One number was one direction

A ratio that squares what it measures established that a depolarisation ratio of exactly three quarters is a statement about a species rather than about a polarisability, and then broke the symmetry to see how the exactness goes. Stretching one bond by a stated amount, the departure from three quarters goes as the square of the distortion, at an exponent of 1.9994 against an exact 2.

It ended by naming what the single number leaves out:

The departure goes as the square of one particular symmetry-lowering change, and a molecule can be distorted in as many ways as it has non-symmetric coordinates. Computing the coefficient for each of them separately would turn one number into a vector — and the interesting part is whether the ratio is more sensitive to the distortions a molecule is soft along.

It is a vector, and it is not.

The exponent is two everywhere and the coefficient is not #

Displacing a molecule along each of its non-symmetric normal coordinates and watching a depolarised band gives a parabola through the origin every time — the same exponent, re-found along every direction rather than one.

What differs is the coefficient:

Methane’s E coordinate is the striking row. It is the second softest of its three non-symmetric species and by far the least sensitive — thirty times less than the T₂ bend just below it in frequency and four times less than the T₂ stretch at more than twice the frequency. So even within one molecule the sensitivity does not order with anything obvious about the coordinates: not the frequency, not the species dimension, not whether the coordinate is a bend or a stretch.

A factor of thirty inside methane and sixty-four across the three. So the single-stretch number is a component of a vector, and the size it reports belongs to the direction it was measured along.

And it is not the soft ones #

The natural hope was specific: if the ratio is most sensitive to the coordinates a molecule is soft along, then it is measuring the coordinate the molecule was going to distort along anyway, which would make it a much more useful instrument than a uniform sensitivity.

Ammonia’s stiff pair is five times more sensitive than its soft one. Boron trifluoride’s stiffer pair is more sensitive too, though only by a fifth. Only methane has its softest coordinate most sensitive — and its second-softest is its least sensitive, so even there the ordering is not the softness ordering.

The answer is therefore no, and it is worth saying why it was never likely. The sensitivity is a property of how a coordinate moves the polarisability, and the frequency is a property of how it moves the energy. Those are different tensors of the same displacements, and nothing connects them: a mode can be soft because a bond angle bends easily and be nearly invisible to a polarisability because the bonds keep their lengths.

There is a second reason worth stating, because it makes the result less accidental than a list of three molecules suggests. The departure from three quarters is driven by the mean polarisability derivative becoming non-zero — a trace, which is what symmetry forces to vanish. A trace is sensitive to changes that alter the sizes of the bond polarisabilities and blind to changes that only reorient them. A soft bending coordinate mostly reorients; a stiff stretching coordinate mostly resizes. So the systematic expectation runs the opposite way to that hope, and ammonia’s factor of five is that expectation showing.

The coordinate a molecule was already soft along is a real and useful quantity, and it is a property of the energy surface. What is established here is that the depolarisation ratio does not measure it.

What the amplitude buys #

There is a second question underneath, and it is the one that matters for an experiment. A molecule does not sample its coordinates equally: a soft mode has a larger zero-point amplitude, so it contributes more than its sensitivity alone suggests.

It helps, and it is not enough. Boron trifluoride’s ordering reverses — its soft pair departs by 0.031 against the stiff pair’s 0.012 — because its two frequencies differ by a factor of three and the amplitude carries the square root of that. Ammonia’s does not: its stiff pair still departs by 0.046 against 0.020, because a factor of five in sensitivity beats a factor of two in amplitude.

So the honest summary is that the ratio’s departure at zero-point amplitude is partly about softness, in the way any amplitude-weighted quantity is, and the weighting does not dominate.

It is worth checking that against the amplitudes rather than simply stating it. Ammonia’s two frequencies are 1680 and 3567 cm⁻¹, so their zero-point amplitudes are in the ratio of the square roots — 1.46 — and the squares of those amplitudes in the ratio 2.12. Against a sensitivity ratio of 4.96 the stiff coordinate wins by 2.3. Boron trifluoride’s frequencies are 480 and 1454, a ratio of 3.03 in the squares of the amplitudes, against a sensitivity ratio of 1.21 the wrong way — so the soft one wins by 2.5. The two molecules differ by which of the two factors is larger, and neither factor is reliably the larger one.

The care a degenerate set needs #

One thing had to be got right before any of the above meant anything.

Methane’s three T₂ components come back at 6.80, 3.42 and 3.40 individually. Those are three different numbers for three vectors that are physically indistinguishable: the components of a degenerate set are defined only up to a rotation among themselves, and which three the eigensolver returns is a fact about the eigensolver.

Only the mean of a set is a property of the species, and it is 4.54. Reporting a component would be reporting an arbitrary basis, and the factor of two between the components is large enough that the mistake would have changed every conclusion above.

That is the same care a zero-point correction to a moment needs, and the expression that summarises those corrections inherits it too — same reason, same fix — and the frequency tolerance is the same too: ammonia’s two equal frequencies differ in the fifth figure, which is the geometry’s rounding rather than a splitting, so a tolerance tight enough to be safe elsewhere fails to recognise the pair here.

What this leaves the ratio #

The instrument is not what was hoped and it is not nothing.

The exactness survives untouched. Three quarters is three quarters for every non-symmetric mode of an undistorted molecule, and that was established with no polarisability entering. Nothing here disturbs it.

The departure is quadratic in every direction, which is what makes the ratio a second-order sensor: a distortion of one per cent moves it by a coefficient times ten to the minus four, so the ratio is genuinely blind to small symmetry breaking and that blindness is uniform.

What is not uniform is which distortion it sees, and the vector above is the useful output. A measured departure of a known size does not say how distorted a molecule is; it says how distorted it is along whichever coordinate the ratio happens to be sensitive to, and for methane those differ by a factor of thirty.

So the ratio is a sharp test of whether the symmetry is exact and a poor measure of how far it is from exact — which is the ordinary fate of a quantity that is protected by symmetry at first order. The flat directions of a force field are the same shape of statement about a different object: a quantity that a symmetry protects is a quantity that carries no information about the protection being slightly broken.

What a sensitivity vector is good for #

The vector is not only a correction to the single-stretch number; it is a usable object, and it is worth saying what for.

A measured departure bounds a distortion from below, not above, in the way a slope floor bounds a model’s derivative rather than fixing it. If a depolarisation ratio is measured at 0.751 rather than 0.750, the distortion responsible is at least $\sqrt{0.001/c_{\max}}$ along the most sensitive coordinate and could be much larger along a less sensitive one. With $c$ spanning thirty, the bound is loose by a factor of five and half in the distortion.

A departure of zero bounds nothing. That is the more useful direction and it is where the spread hurts most: a ratio measured at three quarters to a thousandth says the molecule is undistorted to a hundredth along methane’s T₂ bend and only to a twentieth along its E coordinate.

And two molecules cannot be compared. A departure of the same size in methane and in ammonia is not the same distortion, because the coefficients differ. That is a caution rather than a defect, and it is the kind cheap predictors keep needing: a quantity that is an excellent test of an exact statement is often a poor measure of departures from it.

Two exactnesses, and only one of them is fragile #

There are now two statements about the same three quarters and they behave differently under a distortion, which is worth setting side by side.

That the ratio is three quarters is a consequence of a trace vanishing, and a trace vanishes because the mean polarisability derivative of a non-symmetric mode is zero by symmetry. Distort the molecule and the mean stops being zero — but only at second order, because the first-order term is itself forbidden by what is left of the symmetry. Hence the square.

That the ratio is the same three quarters for every non-symmetric mode is a stronger statement, and it survives a distortion no better: each mode acquires its own departure, with its own coefficient, and the modes stop agreeing with each other before any of them leaves three quarters by much.

So a measured spread among a molecule’s depolarised bands is a more sensitive symmetry test than any one of their values, and it is a test nobody makes — because the natural experimental comparison is a band against three quarters rather than a band against another band. A collection of exact statements is worth more than the sum of them when they can be checked against each other, and here the check is free.

What is quoted, and what is computed #

The bond polarisabilities are quoted and are the same ones used for the single stretch, and the observed frequencies the force fields were fitted to. Nothing else.

Everything else is computed. The molecule is displaced along a normal-mode shape by a stated Cartesian amount, its whole vibrational problem is re-solved at the displaced geometry with the same force constants, and every band’s depolarisation ratio is recomputed from the bond-polarisability model. The coefficient is a least-squares fit of the departure to the square of the displacement through the origin, over three displacements.

The degenerate sets are grouped by frequency at a relative tolerance of 10⁻⁴ and averaged, and the components are kept so that the spread inside a set can be shown rather than hidden.

What this cannot say #

The displacement is Cartesian and the comparison across molecules is not scale-free. A sensitivity is per unit of mass-weighted displacement, so comparing methane’s to boron trifluoride’s compares two quantities whose units carry mass in the same way — which is fine — but the magnitudes also carry the bond polarisability model’s own scale, and that is fitted. The comparisons that are safe are within one molecule; the factor of sixty-four across three is the least trustworthy number here.

Only Raman-active depolarised bands are watched. A molecule’s non-symmetric coordinates include ones that make no depolarised band move, and those have no coefficient — they are absent from the table rather than at zero, and the difference matters if the table is read as a complete list of a molecule’s coordinates.

Three molecules, and all of them small and symmetric. Methane, ammonia and boron trifluoride are the ones with force fields, Raman-active depolarised bands and more than one non-symmetric species. A molecule with lower symmetry has many more non-symmetric coordinates and a correspondingly wider vector, and nothing here says whether the spread of sixty-four is typical or is what three high-symmetry molecules happen to give.

And a normal coordinate is not the distortion a molecule undergoes. A real distortion — a solvent, a crystal field, a substituent — is a mixture of normal coordinates, and its coefficient is a quadratic form in the mixture rather than a weighted sum of the numbers above. What the vector supplies is the diagonal of that form.

Half of the off-diagonal is zero before it is computed #

The last limit above says the vector is the diagonal of a quadratic form and a real distortion is a mixture. That is true, and symmetry removes most of the form’s off-diagonal without any calculation.

A cross term between two coordinates contributes to the mean polarisability derivative of a band only if the product of the three species — the two coordinates and the mode — contains the totally symmetric representation. For a mode already fixed, that requires the two coordinates to belong to the same species as each other. Every cross term between coordinates of different species is exactly zero.

So the quadratic form is block-diagonal by species, and the blocks are small: methane’s non-symmetric coordinates split into one E pair and two T₂ sets, boron trifluoride’s into two sets, ammonia’s into two coordinates of a single species. A mixture of an E and a T₂ distortion produces exactly the sum of what each produces alone — no interference, no cancellation, and no way for two individually invisible distortions to combine into a visible one across species.

That answers half of the worry the diagonal raised and leaves the other half sharply. Within a species the cross term need not vanish, and its sign is not fixed by anything above. Two coordinates of the same species could reinforce or cancel, and cancellation is the case that matters: a distortion whose components are each detectable and whose sum is not would be a genuine blind spot rather than a limit of sensitivity. It would live inside one species, in a block of at most two or three coordinates in these molecules — which is a small enough space to search exhaustively, and is what the calculation should do next.

What was checked #

Every molecule has more than one non-symmetric coordinate to compare, which is the check that the extension happened at all.

The sensitivity is not one number, checked within methane at more than a factor of ten and across the set at more than twenty. It is deliberately not checked per molecule: boron trifluoride’s two coordinate sets are within a fifth of each other, which is a real answer and would make a per-molecule check a demand that every molecule behave like methane.

Every coefficient is positive. A negative one would mean a distortion moving a depolarisation ratio towards three quarters, which cannot happen from a symmetric structure and would mean the fit had found a sign rather than a size.

And the refusal is a totally symmetric mode. Displacing along one cannot lower the symmetry, so no depolarised band moves and there is no coefficient to fit — such modes are excluded before the fit rather than returning a coefficient of nothing, and the check confirms that they are absent from the list rather than present at zero.

Still open: the electronic version, and the surviving cross terms #

The obvious open question is the electronic version. The polarisation of an electronic transition is fixed the same way — a rotational average acting on a transition moment rather than on a polarisability derivative — and its possible values are again set by the species. Whether its sensitivity vector is the same shape is a calculation of the same form with a vector operator in place of a tensor, and the interesting part is that a vector has fewer invariants, so there may be fewer directions it can be blind to.

The nearer question is the off-diagonal that survives. The section above removes every cross term between coordinates of different species, which leaves a form that is block-diagonal and blocks of two or three coordinates. Inside a block the sign is open, and it is the sign that decides whether two distortions the ratio can each see separately can cancel into one it cannot. They cost one more displacement each, along the sum of two coordinates of the same species, and the block sizes here make that an exhaustive search rather than a sampling.