pyridine against benzene

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

Five 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.

Six electrons in a ring that is not all carbon

Pyrrole’s π levels beside the cyclopentadienyl anion’s, both with six electrons. The all-carbon ring has a degenerate pair and levels symmetric about α; pyrrole has neither, and its lowest level has dropped to 2.320β. One diagonal entry and two off-diagonal ones did all of it.

Furan against pyrrole, the two differing only in the heteroatom’s diagonal entry. Furan’s lowest level is deeper — 2.633β against 2.320 — and that depth is exactly what keeps the electrons on the oxygen rather than in the ring.

Furan against the same all-carbon parent, so the two donors can be compared against one reference rather than against each other. The parent’s levels are symmetric about α because its matrix has nothing on the diagonal; furan’s are not, and the size of the asymmetry is the fitted parameter rather than a result.

Hückel with a heteroatom

Pyridine’s six π levels beside benzene’s. The graphs are identical — the same six-membered ring, the same six edges — and the matrices differ in three entries: one on the diagonal and the two bonds touching it. Benzene’s levels are symmetric about α because its diagonal is empty. Pyridine’s are not, and the asymmetry is the size of a parameter rather than a result.

Pyrrole against the cyclopentadienyl ring it is built from — a five-membered ring with a nitrogen contributing two electrons, so six π electrons over five atoms. The nitrogen’s parameter is three times pyridine’s, because a nitrogen holding a lone pair in the π system sits far lower than one holding it in the plane, and the charge on it comes out at 1.72: it has given away nearly a third of an electron to the ring rather than taking any.

Acrolein against butadiene — the same four-atom chain with a carbonyl oxygen at one end. The levels slide down and apart, and the charges run 0.77, 1.03, 0.67, 1.53: the oxygen has taken half an electron, and the carbon two bonds away from it is the most depleted atom in the molecule. Conjugate addition attacks that carbon, and the reason is in the fourth column of a four-by-four matrix.

Hypervalency is about the ligands

The three levels, the two that are filled, and the coefficients on each. The circle areas are the coefficients and the colours are their signs — and the middle circle of the second occupied level is not small, it is absent. That is where the charge comes from.

The charges against the ligand’s electronegativity parameter. The line at half an electron is the floor — the value at zero difference, which no choice of parameter can go under — and both curves move away from it in the same direction as the ligand is made more electronegative.

The floor itself, and what it is proportional to. However the ends’ electronegativity is set, the charge they take cannot go below the value the level structure alone produces — and that floor scales with the coupling rather than with any electronegativity difference. A two-centre bond has no such floor, because it has no level with a node in the middle to put a pair into.

Four centres, and the pair that will not localise

The three-centre four-electron system, which is the same argument with one fewer centre and one more pair. Its levels are bonding, non-bonding and antibonding, and the pair in the non-bonding orbital sits entirely on the two ends. Comparing the two cases shows what changes: there, the pair avoids the middle; here, it belongs to the cap.

Hypervalency does not stop at three centres

The charge on each end of a three-centre system as the electronegativity of the ends is raised. At zero difference the ends already take half an electron each — which is the chain’s own doing — and an electronegativity difference adds to it. The two effects are separable and only one of them is about the elements involved.