Molecular Geometry & Bonding
what the shape decides
Concept

Polyene — where it appears

An open chain of alternating double and single bonds. It is the reference a cyclic system is compared against when the question is what closing the ring is worth, and it is already stabilised relative to isolated double bonds.

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

Hückel levels of hexatriene. The orbital energies of the pi system, computed as the eigenvalues of the molecule's adjacency matrix. Degenerate levels share a line. The electrons are placed by aufbau with Hund's rule, so a half-filled degenerate shell shows as two unpaired spins.

Conjugation, and its limits

Every extra double bond in a chain lowers the gap between the highest occupied and lowest empty orbital, which is why long conjugated molecules are coloured. The trend has a limit, and the limit is not where the arithmetic says it should be.

beyond · Delocalisation The two figures, per electron. For each ring, the delocalisation energy against isolated double bonds and against the same ring with one bond deleted, divided by the number of pi electrons. The upper bar of each pair is the usual reference; the lower one is the cycle's own contribution, and the two are nowhere proportional.

A stabilisation is measured from somewhere

Benzene's delocalisation energy is 2β against three isolated double bonds and 1.0121β against the same six carbons with one bond deleted. Cyclobutadiene's is exactly zero against the first reference and −0.4721β against the second, so one of the two references records the most famously destabilised ring in the subject as neutral.

beyond · Delocalisation One pi energy, two delocalisation energies. For each of six rings: the computed pi energy, the energy of the same atoms with one bond deleted, and the delocalisation energy that follows from each of the two reference states. Every entry is an eigenvalue sum; the two right-hand columns differ only in what was subtracted from the second column.

The frame that was allowed to relax

Four changes of frame were tested on nine quantities and none of them could move a bond order, because a bond order is a property of the eigenvectors and every change left the eigenvectors alone. Letting the geometry answer back does move them — butadiene's central bond falls from 0.4472 to 0.3676 — and it moves naphthalene's the other way, because its weakest bond is the one two rings share and relaxation strengthens it.

bonding · Delocalisation

Named alongside it

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

ConjugationDelocalisationHückel theoryBenzeneBond alternationConventionEigenvalueHOMO–LUMO gapReference stateAntiaromaticityAromaticityBond length

All concepts