Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 5

Page 5 of 36, continuing through the fields in the same order.

Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search

Beyond the octet

Hypervalency without d orbitals, delocalisation, aromaticity, and the structures the first-year rules quietly cannot describe.

sulfur hexafluoride — Oh. 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.

Hypervalency without d orbitals

Sulfur hexafluoride is not d²sp³ hybridised. The d orbitals are far too high in energy to contribute meaningfully, the bonding is three-centre four-electron, and the textbook account has been known to be wrong for fifty years.

5 figures 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.

7 figures 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.

7 figures 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.

8 figures Rings of 6, 10, 20, 60 and the band at 2000. The Hückel levels of rings of 6, 10, 20, 60 atoms, all of them inside the same interval from −2 to +2, beside the density of states of a ring of 2000. The histogram is the computed levels; the line through it is the closed-form density, which diverges at both band edges.

The band limit

Every level of a ring of n atoms lies between −2β and +2β, however large n gets. The levels do not spread out as the molecule grows; they crowd into a fixed interval — and that crowding, computed, is a band with its density of states diverging at both edges.

5 figures Hückel levels of three-centre four-electron. 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.

Three-centre bonding, computed

Three orbitals in a line and four electrons: a bonding level, a level with exactly zero amplitude on the central atom, and an empty antibonding one. The middle atom never exceeds an octet, and the ligands carry the charge — which is why every molecule that needs this arrangement has electronegative ligands.

5 figures 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.

7 figures pyrrole against cyclopentadienyl anion. The Hückel levels of pyrrole beside those of cyclopentadienyl anion, 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.

Six electrons in a ring that is not all carbon

Pyrrole and the cyclopentadienyl anion hold the same six π electrons in the same five-membered ring. The anion spreads them perfectly evenly; pyrrole's nitrogen keeps 1.720 of them and furan's oxygen keeps 1.791 — the more electronegative atom donates less, and the bond orders to it fall with it.

5 figures π bond orders in benzene. 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.

What one pair can hold together

Put a single electron pair into a ring of any size and it supplies a total bond order of exactly two and a π energy of exactly 4β — three atoms, eight atoms or six hundred. Spreading a pair over more centres divides the bonding among them; it neither creates nor destroys any.

8 figures 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.

6 figures 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.

6 figures How often two electrons are in the same place. Double occupancy per site in the exact ground state of a chain of 4, against the repulsion. It starts at the independent-electron quarter and falls to nearly nothing. The second curve is how the neighbouring spins line up in the same wavefunction, which is already -0.0750 at no repulsion at all — that part is exchange — and deepens as the electrons are kept apart.

The hole that is not repulsion

Two electrons in a bond keep out of each other's way, and the obvious reason is that they repel. Setting the repulsion to zero and computing the spin correlation exactly gives −0.125 rather than nothing, and the number is reproduced to nine decimal places by a determinant with no repulsion in it at all.

5 figures