Molecular Geometry & Bonding
what the shape decides
Concept

Resonance — where it appears

Writing a wavefunction as a mixture of several structures, none of which the molecule alternates between. The weights of the structures are not observable, and the standard conventions for computing them disagree by factors.

Named by 5 essays across 2 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 Splitting goes with overlap. For each pair, the atomic levels on the outside and the combinations they form in the middle, with the splitting drawn in proportion to the computed overlap. A pair that symmetry forbids does not split at all, because its overlap is exactly zero.

Molecular orbital and valence bond

Two frameworks, taught as rivals, describing the same molecules. One starts from delocalised orbitals and localises; the other starts from localised bonds and delocalises. Pushed far enough they meet.

bonding · Models 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 The ionic weight against how much ionic structure is in the wavefunction. The percentage each convention calls ionic, at a fixed structure overlap, as the amount of ionic structure in the wavefunction is raised from none to the molecular orbital value. They meet at both ends of the sweep and disagree everywhere between. Every curve is a weight and every set sums to one.

A weight that depends on how it is weighed

The ionic character of a two-electron bond is quoted as a percentage. For one wavefunction at hydrogen's bond length, three conventions in the literature give 18.73, 34.74 and 5.88 per cent — a factor of six — and on a wavefunction with no ionic structure in it at all, one of them still reports a quarter.

bonding · Models seven sets of parameters, one spectrum. Benzene's π levels from seven sets of the three Hückel parameters, each fitted to reproduce the two measured ionisation energies exactly. The two occupied levels sit at the same energy in every column, because that is what was fitted. The empty level moves from -3.15 to 4.69 electronvolts across the family, and the resonance integral from -3.05 to -7.66.

One spectrum, a line of models

Hückel theory has three parameters and benzene's photoelectron spectrum supplies two numbers, so the fit has a curve of solutions rather than a point. Along it the resonance integral runs from −3.05 to −7.66 electronvolts, the empty level moves by eight, the terminal-to-central bond order ratio in butadiene goes from 2.000 to 2.671 — and the delocalisation energy is 6.100 electronvolts in every member.

bonding · Models

Named alongside it

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

DelocalisationBasisHückel theoryMolecular orbitalOverlap integralBenzeneBond orderConventionEigenvectorModel limitπ systemsValence bond

All concepts