benzene — molecular orbital 1

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

14 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.

Conjugation, and its limits

Hexatriene’s highest occupied orbital. The coefficients are largest on the terminal carbons and smallest in the middle, and they alternate in sign — so the orbital an electrophile has to reach has most of its amplitude at the two ends of the chain.

Nodes

The same count in a molecule rather than an atom. Benzene’s second π orbital has one nodal plane through the ring, and its partner has one at right angles to it — the node count rises by one for each step up in energy exactly as it does across an atomic shell, and it is read off the eigenvector rather than assigned.

A chain’s third orbital, with two nodes. The nodes fall between atoms rather than on them, which is what a node is in a discrete system — a sign change between neighbouring coefficients — and counting them gives the same integer that ordering the levels by energy gives.

Benzene’s lowest pi orbital. Every coefficient has the same sign, so no bond changes sign and the count is zero — and this is the orbital that carries the ring current. The circle on each carbon has area proportional to the square of the coefficient, which is the density that orbital puts there.

Delocalisation is stabilising, and other things that are false in general

One of cyclobutadiene’s two non-bonding orbitals. Its coefficients are equal in magnitude on two opposite carbons and zero on the other two — so it is bonding across no bond and antibonding across none either, which is exactly what an energy of α means. The other member of the pair is the same picture rotated by ninety degrees.

For contrast, one of benzene’s degenerate pair at α+β. Every coefficient is non-zero, two bonds carry a sign change and four do not, so the orbital is net bonding — which is why benzene’s six electrons all go into orbitals that are worth something and cyclobutadiene’s last two do not.

Hückel theory and what it gets right

The lowest of benzene’s six pi orbitals. Every coefficient has the same sign, so no bond changes sign along it and the node count is zero. The circle on each carbon has area proportional to the square of its coefficient, which is the density that orbital puts there. The slider walks up the levels, reporting energy and node count together.

Aromaticity as a computed shell closure

One of the two non-bonding orbitals of the planar ring. Its coefficients vanish on four of the eight carbons and alternate in sign on the other four, so it is bonding across no bond and antibonding across none either — which is what a non-bonding orbital looks like when it is drawn rather than named.

Three-centre bonding, computed

The non-bonding orbital, with each coefficient drawn at its computed size. The central circle is absent because the coefficient there is zero — a consequence of the mirror symmetry rather than of the parameters, and unchanged by any value of h or k. Two of the system’s four electrons occupy this orbital, and none of that pair is on the central atom.

Bond order from the eigenvectors

Naphthalene’s lowest pi orbital, with the coefficients printed on each carbon. They are not all equal — the ring-fusion carbons carry less amplitude than the others — and the unequal coefficients are exactly what produces three different bond orders from ten identical atoms.

Three shapes from one search

The share of four thousand starts landing on each description, in rank order, on a logarithmic axis, for three cages and fillings. The heavy segment on each curve is its largest step.

The number of distinct descriptions four hundred starts find, for five deltahedra at every even electron count. Most entries are one.

For each case: how many descriptions, the largest ratio between two consecutive shares, and where in the ranking it falls.

Fifty descriptions of one molecule

Three cages: how many descriptions the search finds, what the commonest takes, how far the whole set spreads in the functional, and whether the commonest is the best.

The eight most-reached of the nine-vertex cage’s fifty descriptions, with the functional each achieves.

The whole spread of the functional across every description each cage’s search finds, relative to the best.

Counting was right except where it mattered

Every cage-and-filling pair, by the number of distinct descriptions its localisation finds and by how far apart they are in the functional.

Of forty-eight pairs, how many find one description, how many find more, and of those how many find descriptions that genuinely differ.

The thirteen cage-and-filling pairs whose localisation finds more than one answer.

A second criterion left a gap too

Every non-zero spread under each criterion on one logarithmic axis, with the largest empty stretch shaded. One is a factor of two hundred; the other is seven decades.

Each pair’s spread under one criterion against the other, with each criterion’s own line drawn through its own gap. The shaded quadrants are the disagreements.

The six pairs the criteria classify differently, with both spreads and both verdicts.

The cage is on both sides

Each cage’s automorphism count against how many of its fillings the two criteria disagree about. The largest group and the second largest sit at opposite ends of the result.

The two relative spreads for each of the six, in each criterion’s own units. In every one, one of them is at the floor.

Every pair’s two spreads against each other, with both thresholds drawn. The agreements occupy two corners; the six disagreements sit on the axes.

Six disagreements and three calculations

Every cage-and-filling pair the family survey covers, with the pairs that hand the localisation the same set of orbitals joined.

For the twelve-vertex cage, the number of electrons against the number of orbitals the localisation treats as occupied.

Per cage: the electron counts enumerated, the distinct orbital sets those produce, the open-shell rows, and the criterion disagreements counted both ways.

The basis a diagonaliser happened to return

The twelve-vertex cage at twelve electrons: its Hückel levels, with each degenerate shell drawn as its members and the occupied ones filled.

Every input whose occupied set takes some members of a degenerate shell and leaves others, with how much of the shell it takes and how far each criterion’s best functional moves.

For each affected input, the best Pipek–Mezey functional reached from each of five orientations of the cut shell, each row scaled to its own range.