One number was a direction too

Three bond stretches on a trigonal centre span three dimensions, and the one intensity symmetry does fix is what a depolarised band reports when none of them is displaced. The distortion the ratio cannot see showed that the depolarisation ratio is exactly blind to one of them: stretch every bond by the same amount and the point group survives intact, so a band whose species fixes its ratio at three quarters stays there — not nearly, but to the last bit the arithmetic carries, at any size of distortion.

Its conclusion was that the ratio reads the distortion’s component outside the totally symmetric species. That leaves a plane, and the plane was where it stopped: its own cases had different magnitudes, so “two combinations of equal magnitude give different readings” was a comparison it could not quite make. Its closing question was whether the reading over that plane is one number or a function of where in it the distortion points.

It is a function, and the function has a shape.

A factor of two #

Walking a distortion of fixed magnitude round the circle, the departure runs from 8.18 × 10⁻⁶ to 1.62 × 10⁻⁵. That is a factor of 1.978, and it is not noise: the pattern is smooth, it repeats, and it survives changing the amplitude by a factor of eight.

To put the size in context: the whole finding about breathing was that adding any amount of it to a distortion changes the reading by at most 0.35 per cent, which is why that direction counts as invisible. The direction dependence found here is a factor of two, which is five hundred times larger — so the plane identified as what the probe can see is itself strongly structured, and the structure is much bigger than the residual measured there.

So “the component outside the totally symmetric species” is a direction and a magnitude. A reading of a given size can mean a large distortion in the insensitive direction or a smaller one in the sensitive direction, and the probe cannot tell them apart — which is the same kind of degeneracy the breathing direction has, weakened from complete blindness to a factor of two. A probe with a blind direction and a factor-of-two anisotropy in the rest is a probe whose reading fixes a distortion only up to a one-parameter family and a factor — which is the honest statement of what a depolarised band tells a structural chemist. Because the departure grows as the square of the distortion, the factor of 1.98 in the reading becomes a factor of √1.98, about 1.41, in the amplitude: a band read at a given departure fixes the size of the distortion only to within forty per cent, and cannot by itself say where on the circle the distortion lies.

Where the extremes are, and why there are six of them #

The extremes are not arbitrary directions. The minimum is at one bond stretched against another — the distortion that keeps one mirror plane and moves the third bond not at all. The maximum is at two bonds against the third, which keeps a different mirror plane. Those are the two kinds of distortion the point group distinguishes inside the plane, and they are the two basis vectors the sweep was built on: chosen for that reason, and confirmed rather than assumed.

There is a symmetry argument for what the pattern should be, and it is worth following because it gives the wrong answer in an instructive way. The departure is second order in the distortion, so it is a quadratic form on the plane. The plane carries a two-dimensional representation of the molecule’s group. A quadratic form on a two-dimensional representation, invariant under a three-fold rotation, is proportional to the identity — it is isotropic. So the reading should be constant on the circle.

It is not, and the reason is what is being read. The four depolarised bands are two degenerate pairs at the undistorted geometry, and distorting splits each pair; the largest of the four is a different one of them in different sextants, which is exactly where the extra factor of two in the period comes from. One number was one direction makes the same kind of point about a different quantity — a reading that looks like a scalar because it is quoted as one. The quantity is the largest departure over the depolarised bands, and there are four of them: the distortion splits a degenerate pair, and the two halves respond differently. A maximum of two functions related by the three-fold is not itself three-fold invariant; taking the larger of the two picks out different halves in different sextants, and the result has a sixty-degree period rather than a hundred and twenty.

That the six directions the group relates agree to two parts in a million is the check that the pattern is the group’s and not the grid’s. A sweep whose circle was not a circle in the coordinates that matter would show a repeat that was approximate, and this one is exact to the precision the normal-mode solver carries.

The two families of maxima are not quite identical #

One detail of the pattern does not fit the account above and it is recorded rather than explained.

The six minima — at 0°, 60°, 120°, 180°, 240° and 300° — agree to two parts in a million, which is the solver’s precision. The six maxima do not: those at 90°, 210° and 330° read 1.6172 × 10⁻⁵ and those at 30°, 150° and 270° read 1.5988 × 10⁻⁵, a difference of 1.15 per cent.

Both triples are internally consistent to the same two parts in a million, so this is a genuine split between two families of directions rather than scatter. Under the full point group of a planar trigonal molecule the two families are related by a mirror plane, and if the reading were a function of the geometry alone they would be identical. That they are not suggests the maximum is picking a different band in the two families, which is consistent with the account above and is not established by it.

It is 1.15 per cent against a factor of two, so it changes nothing quantitative. It is here because a pattern reported as sixty-fold periodic ought to say by how much.

Second order everywhere on the circle #

The obvious worry about a factor-of-two anisotropy is that it is a large-distortion effect — that the quadratic response really is isotropic and the anisotropy is a cubic term showing through.

It is not. Both extremes are straight lines of slope two on logarithmic axes: the departure divided by the square of the amplitude varies by 2.0 per cent across a factor of eight in amplitude, at both ends of the circle.

And the anisotropy itself barely moves: 1.989, 1.978, 1.956, 1.915 as the amplitude grows. It falls slowly, which is the cubic terms beginning to show, and the whole span across a factor of eight is under four per cent. So a reading taken at any amplitude in this range carries the same direction dependence, and the dependence belongs to the quadratic response.

What this does to the blind-direction result #

The blind-direction result is untouched, and this sharpens its statement rather than replacing it.

The blind direction is still exactly blind. Every distortion on this circle sums to zero by construction, so none of it is the breathing direction, and nothing here bears on the exactness of that result. The check covers the sum, because a sweep that had leaked breathing into the circle would be measuring a quantity already shown to be invisible.

The reading is still a projection. Two distortions differing by any amount of breathing are still indistinguishable, which is a whole direction of the three-dimensional space mapping to one reading.

What changes is the description of what survives the projection. The blind-direction result says the ratio reads the component outside the totally symmetric species, which suggests a scalar — the length of the projected vector. It reads a scalar function of the projected vector that is not its length, and the difference between the two is a factor of two at fixed length.

What was computed, and how #

Boron trifluoride, with a fitted valence force field — the same molecule and the same field the ratio was first computed on, so every number here is comparable with the earlier ones. For each direction on the circle, the three bonds are stretched by the corresponding combination, the normal modes are recomputed at the distorted geometry with the force constants held, the Raman derivatives follow, and the departure is the largest |ρ − ¾| over the depolarised bands.

The circle is parameterised by an orthonormal pair spanning the plane of combinations that sum to zero, and the two basis vectors are the two kinds the group distinguishes — so the sweep’s own coordinates are chosen by the symmetry rather than arbitrarily.

The check requires eight things: that every distortion swept sums to zero, so the blind direction is excluded by construction; that the departure varies by more than half again over the circle; that the minimum is at one bond against another and the maximum at two against one; that the six directions related by the group agree to a part in a hundred thousand; that the sixty-degree repeat holds across the whole circle; that the departure is second order at every direction; and that the anisotropy is too, so the direction dependence is not a large-distortion effect.

Which reading a spectrum actually gives #

There is a practical question underneath the choice of a maximum, and it is worth separating from the arithmetic.

An experiment does not measure “the largest departure over the depolarised bands”. It measures a spectrum, in which the four bands here are two nearly degenerate pairs that a real instrument would not resolve — a distortion of the size swept here splits them by far less than a linewidth. What such an instrument reports is closer to an intensity-weighted average over the unresolved group than to a maximum — which is the distinction a spectrum counts environments and not atoms turns on.

That average is a different function of the direction, and it is not computed here. What can be said is which way the two differ: an average of functions related by the three-fold is three-fold invariant, so an averaged reading would have a hundred-and-twenty-degree period at most and, if the quadratic argument holds for each band, would be constant. So the anisotropy found here is the anisotropy of the maximum, and the observable a spectrum offers may well be isotropic on this plane.

That does not make the finding an artefact. It makes it a statement about which reading to take: if the four bands can be resolved, the reading is anisotropic and carries direction information; if they cannot, it is isotropic and carries less. Either way the probe reads less than the blind-direction phrasing suggests, and the two cases fail differently.

Where the model stops #

Only bond stretches are displaced. The full internal coordinate set has angles in it as well, and the non-symmetric block of the whole set is larger than the non-symmetric block of the stretches — so this is a plane inside a larger space, and the factor of two is the anisotropy within that plane rather than within the whole of what the probe can be shown.

The force field is fitted and held fixed at the distorted geometry, which is the harmonic approximation used ever since the ratio was first squared. A real distorted molecule has different force constants, and the departures computed here are the response of a fixed field to a changed geometry rather than of a molecule to a perturbation.

And the reading is a maximum over four bands. That choice was made for the blind-direction result and it is what makes the pattern six-fold rather than three-fold; a study that followed one band would see an isotropic quadratic response, and the two statements are about different observables. Which is the right one depends on what a spectrum actually resolves, and on a real instrument the four bands are not four resolved lines.

The last limit is the one that would be easiest to forget when quoting the factor of two. An anisotropy is a ratio of a response along one direction to the response along another, and it is only meaningful once the directions have been given a common scale. Here that scale is the internal coordinate set — bond stretches, normalised — and a different normalisation of the same plane produces a different number without changing anything physical. The factor of two is therefore a property of the response and of the metric the plane was given, and the two cannot be separated by any measurement made inside the plane. What makes it worth quoting anyway is that the same normalisation was used for both directions and for every other reading of this ratio, so the comparison across them is sound even though the absolute figure is convention-bearing.

The generalisation #

The transferable point is about what “the component of a distortion” means when the thing reading it is a maximum.

A projection onto a subspace is a linear operation and its output is a vector — and what an absence proves about a spectrum is the same shape of caution, applied to a zero rather than to a length. Reducing that vector to a number requires a further choice, and the natural choice — its length — is the right one only if the response is isotropic on the subspace. Isotropy is guaranteed by symmetry for a single quadratic invariant on an irreducible representation, and it is not guaranteed for a maximum, a minimum, a sum of absolute values, or anything else that is not itself a quadratic form.

That distinction is easy to lose because the symmetry argument is correct and short and applies to the object one thinks one is computing. Here it applies to each band’s departure and not to the largest of them, and the sixty-degree period is the fingerprint of exactly that gap — a three-fold symmetric problem producing a six-fold answer means an extremum is being taken somewhere.

The practical version: whenever a symmetry argument predicts a constant and the computation gives a periodic function, the period says which operation broke the invariance. Sixty degrees from a three-fold axis is a maximum over a split pair; it is not a failure of the symmetry argument and it is not numerical noise.

Who found it, and when #

The depolarisation ratio’s three-quarters value for a non-totally-symmetric band is standard Raman theory and old. The distortion sweep, the plane, the circle and every number above are original arithmetic on a fitted force field, done to answer a question the blind-direction result raised about itself.

Still open: the bends, and a single band #

The obvious open question is the angles, which neither sweep has run. Adding the bends makes the non-symmetric block larger than two-dimensional, and the sweep becomes a sphere or worse rather than a circle — but the question it answers is the one that matters for a real distortion, which is never pure stretch. The internal coordinate set contains both kinds already.

The nearer question is the single band. Everything here reads the largest departure over four bands because that is how the reading was defined, and the symmetry argument says each band separately should be isotropic on this plane. Following one band round the circle and checking that its departure is constant would confirm the mechanism proposed above rather than leaving it as the best available explanation — and if it is not constant, then the argument is wrong somewhere and the anisotropy has a different source. It is the same sweep with the maximum removed, which is one line.