Two rules that share no arithmetic

The rule that decides whether a ring is aromatic is a counting rule. Fill the levels, look at the top occupied shell, and ask whether it is full: it is full exactly when the electron count is $4n + 2$. The rule is a group theorem rather than a calculation — the levels of a cyclic system are twice the real part of the characters of its own rotation group, so the pattern of one level, then pairs, then possibly one more is fixed before any matrix is diagonalised.

That is still an argument about a level diagram. What a chemist actually measures, when the question is whether a ring is aromatic, is usually none of those things. It is a magnetic property: the ring current, seen as an unusual chemical shift in a nuclear magnetic resonance spectrum, or as a susceptibility that is too large for the atoms present — a quantity of the same family as the moment a magnetochemist fits and subject to the same kind of doubt.

So there are two rules in circulation and they are usually treated as one. This essay computes the second one from scratch and asks whether it separates the same rings.

A flux is a shift of the label #

Putting a magnetic field through a ring does one thing to a Hückel calculation: it multiplies each resonance integral by a phase factor, and around a closed loop those phases add up to $2\pi$ times the flux measured in quanta. That is London’s construction, and for a regular monocycle that factor can be shared equally among the bonds — which means the whole effect is to shift the label $k$ by the flux. The expression derived from the characters of $C_n$,

$$x_k = 2\cos\!\left(\frac{2\pi k}{n}\right),$$

becomes

$$x_k(\varphi) = 2\cos\!\left(\frac{2\pi (k + \varphi)}{n}\right),$$

with no linear algebra anywhere in it. It is the Hückel matrix with one number changed on each bond, and the number is a phase factor.

There is an immediate consequence, and the character argument comes within a word of it. A Möbius ring — one resonance integral given the opposite sign, which is what a half twist in the ribbon of p orbitals does — has eigenvalues $2\cos((2k+1)\pi/n)$, and in character terms that expression is a projective representation, in which the generator carries $\omega^{k + 1/2}$ rather than $\omega^k$. Half-integer $k$. But half-integer $k$ is exactly what $\varphi = \tfrac{1}{2}$ produces, and the ring with a twist in it meets that expression from the other side.

What benzene does, and what cyclobutadiene does instead #

The total π binding at flux $\varphi$ is the sum of the occupied $x_k(\varphi)$, and for benzene it has a closed form. Filling $k = 0, \pm 1$ with six electrons and adding the cosines gives

$$X(\varphi) = 8\cos\!\left(\frac{\pi\varphi}{3}\right),$$

so $X(0) = 8$, which is the familiar delocalisation energy, and $X''(0) = -8\pi^2/9 = -8.7730$. Since β is negative, a binding that falls is an energy that rises: benzene’s energy goes up when a flux is put through it, in either direction, and a system whose energy rises in a field pushes the field out. That is diamagnetism, and this is where a ring current comes from.

Cyclobutadiene does something else entirely. At zero flux its four electrons meet a half-filled degenerate pair. Turn the flux on by any amount and the pair splits — one member goes up, the other down — and the electrons follow the lower one. Turn it on the other way and they follow the other member. So the binding rises whichever way the flux is turned, and it rises linearly:

$$X(\varphi) = 4 + 2\pi|\varphi| + O(\varphi^2).$$

The corner, and the measurement that cannot see it #

A magnetic susceptibility is a second derivative of the energy with respect to the field, and cyclobutadiene’s energy does not have one at zero field. The two one-sided slopes are different numbers, and the difference is exactly

$$\frac{16\pi}{n}\sin\!\left(\frac{\pi e}{2n}\right),$$

which is $4\pi = 12.566371$ for four electrons in a four-ring, $2\pi$ for eight in an eight-ring, and $7.255197$ for four electrons in a six-ring. The computation agrees with that expression to four decimal places in every case tested.

This is worth being careful about, because the received statement is that an antiaromatic ring has a large paramagnetic susceptibility. Within this model an undistorted one has none at all. Its energy falls whichever way the field is turned, at a finite rate, from zero field — which is a stronger statement than a large susceptibility and a different kind of statement.

Measuring it takes some care, and the care is the reason this section exists. A centred difference stencil through a corner returns the average of the two slopes, which is zero, and so reports a smooth extremum where there is a kink. A calculation done that way reports cyclobutadiene as having no ring current — a plausible number, produced by an instrument that cannot see the thing it is pointed at. The slopes have to be measured on each side separately, at two step sizes, and what distinguishes a real corner from a smooth function is that halving the step halves the apparent kink in one case and leaves it alone in the other.

Every ring, every count #

The mechanism suggests a rule, and the rule can be tested exhaustively at this size. Every ring from four sites to eighteen, at every even electron count it can hold — eighty cases — is classified by whether its energy rises with flux or has a corner. Beside it, the same eighty rings are classified by whether the electron count closes a shell, which is the shell-closure calculation.

Eighty of eighty. That is a two-route agreement of the strongest kind, and the two routes really do share nothing but the ring. The shell closure obeys Hund’s rule, needs a tolerance to decide when two levels are degenerate, and asks whether the highest occupied shell is full — the shell-closure argument. The magnetic classification differentiates a sum twice and looks at a sign; it has no shells in it, no aufbau, and no degeneracy tolerance anywhere.

Both verdicts occur in the census, which is the part that stops the agreement being trivial: forty-four of the eighty are diamagnetic and thirty-six have corners. A rule that said yes everywhere would agree with anything.

Opening a gap #

An undistorted four-ring is not a molecule. Cyclobutadiene distorts to a rectangle, and the reason it distorts is a π effect rather than a σ one — the same analysis that shows benzene’s hexagonal shape is the σ frame’s doing shows that a four-ring’s π system is distortive.

So the honest question is what the distorted ring does in a field. Alternating the bonds opens a gap, the degenerate pair separates, and the corner has nothing to sit on.

The susceptibility that appears is inverse in the gap. Multiplying the two together over a twentyfold range of gaps gives 39.1, 38.7, 37.4, 35.2 and 30.3 — constant to within a factor of 1.3 while the susceptibility itself changes by a factor of 26. So the paratropic response of an antiaromatic ring is a statement about how small its gap is, and the received account, which speaks of it as a property of the electron count, is naming the cause of the small gap rather than the mechanism.

Bigger rings, and which quantity grows #

For the rings that do have a susceptibility, the size of it moves in a direction that repays attention. The response to a flux falls as the ring grows: the second derivative of the binding with respect to flux is −8.773 for benzene, −5.110 for a ten-ring, −3.621 for a fourteen-ring and −2.807 for an eighteen-ring. Each extra pair of sites makes the ring less sensitive to a given amount of flux.

The response to a field does the opposite, because a field makes a flux by multiplying itself by an area, and the area enters twice. The susceptibilities are 59.2, 302.5, 851.4 and 1828.1 in the same units — a factor of thirty-one across a series whose sensitivity to flux fell by a factor of three.

That is the whole reason a large aromatic ring is magnetically conspicuous, and it is an argument about geometry rather than about electrons. The current per unit flux is falling; the flux a given field produces is rising faster.

Fused rings are not separate ones #

A monocycle is the case the closed form covers. Fused rings need the matrix route, in which each bond carries its own Peierls phase computed from the coordinates, and the flux through each ring is that ring’s own area. The two routes agree on every monocycle to five significant figures, which is what makes the second one usable.

Naphthalene’s susceptibility is 129.40 against two benzenes’ 118.44, a nine per cent excess, and the excess per ring keeps rising along the series. Nothing about that is available from a count: both molecules satisfy the counting rule, both have closed shells, and the delocalisation energies do not scale the same way either. It is a fact about a path an electron can take that goes round one ring and comes back through the other.

The third rule, which is neither of these #

The two rules compared here are a shell closure and a susceptibility, and they agree on all eighty cases. Neither of them is what a chemist actually reads off an instrument. What is read is a chemical shift — a proton appearing at 7.3 parts per million rather than the 5.3 an ordinary alkene proton gives — and the step from a molecular susceptibility to that number is a step this essay has not taken and should not be assumed to be free.

A susceptibility is one number for the whole molecule: the second derivative of its energy with respect to a uniform field. A chemical shift is local. It is the field the circulating electrons produce at one nucleus, and that field depends on where the nucleus stands relative to the ring. In the crude and durable model of it, the current is replaced by a magnetic dipole at the ring’s centre, and the shift it produces at a point a distance $r$ away, at an angle $\theta$ from the ring axis, goes as

$$\frac{1 - 3\cos^2\theta}{r^3}.$$

That expression changes sign. A nucleus in the plane of the ring and outside it has $\theta = 90°$ and picks up one sign; a nucleus on the axis above the ring picks up the other; and the surface where the two meet is a cone at 54.7°, on which the same current produces no shift at all.

The consequence is a case that decides the point. In an annulene of eighteen carbons some protons point outward from the ring and some point inward, into the middle of it. They sit in the same molecule, in the same field, driven by the same current. The outward ones appear near 9 parts per million — strongly deshielded, the classic aromatic signature — and the inward ones appear near −3, which is upfield of tetramethylsilane and off the end of the scale a hydrocarbon proton is supposed to occupy. One molecule, one susceptibility, two shifts of opposite sign.

So the third criterion is the first two multiplied by a geometric factor that can be either sign, and it is not a count. “Aromatic means downfield” is a statement about protons standing outside the ring in its own plane, which is where the protons of benzene and of every ring in a first course happen to stand. It is a fact about the usual position of a hydrogen, not about the electrons.

This matters for the essay’s own claim rather than only for the practice. Two rules agreeing in eighty cases is evidence that a shell closure and a magnetic response are the same fact. It is not evidence that either of them is the quantity being measured, and the measured one carries a factor neither calculation contains. The agreement here is between two things a Hückel model can compute; the third rule is the one in circulation, and the reason it usually agrees is that most of the protons anyone looks at are in the same place.

What this cannot say #

The usual limits, and two that belong to this argument in particular.

One electron, and no repulsion. Everything here is a Hückel calculation, so the electrons do not see each other. Real ring currents are computed with repulsion in them and the numbers move; the classification does not, because it depends on the level pattern near the top of the occupied set and that pattern is a symmetry statement — the same kind of protection that makes a degeneracy a group theorem rather than an accident of the numbers.

The current itself is not drawn. What is computed here is a derivative of a total energy, which is a susceptibility. The circulating current density that produces it is a different object, defined at every point in space, and nothing here could draw it. The same distinction separates a density of states from a spectrum: a summed quantity and the distribution it was summed from are different objects. A statement about the sign and size of a susceptibility is not a picture of a current.

The geometry is idealised. Every monocycle here is regular and every bond in it is the same length, which is what makes the closed form available. Real large annulenes are not regular — a ring large enough cannot hold still — and what a ring’s shape costs is the σ frame’s business rather than the π system’s.

And the area convention has to be stated. A flux is a field times an area, so every susceptibility here is quoted for rings of unit bond length. A common alternative puts rings on a unit circle instead, which is a different area by a factor that depends on the ring size — invisible in every level and a factor of two in a four-ring’s susceptibility. Mixing the two conventions is an easy way to get a susceptibility wrong by a size-dependent factor, and the closed form and the matrix route agree only when one convention serves both.

What the classification requires #

Six claims, of which the last two are the ones that could have failed quietly.

A twist is half a flux quantum, for every ring from three to ten, to $10^{-8}$.

The matrix route reproduces the closed form, level by level, at a flux of 0.137 of a quantum, for six ring sizes.

Nothing depends on where the origin is. Shifting the coordinates of a ring by an arbitrary vector changes no level, which is the gauge invariance the Peierls construction has to have and the thing a sign error in it would break.

The census agrees in all eighty cases, and both verdicts occur.

The corner is the closed form, $16\pi\sin(\pi e/2n)/n$, in six cases spanning three ring sizes.

And the refusal: a centred difference through the corner reports no slope at all. That is stated as a fact about the stencil rather than left as a caution, because it is the measurement that a reasonable person makes first and it returns a wrong answer that looks right.

Still open: fused rings, and the corner #

The obvious open question is the ring that is not a ring. A fused system’s response was computed here from the areas of its rings, and the natural next question is whether the current divides between them the way a current in a circuit divides between two resistors — a question with a definite answer, since the derivative of the energy with respect to each ring’s flux separately is computable and the two need not be equal.

The nearer question is about the corner. Every ring on the far side of the census has one, and every real molecule of that kind distorts, so the corner is never observed. What is observed is the large paratropic response of a distorted ring, and the relation between the two — a susceptibility that diverges as the gap closes, and a gap the molecule chooses by a competition with its σ frame — means the measured number is a report on the distortion. That is the same shape of trap as a stabilisation measured from somewhere, one step further out: there the reference was chosen, here the geometry is. Reading it as a report on the electron count is the mistake that the two agreeing rules at the top of this essay make easy.