A distortion needs two states

The argument begins with a chain that cannot stay even and continues with a copper complex that is never quite octahedral. Both are the same shape of argument: an electronic energy that falls linearly in a distortion, against an elastic cost that rises quadratically, so that no stiffness whatever holds the symmetric structure. The gain is steeper than the cost near the origin and that settles it.

Both also need a degeneracy. A half-filled degenerate level for the chain, a partly filled degenerate pair for the copper. A closed shell gains exactly nothing at first order — that zero is computed, and it is a real one rather than a small number.

The question here is what happens when the first-order term is zero and the molecule distorts anyway.

Two quadratics #

Mixing an excited state into the ground state along a distortion lowers the ground state. The lowering is second order in the distortion and inversely proportional to the gap:

$$\Delta E = -\frac{\lambda^2 Q^2}{\Delta}$$

where λ measures how strongly the distortion couples the two states. The elastic cost of the same distortion is $\tfrac{1}{2}kQ^2$.

Both are quadratic. That is the whole structure of the problem and it is different from the first-order case in a way that matters: there, one term was linear and the other quadratic, so the linear one always won near the origin and the only question was how far the distortion went. Here the two terms have the same shape, and which one wins is a comparison of two numbers.

The symmetric structure survives while

$$\Delta > \frac{2\lambda^2}{k}$$

and distorts when it does not. That is a critical gap — a property of the electronic structure — rather than a critical coupling or a critical stiffness.

The selection rule underneath it #

Before any of the arithmetic there is a question that can be answered with no numbers at all: which excited states can be mixed in by which distortion.

A distortion can mix two electronic states only if the product of the two states’ symmetry species with the distortion’s own contains the totally symmetric species. That is the same theorem that decides whether a transition is allowed, applied to a vibration instead of a photon — and here it is a hard rule rather than an approximation, because a matrix element that symmetry forbids is exactly zero.

So the second-order effect has two conditions and they are of different kinds. The symmetry condition is exact and cheap. The energy condition is a comparison of a gap against a number built from a coupling and a stiffness, and it is the one that decides real cases.

What the critical gap is worth #

The critical gap goes as the square of the coupling and inversely as the stiffness, so it varies over a wide range for chemically ordinary values of both.

That range is what makes the effect a real chemical explanation rather than a curiosity. Molecules with a large gap and stiff bonds keep whatever symmetry they have. Molecules with a small gap and a soft mode do not, and the second condition is as important as the first — which is why the effect is common among heavy-element compounds, where bonds are softer, and among species with low-lying charge-transfer states.

The first-order case, for contrast #

The distinction between the two effects is easiest to see with the first-order case beside it.

The first-order case is worth having beside it, because the two are usually named together and behave quite differently near the origin.

And the two of them together, where the interesting behaviour is: neither term acting alone produces what the pair of them produce.

The two cases differ in what has to be true for a distortion to happen.

First order needs a degenerate electronic state, and then always wins. The molecule’s stiffness sets how far the distortion goes and cannot prevent it.

Second order needs a low enough excited state of the right symmetry, and then wins or loses on a comparison. The molecule’s stiffness can prevent it entirely.

So the two are not a rule and its refinement. They are two different mechanisms with different preconditions, and the second one is the reason a great many closed-shell molecules are not the shape their symmetry would allow.

The same argument, in the field next door #

The chain that cannot stay even is this argument with the gap set to zero.

A half-filled chain of equal bonds has its highest occupied and lowest empty levels touching — the gap is nothing at all — so however small the coupling and however stiff the springs, the criterion Δ > 2λ²/k fails. That is why the alternation always wins there and the elastic cost never rescues the even chain, and it is why the gain in that case is not quadratic but goes as δ²ln δ, which is steeper than any parabola near the origin.

So the two results are one result at two values of the gap. A metal — a system with no gap — always distorts. An insulator with a large gap does not. A semiconductor with a small one may, and the criterion says by how much its gap would have to shrink.

That connection is worth having because it crosses a field boundary this collection usually keeps: the chain is in the solids field and the complex is in the applied one, and the arithmetic underneath them is the same competition between a second-order electronic gain and a first-order elastic cost. What differs is only whether the denominator is small or zero.

The numbers in the distorting case #

The abstract criterion is worth turning into a case with numbers in it.

At λ = 0.5 and k = 0.5 the critical gap is 1.00. Take a molecule with a gap of 0.7 — comfortably below it, and with no degeneracy anywhere. The minimum of the total energy sits at Q = 0.850 and is 0.0625 below the symmetric structure.

Two features of those numbers are worth noticing. The distortion is large: at Q = 0.85 the mixing between the two states is substantial and the two-level treatment is being used well outside the regime its second-order expression describes, which is why the curve here is the exact one. And the gain is small: 0.0625 against an elastic constant of 0.5, so the energy released is about an eighth of what it would cost to make the same distortion in a molecule with no electronic gain at all.

That combination — a big displacement for a small energy — is characteristic of second-order distortions and is why they show up as flat, anharmonic potential surfaces rather than as sharp minima. A molecule sitting in one of them is easily pushed back through the symmetric structure, which is what makes such distortions temperature-dependent in a way first-order ones are not.

What this explains that the first-order effect cannot #

Several standing puzzles in structural chemistry are second-order cases, and it is worth naming a few so the abstraction has somewhere to land.

Lone-pair distortions. Compounds of tin(II), lead(II), antimony(III) and iodine(V) are distorted where the isoelectronic closed-shell ion would be symmetric. The distortion mixes the filled s orbital with an empty p orbital, which are of different parity — so a distortion of odd parity is what couples them, and that is exactly the T1u case computed above.

Ferroelectrics. Barium titanate has a d⁰ titanium in an octahedral cage of oxides and its titanium sits off centre. There is no degeneracy anywhere; the distortion mixes the filled oxygen-based orbitals with the empty metal d set through a T1u displacement, and the whole ferroelectric transition is that competition being won at low temperature and lost at high.

Bent versus linear triatomics. Whether an AH₂ molecule is bent depends on how far its highest occupied orbital lies below the orbital that a bend mixes it with. Water is bent and beryllium hydride is linear, with the same number of heavy-atom valence orbitals and different gaps.

None of these is computed here, and the model above is not the calculation any of them needs. What is computed is the shape of the competition and the condition it turns on, which is what the three cases have in common.

Raising the spring constant on a first-order case moves the minimum in and reduces the gain, and it never reaches zero: a first-order term has a linear part, and a linear term always wins near the origin against a quadratic one however large the quadratic coefficient is. That is the difference in kind — a stiffness can limit a first-order distortion and cannot forbid one.

Who worked this out, and when #

The second-order effect is Öpik and Pryce’s, from 1957 — the same paper that put the first-order Jahn–Teller effect on a quantitative footing also observed that a non-degenerate state has a quadratic term and that the quadratic term can win.

What turned it into a tool of structural chemistry was Bader’s work in the early 1960s and Pearson’s in the 1970s, which recast it as a statement about the gap: a molecule distorts when its highest occupied and lowest unoccupied orbitals are close together and of the right relative symmetry. In that form it is the second-order Jahn–Teller rule, and it appears in inorganic textbooks as an explanation for lone-pair distortions and for the shapes of main-group halides.

The framing as a competition between two quadratics — which is what makes the criterion a critical gap rather than a critical anything else — is the part most often left out. Stated as a small gap causes a distortion the rule is a tendency; stated as Δ > 2λ²/k it is a comparison that can be made, and the reason a great many small-gap molecules are perfectly symmetric is that they lose the comparison.

What the model leaves out #

Two levels. A real molecule has many excited states and each contributes. The sum is what matters, and one state dominating is an assumption rather than a result — though it is a good one when one state is much closer than the rest, which is the case the effect is invoked for.

λ and k are parameters. Neither is computed here. λ is a matrix element of the derivative of the Hamiltonian between two states, and computing it needs the states; k is a force constant, and this collection’s force constants are fitted to observed frequencies. So the figures are a map of the competition rather than a prediction for any molecule.

The distortion is one coordinate. A real molecule distorts along whichever combination of modes gains most, and the coordinate that wins need not be a single symmetry species — though it must be built only from species the selection rule allows.

No electron repulsion. The two-level model has one electron’s worth of physics in it. The excited state of a real closed-shell molecule is a many-electron state, and its energy relative to the ground state includes exchange terms this model has nowhere to put.

What was checked, and what refused #

Above the critical gap the symmetric structure is the minimum and below it is not, checked on both sides at a gap half and one and a half times the critical value.

The exact two-level curve reduces to −λ²Q²/Δ at small distortion, to a part in ten thousand. That is the check that what is being solved is the thing the textbook expression approximates rather than a different problem: the exact form is followed all the way out, and its leading term must be the familiar one.

A T1u distortion can mix A1g with T1u and cannot mix it with Eg, computed from the group’s characters. Both halves are checked, because a selection rule that permits everything would pass the first check alone.

The level scheme a first-order distortion acts on is an octahedral d shell split into two sets with a degenerate pair at the top. A configuration that fills that pair unevenly has a linear term and distorts; one that fills it evenly has none, and needs the second-order mechanism or nothing.

A gap as a stability criterion #

The condition is stated as a threshold on where the nearest excited state of the right symmetry lies, and read in the other direction it becomes a general statement about which closed-shell molecules are rigid — one that needs no calculation and uses a quantity that is measured routinely.

The threshold has the gap in a denominator: the distortion is favoured when the coupling squared over the gap exceeds the elastic cost. A large gap therefore protects a structure, whatever the coupling is, and a small one exposes it.

That gives a rule with an optical measurement on one side of it. A closed-shell molecule whose lowest electronic transition lies high in the ultraviolet has no excited state close enough to mix in, and its symmetric structure is safe. One whose lowest transition is in the visible has a state a few electronvolts away, and the same coupling that was harmless in the first molecule may not be in the second.

Two consequences follow that are worth having.

Colour and rigidity are related. A coloured closed-shell molecule is one with a low-lying excited state, which is the ingredient a second-order distortion needs. That does not make every coloured molecule distorted — the symmetry has to be right, and one species in ten qualifies — but it says where to look, and it explains why so many of the molecules with unexpected geometries are the ones that absorb in the visible.

And a stability argument needs the symmetry as well as the gap. A small gap to a state of the wrong species is no threat at all: the mixing vanishes identically, and the structure is as safe as if the state were not there. So the criterion is a conjunction, and either half alone is uninformative — which is why this molecule has a small gap and might distort is not an argument, and this molecule has a small gap to a state of the species that couples to this coordinate is one.

That is a more useful form of the theorem than the threshold itself, because both of its ingredients are available before any energy is computed. The gap comes from a spectrum, the species from the point group, and the two together say whether a calculation is worth doing.

There is one asymmetry in that rule worth stating, and it is the usual one for a symmetry argument. The refusal is exact and the permission is not. A state of the wrong species contributes nothing, identically, whatever the gap; a state of the right species with a gap below the threshold contributes something, and whether that something is enough depends on a coupling and an elastic constant that are numbers rather than integers. So the criterion rules molecules out with certainty and rules them in with a calculation, which is the same division drawn everywhere symmetry is used.

Why the effect is called second order #

The name is a statement about perturbation theory and is worth unpacking once, because it explains why the two mechanisms have such different characters.

Expand the energy in powers of the distortion. The first-order term is the expectation value of the change in the Hamiltonian in the unperturbed ground state. For a non-degenerate state that expectation value is zero for any distortion of non-totally-symmetric species — the integrand is odd — so the first-order term vanishes by symmetry alone, whatever the molecule is.

For a degenerate state the same expectation value need not vanish, because the perturbation can be diagonalised within the degenerate set and its eigenvalues need not be zero. That is the Jahn–Teller theorem, and its content is that a first-order term exists at all.

The second-order term is a sum over excited states of a squared matrix element divided by an energy denominator, and it is always negative for the ground state — mixing anything in can only lower the lowest state. So a closed-shell molecule’s electronic energy always falls under any distortion the selection rule permits. The only question is whether it falls faster than the springs rise, and both go as Q².

That is why the criterion involves the molecule’s stiffness and the first-order one does not, and why the second-order effect can be argued about while the first-order one cannot.

Still open: computing the coupling #

The natural open question is to compute λ rather than to parameterise it, and most of what that needs exists. λ is a matrix element of the change in the Hamiltonian between two states, and matrix elements between orbitals are already computed. What is missing is the many-electron excited state, which is the standing gap in this whole field.

The nearer question is what happens when the two effects act together. A partly filled degenerate level distorts at first order, and the distorted structure then has a smaller gap to its own excited states — so a second-order term acts on top of a first-order one, in the same coordinate. Which of the two sets the final geometry is a question with an answer, and it is not the one either effect gives on its own.