Two distortions in one coordinate

The second-order Jahn–Teller effect can be turned into a number. A closed-shell molecule distorts along a coordinate whenever an excited state of the right symmetry lies close enough, and close enough is a threshold: the symmetric structure survives only while the gap exceeds $2\lambda^2/k$. The case that criterion leaves out is a partly filled degenerate level, which distorts at first order; the distorted structure then has a different gap to its excited states, and a second-order term acts on top of a first-order one in the same coordinate.

That is the case here, and the answer is not either of the two obvious guesses. It also settles a question left standing on the solid-state side, where a chain cannot stay even for a first-order reason and the filling chooses the distortion for a related one.

Three states, one coordinate #

The smallest system with both effects in it has three electronic states: a degenerate pair that the distortion splits at first order, and one excited state at $\Delta$ that the distortion mixes into the lower member of that pair. Writing the coordinate as $Q$,

$$\mathbf{H}(Q) = \begin{pmatrix} -\lambda_1 Q & 0 & \lambda_2 Q \\ 0 & +\lambda_1 Q & 0 \\ \lambda_2 Q & 0 & \Delta \end{pmatrix},$$

with $\tfrac{1}{2}kQ^2$ added to whichever branch is occupied. The second and third states form a two-by-two block, so the whole thing has a closed form and nothing here is iterative.

Both pieces are checked against what they are already known to give. With $\lambda_2 = 0$ the minimum is at $\lambda_1/k$ and the stabilisation is $\lambda_1^2/2k$, which is the elementary first-order result. With $\lambda_1 = 0$ the structure distorts exactly when $\Delta$ falls below $2\lambda_2^2/k$, which is the second-order criterion, and does not when it does not.

The case worth computing #

The interesting regime is the one where the second-order effect is switched off by its own criterion. Take $\lambda_2 = 0.5$ and $k = 0.5$, so the critical gap is exactly 1, and set the gap to 1.5 — half again above it. On its own the second-order term leaves the symmetric structure alone, and the second-order analysis says so.

Now turn on a first-order coupling of $\lambda_1 = 0.3$, which by itself moves the minimum to $Q = 0.600$ and gains $0.0900$.

Together they distort the molecule to $Q = 1.075$ — seventy-nine per cent further — and stabilise it by $0.18031$, which is more than twice the first-order value. The excess over the sum of the two separate stabilisations is $0.09031$: the entire second-order contribution is an excess, because the second-order term’s own contribution when acting alone is zero.

So a molecule can be, in the standard language, not second-order Jahn–Teller active, and the second-order coupling can still supply half of its distortion energy. The activity criterion is a statement about the symmetric structure and about nothing else.

The gap opens, and the effect gets stronger #

The mechanism is not the one that suggests itself, and it took a rewrite of a docstring to notice.

A second-order term is inversely proportional to the gap it divides by, so the natural story is that the first-order distortion closes the gap and thereby strengthens the second-order term. That story is wrong here, and it is wrong in a way that is fixed by which branch is which. The first-order term pushes one member of the degenerate pair down; that is the branch the electrons occupy and the branch the second-order term acts on; and pushing it down moves it away from the excited state above. At the minimum the gap has opened from 1.5 to 1.8225, by twenty-two per cent.

The two reinforce anyway. The mixing between two states goes as the coupling divided by the gap, the coupling here grows as $\lambda_2 Q$ and the gap only as $\lambda_1 Q$ — so the ratio grows, and it grows for any $\lambda_2$ and $\lambda_1$ whatever. The reinforcement is not a property of the parameters chosen.

What this changes about a familiar argument #

The received way to argue about a distortion is to ask which effect applies. Is the ground state degenerate? Then it is first-order Jahn–Teller and it must distort. Is it closed-shell with a low-lying excited state of the right symmetry? Then it is second-order and it may. The two are treated as alternatives, and the second is presented as a weaker version of the first.

The arithmetic here says they are not alternatives and the second is not weaker. When both are permitted — which is a symmetry question with a definite answer, since the two conditions are conditions on different pairs of states — the geometry is set by both, and by more than both.

Which of the two “sets the geometry” therefore has no general answer, and the parameters decide. In the case computed here the first-order coupling is smaller than the second — 0.3 against 0.5 — and the distortion is nearly twice what the first-order term alone would give. Weight them the other way and the answer is different, and the point is that neither is a correction to the other.

The selection rules are separate, and both are needed #

The two effects are permitted by different conditions and it is worth being explicit about that, because it is the part symmetry settles exactly.

A first-order term is non-zero when the symmetric square of the degenerate state’s representation contains the distortion’s species. That is Jahn and Teller’s theorem, and this collection computes it from the group rather than quoting it.

A second-order term is non-zero when the product of the ground state, the excited state and the distortion contains the totally symmetric representation. That is a different condition on a different set of states, and this collection computes that too.

So there is no reason for a molecule to satisfy both, or neither, and both cases occur. The interesting molecules are the ones that satisfy both in the same coordinate, which is what makes the interference above a real situation rather than a constructed one. Copper(II) is the standard example of the first alone; a molecule that will not hold still is what happens when the resulting minima are equivalent and shallow.

The same interference on a chain #

The molecular case above has a solid-state twin, and putting them side by side is what makes the structure of the argument visible.

An evenly spaced chain at half filling has a degenerate pair at the Fermi level and distorts for a first-order reason: alternating the bonds splits the pair, the occupied half goes down, and the gain is first order in the distortion while the elastic cost is second. That is the Peierls argument, and it is exactly the first-order Jahn–Teller argument in a chain.

A chain that is not half filled has no degeneracy at the Fermi level and does not distort at first order — and which distortion it chooses instead depends on the filling, which is a second-order statement. So the two effects are the same two effects, and the same question about their interference is available there.

What the molecular case adds is a system small enough for both terms to be written down exactly, which a chain is not — its second-order term is a sum over every unoccupied state and has no closed form.

Which of the two is bigger, as a map #

The parameters decide, and it is worth seeing the decision as a surface rather than as a single case, because the language of the field treats it as a classification.

Turning the first-order coupling up from zero at a fixed second-order one traces a curve on which the answer changes character. At $\lambda_1 = 0$ the molecule is symmetric — the second-order term cannot act on its own at this gap. At any $\lambda_1$ above zero it distorts, and it distorts further than the first-order term alone would; the extra is largest, in proportion, at small $\lambda_1$, where the first-order distortion is small and the second-order term has the most left to add.

That last clause is the one that inverts the usual intuition. A second-order effect is described as a small correction to a first-order one, and here the fractional correction is largest exactly where the first-order effect is weakest. A molecule with a barely-split degenerate level and a moderately low-lying excited state is the case where the received account is furthest wrong.

What a spectroscopist would see #

None of the above is directly observable, and it is worth saying what would be.

A geometry. The distorted structure is 79 per cent further from the symmetric one than a first-order analysis predicts, which for a real molecule is the difference between a small distortion and an obvious one. A structure determination sees that and cannot say which term produced it.

A vibrational frequency. The curvature at the minimum is different too, so the mode along which the molecule distorted has a frequency that neither term alone predicts. That is measurable and is the quantity most often used to argue about which effect is acting.

And a stabilisation that is not additive. The energy gained is twice what the first-order term gives and the second-order term gives nothing alone, so any argument that partitions a measured stabilisation between the two is partitioning a quantity that does not decompose.

None of the three distinguishes the two terms. A geometry, a frequency and a stabilisation are all quantities the two effects contribute to together, and the table of selection rules above is the only thing here that separates them — it says which distortions each term is allowed to drive, and in an octahedron exactly one of the ten species satisfies the second-order rule. That is what makes the second-order effect rare enough to be treated as an exception, and it is also why the cases where it coincides with a first-order coordinate are worth identifying rather than dismissing.

Why a criterion applied one effect at a time is not conservative #

The result — a second-order effect too weak to distort a molecule on its own, nearly doubling a distortion when a first-order effect is already acting — has a methodological form that is worth extracting, because it invalidates a habit rather than a number.

The habit is to check stability effect by effect. Is there a degeneracy? Then a first-order distortion is available; check whether it wins. Is there a low-lying state of the right symmetry? Then a second-order one is available; check whether it wins. Two checks, two verdicts, and a structure declared safe if both come back negative.

That procedure is not conservative, and this is the counterexample. The second effect was checked on its own, its gap came out half again above the critical value, and it was correctly reported as unable to distort the molecule. It then contributed more than the first effect did.

The reason is that the two are not two coordinates. They act along the same displacement, so the energy along that displacement is one function with contributions from both — and the question does the molecule distort is a question about the sign of one curvature, not about two independent verdicts.

Adding two effects that are individually insufficient can produce a total that is sufficient, and there is nothing surprising about that once it is put in those terms. What is surprising is only that the checks are usually not put in those terms.

So the repair is to compute the curvature rather than the verdicts. Evaluate the total second derivative along the coordinate, with every contribution in it, and ask its sign — which is one calculation rather than two, needs no threshold, and cannot be defeated by two terms that are each below a threshold and sum above it.

The general form applies well outside this molecule. A stability analysis performed contribution by contribution answers a different question from the one it appears to answer, and the difference matters exactly when several contributions act along one direction — which, for a molecular distortion, they usually do, because the symmetry that selects the coordinate selects it for every effect at once.

That last clause is the part that makes the failure systematic rather than occasional. A distortion coordinate is picked out by belonging to a particular symmetry species, and every mechanism that can act along it must belong to the same species — that is what selecting it means. So the effects that share a coordinate are not a chance collection: they are exactly the set the symmetry admits, and a molecule with one such effect is likely to have others, because the same species was available to all of them.

Which turns the warning into a prediction. Wherever a first-order distortion is found, look for a second-order contribution along the same coordinate, because the symmetry that permitted the first has already permitted the second — and the measurement here says it can be worth as much again.

Where the model stops #

The parameters are parameters. $\lambda_1$, $\lambda_2$ and $k$ are not computed here from anything. Computing $\lambda$ would need matrix elements of the change in the Hamiltonian between two states, of which the one-electron case is computable and the many-electron excited state is not.

Three states is the smallest interesting number and not the right one. A real molecule has many excited states of the right symmetry and the second-order term is a sum over all of them. The sum is dominated by the nearest, which is why a three-state model is instructive, and it is not the same as the sum.

The elastic term is harmonic. A distortion of 1.075 in these units is not small, and a real σ framework’s restoring force stiffens. So the numbers above are a model’s numbers and the statement they support is qualitative in size and exact in structure: the reinforcement is a theorem about the two terms, the magnitude is not.

And the branch choice is an assumption. Which member of the split pair the second-order term couples to is put in here. In a real molecule both members couple to different excited states with different strengths, and the essay’s result — that a first-order distortion opens the relevant gap — depends on that choice. The opposite choice gives a closing gap and a stronger reinforcement, so the case computed is the conservative one.

A last practical note, and it is about the language rather than the physics. The phrase second-order Jahn–Teller is used for two different things in the literature: the effect on a closed-shell molecule, which is what has a threshold, and the second-order term in an expansion about a degenerate state, which is what is added here. They are the same term in the same expansion; what differs is whether there is a first-order term beside it. Treating them as two effects is what makes the interference above look surprising, and it is not.

What the calculation requires #

Both pieces, against their own closed forms. First order alone at $\lambda_1/k$ and $-\lambda_1^2/2k$; second order alone distorting below $2\lambda_2^2/k$ and not above.

And a control that the second-order term was actually on: switching $\lambda_2$ to zero must reproduce the first-order answer exactly, which is the check that would fail if the coupling were being dropped somewhere.

The distortion at least a fifth larger with both terms than with the first alone, and the stabilisation larger, and the excess over the sum of the parts positive.

And the gap measured at the minimum, and required to have opened — which is the check that matters most, because the natural expectation is that the gap closes, and that expectation is wrong.

Still open: separate coordinates, and a warped ring of minima #

The natural open question is the coordinate that is not shared. Everything above puts both effects on one coordinate, which is the case where they can interfere; when the first-order term acts on one coordinate and the second-order on another, the molecule distorts along a direction that is neither, and how far along each is a two-dimensional minimisation with a cross term in it.

The nearer question is what the interference does to a barrier rather than to a minimum. A first-order Jahn–Teller distortion has a ring of equivalent minima and a warping term that picks some of them out; a second-order term acting in the same space is not isotropic in that ring, so it should change which of the equivalent structures wins. That is a question about the shape of a surface rather than about the depth of a well, and it is the version of this problem that a real spectrum can distinguish — because a warped ring of minima is what a fluxional molecule averages over.