The same fraction at six times the barrier

The reduction leaves a term out, and the term’s size was established by inversion. One electronic potential has to serve NH₃ and ND₃, the barrier fitted to each of them separately differs by ninety-seven wavenumbers, and a constraint imposed as the limit of a stiff potential leaves behind exactly the sort of quantity that could supply the difference: the zero-point energy of the five degrees of freedom the constraint froze, which depends on the masses where an electronic potential does not. Sized against the transverse zero-point energies of the two molecules, the correction would have to be about five per cent of the difference between them.

Five per cent of something is not a mechanism. It is a number that happens to be small, and a number that happens to be small is the easiest kind of agreement to get by accident — which is why the closing paragraph of that argument named a second molecule rather than a refinement. Phosphine’s barrier is six times ammonia’s. If the same fraction comes out of the same arithmetic there, the term is behaving like a property of the reduction; if it comes out wildly different, five per cent was a coincidence of two numbers that happened to be in the right ratio.

Phosphine has been in this argument since the essay that took the mass account apart, and it arrives here with one disqualifying feature: nothing about its inversion has ever been measured.

What is measured about phosphine, and what is not #

The four fundamentals are. PH₃ shows its symmetric stretch at 2321, its umbrella at 992, its degenerate stretch at 2327 and its degenerate bend at 1118 wavenumbers; PD₃ shows the same four at 1694, 730, 1698 and 806. That is eight measured bands, exactly as ammonia supplies eight, and half the sum of the five transverse ones is the quantity the reduction throws away.

So the supply side of the argument transfers without qualification. Phosphine’s transverse zero-point energy is 4606 wavenumbers and deuterated phosphine’s is 3351, a difference of 1255 — against ammonia’s 1783. Both sums are half the sum of five frequencies, which is the same construction the zero-point correction to a bond angle uses and the same one that makes an average bond length depend on which isotope is in it: a frozen degree of freedom still carries energy, and how much depends on the mass sitting in it.

That the two differences are the same order is worth pausing on, because nothing arranged it. The barrier changes by a factor of six between the two molecules and the apex height doubles, but the modes carrying the isotope difference are the ligands’ own motions, the ligands are the same three hydrogens in both cases, and replacing them with deuterium does nearly the same thing to a P–H stretch as to an N–H one. Phosphine’s difference is seventy per cent of ammonia’s, and the seventy per cent is mostly that its stretches sit a thousand wavenumbers lower to start with.

What is not measured is the splitting. No inversion doublet has ever been observed in either phosphine, and the barrier itself is a quoted estimate rather than a fit — published values cluster near a hundred and fifty kilojoules a mole, and there is no inversion spectrum to pin one down with. So the demand side of the argument cannot transfer at all. Ammonia’s five per cent was obtained by asking what the correction would have to be to close a gap between two measured numbers, and phosphine has no measured numbers to leave a gap.

The question therefore has to be turned round once more. Not what would the correction have to be, which is unanswerable here, but what would the same correction do — which is answerable, and is the comparison that decides whether the five per cent was a mechanism.

A solver that fails by reporting numbers #

Answering it needs phosphine’s splitting, and the first attempt to compute one runs into something more interesting than a limit.

The splitting is a difference of two eigenvalues that agree to fifteen figures, and the solver finds each of them by bisecting a Sturm count. That the splitting is an area under a barrier rather than a height is what makes it so small in the first place, and what makes the difference so much harder to compute than either level. Each is therefore bisected to the last representable bit of itself — and that is the property which made the method the right one for this argument in the first place, because a difference of two levels computed independently to machine precision beats a whole spectrum computed to a tolerance.

Raise the barrier and watch it work. At ammonia’s own 2,020 wavenumbers phosphine’s splitting comes out at 1.8 × 10⁻⁵ wavenumbers; at 3,000 it is 3.6 × 10⁻⁷; at 4,000 it is 1.2 × 10⁻⁸. Then at 5,000 it is 7.8 × 10⁻¹⁰, at 6,000 it is 3.9 × 10⁻¹⁰, at 7,000 it is exactly zero, and at 8,000 it is 7.8 × 10⁻¹⁰ again.

A splitting that rises when the barrier rises is not a splitting. Nothing about the physics permits it: raising a barrier can only make tunnelling through it less likely, and the whole argument rests on that monotonicity. The values past 4,000 are the last few bits of two eigenvalues, appearing and disappearing as the arithmetic rounds one way or the other, and the tell is that they move when the grid is refined at a fixed barrier — 7.8 × 10⁻¹⁰ at 2,200 points, 3.9 × 10⁻¹⁰ at 2,400 and nothing at all at 2,600. That is what noise does. An answer does not.

The failure is quiet in the way that matters. Nothing throws, nothing warns, and the numbers are of a plausible size for a molecule that famously does not invert — a reader looking at 3.9 × 10⁻¹⁰ wavenumbers has no reason to doubt it. What gives it away is only the sequence, and the sequence is visible only because the barrier was swept rather than evaluated once. An eigenvalue sliding towards the limit of double precision is a shape an earlier essay met before, in a basis set whose overlap matrix goes singular; there the smallest eigenvalue halves with every function added and the failure announces itself by the halving. Here there is no trend to extrapolate, because the quantity that fails is a difference and a difference has no scale of its own.

The floor was in the wrong place, and by four orders of magnitude #

This argument has always reported a floor alongside a splitting, precisely so that an underflow could not be read as a zero. The floor was computed as a small multiple of the level itself, on the reasoning that two eigenvalues near E cannot be separated below about one part in 2⁵² of E.

The reasoning is half right, and the half that is wrong is the conclusion. A Sturm count is not a comparison of two numbers near E; it is a recurrence run down every one of two thousand four hundred grid rows, and what limits it is the rounding accumulated in that recurrence. Rounding is proportional to the largest quantity in the matrix, and the largest quantity in this matrix is the kinetic term — tens of hartree on a grid this fine, where the levels are thousandths of one.

The two prescriptions differ by four orders of magnitude. For phosphine’s own well the level-based floor is 1.3 × 10⁻¹² wavenumbers and the norm-based one is 1.7 × 10⁻⁹, and every spurious value in the sweep above sits between them: accepted as a measurement by the old floor, refused by the new one.

No value published here moves. Every splitting reported as a number rather than as a bound is seven orders of magnitude or more above the new floor — ammonia’s ground doublet clears it by a factor of ninety million, and the smallest genuine splitting anywhere in the argument, the five-millionths of a wavenumber an apex-only mass gives, clears it by three thousand. What moves is the bound quoted where a splitting underflows, and it moves in the direction that matters: it was claiming four orders of magnitude more precision than the arithmetic has.

That is the shape of defect this argument keeps finding in its own instruments, and it has the same cause each time. A quantity was derived from the thing being measured rather than from the thing doing the measuring. The mass constructions were three defensible answers to a question the measurement does not ask; the orderings of the kinetic operator were five; and a resolution floor is the same kind of thing one level further down, a choice nobody thought of as a choice.

The action, calibrated where a splitting can still be seen #

The quantity that does not underflow is the action under the barrier, because it is an integral rather than a difference of near-equal numbers. Exponentiated, with the harmonic frequency of one well as a prefactor, it gives the standard one-dimensional tunnelling estimate — and an estimate is exactly what it is.

So it is calibrated rather than trusted. There are four ammonia isotopologues whose splittings the solver reports comfortably above its floor, and running the formula on all four gives two quite different error figures.

Against a splitting it is low by 5.64 per cent for NH₃, 5.97 for NH₂D, 6.26 for NHD₂ and 6.51 for ND₃ — a drift with the mass, not a constant offset. Against a ratio of two of those splittings it is out by 0.36 per cent, 0.67 and 0.94. The ratio is seven times better than either of the values that make it, and the reason is structural rather than lucky: the prefactor is the crude half of the formula, it is nearly common to two isotopologues of one molecule, and the only thing that changes it between them is the harmonic frequency at the minimum.

Phosphine is being asked for a ratio. That is the calibration’s own verdict on whether the instrument is fit for the question, and it happens to be favourable — which is worth saying plainly, because the same calibration would have refused a question about either splitting on its own to better than five per cent.

With the instrument calibrated, phosphine’s two splittings come out at 3.4 × 10⁻¹⁷ and 1.1 × 10⁻²⁴ wavenumbers. Both are far below anything an experiment reaches and both are equally far below what the solver can represent, which is why they are reported here as computed quantities with a stated method rather than as predictions anybody could check.

The gap between them is the arresting number. Ammonia’s two isotopologues differ by a factor of fifteen; phosphine’s differ by a factor of thirty million. An isotope substitution changes the action by about the same fraction in both molecules, and phosphine’s action is seven times larger, so the same fractional change lands in an exponent seven times bigger.

Ten times less against seven times more #

Now the correction. A transverse zero-point term raises each molecule’s effective barrier by a fraction of its own transverse zero-point energy, and since that energy differs between a molecule and its deuterated twin, the two get different effective barriers out of one electronic potential. Ammonia’s two fits say the difference is ninety-seven wavenumbers, which is 5.44 per cent of its 1783.

Taking the same 5.44 per cent of phosphine’s 1255 gives sixty-eight wavenumbers. Against a barrier of 12,300 that is 0.56 per cent, where ammonia’s ninety-seven against 2,117 was 4.58 per cent — nearly ten times smaller a perturbation, because the barrier grew by six and the zero-point difference shrank.

The naive expectation from that alone is that the correction does almost nothing to phosphine. It does not hold, and the reason is the action.

A splitting falls as the exponential of the action, and the action goes as the square root of the barrier at fixed geometry, so a small fractional change in the barrier changes the action by half the action times that fraction. The local power of the splitting in the barrier is not a constant — it runs from −2.5 to −6.2 across a sweep, and the exact statements available are about the action rather than about any power law fitted to the splitting — so it matters that this argument is conducted in the action and converted at the end. The quantity that decides what a correction is worth is therefore a product — and the two factors in it move opposite ways between these molecules. Ammonia: 4.58 per cent against an action of 8.86. Phosphine: 0.56 per cent against an action of 60.76. The products are 0.203 and 0.169.

So the answer is the one the earlier calculation hoped for and could not get. The correction moves ammonia’s predicted isotope ratio by 21.4 per cent and phosphine’s by 16.1 — within a quarter of each other, on two molecules whose barriers differ by a factor of six and whose splittings differ by twenty-four orders of magnitude.

That is what a property of the reduction looks like. A coincidence of two numbers would have no reason to survive the transfer, and this one survives it because the transfer is governed by a product whose factors compensate rather than by either factor alone.

What the numbers are and are not evidence of #

The compensation is not exact and should not be read as an identity. Sixteen against twenty-one per cent is a quarter apart, which is the size of discrepancy the square-root approximation to the action’s barrier dependence would itself produce, and the two molecules also differ in where their states sit relative to their barriers. Ammonia’s ground state sits a quarter of the way up its barrier; phosphine’s sits a twentieth of the way up its own, so the turning points are much deeper in the wings and the part of the potential the action integrates over is a different part of it. What the agreement establishes is an order of magnitude, and an order of magnitude is what the question was about.

The barrier is the weak number and it is phosphine’s. Ammonia’s 2,020 is fitted to the spectrum being computed; phosphine’s 12,300 is an estimate with a wider spread among published values than ammonia’s has, and nothing here can tighten it. Moving it by ten per cent moves the fractional perturbation by ten per cent and the action by five, so the product moves by about five — inside the quarter already quoted, which is the only reason the comparison is worth making at all.

The fraction carried across is ammonia’s, not phosphine’s. Nothing in this argument computes a required correction for phosphine, because nothing can: the requirement was read off a gap between two measured splittings and phosphine has none. What is computed is the effect of a correction of assumed relative size, and the claim is about the effect’s insensitivity to the molecule rather than about the size being right.

And the transverse frequencies are fundamentals used as harmonic frequencies. Both molecules’ are anharmonic by a few per cent, so both zero-point sums are good to a few per cent, which is the same order as the fraction being taken of them. The comparison survives because it is a comparison of two similarly-biased quantities and the bias largely cancels in their ratio — the same argument that makes the action usable for a ratio, applied one level up.

The habit underneath both halves #

Two things in this essay are the same move made twice, and it is worth naming because it is the move that produced the correction to the floor as well as the answer about phosphine.

An instrument’s error on a difference is not its error on a value. The action formula is six per cent wrong about every splitting and one per cent wrong about their ratios; the Sturm bisection is exact to the last bit about every level and blind below 10⁻⁹ about their differences. Each instrument has two accuracies, they differ by an order of magnitude in opposite directions, and reading either number as the accuracy is how a argument ends up quoting a bound four orders of magnitude too tight.

And a floor belongs to the apparatus, not to the answer. The old floor was a fraction of the eigenvalue, which is a property of the molecule; the right one is a fraction of the matrix, which is a property of the grid and the mass. The first will always look more physical and will always be wrong, because the thing that cannot see a small difference is the arithmetic and not the physics.

Who measured what, and when #

The fundamentals of both phosphines are the gas-phase infrared bands, long-standing and quoted in every compilation; phosphine’s structure is from its microwave spectrum. The barrier is a computed and semi-empirical estimate rather than a measurement, as the essay that first used it said. The action formula for a symmetric double well is standard semiclassical tunnelling theory, of the 1930s; the Sturm-sequence bisection for a symmetric tridiagonal eigenvalue is Givens, 1954.

What is computed here is the calibration of the action against four resolvable isotopologues, the norm-based floor and the sweep that exposed the old one, and the transfer of the transverse zero-point correction to a second pyramid.

The number worth carrying is 0.169 against 0.203 — two products, on two molecules six times apart in barrier and twenty-four orders of magnitude apart in splitting.

Still open: the sign, and a molecule with a measurable line #

The obvious open question is still the sign, and it is now a sharper question than it was. The correction has to raise NH₃’s effective barrier above ND₃’s, which means the transverse zero-point difference must be larger at the planar geometry than at the pyramid, which means the transverse frequencies must stiffen as the molecule flattens. That is a definite direction and it is a chemically ordinary one — an X–H bond shortens and stiffens as its central atom goes from pyramidal to planar. What nothing above establishes is whether all five modes have to move that way or whether one of them can do it alone, and the five are not interchangeable: they carry very different shares of the isotope difference, so the fraction each would have to supply on its own is very different. Decomposing the 1783 band by band is arithmetic on numbers already quoted here, and it turns one required fraction into five.

The nearer question is arsine, which the mass-ceiling calculation already carries a mass for and which would put a third point on the comparison. Its fundamentals are measured, its barrier is estimated on the same footing as phosphine’s, and its action would be larger again — so if the product stays near 0.2 across three molecules spanning a factor of ten in barrier, the compensation is a property of the family rather than of a pair. The objection to doing it is the one this essay has already run into twice: three quoted barriers of unequal quality, and a conclusion that depends on their ratios.