The sum was flat all along

One symmetry argument showed the depolarisation probe exactly blind to the totally symmetric direction, leaving a plane. A sweep of fixed magnitude walked a distortion round a circle in the plane of bond stretches that sum to zero, and found the depolarisation reading varying by a factor of 1.978 with a sixty-degree period.

It also gave an argument for why that should not happen and an explanation for why it does. The argument: 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, hence isotropic. The explanation: the quantity being read is the largest departure over four depolarised bands, and a maximum of functions related by the three-fold is not itself three-fold invariant.

The explanation was the best available and it was not tested. Testing it is one line — take the sum over the bands instead of the largest — and the result is decisive.

Four parts in ten thousand #

The sum over the four depolarised bands varies by 0.038 per cent round the circle. The maximum varies by 97.8 per cent. The ratio of the two departures from constancy is 2591.

That is the argument confirmed rather than merely consistent with. A sum cannot be affected by an ordering — reordering a set does not change its sum — so if each band’s departure is an isotropic quadratic form the sum must be exactly constant. It is constant to four parts in ten thousand, which on a quantity assembled from a normal-mode diagonalisation, a polarisability derivative and a ratio is as close to exact as the numerics get.

The reason the sum has to be flat at second order is short enough to give in full. The sum over the bands is unchanged by the molecule’s threefold rotation, so its quadratic part is a symmetric two-by-two matrix on the plane that commutes with a rotation through 120°. The only symmetric matrices that commute with a rotation other than a half-turn are multiples of the identity, so a threefold axis leaves no room for an anisotropic quadratic term, and any direction dependence in the sum has to come from higher orders.

It is worth saying why 0.038 per cent counts as flat rather than as a small anisotropy. The six directions related by the three-fold rotation agree with each other to two parts in a million, which is the precision the normal-mode solver carries — so anything at the level of a few parts in ten thousand is well above the noise and is a real, if tiny, residual. What makes it flatness is the comparison: 0.038 against 97.8 is not a smaller version of the same thing, it is a different regime, and the factor of 2591 between them is the measurement.

So the six-fold pattern belongs entirely to the extremum. Nothing in the physics of the distorted molecule prefers one direction in the plane to another; the reading does, because it reports the largest of four things and which is largest changes as the circle is walked.

What this does and does not overturn #

The maximum-based reading made three claims and the sum touches them differently.

That the reading varies by a factor of two round the circle. True and unchanged. Every number comes back identically, because the sweep is the same and only what is done with the four departures afterwards changes.

That the variation means the probe reads a direction as well as a magnitude. This is the claim that gives way, and it gives way in a specific place: the probe — the physical response of the molecule — does not. The reading defined as the largest departure does. Whether that distinction matters depends on whether the four bands are resolved, which the last section takes up.

That the anisotropy is the extremum’s. Proposed there as the best available explanation, established here.

So the arithmetic behind the factor of two is right, the headline is right about the quantity defined, and the interpretation of that quantity as a property of the molecule is what the sum withdraws.

An ordering is not a band #

The obvious follow-up is to remove the maximum and follow one band, and it does not work — for a reason worth stating, because it looks like it should.

Ordering the four by size at each direction and following the k-th gives a curve, and the curve is anisotropic: the largest varies by 1.978, the second by 2.016, and the two lowest reach exactly zero somewhere and so have no bounded ratio at all.

That is because an ordering crosses between bands. Two bands that cross exchange places in the ordering, so “the largest” is a different physical band before and after the crossing, and the curve made by following the ordering has a kink at every crossing. Only a symmetric function of the four — a sum, a mean, a product — is free of that, which is why the sum is the right test and following a band is not.

Following a genuine band through the sweep is possible in principle: one would track eigenvectors rather than eigenvalues and match them by overlap from one direction to the next. That is more work than the question needs, because the sum answers it exactly and needs none — and a tracking that failed at a crossing would produce the same kink for a different reason, with nothing to distinguish the two. What an absence proves applies to a kink as well.

Two exact zeros #

There is something in the ordered bands that is not an artefact of the ordering, and it is exact.

In the six directions where one bond is stretched against another, the two lowest bands read exactly zero — not small, zero to the last bit the arithmetic carries. Those distortions keep a mirror plane, and a band whose species is fixed at three quarters by that plane stays there exactly, which is the same mechanism that fixes the totally symmetric direction.

So the minimum of the six-fold pattern is not a coincidence of where the ordering happens to sit. It is at the six directions where a symmetry survives, and it is exact there because a surviving symmetry is exact. The maximum’s minimum and the two lower bands’ zeros are at the same six directions, and that is why the pattern’s period is what it is.

Why an extremum breaks a symmetry #

The mechanism deserves a sentence in general terms, because it is not special to this problem and it is easy to state wrongly.

A symmetry operation acts on the distortion and permutes the bands. If each band’s departure is invariant under the subgroup that fixes it, and the operation maps band one to band two, then the set of four departures is the same before and after — but the largest of the set is a different element of it. The set is invariant; the labelling is not; and any function that depends on the labelling inherits the failure.

Sums, means and products do not depend on the labelling. Maxima, minima, medians and “the k-th largest” all do. So the rule is not that extrema break symmetries — it is that label-dependent functions do, and an extremum is the commonest label-dependent function there is.

The period follows from how many things are permuted. Four bands in two pairs, each pair permuted by a three-fold, gives a maximum whose label changes twice per rotation of a hundred and twenty degrees — hence sixty. If there had been three bands in one orbit the answer would still have been a hundred and twenty, and the discrepancy would have been invisible.

That last point is the uncomfortable one. The mechanism here was detectable because the period came out wrong; had it come out right, the reading would have been anisotropic for the same reason and nothing would have flagged it.

What an instrument would report #

The practical consequence follows and it points away from the maximum-based reading rather than towards it.

A real Raman spectrum does not resolve these four bands — a spectrum counts environments and not atoms, and it counts resolved lines and not eigenvalues. They are two nearly degenerate pairs at the undistorted geometry, and a distortion of the size swept here splits them by far less than a linewidth — so what an instrument reports is an intensity-weighted combination over the unresolved group, which is much closer to a sum or a mean than to a maximum.

Both of those are flat. The mean varies by the same 0.038 per cent as the sum, since the four are the same four. So a measurement that cannot resolve the bands carries essentially no information about the direction of the distortion within this plane — and the direction dependence of the maximum is available only to a reading that can pick out the largest of four unresolved lines, which no experiment can.

That does not make the maximum wrong. It makes it a property of the model’s output rather than of anything measurable, which is a distinction worth drawing whenever a computed reading is compared with experiment.

The blind direction, seen from here #

There is a tidier way to state the whole of what is now known about the probe, and it is worth setting down because three arguments arrive at it by three routes.

The three stretch coordinates split into a totally symmetric direction and a two-dimensional plane. On the symmetric direction the depolarised bands’ ratios are fixed at three quarters exactly, by symmetry, at any amplitude — the symmetric-direction result. On the plane each band’s departure is an isotropic quadratic form, by the same kind of argument — the result here. Put together: the response of any depolarised band to a stretch distortion is a single number times the squared length of the distortion’s projection onto the plane, and nothing else.

That is a complete description of the probe on this coordinate space, and it is much simpler than the six-fold rose the maximum draws. The rose is what happens when four such responses are combined by taking the largest, and the underlying object has no rose in it at all.

It also explains the two exact zeros without any further work: a band whose coefficient happens to vanish reads exactly three quarters everywhere on the plane, and two of the four do so in the directions where an extra mirror survives.

What was computed, and how #

The sweep is unchanged: the same molecule, the same fitted valence force field, the same circle in the same plane, the same twenty-five directions. What changes is what is done with the four departures at each direction — ordered rather than maximised, and summed.

The check requires five things: that there is more than one band to follow; that the sum is constant to better than half a per cent; that the maximum’s departure from constancy exceeds the sum’s by more than a hundredfold; that following one ordered band is still anisotropic, so the mechanism is the ordering; and the negative — that the sum is not exactly constant, recorded rather than rounded away, because a residual of four parts in ten thousand means something beyond a three-fold-invariant quadratic form is present at some level.

Where the model stops #

The residual is real. Four parts in ten thousand is far above the machine’s precision and above the two parts in a million the three-fold directions agree to, so the sum is not exactly constant and the quadratic-form argument is not exactly the whole story. What is left is presumably the fourth-order term — the same kind of residual the one intensity symmetry does fix carries when the geometry is not exact — the departure is second order to two per cent over a factor of eight in amplitude, so a small cubic or quartic contribution is expected and would not be three-fold invariant in the same way. That is a guess and the residual is a number; the assertion records the number.

Every caveat about the model applies here unchanged: only stretches are displaced, the force field is fitted and held fixed at the distorted geometry, and the ratio is a squared quantity whose sensitivity to structure carries those caveats.

And “the sum over the bands” is not itself an observable either. It is the right quantity for testing the symmetry argument and it is a stand-in for what an unresolved measurement gives; the actual intensity-weighted combination has weights not computed here.

And there is a limit in the sampling that the residual figure does not carry. The three-fold invariance is checked at the sampled directions, which are the high-symmetry ones, and a departure that vanishes at those directions and not between them would be invisible to every number quoted here. The two parts in a million is therefore a statement about agreement at the sampled points, not a bound on the whole plane, and the distinction is the same one that applies to every symmetry result computed at chosen directions rather than integrated over. Filling the plane in would either raise the residual — in which case the sampled directions are special and the flatness is partly an artefact of where it was looked for — or leave it where it is, which is the outcome the quadratic-form argument predicts and which would be worth having as evidence rather than as expectation.

The generalisation #

Two things, and they are about how to test an explanation rather than about molecules.

The first: when a symmetry argument predicts a constant and a computation gives a periodic function, the fix is to find the quantity the argument actually governs and check that. Here the argument governs each band’s departure, and the way to reach it without following bands through crossings is to take a symmetric function of them. That took one line and turned a plausible story into a measurement.

The second: the period is a fingerprint. Three-fold symmetry producing a six-fold answer says an extremum is being taken over an even number of things related by the three-fold, and the doubling is the signature. If the answer had come out three-fold the explanation would have to have been different. So the shape of the discrepancy identified the operation that caused it, before any of this was computed — and that is worth looking for whenever a symmetry prediction fails by a pattern rather than by an amount.

Who found it, and when #

The isotropy of a three-fold-invariant quadratic form on a two-dimensional representation is elementary group theory. Everything computed here is new arithmetic on a fitted force field, done to test an explanation that had been proposed and not checked. It is another case where an explanation offered with a finding turned out to need its own control, and the finding survived it.

Still open: the residual, and the weights #

The obvious open question is the residual. The sum is flat to four parts in ten thousand and not to machine precision, and the natural suspect is the quartic term in the distortion. Repeating the sum at four amplitudes and asking whether the residual scales as the fourth power would settle it: a fourth-order origin means the residual falls by 256 when the amplitude is halved, and anything else means the suspect is wrong. It is the same sweep at four sizes, which is already done for the maximum.

The nearer question is the weights. What an unresolved measurement reports is an intensity-weighted combination of the four bands, not their unweighted sum, and the Raman intensities are computed at every direction already. Forming that combination and asking whether it too is flat would say whether the flatness is a property of the physics or of the equal weighting — and the answer matters, because it decides whether the maximum’s factor of two is unobservable in principle or merely unobserved.