actually does it: the singlet–triplet splitting of two sites holding two electrons, against the on-site repulsion, from an exactly diagonalised model with no magnetic term in it whatever. The second curve is the same quantity multiplied by U, which settles on −4t² — the signature of a second-order effect.

Every state of the two-site model against U. At U = 0 the lowest is at −2t, which is two electrons in a bonding orbital — the Hückel answer. The dashed line is the parallel-spin state, flat at zero because it cannot hop, and the gap between the two is the coupling.

The same competition on four sites taken to a larger repulsion. The singlet drops away from the triplet the moment the electrons repel and the separation goes on growing, so what couples the spins is not a magnetic term arriving at some threshold — it is present at every repulsion above zero and absent only at zero.

The smallest many-electron calculation

The whole spectrum of two sites holding two electrons, against the on-site repulsion. Four states, computed exactly. At U = 0 the ground state is −2t, which is the one-electron answer; as U rises it climbs, and the state with parallel spins stays flat at zero because it cannot hop at all.

The four-ring taken to a larger repulsion than the figure above. The singlet drops away from the triplet the moment the electrons repel and goes on dropping, so the degeneracy the one-electron picture had to make a choice about is resolved at every repulsion greater than zero — and the exact calculation never has to choose.

The two-site case out to a repulsion of twenty-four, where the two singlets and the triplet have almost separated into their strong-coupling limits. Every level here is exact and checked against a closed form, and the whole diagram is four numbers per repulsion — which is what makes this the smallest calculation that has electron–electron repulsion in it at all.

Where molecular orbital theory dissociates

The four states of the dimer against U. The triplet sits flat at zero — two electrons on different sites with parallel spins cannot be on the same site, so the repulsion never touches it — and the lower singlet approaches it from below as the repulsion grows. The gap between them is the exchange coupling.

The moment a fit invents

Where the coupling in this model comes from: the singlet–triplet splitting of two sites against the repulsion between the electrons, with the product settling on −4t². That number is the J the fits above are trying to recover, and it was computed from hopping and the Pauli principle with no magnetism in it at all.

Two wrong numbers and a right difference

A quantity that survives all of this, for contrast: the singlet–triplet splitting of a dimer in the large-repulsion limit, which approaches −4t²/U exactly. It is a difference between two states of the same system and it is computable to any accuracy, because both states can be described — which is the condition this essay is about, met rather than violated.

The model is what is fitted