Hückel levels of 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.

23 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

Where the series starts: two carbons, two levels, one occupied. The gap between them is 2β, and it is the largest gap any conjugated chain has — every carbon added from here narrows it.

Hexatriene’s six pi levels. Three occupied, three empty, and the gap between the highest occupied and the lowest empty is 0.890β — smaller than butadiene’s 1.236β and very much smaller than ethene’s 2β.

The same three numbers and nine more, from chains of up to sixty carbons, with the asymptote the algebra predicts drawn beneath them. The computed gap sits just below 2πβ/(n+1) everywhere and closes on it — 95.5 per cent of it at two carbons and 99.99 at sixty — so the closed form is not an approximation being tested here but a limit being approached. Every level was checked against 2cos(kπ/(n+1)) before a gap was read off it.

Where two-centre bonding stops

The radial set of a six-vertex cage — the octahedral B₆H₆²⁻ skeleton — with the one skeletal pair that belongs to this set placed in it. The eigenvalues are 4, 0 three times, and −2 twice, and the nodeless orbital sits four β below the next thing there is: a single totally symmetric orbital, well separated, which is the first half of the pattern Wade’s rule needs. The remaining six pairs live in the tangential orbitals, which are not in this calculation.

The same construction on twelve vertices — the icosahedral B₁₂H₁₂²⁻ skeleton. The eigenvalues are 5, 2.236 three times, −1 five times and −2.236 three times: one nodeless orbital, then a triple, then a quintuple. The nodeless one is 2.76β below the triple, so the separation is a fraction of the octahedron’s four β and the qualitative pattern is identical. That the top level is the only nodeless one is Perron’s theorem about a connected graph, and it is checked here rather than read off the drawing.

An eight-membered ring closed with a half turn in it. The levels now come in degenerate pairs from the bottom up rather than singly, so the count that closes a shell is 4n and not 4n + 2 — and eight π electrons, which leave a flat eight-ring open, close this one. Nothing about the atoms, the connectivity or the electron count has changed; one integral changed sign, and the closure rule inverted.

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

Cyclobutadiene’s four levels: one at α+2β, a degenerate non-bonding pair at exactly α, and one at α−2β. Four electrons fill the lowest and then singly occupy the degenerate pair by Hund’s rule. The computed delocalisation energy against two isolated double bonds is zero.

The control, one ring larger. Benzene’s six levels are 2, 1, 1, −1, −1, −2, and six electrons fill the lowest three exactly — a closed shell, with every occupied orbital below the reference line and none on it. The delocalisation energy against three isolated double bonds is 2β. Same kind of ring, same kind of matrix, and a shell that closes.

And one ring smaller, with a charge instead. The cyclopropenyl cation has three carbons and two π electrons, one level at α+2β and a degenerate pair above it, and its two electrons close the shell — giving a delocalisation of 2β against one isolated double bond, the same figure as benzene’s over twice as many carbons. Delocalisation over three centres is worth exactly what delocalisation over six is, when the count is right.

Hückel theory and what it gets right

Benzene’s six pi levels, computed as the eigenvalues of its adjacency matrix. One orbital at α+2β, a degenerate pair at α+β, a degenerate pair at α−β, and one at α−2β. Six electrons fill the lower three. Nothing about carbon was used to produce this.

Butadiene. Four levels at α+1.618β, α+0.618β, α−0.618β and α−1.618β — the golden ratio and its reciprocal, which arrive here because cos(36°) is (1+√5)/4 and for no deeper reason. Four electrons fill the lower two, and the computed delocalisation against two isolated double bonds is 0.472β.

Two carbons further along the same series. Six levels, three occupied, and the whole set of levels has closed up: the gap between the highest occupied and the lowest empty is 0.890β, against butadiene’s 1.236 and ethene’s 2. Nothing was changed but the length of the chain, and the closing follows a closed form that this arithmetic reproduces exactly.

The band limit

Rings of six, ten, twenty and sixty atoms, with every level drawn at its computed energy, beside the density of states for a ring of two thousand. The band edges at ±2β are the same for all of them. The bars are the computed levels binned; the curve through them is the closed-form density, which diverges at both edges.

Benzene, the smallest case worth calling a ring: six levels, the outer two singles and the inner two pairs, spanning exactly the range every larger ring spans. Everything a band does is already visible here in miniature — the edges, the degeneracies, and the crowding towards the top and bottom.

The small rings, with a coarser band for comparison. Four, six, eight and twelve atoms: the edges are already in the right place at four, and the crowding towards them is visible by eight. Four, eight and twelve all have a degenerate pair at the band centre with two electrons in it; six does not, which is the whole of 4n+2 seen as a band-filling condition. The features converge in a definite order and the gap is last.

Aromaticity as a computed shell closure

Cyclobutadiene: four pi electrons, two in the lowest orbital and two in a degenerate non-bonding pair. The computed delocalisation energy against two isolated double bonds is exactly zero — the ring gains nothing at all from being a ring — and Hund’s rule then puts one electron in each degenerate orbital, so the prediction is a triplet.

The cyclopropenyl cation: three carbons, two pi electrons. Its levels are one at α+2β and a degenerate pair at α−β, so two electrons close the shell — and the computed delocalisation against one isolated double bond is 2β, the same as benzene’s. A three-membered ring is aromatic, which is not the first thing anybody would guess.

The cyclopentadienyl anion: five carbons, six pi electrons. Its levels are one at α+2β and two degenerate pairs, and six electrons fill the lowest three exactly — the same closed shell benzene has, on a ring with an odd number of atoms and a charge. The count is what closes the shell, and the count is all that closes it.

The trans influence is an overlap argument

The three-centre four-electron system’s levels, computed. Bonding, non-bonding, antibonding — and the non-bonding orbital has no amplitude on the centre at all, which is why the four-electron version puts its extra pair on the two ends and gives them a partial negative charge.

Three-centre bonding, computed

The three-centre four-electron system: a ligand orbital at each end, the central atom’s orbital between them, four electrons. The lowest level is filled and bonding, the middle one is filled and non-bonding, and the top one is empty. One electron pair is holding three atoms together, which is what “electron-deficient” means — and the arrangement has more electrons in it than a two-centre bond, not fewer.

The same three centres with one pair rather than two. Only the bonding level is occupied, so there is no charge pushed onto the ends and the arrangement is an ordinary three-centre two-electron bond — the boranes’ case rather than the hypervalent one, from the same matrix with the electron count changed.

An allyl system for comparison, which is the same three-site graph with the ends joined to the middle in the same way and a different set of site energies. Its levels are bonding, non-bonding and antibonding for the same reason, and the non-bonding one has the same node on the central atom — so the feature is a property of the graph and not of what the atoms are.

The vibration that lowers the symmetry

The square molecule’s levels. Two electrons in a degenerate non-bonding pair at exactly α — the arrangement Hückel theory produces and the arrangement the Jahn–Teller theorem says cannot survive. Everything below turns on that pair being half-filled rather than on any energy in this diagram.

A moment counts electrons, not orbitals

The site’s oldest example of the same arithmetic: square cyclobutadiene, whose two non-bonding orbitals are degenerate and hold two electrons. Hückel theory plus Hund’s rule predicts a triplet with two unpaired electrons and a moment of 2.83 — and the molecule is a singlet, for reasons the smallest many-electron calculation computes exactly.

Counting electrons in an extended structure

Cyclobutadiene’s four π levels: one filled, two degenerate with one electron each, one empty. Every carbon has its octet and the system has a half-filled shell — the local count and the global count disagreeing, at the smallest size where they can.

What one pair can hold together

Ring levels for sizes six to sixty, with the density of states of a ring of two thousand behind them. Every level of every ring lies inside the band, and the band is the limit the discrete levels fill in — which is the sense in which a solid is a large molecule.

Hückel with a heteroatom

The checks a hydrocarbon system passes: eigenvalues against the ring closed form, the two trace relations, and pairing in both directions. The trace relations are the ones that had to be generalised when heteroatoms arrived — the sum of the eigenvalues equals the trace of the matrix, which is zero only when the diagonal is empty, and the sum of their squares equals the trace of its square, which is twice the bond count only when every bond is one.

The ring with a twist in it

Four atoms in a ring with one interaction reversed. Two degenerate levels at ±√2 rather than one at +2, two at zero and one at −2, so four electrons fill the lower pair completely. Cyclobutadiene’s own electron count closes a shell in a twisted ring.

The flat version of the hero figure, for comparison. One level at +2, two at zero holding one electron each, one at −2. The π energy is exactly 4β, the delocalisation energy is exactly nothing, and the two unpaired electrons are what the twist removes.

Benzene’s electron count in a twisted ring: two levels at √3 filled, two at zero holding one electron each. The π energy is 6.928β against 8.000β flat, and the closed shell has become an open one.

A cage needs one pair more than it has corners

The six radial orbitals of an octahedral cage — one per vertex, pointing at the centre — as the eigenvalues of the cage’s own adjacency matrix. One level at +4, three at zero, two at −2. Exactly one of the six is nodeless, which is the “+1” of n+1.

The icosahedron’s radial set: one level at exactly +5, three at √5, five at −1 and three at −√5. Every vertex has five neighbours and the nodeless combination sits at exactly five, which is a property of a regular graph rather than of this particular one.

The pentagonal bipyramid, which is the first cage in the series whose vertices are not all alike: two of them have five neighbours and five have four. Its top level comes out at 4.317, between the two degrees and equal to neither, and it is still simple and still nodeless — which is what Perron’s theorem promises for a graph that is merely connected rather than regular.

A band gap is not a bond energy

The two-level case where all of this can be seen at once. The gap between the two levels is the excitation energy; twice the distance from the bonding level up to α is roughly the bond energy; the ionisation energy is measured from the bonding level to somewhere off the top of this picture entirely. Three quantities, one diagram, and the diagram is what makes them look alike.

The same ring, three charges

The cyclopropenyl cation: three π levels at 2β, −1β and −1β, with two electrons in the lowest. The shell is closed, the gap to the next level is 3β, and the delocalisation energy against one isolated double bond is exactly 2β.

The same three levels with four electrons. The extra pair occupies the degenerate antibonding pair one apiece, the total π energy has fallen from 4β to 2β, and the delocalisation energy against the same reference is nothing at all.

The tropylium cation: seven levels, six electrons, a closed shell and a gap of 1.69β. Delocalisation energy 2.988β — the largest of any ring in this collection relative to its reference — which is why a seven-membered carbon ring bearing a positive charge is a stable, crystallisable salt.

The bonds are what is left over

The radial orbitals of a six-vertex deltahedron, solved as a graph. One strongly bonding combination and a set above it — a pattern with no relation to the number of edges, which is why counting edges stops working exactly here.

A third kind of correlation

Why: the half-filled ring of four has a degenerate pair at the top of its occupied set with two electrons to put in it, and no way to put them in that is a single determinant. Everything above follows from that one fact about the level pattern.

A stabilisation is measured from somewhere

The check underneath both columns. Every ring’s computed eigenvalues against the exact expression they must satisfy, with the largest disagreement printed. A delocalisation energy is a difference of two sums of these, so an error of 10⁻¹⁴ in a level is an error of 10⁻¹⁴ in the conclusion.

One spectrum, a line of models

Benzene’s six π levels from seven members of the family. The two occupied levels are at the same height in every column, because those are what was fitted. The empty level is not: it runs from 3.15 electronvolts below the vacuum to 4.69 above it, which is the difference between a molecule that binds an extra electron easily and one that does not bind it at all.

The levels in the units the theory is usually written in, with the reference state marked. In these units the delocalisation energy is 2β by construction — the number 2 is a property of the ring’s eigenvalues and has nothing to do with what β is worth in electronvolts.

Three of the family rather than six, so the movement can be followed. The two fitted levels sit at the same height in every column because that is what was fitted; the empty level moves, and the resonance integral behind the three columns differs by more than a factor of two.

Two rules that share no arithmetic

Cyclobutadiene’s levels, with the half-filled degenerate pair that the corner comes from. Every ring whose energy has a corner at zero flux has this shape at the top of its occupied set, and every ring whose energy is smooth there does not. That is the whole mechanism, and it is a statement about a level diagram rather than about a magnet.

The current does not divide

The levels the whole response comes out of. A flux moves them, the occupied ones move differently from the empty ones, and the second derivative of what is left is the current.