A ratio that squares what it measures

Symmetry fixes exactly one intensity. A Raman band whose mode is not totally symmetric has a mean polarisability derivative of exactly zero, so its depolarisation ratio is exactly three quarters — a number set by the species and by nothing about the strength or the shape of anything, and one of the very few exact numbers in vibrational spectroscopy.

And the escape route shows up by accident. Ammonia’s coordinates as usually stored are rounded, so its three N–H bonds differ in the fifth decimal place, and its E modes came out at 0.74999997 rather than 0.75. A slightly distorted molecule has slightly non-zero mean derivatives for modes that were forbidden, and the ratio comes off three quarters by an amount that measures the distortion.

Done deliberately, with a distortion of known size, the mechanism works and the hope attached to it does not.

What was done #

One N–H bond of ammonia is lengthened along its own direction by a stated amount, from two thousandths of an ångström up to a tenth. Everything else is left alone.

The normal modes are then recomputed on the distorted molecule with the same force constants, which is what makes the distortion structural rather than a change of force field. The polarisability derivatives follow from the bond-polarisability model used throughout, and the depolarisation ratio of each band follows from those.

One thing had to be got right here and is worth recording, because getting it wrong produces a null result rather than an error. Holding the mode shapes fixed and distorting only the geometry does nothing at all. For a fixed E-mode shape the three bond stretches sum to exactly zero, so the mean derivative stays zero however the molecule is bent. The symmetry that keeps a band at three quarters lives in the normal coordinate, not in the polarisability — so the modes have to move for the ratio to move.

What it gives #

Fitted across that range, the departure goes as the distortion to the power 1.9994. It is a square, essentially exactly, and the reason is one line: the mean derivative is first order in the distortion, and the depolarisation ratio depends on the square of the mean derivative — so a linear cause has a quadratic effect.

Writing that out makes the structure visible. The ratio is $3\gamma^2 / (45\bar{\alpha}^2 + 4\gamma^2)$ with $\bar\alpha$ the mean derivative and $\gamma^2$ the anisotropy. At the symmetric geometry $\bar\alpha$ is exactly zero and the expression collapses to $3/4$ whatever $\gamma$ is — which is why the number is exact and why it carries no information about the molecule. Away from it, $\bar\alpha$ grows linearly and the departure is $45\bar\alpha^2/(4\gamma^2)$ to leading order: a ratio of two model quantities, one of which is the thing being measured.

The undistorted row is worth reading as data rather than as a zero. Its 3.1 × 10⁻⁸ is the rounding in the stored coordinates, whose three bonds differ by 4.5 × 10⁻⁵ Å — and the quadratic law says a distortion of that size should give about 10⁻⁸, which it does. The accidental demonstration is the first point on the same curve.

Why that makes it a weak probe #

The square is the whole practical difficulty.

A depolarisation ratio is measured by taking the intensity in two polarisations and dividing, and a careful measurement gets three decimal places. Three decimal places on the ratio is 10⁻³ on the departure, which by the quadratic law is a distortion of about 0.015 Å.

That is not nothing — it is larger than the difference between many pairs of bonds a structural argument turns on — but it is much worse than what any diffraction or rotational measurement gives, and it is worse than the ratio’s own precision suggests. A quantity that responds to the square of the thing sought halves the significant figures available.

So the ratio is not a structural instrument competing with the ones that exist. What it is good for is a presence test: whether a molecule is distorted at all, at the level of a hundredth of an ångström, from a measurement that needs no structure model to interpret.

And the size is a model number #

The natural hope is for a structural measurement with no model in it. The mechanism is real and that description is not, and the difference is worth being precise about.

Symmetry decides that the ratio moves. A distorted molecule has no species to forbid a mean derivative, so the ratio must come off three quarters, and this is as exact as the original statement.

The parameters decide how far. Recomputing the same distortions with bond polarisabilities chosen to be absurd — parallel and perpendicular components swapped, derivatives given the wrong signs — gives a departure of 6.5 × 10⁻⁵ at a tenth of an ångström instead of 4.2 × 10⁻². A factor of 650 for the same structural change.

Both runs are quadratic, both are exactly three quarters undistorted, and both would let a spectroscopist conclude “this molecule is distorted”. Neither lets them say by how much without a polarisability model, and a polarisability model is exactly the thing this collection is careful about elsewhere.

That is the same division what an absence proves draws. A zero is a symmetry statement and survives any parameters; a small number is a magnitude and does not. The exact three quarters is on the first side of that line, and the departure measured here is on the second.

What a spectroscopist should take from it #

A ratio at exactly three quarters is evidence of symmetry, and it is strong evidence, because nothing but symmetry produces an exact number and no accident of parameters reproduces one.

A ratio a little off three quarters is evidence of a distortion, and the amount is a hundredth of an ångström per thousandth of a ratio, on this molecule with these parameters. It is not a general conversion factor, and the calculation that would supply one for a different molecule is the one run here rather than an entry in a table.

And a ratio well off three quarters means the assignment should be checked first. A departure of 0.042 needed a distortion of a tenth of an ångström, which is enormous — larger than the difference between a single and a double bond. A band at 0.70 is far more likely to be a misassigned totally symmetric mode than a hugely distorted molecule.

That last is worth stating as a rule because the arithmetic makes it lopsided. The quadratic law means a departure of 0.04 costs a hundred times the distortion a departure of 0.0004 does, so the range of departures a distortion can plausibly produce is narrow and sits very close to three quarters. Anything further away is telling a different story — a misassignment, a resonance, or a mode that was never depolarised. What an absence proves is the same asymmetry read for a zero rather than for a number: an exact statement is strong and a small deviation from one is weak.

The two exact numbers symmetry supplies #

It is worth putting the two results side by side, because between them they say something about what symmetry supplies to a spectrum.

A count is exact and survives everything. How many bands there are, which are infrared active, which are Raman active, whether the two sets overlap — all of that is a count over a character table and none of it moves when a parameter does. Two structures give two spectra is a statement of that kind and is as reliable as arithmetic.

One intensity is exact and survives everything. Three quarters, for any non-totally-symmetric mode of any molecule, whatever it is made of.

And nothing else is. Every other intensity is a magnitude that depends on a model, and the departure from the one exact intensity turns out to be a magnitude too. Symmetry hands over a number and a set of counts, and hands over nothing about how large anything is — which is a smaller endowment than it is usually credited with, and an unusually trustworthy one.

What is quoted, and what is computed #

The force constants are fitted to measured frequencies elsewhere, and the bond polarisability parameters are quoted from the standard bond-polarisability tables — the one place this essay leans on a model, and the place the factor of 650 above measures the cost of.

Everything else is computed: the distorted geometries, the normal modes on each, the polarisability derivatives by finite difference, the invariants, the ratios and the exponent.

The molecule’s point group is not used anywhere in the calculation. The distinction between a band that should sit at three quarters and one that should not comes from the species of each mode, which is computed from the coordinates rather than assigned — so a distorted molecule does not have its symmetry assumed back into it.

Most of the sensitivities are exactly zero #

The question of which distortion has an answer before any of it is computed, and the answer is that almost none of them do anything at all.

The departure exists because a distortion gives a forbidden mode a non-zero mean polarisability derivative. That derivative is an integral over the molecule, and it survives only if the product of the distortion’s symmetry species with the mode’s species contains the totally symmetric representation — which, for real representations, means the two species must be the same. A distortion of one species cannot unlock a mode of another.

So the sensitivity is not a vector with a number in every slot. It is a vector that is zero everywhere except along the coordinates that share a species with the band being watched, and the pattern is readable off the character table.

Ammonia makes the point cleanly. Its six vibrations are two of species A₁ and two doubly degenerate ones of species E, and the depolarised bands are the E pair. So:

An A₁ distortion moves nothing, however large. Stretch all three N–H bonds together, or open the umbrella to any angle short of planarity, and the molecule is still C₃ᵥ. The E modes are still degenerate, their mean derivatives are still required to vanish, and the ratio is still exactly three quarters. A distortion of half an ångström, ten times the largest one in the table above, gives a departure of zero rather than a departure of four — not small, zero, and for a reason no calculation is needed to see.

An E distortion moves it, and is the only thing that does. Stretching one bond is not an E distortion; it is a combination, one third A₁ and two thirds E, and only the second part contributes. That is why the coefficient measured above is what it is: the quoted displacement includes a component that does nothing.

Two things follow that are worth more than the numbers.

The probe is selective rather than weak. The quadratic makes this a poor way to measure a distortion. The selection rule sharpens that into something more useful: the ratio does not measure the distortion, it measures the component of the distortion in one particular species, and it reports exactly zero for everything else. An instrument that ignores most of what is done to it is a filter, and a filter with a known pass-band is worth having even when it is insensitive.

And the pairing is one to one. Every depolarised band watches its own species. Watching several bands at once therefore resolves the distortion into its symmetry components — which is a different measurement from resolving it into bond lengths and angles, and is the one a spectrum is naturally able to make. The bond lengths were never what the experiment saw; the species were.

What this cannot say #

One distortion of one molecule. A single bond stretched on ammonia, which is the simplest symmetry-lowering change there is. A bend, a twist or a distortion that preserves a subgroup would all give different coefficients, and possibly different exponents where the leading term happens to vanish — a distortion that keeps the molecule in a subgroup with the mode still non-totally-symmetric would give no departure at all, which is the mutual-exclusion argument in a different costume.

A bond-polarisability model. The polarisability is a sum over bonds with two parameters each, which is a crude description of an electron distribution and is the same kind of additive assumption refused for dipoles for the dipole.

No isotopic or thermal averaging. A measured ratio is an average over a thermal population of a molecule that is vibrating, and the vibration is itself a distortion — of about the size the table above is measuring. Whether a thermally averaged ratio sits at three quarters or a little below it is a question this essay’s own mechanism raises and does not answer, and a spectrum that changes when only a mass does is where the amplitude of that vibration enters.

No rotational averaging beyond the standard one. The three-quarters result and its departures are for the usual isotropic average over molecular orientations, which is what a gas or liquid gives and not what an oriented sample gives.

And the modes are harmonic. A real distorted molecule is anharmonic, and the amplitude of a mode is not small compared with 0.1 Å — so at the large end of the table the distortion and the vibration are comparable, which is a regime the harmonic modes do not describe.

What was checked #

Undistorted, the depolarised bands are at three quarters to better than 10⁻⁵ — with the residual named as the rounding in the stored coordinates rather than left as noise.

A larger distortion moves the ratio further, at every step of seven, which is the shape of the effect and not only its existence.

A distortion a laboratory could measure gives a departure a laboratory could measure, above 10⁻³.

And the departure goes as roughly the square of the distortion, between exponents of 1.5 and 2.5. Measured: 1.9994.

An undistorted molecule is depolarised exactly under absurd bond parameters too, which is the symmetry half of the argument, checked against a model chosen to be wrong.

And a distorted one is not, under those same parameters — so the departure is not an artefact of the real parameters either.

While how far it moves is a property of the bond parameters and not of the distortion alone, by more than a factor of e. Measured: 650, which is the limit on the whole idea and is checked rather than mentioned.

And the refusal is a molecule with no bonds, which cannot have one stretched and is refused rather than returned unchanged. It is the tripwire a distortion calculation needs: one that silently returned the molecule it was given would produce a table of zeros that looked exactly like a symmetry result.

Still open: an electronic transition’s polarisation #

The obvious open question is the other exact ratio in the same family. 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 it degrades under a distortion at the same rate, or at a different one because the moment is a vector rather than a tensor, is a calculation of the same shape with a different operator in it.

The nearer question is about which distortion. The departure here 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 — a sensitivity per coordinate — and the interesting part is whether the ratio is more sensitive to the distortions a molecule is soft along, which is already measured. If it is, the ratio is measuring the coordinate a molecule was going to distort along anyway, which would make it a much more useful instrument than a uniform sensitivity would.