Molecular Geometry & Bonding
what the shape decides
Concept

Antiaromaticity — where it appears

The destabilisation of a cyclic conjugated system whose electron count leaves a degenerate shell half filled. It is a statement about a level pattern rather than about a ring, and its size depends sharply on what the stabilisation is measured against.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

Hückel levels of cyclobutadiene. 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.

Delocalisation is stabilising, and other things that are false in general

Spreading electrons over more centres is supposed to lower their energy. Cyclobutadiene is delocalised in exactly the same sense as benzene, gains exactly nothing by it, and the arithmetic says so before any experiment does.

wrong · Aromaticity Which rings close a shell. Each ring is filled with its own number of pi electrons and asked whether the highest occupied shell came out full. Of the rings drawn here, C6 close — at 6 electrons — which is Hückel's 4n+2, produced here rather than recalled.

Aromaticity as a computed shell closure

Hückel's 4n+2 rule is not a rule. It is what comes out when every ring size is filled and asked whether its highest occupied shell came out full — and the calculation that answers does not know the phrase.

bonding · Aromaticity cyclobutadiene: what alternation costs and gains. The π energy of cyclobutadiene as its bonds are alternated by β(1 ± δ), the elastic cost of alternating them at a stated stiffness, and the sum. The π curve falls linearly in δ, which is measured here and is the whole difference between a molecule that distorts and one that does not. The stiffness is a stated parameter, so where the minimum falls is not a claim; which power of δ each curve goes as is.

The vibration that lowers the symmetry

A molecule in a degenerate electronic state distorts until the degeneracy is gone. Which distortion it needs is a direct product; whether it wins is a race between a π energy falling linearly and a σ frame resisting quadratically, and both powers are measured here.

beyond · Aromaticity Levels of a ring closed with a half turn in it. The orbital energies of a ring with one resonance integral reversed in sign, which is what half a turn in the ribbon of p orbitals does to it. The levels come in degenerate pairs from the bottom up rather than singly, so the count that closes a shell is 4n rather than 4n + 2. 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.

The ring with a twist in it

Reverse the sign of one resonance integral in a ring and its levels stop being one nodeless orbital above degenerate pairs and become degenerate pairs all the way up. The count that closes a shell changes from 4n+2 to 4n, and every aromatic ring size and antiaromatic ring size exchange places.

beyond · Aromaticity Hückel levels of cyclopropenyl cation. 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.

The same ring, three charges

Three carbons in a ring are aromatic with two π electrons, a doublet with three, and a triplet with a delocalisation energy of exactly zero with four. Nothing about the molecule changed but the count, and adding electrons to a π system can make its π binding energy fall.

bonding · Aromaticity The characters of C6, and the levels they are. The 6 representations of the ring's rotation group, drawn as points on the unit circle at 2πk/6. Each level is twice the horizontal coordinate: a representation and its complex conjugate have the same real part, so they are degenerate, and the levels pair up automatically. Only k = 0 — and k = n/2 when n is even — lands on the real axis, so one level is unpaired at the bottom and the shell closes at 4n + 2. Nothing here has been diagonalised.

The ring's levels are its group's characters

The Hückel energies of a cyclic system are twice the real part of the characters of its own rotation group, so the level pattern — one level, then pairs, then one more if the ring is even — is a group theorem rather than a calculation. Hückel's 4n+2 rule is a statement about the representations of a cyclic group and about nothing else.

symmetry · Aromaticity 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 Two kinds of answer to a flux, and only one of them is a curve. The π binding of three rings against the magnetic flux through them, in units of beta and of the flux quantum, measured from each ring's own value at zero flux. Beta is negative, so a binding that FALLS is an energy that rises: benzene's does, which is what a diamagnetic ring current is. Cyclobutadiene's rises in both directions from a corner — its energy has no second derivative at zero field at all, and the two one-sided slopes differ by 12.57.

Two rules that share no arithmetic

Hückel's rule is a statement about a gap. Put a magnetic flux through the same rings instead and ask which of them push the field out, and the answer is the same set — eighty cases, no exceptions — although the second calculation counts nothing and has no shells in it. And the rings the rule excludes turn out to have no magnetic susceptibility at all: their energy has a corner at zero field.

symmetry · Aromaticity

Named alongside it

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

DegeneracyHückel theoryAromaticityHückel's 4n+2 ruleShell closureDelocalisationConjugationEigenvalueBenzeneIrreducible representationsReference stateBond order

All concepts