Molecular Geometry & Bonding
what the shape decides
Figure

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.
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.

One of the figures on hückel systems: Adjacency matrices diagonalised: levels, coefficients, bond orders, and the shell closures that decide which rings are stable.

Three essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

In the essays

A stabilisation is measured from somewhere

Six rings, each with its π energy computed from its own adjacency matrix, and the same six with one bond deleted. The two right-hand columns are delocalisation energies: the first subtracts a count of isolated double bonds, the second subtracts the open chain. Nothing else differs between them — the same eigenvalues, the same electron counts, the same code.

Benzene’s six levels beside the six levels of the same six carbons with one bond removed, both filled with six electrons. The closed ring’s occupied sum is 8.0000β and the open chain’s is 6.9879β, so closing the ring is worth 1.0121β. Three isolated double bonds would have put the difference at 2.0000β.

Cyclobutadiene, closed and cut open. The ring’s four electrons give exactly 4.0000β, which is precisely two ethenes; butadiene’s four give 4.4721β. Closing the ring is therefore worth −0.4721β — it costs energy — while the isolated-bond reference reports 0.0000β.

Which numbers carry a frame

Nine quantities against four changes of frame, with a filled mark where the number moves. Three quantities survive all four and every one of them is a property of the eigenvectors. The two leftmost columns are exact symmetries, so a filled mark there means the row is a coordinate rather than a quantity; the two rightmost are conventions, so a filled mark means the row needs its convention quoted beside it.

The two references side by side and per electron, for the same six rings. The upper bar of each pair is the usual reference and the lower is the other, and the divisor is the electron count in both — so what separates the pairs is the reference alone, and what separates one pair from the next is the molecule. Reading the figure by pairs and reading it by rows answer two different questions, which is what having two conventions on one number means in practice.

Benzene’s delocalisation energy and cyclobutadiene’s, measured from two references. Against isolated double bonds the second is exactly zero, so no ratio exists; against the ring cut open it is −0.472 and the ratio is a finite number of the wrong sign. Both columns are the same two molecules and the same solver, and only the second operand of a subtraction has changed.

The frame that was allowed to relax

The reference problem: the same molecules’ delocalisation energies against several constructions of what they are being compared with. Adding relaxation makes it worse rather than better, because a relaxed molecule should be compared with a relaxed reference and nothing says how to relax an isolated double bond.

Every figure · Every orbital, by what it encloses · All essays