Molecular Geometry & Bonding
what the shape decides
Concept

Error cancellation — where it appears

The reliance on two errors in a calculation being nearly equal, so that a difference between two computed quantities is better than either. It is why relative energies are quoted where absolute ones are not, and it fails silently once the two systems stop resembling each other.

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

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 energyMean-field approximationTransferabilityElectron correlationExact diagonalisationHubbard modelMany-electron wavefunctionsModel selectionReference stateSymmetry breaking

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