π bond orders in 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.

Twelve 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.

Delocalisation

An open chain for contrast, where the orders are not all alike: 0.894 at the ends and 0.447 in the middle. Delocalisation is not a property that a molecule either has or lacks — every conjugated system has it, and what benzene has in addition is that its six orders come out equal by symmetry.

Two carbons further along the same chain, where the alternation narrows towards the middle: 0.871, 0.483, 0.785, 0.483, 0.871. The contrast between formal single and formal double is largest at the ends, and an infinite chain would have none of it — which is the limit the ring reaches by having no ends at all.

Benzene’s bond orders, drawn with each bond’s thickness set by the computed value. All six come out at exactly two thirds, which is the arithmetic form of the measured fact this essay opened with.

Conjugation, and its limits

Hexatriene, where the alternation narrows toward the middle: 0.871, 0.483, 0.785, 0.483, 0.871. The contrast between formal single and formal double is largest at the ends and smallest in the centre, which is the trend that would vanish entirely in an infinite chain.

Bond order from the eigenvectors

Naphthalene’s pi bond orders, each bond drawn at a thickness set by its computed value. Three distinct numbers, in an order no set of equally weighted Kekulé structures gives, and matching the measured bond lengths.

Benzene: six bonds, one number. The uniformity is checked rather than reported — a computation that returned six different values here would be wrong — and the check is worth having even though the result was never in doubt.

Butadiene’s three bonds. The two outer bonds come out at 0.894 and the central one at 0.447 — so the pattern is high, low, high, which is what the drawn structure of two double bonds separated by a single bond suggests. The numbers are not what it suggests.

What one pair can hold together

Benzene’s ring with a single π electron pair in it rather than six. Every bond order is 0.3333, every carbon carries 0.3333 of an electron, and the six bond orders sum to exactly 2.

The smallest case: three atoms, one pair, bond orders of 0.6667 on each of three links, total 2. The same pair in benzene gives 0.3333 on each of six links, and the total is the same.

An eight-membered ring with one pair, with the bond orders drawn. Every bond carries a quarter, eight of them, and the total is exactly two — the same total the three-ring gives with the same pair over three bonds. The size of the ring changes how the two is divided and not that it is two.

Hypervalency does not stop at three centres

The bond orders of a neutral polyene chain at its normal filling, where the alternation has a different origin — the σ frame’s own preference, and the pairing of the π levels. Comparing with the charged chains above separates two effects that look the same in a picture: one is the filling and one is the framework.

Which numbers carry a frame

Butadiene’s bond orders, which are the numbers in the third and fourth rows above. The terminal-to-central ratio of exactly 2.000 is a pure eigenvector quantity, so it survives every change of frame this essay makes — although it does not survive keeping the overlap in the secular equations, which is a change to the model rather than to the frame.

The frame that was allowed to relax

Butadiene’s bond orders in the uniform calculation: 0.8944 at the ends and 0.4472 in the middle. Every number in this collection until now has come from a matrix with every off-diagonal element equal to one, which is a geometry in which all three bonds are the same length.

Butadiene’s three bond orders before and after. The two end bonds rise from 0.8944 to 0.9300 and the central one falls from 0.4472 to 0.3676 — a fifth of its value. That is the fifth column doing something none of the previous four could.

Benzene, unmoved by the relaxation. Every bond order is where the uniform calculation put it, because the matrix the feedback converges to is the matrix it started from — the symmetry that makes the six bonds equivalent is a symmetry of the fixed point as well, and a feedback cannot break a symmetry the matrix has.

An anomaly that is not the first of a series

How far the relaxation moves the most interior cross bond, against the number of rings, at three couplings. Every curve falls, and every one crosses zero.

Every cross bond of four acenes, from one end to the other, unrelaxed dashed and relaxed drawn. The ends do the same thing at every length; the middle changes its mind.

The molecule the anomaly was found in, relaxed. Its shared bond is the shortest thing in the molecule and it strengthens; every longer acene’s does the reverse.

The floor was in the bookkeeping

The change every ring’s bonds undergo when one carbon of the end ring is given a site energy, on a chain of twelve fused hexagons. The upper curve is what the relaxation as written reports. The lower one is the same molecule.

The floor at five strengths of the geometric feedback, beside the residual each fixed point converged to. The floor tracks the model’s parameter; the residual does not move at all.

The fitted reach against the number of rings, bare and with a gap held open. The bare molecule’s reach grows at every step, and its gap is closing at the same time.

The reach is the molecule's

The response of each ring to a heteroatom on the fifth ring of twelve, on a logarithmic scale, with the two rings that share the peak marked.

The step from each ring to the next, taken in both directions from an interior heteroatom and away from an end one.

The height of the response at its peak, against which ring carries the heteroatom.

A bend is not an end

The response on the heteroatom’s own ring, on a straight chain of twelve and on one with a kink in the middle, as the heteroatom is moved along.

The bent chain’s response divided by the straight chain’s, for each placement of the heteroatom.

The fitted decay length on each side of the heteroatom, on the bent chain and on the straight one.

An angular ring rescales what lies beyond it

Each ring’s response as a fraction of the straight chain’s, with the heteroatom on ring 0 and the first one, two, four or ten interior rings angular.

Against the number of angular rings from the heteroatom’s end: the fitted decay length and the responses on the third and sixth rings, each as a fraction of the straight chain’s.

Each ring’s response as a fraction of the straight chain’s, with a single angular ring at ring 1, 3, 5 or 7.