Correlation energy — where it appears
Named by 19 essays across 3 fields — each of them below, with the objects they name alongside it.
A third kind of correlation
The two-site model gave an exact identity — the power of the correlation energy in the repulsion equals the occupation of the bonding natural orbital — and asked whether anything like it survives with more orbitals. It does not, and the way it fails is better than the identity was: a ring of four gives a power of one where every closed-shell system gives two, at repulsions fifty times weaker than the hopping, because its reference was never a single state.
A mean field cannot get out of the way
The energy a mean field misses grows without limit as the repulsion rises — 27.82 at U = 32 for four electrons on four sites, against an exact energy of −0.2946, so the error is ninety-four times the answer. The rate it grows at is not an energy at all: it is n²/4N, a count of the coincidences a spread-out density cannot avoid.
The reference decides the correlation
The correlation energy of one exact state, measured against two references that are both called Hartree–Fock, is −27.82 and −0.107 at the same repulsion — a factor of 259, on a system whose exact energy is a single smooth curve. Below the instability at U = 2 the two agree to the last bit; above it one grows without limit while the other falls, and the reference that reports almost no correlation has ⟨S²⟩ = 1.99 where a singlet is zero.
Two wrong numbers and a right difference
A mean field gets the total energy of a four-site system wrong by 12.11 and of two two-site systems wrong by 12.49, and 96.9 per cent of that error cancels out of the difference between them. The residue is 0.38 — and the reaction energy it is a residue of is 0.09, so the cancellation improves and the answer gets worse at the same time. Change the pair being compared to a singlet and a triplet and nothing cancels at all: the sign goes.
The correction that was computed somewhere else
Every composite method rests on one assumption: that an expensive correction computed on a small case can be added to a cheap calculation on a large one. Tested on four sites where both answers are exact, it removes 99.74 per cent of the cheap method's error near the reference and −1450 per cent of it further away — at which point the recipe is fifteen times as wrong as the calculation it was improving.
Where the electrons are, without subtracting anything
A correlation energy is an exact energy less a mean-field one, so the answer depends on which mean field was picked — by a factor of 259. The pair distribution says the same thing with no subtraction in it, and two systems matched to the same correlation energy turn out to have electrons in visibly different places.
The warning a cheap calculation gives
A correlation correction computed on one system and carried to another works until it does not, and nothing in the scheme says in advance which. The mean field's own symmetry breaking says: it collapses at a definite field, and the transfer fails where it goes. Across thirty-two systems the two rank together at 0.90.
Half of it is given back at one bond
The correlation hole of a ring removes 1.2653 pairs from zero separation and puts 1.1126 of them one site away. With an on-site repulsion that second number costs nothing, because the interaction is zero there — but with anything that reaches a neighbour it costs 44 per cent of what the hole saved, and with a Coulomb tail 46.9.
The give-back that turned into a saving
Take a wavefunction optimised for an on-site repulsion, weight its correlation hole with an interaction that reaches one neighbour, and the enhancement at one bond gives back forty-four per cent of the on-site saving. Putting that neighbour term into the Hamiltonian and solving exactly does not shrink the give-back. It reverses its sign.
The half of the square a ring of four cannot show
There is a warning that says in advance whether a transferred correction will hold: the mean field's own symmetry breaking, which collapses at a definite site-energy modulation and takes the transfer with it. The other axis of the same square has a threshold too — on the other side — and the ring of four it was all measured on is the one system with no threshold to find.
A sign change is not always a zero
The solved give-back changes sign somewhere between a neighbour repulsion of two and one of four, and a bisection looks like the way to find the value where the structure beyond contact contributes exactly nothing. It changes sign twice. One crossing is that value; at the other the quantity the fraction is a fraction of has vanished instead, and a bisection reports the two in identical words.
The second number is the error, rearranged
A cheap diagnostic for a composite method leaves a scatter it cannot explain, and the number that ought to close it is the change in the correlation energy, already computed at every point, so the test is arithmetic rather than a calculation. It is arithmetic, and the arithmetic is the answer. The composite's error is that change with a sign on it.
Five failures in five different places
Seven quantities a mean field produces for nothing have been tried as diagnostics, and none of them is usable. The question left is whether that is one finding or seven — whether the same awkward corner of the square breaks every candidate, or each is broken somewhere else. Each is broken somewhere else. Five candidates, five failing pairs, ten systems, and not one of them appearing twice.
The arms were the kindest part of the square
Seven cheap quantities have been tried as diagnostics for when a transferred correlation correction will fail, and every one of them was tested on systems lying along one of two arms — a repulsion swept at zero site energy, or a site energy swept at one repulsion. Filling in the square makes every one of them worse, by up to a factor of twelve, so the diagonal is not a region where they might work but the region where they fail hardest. And a grid asks a question a cross cannot: whether a candidate even sorts the systems the way the error does. One of them does not.
A blank is not a pass
Two of the seven cheap diagnostics for a transferred correlation correction could not be scored at all, because a scoring rule that only compares systems whose diagnostic values agree to fifteen per cent finds no such pair for them. The census reported them as blanks, which is correct and reads as the best entry in the table. Dropping the band and reporting the gap beside the ratio scores both — and the two turn out to be unscoreable for opposite reasons, one because it separates the two families as a monotone function of something that already failed, and one because it is unrelated to anything.
The worst system in the square was the solver's
Every score in the transferability square rests on an unrestricted mean field, and every one was computed from a single conventional start. On a band of nine systems, each with a site energy one or two below the repulsion, that start lands above the lowest solution the method has — by 1.12 at a repulsion of six and a site energy of four, which was the largest error in the whole square. Recomputed on the lowest field, the factor of forty-six that first showed the polarisation to be no rule falls to 2.61 and passes; the grid still fails every candidate, by a factor of fifteen rather than forty-eight; and the typical comparison, which no worst case could show, is far better than chance for every candidate but one.
A cancellation between two separations
The give-back's numerator has one term in it, so its root is the repulsion at which the pair count at one bond returns to its free value. Let the interaction reach two bonds and the numerator has two terms with opposite signs, and its root becomes a cancellation between an enhancement at one separation and a depletion at the next — at a repulsion where neither is at its free value. Which side of the old zero the new root falls on reverses between a reach of a third and a half, because whether two bonds are enhanced or depleted is the interaction's own doing.
The worst case measured the reference
Every score on the transferability square carries one correlation correction, fitted at a repulsion of eight and no site energy. Carried instead from seven other systems — three beside the band where the mean field's start misses, four on it — the polarisation's worst error ratio on the lowest mean field runs from 9.7 to 213, where fixing the start had moved it only from 48 to 14.7. Its median stays between 1.21 and 1.55. A worst ratio divides by the smallest comparable error it can find, and the reference decides where that is. And a reference fitted to the wrong solution passes the census at 2.05 while every system's error is above a half.
A root that moves when nothing else does
Give the ring of six an interaction that falls off as one over the distance and the give-back's numerator has three terms. It still has one root, and at that root every separation's pair change has the opposite sign from the two-bond case. But the root has a weakness the earlier essays never tested: lower the interaction by the same amount at every separation and not one state changes, while the root moves — by as much as the reach of the interaction did, and out of existence altogether if the zero is put at one bond.
Named alongside it
The objects these essays reach for when they reach for this one.
Exact diagonalisationHubbard modelReference stateElectron correlationModel limitOn-site repulsionSymmetry breakingHartree–FockComposite methodMany-electron wavefunctionsMean-field approximationTransferability