Molecular Geometry & Bonding
what the shape decides
Concept

Mean-field approximation — where it appears

A description in which each electron moves in the averaged field of all the others, so a many-electron problem becomes a set of one-electron ones. It is exact when the true state is a single determinant, and what it misses is the correlation energy, defined as the difference from the exact answer.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The slope goes to a half, and a window fit stops short of it. The local slope of the alternation against the reduced temperature, between each neighbouring pair of points, on a ring of 40 at K = 1.6. It rises monotonically from 0.4115 to 0.5053 as the transition is approached, crossing a half at about a part in a thousand of the reduced temperature. The fitted 0.44 is the average of the left-hand end of this curve; the exponent is one half, which is what a free energy analytic in one order parameter is obliged to give.

The exponent was the window's

A fit over the last decade before a distortion vanishes gives an exponent of 0.44, and running it on larger rings should say whether the number belongs to the transition or to a forty-site ring. It belongs to neither. The local slope runs to 0.5020 as the transition is approached, and 0.44 is what a fit over that particular decade returns — on every ring size and every stiffness, because the whole curve is one curve.

wrong · Metal The second number is the first one, rearranged. The composite's error against the change in the correlation energy, at every point on both axes of the square. They lie on the diagonal because they are the same quantity: the composite is the target's mean field plus the reference's correlation energy, so its error is the reference's correlation energy minus the target's. The largest departure across 20 points is 2.2e-16, which is the arithmetic's own precision and not a measurement.

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.

wrong · Approximation Five candidates, five failures, five different places. Every point of the square, with each cheap diagnostic's failing pair joined by a line. The five tested candidates fail on five different pairs involving 10 different points — no line shares an end with another. Had they all failed on one corner the lines would have converged on it, and the honest conclusion would have been that composites are safe away from that corner.

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.

wrong · Approximation

Named alongside it

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

Composite methodCorrelation energyError cancellationTransferabilityBand gapClosed formConventionCritical exponentElectron correlationExact diagonalisationFree energyHubbard model

All concepts