Molecular Geometry & Bonding
what the shape decides
Concept

Benzene — where it appears

The six-carbon ring whose π system holds six electrons in a closed shell. It is the standing example of a delocalised molecule, and the size of its stabilisation depends on which reference the comparison uses.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

benzene — D6h. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

Delocalisation

Benzene does not alternate between two structures. It has one structure, and the two Kekulé forms are basis functions in a description of it — which is a different and much less exciting claim than the one usually made.

beyond · Delocalisation 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 and C10 close — at 6 and 10 electrons — which is Hückel's 4n+2, produced here rather than recalled.

Where two-centre bonding stops

A bond between two atoms is a special case, not the general one. Rings, clusters and metals are held together by orbitals spread over many centres, and the arithmetic that describes them is the arithmetic already used for benzene.

beyond · Multicentre 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 Hückel levels of benzene. 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.

Hückel theory and what it gets right

A conjugated system's orbital energies are the eigenvalues of a matrix of ones and zeroes. Nothing about carbon enters, no geometry enters, and the results that survive are exactly the ones that depend on neither.

bonding · Delocalisation 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 π bond orders in naphthalene. Each bond drawn with a thickness set by its computed pi bond order, and the number printed beside it. The orders come from the eigenvectors and cost nothing extra once the matrix is diagonalised.

Bond order from the eigenvectors

Having diagonalised the matrix, the coefficients are already there. Two sums over them give every bond's order and every atom's charge, and naphthalene's three kinds of bond come out in the order the measurements find them.

bonding · Delocalisation pyridine against benzene. The Hückel levels of pyridine beside those of benzene, which is the same graph with one diagonal entry and the bonds touching it changed. The parent's levels are symmetric about α because its matrix has nothing on the diagonal; the substituted system's are not, and the asymmetry is the size of the fitted parameter rather than a result.

Hückel with a heteroatom

A hydrocarbon's Hückel matrix contains no fitted number at all. Put one nitrogen in the ring and two arrive at once, the levels stop being symmetric about α, and the delocalisation energy stops being quotable — all from changing three entries in a matrix.

bonding · Delocalisation benzene: what alternation costs and gains. The π energy of benzene 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 quadratically 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 hexagon is the frame's doing

Benzene's delocalisation energy is quoted as 2β and read as the reason its bonds are equal. Let the bonds alternate and the π energy goes down, not up — at every ring size, for 4n+2 as much as for 4n. The π electrons are not what keeps the hexagon regular; the σ frame is, and the margin between them is uncomfortably thin.

beyond · Delocalisation Face to face repels, edge to face attracts. Two benzene molecules 5 Å apart, one turned against the other, with only the quadrupole interaction between them. Stacked, the two negative faces meet and the energy is +8.8 kJ/mol; perpendicular, one molecule's positive rim meets the other's negative face and it is -4.6. The sign changes on the way, so the preference is not that one arrangement is weaker — it is that they are opposite. This is a molecule with no dipole moment at all.

Zero dipole is not no interaction

Benzene's dipole moment is exactly zero at every origin, and its quadrupole moment is large enough to decide a crystal structure. Two benzenes face to face repel by nine kilojoules a mole; edge to face they attract, and the sign changes on the way between.

shape · Dipole 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.

DelocalisationHückel theoryπ systemsReference stateAromaticityConjugationDegeneracyAntiaromaticityCharge densityConventionEigenvalueAdjacency matrix

All concepts