The band that cannot supply what it changes

The correction this argument has been sizing for three essays is the zero-point energy of everything the reduction froze. Ammonia has six vibrations, the model keeps one of them, and the other five carry energy that changes as the molecule flattens — which makes the term isotope-dependent, where an electronic potential is not, and therefore makes it the natural candidate for the one defect four wells and five mass constructions could not shift.

It was sized against the sum. Half the transverse frequencies add to 6740 wavenumbers for NH₃ and 4957 for ND₃, the difference is 1783, and the gap between the two fitted barriers is ninety-seven — so the correction has to be 5.44 per cent of that difference. Carrying the same fraction to phosphine showed it does nearly the same thing at a barrier six times larger, which is what makes it a mechanism rather than an accident.

A sum is where a question can hide, and this one hid a factor of two. Five frozen degrees of freedom are not five copies of one thing. They are three bands — a symmetric stretch, a doubly degenerate stretch and a doubly degenerate bend — with very different frequencies and therefore very different shares of the isotope difference. The correction is a statement about the force field changing along the path, and a force field changes one band at a time.

Which band carries the difference, and it is not the one with the most modes #

Each band’s contribution is half its frequency times its degeneracy, differenced between the two molecules. The symmetric stretch is one mode at 3337 against 2420, contributing 458.5 wavenumbers. The degenerate stretch is two modes at 3444 against 2556, contributing 888. The degenerate bend is two modes at 1627 against 1191, contributing 436.

So the shares are 25.7, 49.8 and 24.5 per cent. The degenerate stretch carries half the isotope difference on its own — two modes rather than one, and the highest frequencies in the molecule, so the largest absolute drop on deuteration. The bend is two modes as well and carries half as much, because its frequencies are half as high.

That ordering is worth holding onto, because the intuition about which band changes along the inversion path runs the other way. The umbrella mode is the reaction coordinate; the two degenerate bends are its nearest relatives in the force field, they involve the same angles opening and closing, and they are what anybody would expect to be most affected when the pyramid goes flat. The stretches are a bond length away from the coordinate being driven.

There is a second reason the degenerate stretch dominates that has nothing to do with chemistry. A band’s contribution to the isotope difference is its degeneracy times the drop in its frequency on deuteration, and the drop is the frequency times one minus the isotope ratio. Since every one of these ratios is nearly the same — a fact this essay comes back to and makes the most of — the ranking of the bands by contribution is just their ranking by degeneracy times frequency. Which is to say: the isotope difference is essentially the transverse zero-point energy itself, rescaled. Nothing in the substitution reorders anything.

What each band would have to do on its own #

Turn the requirement into a statement about a band. If band i stiffens by a fraction f of its own frequencies between the pyramid and the plane — the same fraction in both isotopologues, because what changes is the force field and the isotope dependence is already in the frequency — then each molecule’s effective barrier rises by f times that band’s zero-point energy. The two molecules get different rises out of one electronic potential, which is the whole point, and the gap closes when f is the gap divided by that band’s share of the isotope difference.

For the symmetric stretch that is 21.16 per cent. For the degenerate stretch, 10.92. For the bend, 22.25. Each of those is a fraction of the band’s own frequency, so in wavenumbers they are quite different amounts of stiffening: 706 on the symmetric stretch, 376 on each component of the degenerate one, 362 on each bend. The absolute amounts are all of the same order, which is the first hint of what the last section of this essay is about.

None of those is five per cent, and the smallest is twice it. The 5.44 per cent quoted before is not the requirement on a band at all; it is the requirement on the whole transverse set moving together in the same proportion, which is the one arrangement in which every share is used at once and is therefore the arrangement that asks least of any individual band. Read as a statement about a frequency, it was the most generous reading available.

And the band that changes most can supply least. The degenerate bend carries a quarter of the difference, so it needs 22.25 per cent — twice what the degenerate stretch needs, and more than four times the number quoted. A quantity that is obviously sensitive to the distortion turns out to be the wrong lever precisely because it is low in frequency, and low frequency is why its isotope shift is small.

The sign was not blocked after all #

One thing this decomposition settles rather than complicates. The essay below called the correction’s sign genuinely blocked, on the grounds that the measured fundamentals are all at the pyramid and say nothing about the plane.

They do not have to, and the reason is that a sign is a cheaper thing to establish than a size. The direction is fixed by the direction of the gap, which is measured. The deuterated molecule’s fitted barrier is the lower of the two, and the deuterated molecule’s transverse zero point is also the lower of the two, so the rise that has to be added to each molecule’s barrier must be larger for the light one — and since the rise is a positive fraction of a positive zero-point energy, that fraction is positive. Every band, every construction of the mass, both wells: the transverse frequencies must go up as the pyramid flattens.

That is a chemically ordinary direction for the part of the set that matters. An N–H bond shortens as its nitrogen goes from pyramidal to planar and a shorter bond is a stiffer one, so the stretches stiffen. The bends are the awkward half: flattening takes the molecule towards the geometry where the umbrella coordinate has no restoring force at all, and its two degenerate relatives soften along with it.

So the realistic arrangement is not one band and not all five in proportion. It is the stretches stiffening while the bend softens, and then the bend’s contribution has the wrong sign and becomes a debt. With the bend neutral the stretches owe 7.20 per cent. With the bend softening by four per cent they owe 8.50; by eight, 9.79.

Nine per cent is a great deal for a stretching frequency. The N–H stretch across the nitrogen hydrides spans something like that from its highest to its lowest, and the change being asked for here is between two geometries of one molecule, a fifth of an ångström apart in apex height. The mechanism remains the only candidate on the table, and what the decomposition does to it is remove the comfortable reading: 5.44 per cent was an ordinary amount for five frequencies to change, and 7 to 10 per cent in the stretches alone is at the edge of what the chemistry supplies.

It is worth being precise about how much that costs the argument, because it is less than it sounds. The requirement scales with the gap between the two fitted barriers, and that gap is a property of the mass construction: the bond-conserving mass wants ninety-seven wavenumbers and the constant planar mass wants a hundred and nine, so the requirement moves by thirteen per cent between two constructions this argument cannot choose between. A gap smaller than ninety-seven would relax everything above proportionally, and the constructions that give a smaller gap are the ones a better reduction would be expected to produce. What the decomposition rules out is not the mechanism; it is treating five per cent as the number to check against a force field.

The one number that does not depend on the choice #

Everything above is underdetermination, and underdetermination is usually where an argument stops. Here it does not, because one quantity comes out the same whichever band is supposed to be doing the work.

The bare electronic barrier is the fitted one less the zero-point rise at the pyramid — f times that band’s zero-point energy in the light molecule. Put the required f into that and the band’s zero-point energy cancels almost entirely:

$$B_{\text{bare}} = B_{\text{fit}} - \frac{\Delta B}{1 - \nu_i(\text{heavy})/\nu_i(\text{light})}$$

The subtracted term depends on that band’s isotope frequency ratio and on nothing else about the band. Not on its frequency, not on its degeneracy, not on its share.

And every one of those ratios is nearly the same. Ammonia’s three are 0.7252, 0.7422 and 0.7320 — a spread of 2.34 per cent, all sitting just above $1/\sqrt{2}$. The reason is the one the whole umbrella problem turns on: in every transverse mode of a pyramidal hydride the hydrogens move and the central atom nearly does not, so the effective mass of each mode is close to a hydrogen’s, and deuterating it divides every frequency by close to $\sqrt{2}$.

Phosphine’s four bands do the same thing, at 0.7298, 0.7297 and 0.7207 — a spread of 1.26 per cent. So this is a property of a pyramidal hydride and not of ammonia, which is what makes it worth stating as a mechanism rather than as a coincidence of three numbers.

The barriers that come out are 1763.9 wavenumbers if the symmetric stretch does it, 1740.6 if the degenerate stretch does, 1754.9 if the bend does, and 1750.1 if all five move together. A factor of 2.04 in the requirement becomes 1.33 per cent in the barrier — the underdetermination is divided by a hundred and fifty on its way to the only quantity a calculation of the potential energy surface could be compared against.

Four numbers called the barrier #

That leaves this argument with four quantities all called ammonia’s inversion barrier, and it is worth setting them out because they are routinely quoted as though they were one.

The two fits, 2116.9 from NH₃ and 2019.9 from ND₃, are what this model needs to reproduce each isotopologue’s measured splitting. They differ by ninety-seven wavenumbers and they cannot both be right, which is the defect the whole correction exists to explain. The published 2020 is a fit of the same kind through a different well, and lands close to the deuterated one by arithmetic that has nothing to do with the deuterated molecule.

The bare electronic barrier, 1752 give or take twenty, is the new one. It is lower than all three, by about 365 wavenumbers, and it has to be: a zero-point rise adds to a barrier, so taking the rise out leaves less. It is also the only one of the four that means what “the barrier” is usually taken to mean — the height of the electronic potential energy surface at its saddle, with no nuclear motion in it at all.

Whether 1752 is right is a question for a calculation of that surface and not for anything here. What is established is that the number this argument can offer is not the fitted one, is 17 per cent below it, and does not move when the mechanism’s least determined feature is varied across its whole range.

There is a check available on the 1752 that costs nothing, and it is the one the whole argument started from. A splitting is an area under a barrier and not a height, so a barrier 365 wavenumbers lower does not produce a splitting seventeen per cent larger — it produces one larger by a factor set by the action, and the local power of the splitting in the barrier is nowhere near constant. Solving the bare well and reading its splitting off would give a number several times the measurement, which is correct and is not an inconsistency: the bare barrier is not a barrier any nucleus ever moves on, because the transverse zero point is part of the potential the nuclei see. The 1752 is a quantity for comparison against electronic structure, and the 2117 is the quantity for comparison against a spectrum. Quoting either one where the other belongs is the confusion this figure exists to prevent.

What the arithmetic assumes #

One band at a time is a fiction and is used as one. A force field does not change one band while holding two others fixed; the normal modes themselves rearrange as the geometry changes, and at the planar geometry the correlation between a pyramidal band and a planar one is a projection rather than an identity. The single-band requirements are therefore bounds on a family rather than predictions, which is exactly how they are read above: the point is the factor of two between them and the invariance of what they imply, neither of which needs the fiction to be literal.

The fraction is taken as constant in both isotopologues. That is the content of “the force field changes”: a fractional change in a harmonic frequency at a geometry is a fractional change in a force constant divided by a mass that is not changing with geometry. It fails at second order, because the normal-mode composition of a band differs slightly between isotopologues, and the size of that failure is the 2.34 per cent spread in the ratios rather than anything larger.

The shape along the path is nowhere in this. Only the endpoints enter: the zero-point energy at the pyramid and at the plane. That is enough for the barrier, which is a difference of two endpoints, and it is not enough for the splitting, which is an integral along the path. A correction that arrives all at once near the top of the barrier and one that grows smoothly from the pyramid give the same 1752 and different splittings, and nothing here distinguishes them.

And the fundamentals are used as harmonic frequencies. Both molecules’ are anharmonic by a few per cent, so every zero-point sum here carries that error. It cancels well in the ratios, which are ratios of two similarly anharmonic quantities, and it does not cancel in the sums — so the 1783 and the 1255 are good to a few per cent and the 0.7252 is good to a great deal better. That is the same division of labour the action turned out to have when it was calibrated against four isotopologues: an instrument’s error on a ratio and its error on a value are two different numbers, and which one applies depends on which the argument uses.

A sum is a place to hide a factor of two #

The habit this essay is an instance of: when a model needs a quantity that is a sum, ask what the requirement is on each term, because the sum always asks least.

The correction was sized against five modes moving in proportion, which is a defensible thing to compute and is the most forgiving arrangement in the family. Every other arrangement asks more of something, and the arrangement the chemistry actually suggests — stretches up, bends down — asks nearly twice as much. Nothing about the first calculation was wrong; what was wrong was reading its answer as the size of the required change rather than as the smallest one.

The corollary is the more useful half, and it is the opposite lesson. Underdetermination is about a quantity, not about a model. Three bands and no way to choose between them is a genuine failure to determine the mechanism, and it coexists with a perfectly determined barrier, because the barrier depends on the one feature the three bands share. Finding which quantity survives a sweep is worth as much as narrowing the sweep, and it is usually cheaper.

This argument has now done that three times and the pattern is consistent enough to be worth stating. Five orderings of the kinetic operator spanned half a per cent in the splitting and the barrier fitted through them barely moved. Three mass constructions spanned forty-three per cent in the splitting and the direction of every isotope prediction was the same under all of them. Three bands span a factor of two in required stiffening and one and a third per cent in the barrier. Each time the sweep was run to bound an error and each time the useful finding was a quantity the sweep could not touch.

Who measured the bands, and when #

The eight fundamentals of NH₃ and ND₃ and the eight of PH₃ and PD₃ are long-standing gas-phase infrared measurements, quoted in the standard compilations. The barriers fitted here are this model’s own; the published 2020 is Swalen and Ibers, 1962, fitted to the inversion spectrum through a harmonic well with a Gaussian hump. That the transverse modes of a hydride are hydrogen motion against a nearly stationary heavy atom, and that their frequencies therefore scale as the inverse square root of the ligand mass, is the oldest observation in vibrational spectroscopy.

What is computed here is the band-by-band decomposition of the required correction, the direction it fixes, and the bare barrier it implies together with that barrier’s independence of which band supplies the term.

The numbers worth carrying are 2.04 and 1.33 per cent — the spread in what the mechanism demands, and the spread in what it determines.

Still open: the second line, and a fourth band #

The obvious open question is what the correction does to the excited doublet. Everything above uses one measured splitting per molecule, and each molecule has two: the ground-state interval and the first excited one, 35.81 and 3.35 wavenumbers. A barrier shift fitted to close the ground-state gap is then a prediction about the excited-state gap, and the excited state sits much higher in the well where the barrier is narrower and the correction’s shape along the path matters more. Whether it closes the second gap too, over-closes it, or leaves it untouched is four solves on wells this argument already has — and it is the one test available that the correction cannot pass by construction, since nothing in it was fitted to a second line.

The nearer question is a band the decomposition does not have. Ammonia’s degenerate stretch and degenerate bend are each two modes treated as one band, and at the planar geometry they remain degenerate, so treating each pair as a unit is exact there. It is not exact in between: a pyramid on its way to flat passes through no symmetry lower than its own threefold axis, which keeps the pairs together, but the partly deuterated molecules have no threefold axis at all and their six frequencies are six. Four of this argument’s isotopologues are mixed, their fundamentals are measured, and whether their transverse sums sit on the line the two pure ones define is a test of whether a band is the right unit — arithmetic on published numbers, with nothing to solve.