How often two electrons are in the same place

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.

13 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

The virtual excursions, counted. Double occupancy in the exact ground state falls from the independent-electron quarter at U = 0 to almost nothing at large U — but not to nothing, and what remains is exactly the residual hopping that produces the coupling. The coupling and the residual double occupancy are the same phenomenon measured two ways.

The hole that is not repulsion

Two questions asked of the same wavefunction as the repulsion is turned up. How often two electrons are on one site, falling from the independent-electron quarter to nothing; and how the neighbouring spins line up, which starts at −0.075 with the repulsion at zero and deepens from there. The left-hand edge of the second curve is the whole subject of this essay.

How many pairs the correlation moves and where it moves them to, which is the quantity the essay is about. At U = 0 the exact ground state already keeps the electrons apart more than a product of independent orbitals would — the hole is there before any repulsion is switched on, and it is the Pauli principle rather than the repulsion that put it there.

The same hole priced three ways. What it is worth depends on what it is charged against, and the part that is genuinely repulsion — the part that vanishes at U = 0 — is only one of the three. The other two are there at zero repulsion and stay there, which is the separation this essay exists to make.

Where molecular orbital theory dissociates

The probability that both electrons sit on the same site, in the exact ground state of a two-site model, against the on-site repulsion. It falls from a half at no repulsion to 0.0039 at U = 32t. The molecular orbital description gives exactly 0.5 for every point on this axis and the valence bond description gives exactly 0.

That correlation, as the thing it is: a hole. For a ring of six at a repulsion of eight, the change in the number of electron pairs at each separation against the same ring with no repulsion. It removes 1.2653 pairs from separation zero — two electrons on one site — and puts 1.1126 of them at separation one, which is the neighbouring site. Almost everything the correlation does is a move of one step, and a product state has no way to make that move at all, because in a product the two positions are independent by construction.

The first of those rows drawn, with the price beside it: double occupancy per site in the exact ground state of a chain of four against the repulsion, starting at the independent-electron quarter and falling to nearly nothing. The second curve is what the state actually pays in repulsion, which is the product of the two and is not monotone in either — it rises while the double occupancy is still large and falls once the electrons have finished getting out of each other’s way.

Two kinds of correlation, and only one is small

How many pairs the correlation actually moves, and where it moves them to. This is the quantity behind both kinds: the electrons stop meeting, and where the density they vacate goes is what distinguishes a correlation that lowers the energy from one that changes the description. The gap between the two is what the rest of this essay measures.

The occupations of the two natural orbitals against the repulsion. At zero they are 2 and 0, which is a single determinant exactly. As the repulsion grows they converge on 1 and 1, which is a state no single determinant has. The entropy of the occupations rises to ln 2 — one bit, the two determinants of the singlet.

The local power of the correlation energy in the repulsion, measured as a logarithmic derivative, with the bonding occupation drawn over it. At weak repulsion the power is two; at strong repulsion it is one.

Two pictures, one plane

The same quantity computed the other way: the expectation of the double-occupancy operator, summed over sites, in the exact ground state. For two electrons that number IS the ionic weight, which is why the curve here and the curve two figures up are the same curve.

A better energy is not a better answer

The observable being got wrong. Double occupancy in the exact ground state of the two-site system, against the repulsion — a number the trial functions above miss by up to half while their energies are nearly right.

A method that is not additive

The quantity being thrown away, for one dimer: how much of the time both electrons sit on the same site, against the repulsion. At U = 0 it is a half and at U = 8t it is a twentieth. Two dimers’ worth of that, squared, is what the truncation forbids.

The same quantity on four sites rather than two, which is where the replicated units start interacting with the truncation rather than only with each other. The curve has the same shape and a different scale; what the truncation removes are the configurations in which two units are excited at once, and there are none of those until there are two units.

Where the electrons actually are, with no subtraction anywhere in it. This is the quantity a truncation gets wrong first and an energy reports last: an energy is second order in a wavefunction’s error where a property is first order, so a description that is visibly wrong here can still produce an energy that looks nearly right — which is the mechanism by which a size-inconsistent method passes for healthy.

Koopmans' theorem is exact for nothing

What the correlation term is made of. The double occupancy of the four-site chain falls as the repulsion rises — the electrons avoid each other — and the energy that avoidance saves is precisely what a mean field cannot represent, because a mean field has each electron moving in an average rather than dodging a particular one.

A mean field cannot get out of the way

The same quantity, with what it costs beside it. The repulsion actually paid rises to a maximum and then falls, because beyond a certain strength the electrons have already stopped meeting and a further increase has nothing left to charge for. That non-monotonic curve is the exact state’s escape; the mean field’s version of it is a straight line through the origin.

A weight that depends on how it is weighed

The same quantity from the other side: the double occupancy of the exact two-site ground state, which is twice the ionic weight. That it is measurable in the model, unambiguous, and falls as 1/U² is what makes the orthogonal case the one to reason in — and the reason the ambiguity above belongs to the description rather than to the physics.

The warning a cheap calculation gives

What the repulsion is actually doing: the double occupancy falling as the electrons learn to avoid each other. The mean field cannot see that avoidance, which is the error the correction is carrying — and how much of it there is depends on the system in the same way the polarisation does.

Where the electrons are, without subtracting anything

The opposite-spin pair distribution of a half-filled ring of six at six repulsions, each divided by what uncorrelated electrons of the same density would give. Nothing in this picture is a difference between two calculations.

Two rings matched to the same correlation energy per site, to a part in a million, with their opposite-spin pair distributions drawn. They agree about the number and not about the electrons.

The on-site value alone, against the repulsion — the double occupancy, and what the repulsion is charging for it. It is one point of the function above, and the point that carries the whole of the energy.

Half of it is given back at one bond

The same hole weighted three ways. The first row is a hundred per cent at one separation by construction; the other two are not.

The same numbers as a picture. One bar down, one bar up, and two that barely move.

The quantity the on-site energy is proportional to, across the same range: the double occupancy falling towards zero. Everything in the first column of the table above is this number times the repulsion.