The energy is the last thing a wrong wavefunction gets wrong

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.

Four 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.

A better energy is not a better answer

Two errors against the error in the wavefunction that produced them, on log axes. The energy’s line has slope 2.00 and the double occupancy’s has slope 1.01. The two diverge as the wavefunction improves, without limit.

Nearly all of the error cancels, and the answer gets worse. Two trial functions whose energies differ by very little give observables that differ by a great deal, because the cancellation that protects the energy is exactly the cancellation that leaves the wavefunction free to be wrong. That is the essay’s claim as a measurement rather than as an argument from orders.

The same measurement on four sites rather than two. The energy is still the last thing a wrong wavefunction gets wrong, and the margin has widened: with more orbitals there are more ways for a trial function to be wrong in directions the energy does not see. Nothing about the argument is a two-site accident.

Two wrong numbers and a right difference

For each repulsion: the mean field’s error in the four-site total energy and in the two-dimer total, the residue left in their difference, and two percentages. The cancellation rises from 83 per cent to 97 across the figure. The residue as a share of the answer rises from 1 per cent to 423 over the same range.

The exact singlet–triplet gap of the four-site chain against the repulsion, and the mean field’s. They agree at zero repulsion and part immediately. From a repulsion of four the mean field puts the triplet below the singlet, which is not an inaccurate gap but the wrong ground state.

The exact ground state of a four-site ring, the mean field’s, and the composite: the mean field plus a correlation correction computed on the symmetric system and added to an asymmetric one. Where the two systems are the same the composite is exact, by construction and to the last bit. As the sites are made unlike it drifts, and what it drifts by is the change in correlation energy the correction did not know about — which is the residue of this essay, arriving in a scheme built to exploit the cancellation rather than in a reaction that happens to enjoy it.

The correction that was computed somewhere else

Nearly all of the error cancels, and the answer gets worse. That is the mechanism a transferred correction relies on and the reason it cannot be relied on: the cancellation is between two calculations sharing a convention, and a correction computed elsewhere shares neither the convention nor the system.

The exact ground state, the mean field’s, and the composite, at seven asymmetries. The two columns on the right are the errors: the composite’s and the one it is supposed to be improving on. At ε = 0.5 the composite removes 99.74 per cent of the mean field’s error, and at ε = 8 it is fifteen times as wrong as the mean field.

The same table as a fraction: how much of the mean field’s error the transferred correction removes. It goes through zero and keeps going, and once it is negative the recipe is subtracting an error rather than adding a correction. The clipping at −120 per cent is the plot’s, not the arithmetic’s.

Where the electrons are, without subtracting anything

The quantity the argument began with: how far a single determinant is from the exact answer. The answer now is that the gap is real, that its size depends on what it is measured from, and that the thing it is a gap about is better looked at directly.