Molecular Geometry & Bonding
what the shape decides
Concept

Adjacency matrix — where it appears

A matrix with a one wherever two atoms are joined and a zero everywhere else, which is the whole of a Hückel calculation's input. Its eigenvalues are orbital energies and its eigenvectors are the coefficients, so a conjugated system's level pattern is a property of a graph rather than of chemistry.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

⟨x²⟩ is the average number of neighbours. For each structure, the mean coordination counted off the edge list beside the mean of x² measured off the eigenvalues. They are equal by an identity about graphs, not by a limit — the full band width beside them obeys no such rule.

The width of a band is a count of neighbours

The mean of the squared level energies equals the average coordination — exactly, for any structure, with no limit taken and no periodicity assumed. It is the one statement in this field that is arithmetic rather than physics, and the usual textbook formula for band width is a special case of something weaker.

solids · Bands in a solid 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 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 The radial set of a 6-vertex cage. The energies of the 6 radial orbitals of a deltahedron — one per vertex, pointing at the centre — as the eigenvalues of the cage's own adjacency matrix. The top level is nodeless and is the only one that is, which is Perron's theorem about a connected graph rather than an observation. 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.

A cage needs one pair more than it has corners

Every closed borane holds n+1 skeletal electron pairs for n vertices, and the extra one is a theorem about connected graphs rather than an observation about boron. A cage's radial orbitals have exactly one nodeless combination, always, whatever its shape.

beyond · Multicentre ⟨x²⟩ is the average number of neighbours. For each structure, the mean coordination counted off the edge list beside the mean of x² measured off the eigenvalues. They are equal by an identity about graphs, not by a limit — and the binding each bond supplies, in the last column, obeys no such rule and falls as neighbours are added.

The bond that weakens as neighbours multiply

Wrapped structures with two, four and six neighbours per site give a mean of x² of exactly 2, 4 and 6. Their binding per site goes 1.272, 1.611, 1.979 — and their binding per bond falls from 0.636 to 0.330, which is why a metal atom with twelve neighbours has weak bonds and a great many of them.

solids · Cohesion 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

Named alongside it

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

EigenvalueGraphCoordination numberDegeneracyHückel theoryBands in a solidBenzeneConjugationDelocalisationπ systemsTight-binding modelsTrace

All concepts