Concept

Pair distribution — where it appears

How likely two electrons are to be found a given distance apart, counted from one state alone. Measured against the free state it shows where correlation moves pairs, and the changes at every separation add to zero because the number of pairs is fixed.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The same hole, priced three ways. The correlation hole of a ring of 6 at a repulsion of 8, weighted by three interactions. With an on-site interaction the answer is 100 per cent at separation zero — as an identity, since the interaction is zero everywhere else. With one that reaches a neighbour, the enhancement at separation one costs rather than pays, and gives back 44.0 per cent of the on-site saving; with a Coulomb tail, 46.9. Everything beyond one neighbour is worth under a twentieth of the on-site term.

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.

beyond · Correlation

Three terms in the numerator, and one place they cancel. The three contributions to the give-back's numerator on a six-site ring whose interaction falls as one over the distance round the ring — a half at two bonds, a third at the antipode — with their sum. The sum crosses zero once, at V = 5.9198, where the one-bond and antipodal terms are negative and the two-bond term positive: the two-bond enhancement is paying for depletions on either side of it.

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.

bonding · Correlation

Named alongside it

The objects these essays reach for when they reach for this one.

Convention Correlation energy Electron correlation Exact diagonalisation Hubbard model Closed form Expectation value Finite-size effect Long-range interaction Model limit On-site repulsion Probability density

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