The pyramid trades its pairs

A depolarised Raman band sits at exactly three quarters — the one intensity symmetry does fix — and a distortion moves it off by an amount second order in the distortion’s size. Swept round a circle of distortions in the plane of non-symmetric bond stretches, the largest departure over a molecule’s depolarised bands reads a direction as well as a size: it varies by a factor of about two, with a sixty-degree period. Run on ammonia instead of boron trifluoride, the period and the factor come back and every size does not — the pyramid’s departures are a hundred and sixty-nine times the plane’s at the same amplitude.

Then a spectrum of finite resolution was put between the probe and the reader. On boron trifluoride the direction survived until the line width approached the bending pair’s splitting, 2.8 cm⁻¹ at a distortion of two hundredths of an ångström, and not the stretching pair’s 0.53 — because the largest departure belonged to the bending pair. That essay ended on ammonia: its pairs are split by different amounts, its departures are far larger, and the width at which its direction disappears and the precision needed to read it were one calculation away.

The expectation going in was simple. The same distortion splits both of ammonia’s pairs more than either of boron trifluoride’s, so its direction should survive a wider line. It survives a narrower one, and the reason is the same rule that decided the plane, applied to a molecule where it picks the other pair.

One instrument for two molecules #

The comparison is only worth making if the two molecules go through exactly the same calculation, so they do. Each is distorted by stretching its three bonds along a direction in the non-symmetric plane, with an overall amplitude of 0.02 Å unless stated. Its normal modes are recomputed on the distorted geometry with the same force constants, each depolarised band’s polarisability derivative is taken from the bond-polarisability model, and from that come the band’s parallel and perpendicular intensities for ninety-degree scattering. Each degenerate pair is drawn as a sum of Lorentzian lines of one stated width, the parallel envelope’s peaks are found on a grid of four thousand points, and the ratio is read at each peak.

The refusal is built into the comparison. Boron trifluoride run through this calculation must return the finite-resolution essay’s bands and readings exactly — same frequencies, same departures, same reading at zero width and at two — or the difference between the molecules would be a difference between two instruments. It does, to the last bit.

Resolved, the two readings vary by almost the same factor, 1.956 on the plane and 2.007 on the pyramid, which is the part that belongs to the three-fold axis. Widen the lines and they part company. The plane’s variation falls to half its resolved excess at 3.48 cm⁻¹, and the pyramid’s at 2.56. By four wavenumbers the pyramid’s reading varies round the circle by a quarter of a per cent, while the plane’s still varies by thirty-eight.

The pairs swap roles #

Ammonia’s E bending pair sits at 1,680 cm⁻¹ and is split by the distortion into components 6.50 cm⁻¹ apart; its E stretching pair sits at 3,567 cm⁻¹ and is split by 2.12. Boron trifluoride’s are at 480 and 1,454, split by 2.82 and 0.53. Both of ammonia’s splittings are larger than both of boron trifluoride’s. So a naive reading of the finite-resolution result — the direction survives until the bending pair merges — predicts that the pyramid should hold its direction to about twice the plane’s width.

That reading had the rule wrong in a way the plane could not show. What the finite-resolution essay established was that the width that matters is the splitting of whichever pair carries the largest departure. On the plane that was the bending pair, and the bending pair happened to be the wider-split one.

On the pyramid it is the other way round. At thirty degrees round the circle the stretching pair’s upper component sits 2.8 × 10⁻³ from three quarters and the bending pair’s upper component 9.7 × 10⁻⁵ — twenty-nine times smaller. The lower component of each pair sits at three quarters exactly, because thirty degrees keeps a mirror plane and the band antisymmetric under it cannot acquire a mean polarisability derivative; that zero is the same one the plane shows at the same direction, and it is a symmetry fact rather than a coincidence of numbers. At every direction on the circle and every width up to three wavenumbers, the pyramid’s largest departure is its stretching pair’s.

Why the stretches dominate on the pyramid is not derived here, and the obvious story should be flagged as a story. In the plane every bond lies in one plane, and in the pyramid the bonds point out of it at the pyramid’s angle, so a distortion that lengthens some bonds and shortens others meets a different geometry in the two molecules. The stretching modes move the bonds’ lengths directly and the bends move only their angles, and a bond-polarisability model builds its derivatives from bond lengths and directions — which makes it plausible that the stretches pick up a mean derivative faster on a molecule whose bonds do not share a plane. That is a reading consistent with the pyramid’s residual being a hundred and sixty-nine times the plane’s, not a calculation that isolates the cause. What is calculated is the ratio, and the ratio is twenty-nine.

So the splitting that decides ammonia’s reading is 2.12 cm⁻¹, the smaller of its two, and not 6.50. It is still four times the plane’s stretching splitting, but it is below the plane’s bending splitting of 2.82, and that is why the pyramid’s direction goes first.

The merge, drawn #

What a line width does to the stretching pair is easiest to see on the envelope itself.

At half a wavenumber the two components are two clean peaks, 2.12 cm⁻¹ apart, and the ratio read at the upper one is the band’s own. At two wavenumbers they are still two peaks, overlapping heavily, and the departure read at the upper has fallen from 2.8 to 2.3 × 10⁻³. Between three and four wavenumbers the envelope loses its second maximum; from four on it shows one peak, and the ratio read there is within a fraction of a per cent of the pair’s blend, 1.38 × 10⁻³, at every direction on the circle.

That last number explains a feature of the hero figure that would otherwise look like an artefact. The pyramid’s curve does not fall smoothly towards one; it drops to 1.0025 at four wavenumbers and then sits at 1.0045 for every wider line. Once the stretching pair has merged, its blended departure — which barely changes round the circle — is still larger than anything the bending pair carries, so the maximum over pairs locks onto a directionless number and stays there. The plane never showed this, because on the plane the pair that merged first was also the pair that did not matter. On the pyramid the largest-over-pairs reading is not just blurred by a wide line; it is captured by the wrong pair.

Read pair by pair, the bending pair still points #

The maximum over pairs was a convenience inherited from essays that read bands, not spectra — the same essays in which the sum over bands turned out flat and the maximum was the reading left standing. A spectroscopist would not take it: a spectrum reports peaks where it can resolve them, in the way it counts environments rather than atoms, and a peak belongs to one pair. Ammonia’s two pairs sit nearly two thousand wavenumbers apart, so each is read on its own, and the question for each is simply how far its own reading varies round the circle.

Read that way the pyramid gets its wide-line behaviour back, from the pair that was masked. Ammonia’s bending pair varies by 2.051 round the circle resolved, still by 1.81 at five wavenumbers, and by 1.33 at ten — its variation halves only at 8.0 cm⁻¹, where the plane’s bending pair halves at 3.5. The stretching pair alone loses its direction by four wavenumbers, as the envelope showed.

So the question “at what width does ammonia’s direction disappear” has two answers, and they differ by a factor of three. Read the way the plane was read, as a maximum over bands, it disappears at about 2.6 cm⁻¹. Read the way an instrument would read it, pair by pair, the stretching pair’s direction goes at about 2.6 and the bending pair’s lasts to beyond eight. The first answer is the direct continuation of the question the finite-resolution essay asked; the second is the one worth having.

It is not free. The bending pair keeps its direction precisely because it carries so little: its departure is thirty times smaller than the stretching pair’s, and a direction is useless if the variation that encodes it is below what the ratio can be measured to. Which brings the essay to the number the plane never came close to.

The contrast is what a measurement resolves #

A direction is readable when the reading’s variation round the circle — the largest reading less the smallest, in the ratio’s own units — exceeds the error on one measurement of the ratio. That difference is the contrast, and it is the honest single number to compare with an instrument, because it carries both the direction and the size.

Resolved, at 0.02 Å, ammonia’s contrast is 1.39 × 10⁻³ and boron trifluoride’s 7.8 × 10⁻⁶: a factor of 178. The finite-resolution essay estimated in its limits that the pyramid’s larger departures would move the required precision “from two parts in a million to about three parts in ten thousand at a hundredth of an ångström”; computed, ammonia’s resolved contrast at 0.01 Å is 3.4 × 10⁻⁴, which is that estimate confirmed rather than assumed.

Against the two precisions the picture is stark. A depolarisation ratio measured to one per cent of its value — 0.0075 absolute — is a careful measurement; a tenth of that is an excellent one. At 0.02 Å ammonia’s contrast clears the tenth of a per cent at every width up to about two wavenumbers, while it is carried by the stretching pair, and falls away through three and four as that pair merges. Boron trifluoride’s sits two orders of magnitude below the stricter line at every width.

The smallest distortion ammonia could show #

Holding the distortion at two hundredths of an ångström answers a question about one amplitude. The question an experiment actually asks is the other way round: at a precision the instrument has, what is the smallest distortion whose direction the spectrum could show?

Both ingredients scale, and they scale differently — the result the plane first showed and the pyramid repeats. Ammonia’s stretching splitting grows with the first power of the distortion, a slope of 1.010 on logarithmic axes, and its contrast with the second, 2.016. So a larger distortion both raises the signal and relaxes the resolution needed to see it, and the smallest readable distortion rises only slowly with line width.

At one per cent of three quarters, ammonia’s direction becomes readable from 0.046 Å resolved, 0.048 Å at a width of two wavenumbers, 0.055 Å at five and 0.074 Å at ten. At a tenth of a per cent it is readable from 0.015 Å resolved and 0.032 Å at five. Boron trifluoride reaches neither precision at any width anywhere in the range computed, up to 0.16 Å, where its best contrast is 4 × 10⁻⁴.

A twentieth of an ångström is a large distortion — five times the hundredth of an ångström the finite-resolution essay took as a realistic symmetry-lowering perturbation of a bond. But it is a distortion, not an impossibility, and it is read at a precision spectroscopists routinely quote. That is the first time in this line of essays that the probe’s direction has come inside the range of an instrument at all, and it happens on the pyramid and not the plane.

How the claims can fail #

Every statement above is checked wherever its figures are drawn, and each check can refuse. On the plane the bending pair must carry the largest departure; on the pyramid the stretching pair must, at every direction and every width to three wavenumbers. Both of the pyramid’s splittings must exceed both of the plane’s, and yet the pyramid’s variation must halve at the narrower width. Ammonia’s bending pair, read alone, must still vary by more than 1.3 at ten wavenumbers. The resolved contrast must be the pyramid’s by more than a hundred. At one per cent of three quarters the pyramid must be readable from under 0.06 Å at every width to five wavenumbers, and the plane at no sampled amplitude. The pyramid’s splitting must scale with slope one and its contrast with slope two, each within a few hundredths.

The refusal is the one stated at the start: the plane, computed here, must be the finite-resolution essay’s plane exactly.

The readable distortions are interpolated. Contrast is computed at nine amplitudes from 0.01 to 0.16 Å, spaced by a factor of $\sqrt{2}$, at five widths, and the threshold is placed on a straight line between neighbouring amplitudes on logarithmic axes. Because the contrast’s slope is close to two throughout, the interpolation error is a few per cent of the quoted distortion at worst.

Where the model stops #

A harmonic force field for a molecule that inverts. Ammonia tunnels through its planar geometry, and the inversion splits every one of its levels into a pair — by a large amount in the umbrella mode and by much less in the others. Nothing here has that doubling. Each E band computed here is in reality two tunnelling components, and whether the distortion’s splitting or the inversion’s is the larger for a given band changes which lines are merging with which. The harmonic field also has no anharmonic shift of the E pairs with the distortion.

A bond-polarisability model. The pair that carries the departure is decided by how fast each pair’s mean polarisability derivative grows with the distortion, and that is the model’s statement. A polarisability model with angle-dependent terms would give the bending pair more and might move the balance, though a factor of twenty-nine is a wide margin to close.

Lorentzian lines of one width. A gas-phase Raman band has a rotational contour, and ammonia’s is wide because the molecule is light; in a condensed phase the width is set by the environment. The widths above are best read as multiples of the splittings, as they were for the plane.

And a distortion from nowhere. The distortion here is imposed. What produces a stretch-plane distortion of five hundredths of an ångström in real ammonia — a crystal site, a hydrogen bond to one proton, a coordinated metal — also changes the force field and the polarisability, and none of that is in the calculation. The result is that the probe could be read on the pyramid at such a distortion, not that any real environment supplies one.

Which pair matters is a question about sizes #

The plane made the rule look like a statement about frequencies: the direction survives until the wider-split pair merges. The pyramid shows that the frequencies are only half of it. The rule is about which pair carries the signal, and that is a question about sizes — about how fast each pair’s mean polarisability derivative grows with the distortion — which the plane happened to answer in the same way as the frequencies.

The shape of the probe was the group’s and the size was not: the sixty-degree period and the factor of two transfer from the plane to the pyramid, and the magnitudes do not. The finite-resolution behaviour splits the same way. The fact that a merged pair reads its blend, and that the approach to the blend is quadratic in splitting over width, is a statement about line shapes and would hold for any molecule. The width at which the direction goes, the pair that decides it, and whether the direction can be read at all are the molecule’s — and on this pair of molecules they point opposite ways, the pyramid losing its direction sooner and yet being the only one on which it could ever be seen.

The same lesson runs through the whole question of what a ratio measures: a symmetry fixes the zero, and everything that can be measured departs from it by amounts the symmetry says nothing about.

Still open: the inversion doublets, and one band followed #

The obvious open question is the one the harmonic field hides. Ammonia’s E bands are tunnelling doublets, and a stretch-plane distortion competes with the inversion for the same levels: at a small distortion the tunnelling splitting is the larger and the degenerate pairs are split within each tunnelling component, while at a large one the distortion dominates and the inversion is a small perturbation of each distorted band. Where the crossover falls for the stretching pair — the pair that decides — would say whether the 0.046 Å threshold is read on the bands computed here or on tunnelling components with different intensities, and it needs the inversion coordinate added to the same force field, which the double-well calculations of ammonia’s inversion already supply for the umbrella.

The nearer question is the one these essays have carried since they began reading a maximum: follow one physical band round the circle rather than the larger of two. On the pyramid it now has a sharper form. Read pair by pair, the stretching pair’s contrast decides readability and the bending pair’s decides how long a direction survives a wide line. A single band followed by its composition would say whether the two roles belong to the pairs or to particular components within them — and the pyramid, where one component of each pair sits at exactly three quarters at every mirror direction, is where that labelling can be checked against a zero.