The vibration that lowers the symmetry

A molecule in an orbitally degenerate electronic state cannot stay in its symmetric geometry. It distorts, the degeneracy splits, one of the two levels drops, and the electrons that were sharing the degenerate pair are now in something lower. That is the Jahn–Teller theorem, and it is usually presented as a decree.

It is arithmetic, and the arithmetic is worth doing, because it names the distortion and it says why an analogous argument fails for benzene.

Which distortion, from a direct product #

The theorem says a degenerate state is unstable; the group says what it is unstable towards.

The energy of a state changes to first order in a distortion Q through an integral of the state’s density against the derivative of the potential along Q. For the splitting of a degenerate pair, the relevant object is the symmetric product of the degenerate representation with itself, and the distortions that can split the pair are the species appearing in it, other than the totally symmetric one — which changes the size of the molecule and not its symmetry.

For a square molecule with a degenerate eg pair, that product is computable in one line.

So the theorem is not a decree about instability in general: it names a coordinate, and the coordinate is the one that turns a square into a rectangle.

Why cyclobutadiene has to move and benzene does not #

Delocalisation is stabilising, and other things that are false in general computes cyclobutadiene’s square-molecule Hückel problem: four π electrons, levels at α+2β, α, α and α−2β, and a delocalisation energy of exactly zero against two isolated ethenes. Two of its four electrons sit in the degenerate non-bonding pair, one in each, which is the open degenerate shell the theorem needs.

Alternate the bonds — β(1 + δ) on one pair of opposite bonds, β(1 − δ) on the other — rebuild the matrix, diagonalise, and fill. The degenerate pair splits into 2δ above and below zero, the two electrons drop into the lower one, and the π energy gains 4δ in units of β. Linear in the distortion.

An elastic frame resists quadratically, as any Hooke’s law frame does: the cost goes as δ². A linear gain and a quadratic cost cross at δ = 0 with the gain winning, so there is always some distortion that pays, whatever the stiffness. That is the theorem, in one comparison of powers.

Measuring the power, without a tolerance #

The difference between the two cases is a difference of powers, and this site prefers not to test that with a threshold somebody chose.

There is a cleaner discriminator. The alternation is even in δ: sending δ to −δ relabels which bonds are short, which is a rotation of the ring, so a central difference is exactly zero in both cases and says nothing. What separates them is that cyclobutadiene’s energy is linear in |δ| and benzene’s is quadratic, and a forward difference tells those apart by how it behaves when the step doubles.

Double the step. For a linear dependence the difference quotient is unchanged; for a quadratic one it doubles. Cyclobutadiene’s ratio comes out at 1.000 and benzene’s at 2.000, and the test is on the ratio rather than on any threshold — so no tolerance had to be guessed. A distortion applied to the wrong bonds, or a filling that put the electrons in the wrong levels, would fail it.

That second clause is the one worth holding on to. Benzene is not stable because alternating costs it π energy; alternating gains it π energy, at second order, and it is stable because the σ frame’s second-order cost is larger. Nothing about a closed shell forbids the distortion, and the theorem does not claim it does.

What the figure is and is not claiming #

The stiffness is stated, not fitted. Where the minimum falls in the figure depends on a number that was chosen; how the two curves behave near zero does not. The claim is about the powers, and the position of the minimum is illustration.

Hückel is the whole model. Hückel theory and what it gets right sets out what that means: a matrix of ones and zeroes, no repulsion, no geometry beyond which atom is next to which. The bond alternation enters as a change to the off-diagonal entries, which is the standard treatment and is exactly as good as Hückel is.

Cyclobutadiene’s real ground state is more complicated than this. The square molecule’s Hückel prediction is a triplet, by Hund’s rule applied to the degenerate pair; the real molecule is a singlet rectangle. The distortion computed here is the reason the rectangle wins, and electron correlation — which Hückel has none of — is part of why the singlet does. The molecule is fleeting and was characterised only in a matrix at very low temperature, in 1972.

What makes the b₁g label concrete is where the two non-bonding orbitals have their amplitude. They have it on different pairs of atoms — Hückel theory prints the coefficients — so lengthening one pair of bonds and shortening the other treats the two orbitals differently: one becomes bonding across the short bonds and the other antibonding across them. A distortion that moved all four bonds together would leave both alone, which is why the breathing mode a₁g does nothing here and the rectangular mode does everything.

What the distortion does to the levels #

The energy curve is the summary; the levels underneath it are where the mechanism is visible, and they have a closed form for an alternating ring.

For a four-ring with resonance integrals alternating as β(1 ± δ), the eigenvalues come out at ±2 and ±2δ. At δ = 0 that is 2, 0, 0, −2 — the degenerate non-bonding pair — and the four π electrons fill the lowest level and put one in each half of the pair. At any δ ≠ 0 the pair splits to +2δ and −2δ, both electrons drop into the lower one, and the π binding energy rises by 4|δ| in units of β.

Linear in |δ|, with no threshold and no barrier. That is the whole of the first-order Jahn–Teller effect for this molecule, and it is why the theorem admits no exceptions for an orbitally degenerate state: there is no distortion so small that the gain does not exceed a quadratic cost.

None of those four eigenvalues is drawn from a diagram. The diagonalisation reproduces 2cos(2πk/n) for every ring from three to ten — the check is set out in Hückel theory — so the degenerate pair at zero is a computed fact, and the splitting under alternation is the same calculation run on a matrix with two of its entries changed.

Where the theorem does not apply #

Three exclusions, and each is a place the argument’s premise fails rather than a place it is overruled.

A non-degenerate state. Benzene’s ground state, and every closed-shell molecule’s. The theorem says nothing, and whether such a molecule distorts is a quantitative question — the second-order Jahn–Teller effect, in which a low-lying excited state of the right symmetry mixes in and can drive a distortion if it is close enough. That mechanism has a threshold and the first-order one does not.

A linear molecule. Jahn and Teller’s proof is an enumeration over the point groups, and linear molecules are the case where the symmetric product of a degenerate representation with itself contains nothing that lowers the symmetry. Their instability, where it exists, is the Renner–Teller effect and works differently.

Spin degeneracy. The theorem is about orbital degeneracy. A state degenerate only in spin has no distortion available to it, because spin does not couple to the nuclear positions at this order.

Which rings have the open degenerate shell the theorem needs is settled by the count this field began with. Aromaticity as a shell closure fills every ring from three to ten with its own electrons: the ones that close are the 4n+2 cases, and the ones that do not are left with a half-filled degenerate pair — cyclobutadiene at four, cyclooctatetraene at eight. The distortion argument applies to exactly the second set, which is why it is a statement about a count and not about a molecule.

And one case that looks like an exclusion and is not: cyclooctatetraene, which has the open degenerate shell and does not distort in the plane. It escapes by leaving the plane entirely — the tub conformation — which relieves the degeneracy by a route the planar arithmetic here cannot represent. The theorem said it must distort and gave no promise about how.

The same theorem in three other places #

The argument generalises well past this molecule, and the pattern is worth naming.

Octahedral transition-metal complexes. A d⁴ or d⁹ ion in an octahedral field has an unevenly filled eg pair, and the complexes distort — four short bonds and two long, or the reverse. Eg × Eg in Oh contains the same kind of splitting coordinate, and copper(II) complexes are the standard example.

Benzene’s excited states. The ground state is closed-shell and safe; several excited states are degenerate and are not. The distortions they undergo are the reason those states have the vibrational structure they do.

The Peierls distortion. A one-dimensional chain with a half-filled band dimerises, which is the same argument in the limit of infinitely many atoms — the band limit computes that limit for the undistorted chain, and the instability is what stops a chain of equally spaced atoms being a metal.

All four are the same statement: a partly filled degenerate level is unstable towards whatever distortion splits it, and the only question is whether the frame is stiff enough to stop it, which for a linear gain it never is.

For a closed shell it is the whole question, and it has a number in it.

How the distortion changes the spectrum #

A molecule that distorts stops being the molecule whose spectrum was counted, and the change is exactly the kind of thing the essays around this one compute.

Square cyclobutadiene would be D4h: a centre of inversion, therefore mutual exclusion, therefore no band in both spectra. Rectangular cyclobutadiene is D2h, which also has a centre — so mutual exclusion survives the distortion — but the counting changes underneath it. Modes that were degenerate in the square split, so the number of distinct frequencies rises, and species that were forbidden may correlate onto ones that are not.

A distortion is therefore visible as a splitting rather than as a shift, and that is the standard experimental signature of a Jahn–Teller molecule: a band that is one line at high symmetry and two at low. It is the same signature that distinguishes a symmetry-required degeneracy from an accidental one in what an absence proves, used in the other direction.

The trapped-matrix infrared spectrum of cyclobutadiene, measured in 1972, is what settled the rectangular structure — the band count matched D2h and not D4h. Counting bands settled a geometry that no diffraction measurement could reach, because the molecule survives only in an argon matrix at eight kelvin, which is two structures, two spectra’s argument applied to a molecule that barely exists.

The arithmetic of that is worth one sentence with numbers in it. D2h — the group a rectangular cyclobutadiene would have — holds no degenerate representation at all, so a D2h molecule gets one frequency per mode: ethene’s twelve modes give twelve frequencies. D6h does hold them, so benzene’s thirty modes give twenty frequencies. A spectrum counts environments, not atoms does that counting for both. A distortion from D4h to D2h works the same way: every degeneracy the higher group required is gone, and the band count rises by exactly the number of degeneracies lost.

One consequence of that is worth stating in general rather than for the molecule. A degeneracy is not a stable feature of a molecule with an open shell in it — it is a feature of a geometry the molecule will leave. So degeneracies computed from point groups are statements about a structure, and whether a molecule keeps that structure is a separate question with a separate answer.

The double well the two powers imply, and its one parameter-free number #

The comparison of powers decides that a distortion happens. It also fixes the entire shape of what the molecule moves on, and one number about that shape survives having chosen no parameters at all — which is worth extracting, because the stiffness was stated rather than fitted and everything depending on it has to be read as illustration.

Both branches are already written down. The π energy gains $4|\delta|$ in units of β and the frame costs $k\delta^2$, so

$$E(\delta) = k\delta^2 - 4\beta|\delta|.$$

The absolute value is not decoration. Sending δ to −δ relabels which pair of bonds is short, which is a rotation of the ring, so the two signs are the same molecule and the curve must be even. That makes this a double well: two equivalent minima at

$$|\delta^*| = \frac{2\beta}{k},$$

separated by the square geometry at δ = 0.

Now evaluate. At the minimum the energy is $4\beta^2/k - 8\beta^2/k = -4\beta^2/k$, so the barrier back through the square arrangement is $4\beta^2/k$. The π stabilisation actually collected at the minimum is $4\beta\delta^* = 8\beta^2/k$. The two differ by exactly a factor of two, and the stiffness cancels:

$$\frac{\text{barrier}}{\text{π energy gained}} = \frac{1}{2},$$

for every stiffness, every ring, and every molecule whose gain is linear and whose cost is quadratic. Half of what the distortion buys is spent holding the frame away from its own preferred geometry, and the other half is the barrier that keeps the molecule in one well rather than the other.

That is the one statement in this essay’s arithmetic that needs no chosen number, and it is worth more than the position of the minimum precisely because of it. A stiffer frame gives a smaller distortion, a shallower well and a lower barrier, all in the same proportion; nothing about the ratio moves.

It also predicts something checkable in the right direction. A molecule that gains little by distorting has a proportionally low barrier to shifting between its two distorted forms, so a weak Jahn–Teller distortion and a fast interconversion are the same fact, not two facts that happen to travel together. Cyclobutadiene is the case: the bond-shift that exchanges its long and short bonds is fast enough that the molecule’s two rectangles are hard to tell apart, and the process is one of the standard examples of tunnelling by atoms rather than by an electron — a barrier low enough and a coordinate light enough for the carbon skeleton itself to pass through rather than over.

Two honest qualifications, both about the shape rather than the ratio.

The cusp is an artefact. A first-order treatment gives $|\delta|$, which has a corner at the origin, and a real molecule’s curve is smooth there. What rounds it is the second-order term the first-order theorem discards, and rounding lowers the barrier a little without moving the minima. So the one-half is an upper bound on the ratio rather than an identity, and the direction of the error is known.

The barrier is in units of β²/k, which is to say in units of two things not fitted here. The ratio is dimensionless and survives; the height is not a number here and cannot be turned into one without both a resonance integral and a force constant.

What the two together buy is a reading of the earlier figure that does not depend on where its minimum happens to fall. The stiffness chosen for that plot sets the horizontal position of the wells and the depth of them, and it cannot change the fact that the barrier between them is half the depth. A reader wanting to know how deeply a given molecule is committed to its distorted shape needs one measured stabilisation energy and nothing else.

What the model cannot show #

No geometry comes out of this. The distortion is measured in a dimensionless alternation of resonance integrals, not in ångströms, because Hückel theory has no lengths in it. Turning δ into a bond-length difference needs a relation between β and distance, which is a fitted quantity and is not fitted here.

No barrier in energy units, and no dynamics. The section above gets the barrier’s ratio to the stabilisation without fitting anything, and that is as far as the model reaches: whether a molecule distorts statically or moves between equivalent distorted forms depends on that barrier measured against the zero-point energy, and neither of those is a number here.

The elastic term is a placeholder. Real σ frames resist bond alternation with a stiffness that depends on the bonding, and the number here is stated so that the two curves can be drawn on one axis. Nothing in the essay’s conclusions depends on its value.

Correlation is absent throughout. Cyclobutadiene is a molecule where electron correlation matters unusually much, which is exactly what makes it a hard test case for simple theory and an interesting one for a simple model: Hückel gets the instability right and the ground-state multiplicity wrong.

Who found it #

Hermann Jahn and Edward Teller proved the theorem in 1937, and the proof is unusual in form: they enumerated every point group and every degenerate representation and checked, case by case, that the symmetric product always contains a non-totally-symmetric species — with one exception, linear molecules, where it does not. So the theorem is a result about a finite list rather than a general argument, and the exception is real.

Teller’s name is on the other exact identity nearby as well — the Teller–Redlich product rule of the isotope shift is arithmetic, from two years earlier.

Cyclobutadiene took much longer. Predicted unstable in the 1930s, hunted for decades, and finally trapped in an argon matrix at 8 kelvin by Lin and Krantz in 1972, where its infrared spectrum showed the rectangular structure this arithmetic requires.

Where spectroscopy meets bonding #

Molecular spectroscopy turns symmetry counts into frequencies, and it ends in three places: a vibrational spectrum whose positions are fitted and whose shapes and species are not; a rotational spectrum that returns a geometry outright; and a photoelectron spectrum whose bands are ionisation energies rather than orbital energies however often they are called otherwise.

The Jahn–Teller distortion is where spectroscopy runs back into bonding, and it shows the whole apparatus working together: a group theory argument names a coordinate, a Hückel calculation measures a slope, and the comparison of two powers decides whether a molecule keeps the shape it started with.