The distortion the ratio cannot see

One number was one direction turned a piece of symmetry into an instrument. A vibration that is not totally symmetric has a mean polarisability derivative of exactly zero, so its depolarisation ratio is exactly three quarters — a number fixed by the species and by nothing else. Break the molecule’s symmetry and the mean derivative is no longer zero, so the ratio comes off three quarters by an amount that depends on how far the structure has been moved.

That makes a ratio into a probe of structure. Its closing paragraph asked what the probe cannot resolve: inside a block of coordinates of the same species the sign is open, and two distortions the ratio can each see separately might cancel into one it cannot.

They can, and it is not a marginal case: each of the three single-bond stretches moves the ratio by $1.07 \times 10^{-5}$ and their sum moves it by nothing at all. The cancellation is exact rather than approximate, it holds at any size of distortion, and the direction it happens along is not obscure — it is the one that leaves the molecule’s symmetry alone.

What the probe was for #

The instrument is worth recalling before its limits are drawn, because the limits are not a reason to discard it.

A ratio that squares what it measures established the first limitation: the departure is second order in the distortion, so a ratio known to three decimal places gives a distortion to one and a half. That is a weak probe and it was described as one from the start.

What made it worth building anyway is that the zero is exact. The one intensity symmetry does fix is the account of why: almost nothing about a Raman intensity follows from symmetry alone, and this ratio is the exception — three quarters, from the species, with no parameter in it. An instrument whose null reading is a theorem is worth having even when its scale is poor, because a departure from it means something whatever its size.

This essay is about a second thing the null reading does not distinguish.

Three visible things summing to nothing #

Boron trifluoride has three equivalent bonds. Stretch any one of them and the departure from three quarters is $1.07 \times 10^{-5}$; the other two give the same number, because they are the same distortion moved by a symmetry operation.

Stretch all three by the same amount and the departure is zero. Not $10^{-12}$, not below the noise floor of the quadrature — the arithmetic returns exactly zero, because nothing has been broken. Stretching every bond equally takes a planar trigonal molecule to a larger planar trigonal molecule, every symmetry operation still applies, and the theorem that fixes the ratio at three quarters applies with it.

So three distortions, each of which the instrument reads, sum to one it reads as nothing. That is the question that was asked, and the answer is stronger than “can”: it must, and by an identity rather than a coincidence of numbers.

What the reading is actually of #

The more useful statement is not about one blind direction but about what the instrument measures instead.

Take any distortion and add breathing to it — stretch one bond, then stretch all three as well. The reading changes by at most 0.35 per cent, and what remains is second order in the displacement rather than a failure of the statement. The same holds for every combination tried.

So the ratio’s departure is not a function of the distortion. It is a function of the distortion’s component outside the totally symmetric species, and a whole direction of the space of distortions maps to the same reading. An experimenter who measured a departure of $10^{-5}$ on this molecule would learn something real about the non-symmetric part of its distortion and precisely nothing about how much it had breathed.

The three combinations that share a non-symmetric magnitude of 0.816 — one bond, another bond, and two bonds together — give readings within one per cent of each other, although the third moves twice as many atoms as the first. That is the projection made visible: what they have in common is the only thing the instrument responds to.

Worth stating plainly is what this figure does not show. The relation is not a simple power of the non-symmetric magnitude: the two purely non-symmetric combinations, at magnitudes 1.41 and 2.45, differ in reading by a factor of six where their squared magnitudes differ by three. The instrument responds to the non-symmetric part alone, and how it responds within that subspace is not a single number.

This is the same shape what an absence proves draws around a missing band: a null result is only as informative as the set of things that could have produced it. There, a band absent from a spectrum was consistent with several structures. Here, a ratio sitting at three quarters is consistent with no distortion and with an arbitrarily large breathing one, and the two cannot be told apart by that measurement at all.

Not a small effect #

A blind spot in a second-order probe invites an obvious objection: perhaps it is merely a small effect that a larger distortion would expose.

It is not. Over displacements from 0.005 to 0.1 the single-bond departure grows as the square of the displacement — the earlier result, reproduced — and the breathing departure is exactly zero at every one of them. At five times the displacement it is still exactly zero. That is the difference between a quantity that is small and a quantity that is not there, and it is a distinction worth insisting on.

The distinction matters beyond this instrument. A quantity that is small can be made visible by turning something up; a quantity that is zero by symmetry cannot, and the two look identical in any single measurement. Only varying the amplitude tells them apart, which is why the figure above varies it over a factor of twenty rather than checking one displacement carefully.

The reason is worth stating in one line: the theorem fixing the ratio at three quarters is a statement about the species of the mode, and the species is a property of the point group. A distortion that preserves the point group cannot change a species, so it cannot change anything the species determines, at any amplitude whatever.

How big the blind subspace is #

It is worth putting a size on what is lost, because “a whole direction” sounds worse than it is and may be worse than it sounds depending on the molecule.

For boron trifluoride’s three bond stretches the representation is $A_1' + E'$: one totally symmetric dimension out of three. So a third of the stretch-distortion space is invisible to this measurement, and two thirds is not. That is a large fraction and it is the best case among simple molecules, because a high-symmetry molecule has fewer totally symmetric coordinates relative to its size.

A molecule of low symmetry is worse off in exactly the way that matters. In the limit of no symmetry at all every coordinate is totally symmetric, the band is not fixed at three quarters to begin with, and the instrument does not exist. So the probe is sharpest precisely where a structure is already best known, which is a limitation of a familiar kind — two structures, two spectra is the same trade seen from the other side.

One consequence worth drawing before the arithmetic. A measured departure of a given size is consistent with a family of distortions rather than one, and the family is a line — the distortion found, plus any multiple of the breathing. So an inversion from this measurement is underdetermined by exactly one parameter per totally symmetric coordinate, which is the shape how many frequencies, not how many modes draws for the count of observables against the count of unknowns.

What was computed, and how #

The molecule is displaced along a stated combination of bond stretches, its normal modes are recomputed on the displaced structure with the same force constants, and the Raman derivatives are taken from those modes. Holding the force constants fixed is what makes the distortion structural rather than a change of force field, and it is the earlier convention kept unchanged so the numbers are comparable.

The depolarised bands are identified by species — active, and not totally symmetric — and the departure reported is the worst over all of them. Four such bands exist for this molecule.

The refusal is the size. A blind direction that were an artefact of a small displacement would grow, so the check demands that five times the breathing still gives exactly zero. If that ever came back non-zero the blindness would be a small effect after all, and everything above would be a statement about a particular displacement rather than about the point group.

A spectrum counts environments, not atoms makes the companion point about what a vibrational measurement is a count of, and the two limitations compose: a spectrum reports environments rather than atoms, and this ratio reports the non-symmetric part of a distortion rather than the distortion. Neither is a defect — both are what the measurement is — but a reader who wants a structure has to know which projection they are holding.

Where the model stops #

One molecule, and a well-chosen one: boron trifluoride’s three bonds carry a bond-stretch representation of exactly $A_1' + E'$, so the totally symmetric part is one dimension out of three and the blind direction is easy to name. A molecule of lower symmetry has a larger totally symmetric block, and correspondingly more of its distortion space is invisible — but the proportion invisible is not computed here and would need the same treatment on several molecules.

Only bond stretches are displaced. A real distortion moves angles too, and the totally symmetric block of the full coordinate set is larger than the block of the stretches alone. So the blind subspace measured here is a lower bound on the blind subspace the instrument really has.

And the whole argument concerns the depolarisation ratio, which is a weak probe for a separate reason — the departure is second order, so a ratio measured to three decimals gives a distortion to one and a half. This essay adds a direction it cannot see at all, which is a different limitation and does not replace that one.

A last note on why the answer took the shape it did. The question asked about coordinates of the same species, on the reasoning that a block of the force-constant matrix is where a sign is open — and that framing is right for the force constants and slightly wrong for this. What decides the reading is not which block a coordinate sits in but whether it sits in the totally symmetric one, because that is the species the theorem is about. Two coordinates of the same non-symmetric species do not cancel; a coordinate of the symmetric species cancels everything of its own kind. The question found the right place to look by an argument that does not quite name what is found there.

The generalisation #

The habit is to ask, of any instrument, what maps to the same reading.

A measurement that returns one number from a many-dimensional object is a projection, and a projection has a kernel: a set of changes that produce no change in the reading. That kernel is usually not mentioned, because the instrument is described by what it responds to rather than by what it ignores.

Where the instrument is justified by a symmetry argument — as this one is — the kernel is not an accident of sensitivity but is fixed by the same argument, and it can be written down exactly. The theorem that says this quantity is fixed at three quarters unless the symmetry is broken is also the theorem that says any change preserving the symmetry leaves it at three quarters. The second half is the kernel, and it comes free with the first.

The check is cheap in the way symmetry arguments usually are: it costs one extra calculation with the kernel direction added, and the answer is either “the reading did not move” or “the kernel is not what expected”. Both are worth knowing before anything is concluded from a null.

That is worth doing before an instrument is used rather than after. A structural probe with an unexamined kernel will, sooner or later, be used to argue that a structure has not changed — and a structure can change a great deal along the direction the probe cannot see.

One caution about generalising too fast. The kernel identified here is exact because the instrument’s null is exact — the three quarters is a theorem, not a fit. An instrument whose baseline is calibrated rather than derived has a fuzzy kernel instead of a sharp one, and the argument above does not transfer to it unchanged: there, distortions along the kernel would produce a small reading rather than none, and telling that from noise is the whole difficulty. A spectrum that changes when only a mass does is this collection’s example of a reading that moves for a reason having nothing to do with the structure anybody was asking about.

Who found it, and when #

That a totally symmetric distortion preserves the point group, and so preserves every selection rule and every symmetry-fixed intensity ratio, is elementary group theory and is as old as the application of group theory to vibrational spectra. Nobody discovered it here.

What is new here is the measurement rather than the principle: that the departure is invariant to a breathing component to within a third of a per cent, that the invisible combination is invisible at every size tried, and that three individually visible distortions of this molecule sum to one that is not.

What survives #

Three earlier results stand unchanged, and listing them is the fair way to bound what has been taken away.

The exact three quarters is untouched: it is still a theorem, still parameter-free, and still the reason the instrument is worth anything. The second-order growth is untouched — the visible distortions still go as the square of the displacement over a factor of twenty. And the probe still works: two thirds of the stretch-distortion space of this molecule produces a reading, and a departure from three quarters still means the symmetry is broken.

What has changed is the reading of a null. On the earlier reading, a ratio at three quarters meant the molecule was undistorted. It now means the molecule is undistorted or distorted along the totally symmetric direction by any amount whatever, and no measurement of this ratio can separate the two.

Still open: the angles, and the second species #

The obvious open question is the angles. Only bond stretches are displaced here, and the totally symmetric block of the full internal coordinate set — stretches and bends together — is larger than the block of the stretches alone. Computing the departure along a totally symmetric combination that includes angle changes would say how much of the whole distortion space is invisible rather than how much of one corner of it, and the internal coordinate set already contains both kinds.

The nearer question is the second species. The non-symmetric subspace of the stretches is two-dimensional and the reading is a function of the position within it, but not a simple one — two combinations of equal magnitude give different readings. Mapping the departure over that two-dimensional subspace, at fixed magnitude, would say whether it is a smooth function with a preferred direction or something with structure in it. It is one loop over an angle in a plane, on a calculation already set up, and it decides whether “the component outside the totally symmetric species” is one number or two.