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 2 essays across one field — 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

Named alongside it

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

Bond orderClosed formConventionDelocalisationHückel theoryModel limitReference stateApproximationBand gapBond lengthConvergenceDegeneracy

All concepts