The shape was the group and the size was not

The sum over a molecule’s depolarised Raman bands is nearly isotropic over a circle of distortions, and nearly was measured: four parts in ten thousand rather than exactly. The residual is not the quartic term the obvious argument names, and a fit of a cubic against a quartic gives an exponent of 1.248 and a crossover amplitude — no integer, and a pattern left in the residuals of the fit.

Every one of those numbers was measured on boron trifluoride, and the closing question of the essay before it was whether they belong to it or to its point group. Any molecule whose degenerate stretches transform the same way under a three-fold axis should give the same angular shapes; the coefficients that set the sizes are a force field’s business.

Ammonia is the natural second case. The same three bonds to one central atom, the same three-fold axis, and the horizontal mirror gone — a pyramid rather than a plane. It is also a molecule with a force field already fitted here, which matters: a force field is the input every one of these readings depends on, and fitting a new one to make a comparison would put the comparison at the mercy of the fit.

The circle is the group’s #

The maximum over depolarised bands repeats every sixty degrees for both molecules — to 1.8 per cent for the planar one and 2.3 for the pyramid, which is the accuracy of the underlying solve rather than a departure. Its minima sit at multiples of sixty and its maxima between them, in both.

And the anisotropy — largest over smallest round the circle — is 1.9779 for boron trifluoride and 2.0271 for ammonia. Two and a half per cent apart, on two molecules with different masses, different bond lengths and force constants differing by a factor of four.

That is what a consequence of a point group looks like, and it is worth noting that the two molecules are not otherwise alike: ammonia’s bonds are shorter, its central atom is lighter, and its stretching force constant is about a quarter of boron trifluoride’s. None of that reaches the ratio. The reason the maximum has a period at all is that a single band’s reading would be isotropic — a quadratic form on a two-dimensional representation invariant under a three-fold rotation has to be — and what is being read is a maximum over a degenerate pair that the distortion splits. A maximum of two functions related by the three-fold is not itself invariant, and its period is sixty degrees. Nothing in that argument mentions a mass or a force constant, and the numbers agree accordingly.

So half the earlier essay’s finding transfers, and it is the half that was derived rather than fitted.

Everything with a size in it is the molecule’s #

At the same distortion amplitude the pyramid’s smallest departure from three quarters is 1.384 × 10⁻³ against the planar molecule’s 8.177 × 10⁻⁶ — a factor of 169. The sum’s residual at the smallest amplitude is 2.43 × 10⁻³ against 1.05 × 10⁻⁴, a factor of 23.

Two orders of magnitude on the first quantity is the more arresting number and the more expected one: ammonia’s polarisability derivatives are much less symmetric than boron trifluoride’s, and the departure from three quarters is a direct measure of that asymmetry. The departure is second order in the distortion in both molecules, so the factor is a property of the coefficient and not of where on the curve the comparison is made.

That the two ratios differ is the less obvious point and the more useful one. If the pyramid were the planar molecule with one knob turned, both quantities would scale by the same factor. They do not, so the departure and the residual are set by different combinations of the force field — and a probe calibrated on one of them cannot be transferred by rescaling.

And the exponent is neither molecule’s nor the group’s #

The essay below set out three hypotheses with three exponents: zero if the residual is the solver’s own precision, one if it comes from a cubic term in the expansion, two if from a quartic. Boron trifluoride gave 1.2481, which reads as a cubic term with a quartic contribution on top — the reading the two-term fit was built on.

Ammonia gives 0.8155.

Both are nearest to the cubic and they are on opposite sides of it, so a two-term model fitted to each would put the quartic coefficient positive in one case and negative in the other. Which is exactly what happens.

The planar molecule’s cubic and quartic coefficients are 1.014 × 10⁻² and 6.934 × 10⁻², giving a crossover at an amplitude of 0.146. The pyramid’s are 2.845 × 10⁻¹ and −5.886 × 10⁻¹. A negative quartic coefficient has no crossover in it, and reporting one would mean quoting a negative amplitude.

The fit’s own diagnostic caught this in the earlier essay too, and it is worth noticing that it was not decisive there. That essay counted the sign changes the two-term model leaves in its residuals — a model missing a term leaves a pattern, a model that is right leaves none — and found two for the planar molecule, which it read as a term still missing. The pyramid leaves three. So the diagnostic was working and was reporting a mild version of the same failure, at which point the natural response is to look for the third term rather than to doubt the expansion.

The pyramid’s residual is not a power law #

The fit’s own diagnostic says so before any model does. A single power describes boron trifluoride’s residual to a factor of 1.070 — seven per cent at the worst point out of a sixteenfold span in amplitude. It describes ammonia’s to 1.561.

And the sequence is not monotone. Ammonia’s residual runs 2.43 × 10⁻³, 4.37, 6.63, 8.89, 1.10 × 10⁻², 1.17 × 10⁻², then 1.09 × 10⁻², then 1.63 × 10⁻² and 4.62 × 10⁻². It falls between amplitudes of 0.057 and 0.08.

No sum of positive powers of the amplitude does that. So the expansion the essay before it was fitting — quadratic plus cubic plus quartic, with the anisotropy in the higher terms — is not what the pyramid’s residual is, at least not over this range.

There is a mechanism available and it is in the quantity’s definition rather than in the physics. The residual is computed from a maximum over the depolarised bands, and a maximum of several smooth functions is not smooth: it has a kink wherever two of them cross. A fit of smooth powers to a function with kinks in it returns an exponent that describes neither segment, and the non-monotonicity is where a kink falls between two sampled amplitudes. The planar molecule’s bands are further apart in this reading, so its crossings are outside the range sampled; the pyramid’s are inside it.

That is a statement about the instrument and not about ammonia, and it applies retrospectively to the earlier reading: 1.248 is the exponent of a maximum, and a maximum’s exponent is not a term in an expansion. The planar molecule’s residual happens to be smooth over the range sampled, which is why nothing flagged it there.

Why the pyramid is worse behaved, and it is not the pyramid #

There are two candidate causes for the kinks and they are worth separating, because only one of them is about ammonia.

The first is the maximum. The reading is a maximum over four depolarised bands and a maximum is not smooth. Where two bands cross, the function being fitted changes which band it is following, and its derivative jumps. That is a property of the quantity’s definition and it applies to every molecule; what differs between the two here is only whether a crossing falls inside the range of amplitudes sampled.

The second is the leak. A pyramid’s stretch plane is not orthogonal to its bends, so the distortion may be exciting bending character at larger amplitudes, changing the bands themselves rather than just their ordering.

The two make different predictions and one test separates them. Following a single band rather than the maximum removes the first cause entirely and leaves the second untouched. If the pyramid’s residual becomes a clean power law when one band is followed, the kinks were the maximum; if it stays ragged, they are the leak. That is not run here, and it is the open question this essay leaves.

What can be said now is which cause is the more likely, and it is the maximum, on the evidence of the planar molecule. Boron trifluoride has the same four-band maximum and its residual is smooth over this range; it has no leak to speak of, because its stretch plane is orthogonal to its out-of-plane coordinate by symmetry. So the planar molecule has one of the two causes and behaves well, which says the cause it has is not sufficient by itself — and that is an argument about the amplitudes sampled rather than about the mechanism.

What is shared and what is not, stated once #

The division the two molecules make is clean enough to state as a rule for this probe.

Shared, because a three-fold axis fixes it: the existence of the flatness in the sum, the blindness to the totally symmetric direction, the sixty-degree period of the maximum over bands, and the factor of two between its largest and smallest values.

Not shared, because a force field sets it: the magnitude of the departure from three quarters, the magnitude of the residual, the exponent the residual scales with, and whether the residual is a power law at all.

The essay below reported the first group as derived results and the second group as measurements, which was correct, and it did not have a second molecule to say which was which. Having one, the useful reading of 1.248 is not “the exponent” but “boron trifluoride’s exponent over this range of amplitudes, for a maximum over four bands.”

What the two calculations are #

Two fitted force fields, each to its own molecule’s measured frequencies. Ammonia’s is fitted to its four fundamentals and boron trifluoride’s to its four, in the same way, with the same coordinate conventions. The two are therefore comparable in construction and are not comparable in accuracy: ammonia’s umbrella mode is strongly anharmonic and a harmonic field fitted to a fundamental carries that.

The same distortion, defined the same way. The circle lies in the plane of stretch combinations that sum to zero — one bond against another, and two bonds against the third — at a fixed amplitude, with the bond lengths changed and nothing else. For the pyramid that plane is defined by the same three bonds and does not involve the umbrella coordinate at all.

Which is the assumption to be suspicious of. A pyramid’s stretch distortion is not orthogonal to its bending coordinates in the way a planar molecule’s is, so changing three bond lengths in a pyramid moves the molecule in directions the normal-mode analysis resolves into stretches and bends. Nothing here separates that, and it is a candidate for the kinks: a distortion that leaks into the bends at larger amplitude would change which bands are depolarised and where they sit. The probe is blind to the totally symmetric direction by a theorem and blind to nothing else, so a leak into a non-symmetric bend is a leak the reading sees.

And the amplitudes are the earlier essay’s. Nine of them over a sixteenfold span, chosen there and kept here so that the two exponents are fitted over the same range. A range chosen to suit ammonia would give a different exponent, which is the honest form of the finding: the exponent depends on the window.

The bands watched are whichever ones the probe finds depolarised, which is four for each molecule and is not the same four. Boron trifluoride’s are its degenerate stretch and bend pairs; ammonia’s are the same two pairs of its own, which are not at the same frequencies and are not composed the same way. Which bands a spectrum shows and which it forbids is settled by the group and is therefore shared; which mode is which is not.

A derived result and a fitted one look alike in one molecule #

The habit: a quantity computed once cannot be sorted into the part the symmetry fixes and the part the parameters set, and a second case sorts it for nothing.

Everything in the earlier essay came out of one molecule’s force field, and the two kinds of statement sat in the same table: the period of sixty degrees, which a group theory argument gives, and the exponent of 1.248, which nothing gives. Both were reported as measurements because both were measured, and the reader has no way to tell them apart.

The corollary is about which second case to pick. The informative one is not the most different molecule but the one that keeps exactly the structure the derivation uses and changes everything else. Ammonia keeps the three-fold axis, the three equivalent bonds and the degenerate stretch pair, and changes the masses, the force constants, the bond lengths and the point group’s mirror. So what survives is attributable to what was kept, and there is only one thing kept.

What a third molecule would settle #

The division this essay draws rests on two molecules, and two points cannot separate two variables. Ammonia differs from boron trifluoride in its point group, its masses, its bond lengths and its force constants all at once, so everything that transferred is attributable to the one thing kept and everything that did not is attributable to four things changed together.

The angular half of the finding is safe under that, because it is derived: a three-fold axis gives a sixty-degree period whatever else is true, and the two molecules agreeing to two and a half per cent is a confirmation rather than the evidence.

The quantitative half is not. A factor of 169 in the departure and 23 in the residual are two numbers with four candidate causes between them, and nothing here says which. A third molecule with the same group and only the masses changed would separate them completely — deuterated ammonia is exactly that, its fundamentals are measured, and its force field is the same field this essay already uses.

That is the cheapest experiment available and it is not run here. What is worth saying is what it would decide: if the departure moves and the residual does not, the departure is a mass effect and the residual is a force-constant one, which would make the two ratios’ disagreement a statement about which quantity is which rather than a puzzle.

Who measured what, and when #

The depolarisation ratio’s fixed value of three quarters for a non-totally-symmetric band is a symmetry theorem of the Placzek polarisability theory, from the 1930s. Ammonia’s and boron trifluoride’s fundamentals are long-standing gas-phase measurements. The force fields here are fitted to them by this collection, and the probe — the circle in the non-symmetric plane, the sum over bands, the residual and its exponent — is this argument’s own from the three essays before it.

What is computed here is the identical probe on a second molecule, with the angular quantities and the size quantities separated, and the two-term model’s failure on the pyramid diagnosed as a property of taking a maximum.

The numbers worth carrying are 2.027 and 0.816 — an anisotropy that transferred to two and a half per cent and an exponent that did not transfer at all.

Still open: one band rather than the maximum, and a third molecule #

The obvious open question is to stop taking a maximum. The kinks, the non-monotonicity and the failure of the power law are all explicable as the maximum switching bands, and the way to test that is to follow one band — the same band, identified by its composition rather than by its rank — round the circle and up the amplitudes. A single band’s reading should then be smooth, its residual should be a genuine power law, and its exponent should be a term in the expansion rather than an artefact. The band identification is the work: a degenerate pair split by a distortion has no canonical labelling, and an earlier essay ran into it in the sorted readings it already reports.

The nearer question is a third molecule with the same axis. Methane’s stretches are triply rather than doubly degenerate, so it is not the same case; a substituted methane with a three-fold axis — CH₃F, or ammonia’s own deuterated forms — keeps the group and changes the masses only. That is the sharpest available test of the division this essay draws, because a change of mass alone cannot touch the force field, so anything that moves is the mass and anything that does not is the group. Both molecules’ fundamentals are measured and one of them needs no new field at all.