Molecular Geometry & Bonding
what the shape decides
Concept

Self-consistency — where it appears

An iteration in which a calculation's output is fed back into its own input until the two agree. Here it is a geometry allowed to answer back — a strong bond gets a larger resonance integral, which makes it stronger still — and the fixed point is what the molecule settles at.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The shared bond's response falls and reverses, at every coupling. How far the relaxation moves the most interior cross bond of an acene, against the number of rings, at three couplings. Every series falls monotonically and every one changes sign — at λ = 0.2 between 3 and 4 rings, at λ = 0.4 between 3 and 4 rings, at λ = 0.6 between 4 and 5 rings. Where it crosses is a property of the coupling; that it crosses is not. The natural reading — that the effect grows with the number of rings feeding a bond — is refused by every one of these curves.

An anomaly that is not the first of a series

Naphthalene's shared bond behaves differently when the geometry is allowed to answer back, and the obvious reading is that two rings feeding one bond is what does it. Run it up the acenes and the effect falls at every step and changes sign: +0.02036 for two rings, +0.00678 for three, −0.00029 for four, −0.00603 for five. Two rings feeding a bond is not the beginning of anything.

bonding · Delocalisation One heteroatom, and how far the relaxation carries it. The change every ring's bonds undergo when one carbon of the end ring is given a site energy, on a chain of 12 fused hexagons with its gap held open. The upper curve measures each molecule's couplings from its own mean bond order, which is what this collection's relaxation has always done; it stops falling at 4.48e-5 and stays there. The lower curve measures both from the same mean and keeps falling to 7.89e-9. Nothing about the molecule differs between them.

The floor was in the bookkeeping

A heteroatom in one ring of a fused chain is a perturbation with a place, and the relaxation carries it along the molecule. Measured directly, the response stops falling after five rings and sits at four hundredths of a millionth for ever. That floor is not the molecule. It is the relaxation measuring each system's couplings from its own mean, and removing it recovers nine orders of magnitude.

bonding · Delocalisation The conventional start misses the lowest mean field on one band, and nowhere else. A four-site ring at every combination of repulsion and staggered site energy on a 12 by 12 grid. A small dot is a system where the conventional broken-symmetry start reaches the lowest mean-field solution; a circle is one where it lands above it, with area proportional to how far. There are 9, every one with a site energy one or two below the repulsion, and the largest miss is 1.124 at a repulsion of six and a site energy of four.

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.

wrong · Approximation

Named alongside it

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

Bond orderClosed formConventionDelocalisationHückel theoryModel limitReference stateApproximationBand gapBond lengthComposite methodConvergence

All concepts