It was never the mass

Ammonia turns itself inside out about twenty-four billion times a second. Phosphine does not do it at all, in the sense that no inversion splitting has ever been observed in its spectrum, and a phosphine with three different substituents is a stereocentre that can be resolved and put in a bottle, in the sense a symmetry argument makes precise.

The explanation in circulation is the mass. Phosphorus is heavier than nitrogen — thirty-one against fourteen — and a heavier particle tunnels less. The sentence is short, it appeals to something real, and it is wrong by four orders of magnitude.

Three things about phosphine differ from ammonia, and the one-dimensional model built for ammonia’s inversion takes exactly three inputs. Each can be substituted into ammonia’s well on its own, which is the only way to find out what each is worth.

The three inputs, and how far apart they are #

The reduced mass of the umbrella coordinate is not the mass of the central atom. It is the apex moving against a triangle of ligands with the centre of mass held still, which is $3m_L m_X/(3m_L + m_X)$ — and that expression is dominated by the three hydrogens, not by the atom in the middle. Going from nitrogen to phosphorus moves it from 2.4866 to 2.7546 unified mass units, which is 10.8 per cent.

That is the whole of the mass difference the tunnelling problem sees. Doubling the central atom’s mass changes the reduced mass by a tenth, because the light end of a reduced mass is the end that decides it.

The geometry is a different matter. Phosphine’s bond angle is 93.50 degrees against ammonia’s 106.67 — much nearer a right angle, which is what a heavier main-group hydride does, and which the repulsion account of shape predicts in direction and not in size — and its bond is 1.4200 ångström against 1.0124. Neither number is the one the inversion cares about; the one it cares about is the height of the apex above the ligand plane, computed from those two, and it is 0.76815 ångström against 0.38163. Phosphine’s pyramid is 2.013 times taller.

The barrier, quoted near 12,300 wavenumbers against 2020, is 6.09 times higher.

So of the three inputs, one changes by a tenth and two change by factors of two and six. Before any solve, that is already an argument, and the sensitivity exponents sharpen it: the geometry’s sensitivity is 2.56 times the barrier’s, so the one input that doubles is also the one the answer is most sensitive to.

One at a time #

Ammonia’s well gives a ground splitting of 1.3508 wavenumbers. Substituting each of phosphine’s three inputs, alone:

The reduced mass takes it to 0.83664 — a factor of 1.61.

The pyramid height takes it to 1.4011 × 10⁻⁴ — a factor of 9,641.

The barrier takes it to 1.4371 × 10⁻⁵ — a factor of 93,992.

The mass is smaller than either of the others by four orders of magnitude. It is not a small effect on top of two large ones; it is a rounding error beside them, and it is the only one the textbook sentence mentions.

There is a subtlety in that comparison worth stating, because the obvious reading of it is slightly wrong. The mass does little because it barely changes, not because the splitting is insensitive to it. The mass exponent is exactly one unit steeper than the barrier exponent — an exact consequence of rescaling the equation, not an approximation — so per cent for per cent, mass is worse than barrier. Ten point eight per cent applied to the barrier would cost a factor of 1.45; applied to the mass it costs 1.61; applied to the pyramid height it would cost 2.36. The ordering of the sensitivities is the reverse of the ordering of the effects, and the effects win because the changes are so different in size.

The mass account has a ceiling #

The substitutions above say the mass contributes little for phosphine. There is a stronger statement available, and it does not depend on which element is being proposed.

A reduced mass is dominated by its lighter end. Here the lighter end is three hydrogens, and $3m_L m_X/(3m_L + m_X)$ rises towards $3m_L$ as the central mass grows without ever reaching it. Three hydrogens give 3.0235 unified mass units, against ammonia’s 2.4866 — so the whole range available to the central atom, from nitrogen to an infinitely heavy element, is 21.6 per cent.

Substituting the series into ammonia’s well: nitrogen gives 1.35078 wavenumbers, phosphorus 0.83664, arsenic 0.64475, bismuth 0.56902, and an infinite mass 0.52954. The floor is a factor of 2.55.

The fall that has to be explained is five thousand million million. So the mass account is not merely the smallest of three contributions; it is capped at a factor of two and a half, and the cap is fifteen orders of magnitude short. No choice of element rescues it, and there is nothing left to argue about.

The bound is arithmetic rather than a property of this well, and the way to show that is to move the thing it depends on. Deuterating the ligands doubles the ceiling, and the cap quadruples — from 2.55 to 10.2. A statement about the potential would not have responded to the ligand mass that way.

It is also worth reading the series for what it does say. Between nitrogen and bismuth the splitting falls by a factor of 2.37, which is not nothing: a mass effect exists, it points the right way, and in a family of molecules whose barriers and geometries happened to match it would be the thing deciding the answer. The failure is one of magnitude, and magnitude is exactly what a sentence naming a mechanism does not supply.

Where a derivative stops being a guide #

The three exponents at ammonia’s own geometry are −3.5628 for the barrier, −4.5628 for the mass and −9.1256 for the height. Multiplying each by the logarithm of the corresponding change gives a prediction for each substitution, and comparing those against the solves says something about the exponents rather than about phosphine.

For the mass — a change of 10.8 per cent — the extrapolation gives 1.595 against a computed 1.615, which is right to one per cent. For the pyramid height, a change of a factor of two, it gives 592 against 9,641: low by sixteen. For the barrier, a change of a factor of six, it gives 624 against 93,992: low by a hundred and fifty.

Every extrapolation errs in the same direction and the error grows with the size of the step, which is what happens when the exponent itself steepens with the input — as a direct measurement of the exponent shows, the barrier exponent runs from −2.53 to −6.15 across the swept range. A logarithmic derivative is a tangent, and a tangent to a curve that is bending away from it underestimates.

That is worth having as a working rule, because the three quantities involved here are all quoted with sensitivities attached in the literature. A sensitivity supports a correction and not a substitution. Ten per cent is a correction and a factor of six is a different molecule.

The three do not add #

A splitting is an exponential of an action, so if the three inputs acted independently their costs would add in the exponent and the factors would multiply. The one-at-a-time costs in the action are 0.4304, 8.5039 and 12.3896, which sum to 21.324. Substituting all three at once gives 36.242.

The three reinforce each other by seventy per cent, and the mechanism is visible in the integral. The action is taken between the turning points at the state’s own energy, so every input moves the region being integrated over as well as the integrand. Raising the barrier lifts the potential and lowers the state’s position relative to it, which widens the classically forbidden region — and the widened region is then crossed at the larger pyramid height, over a longer distance, with the heavier mass. Each change makes the next one more expensive.

So the decomposition above is a set of one-at-a-time experiments and not a factorisation of the answer, and it is reported as such. That distinction matters more than it looks: a factorisation would license the statement “the mass accounts for such-and-such a share”, and there is no such share. What the substitutions establish is an ordering and a scale — which input, moved alone, does how much — and that is enough to refute an explanation that names the smallest of the three.

A number the arithmetic cannot see #

Solving phosphine’s own well returns two eigenvalues that are equal to the last representable bit. Each is bisected to machine precision on its own, so the finest interval the method can report between two levels near 709 wavenumbers is about 1.3 × 10⁻¹² wavenumbers, and phosphine’s splitting is below it.

That is reported as a bound rather than as a zero, and the distinction is not pedantry. A zero would say the two arrangements never exchange, which is false; the bound says the arithmetic cannot see how often. The quantity that does not underflow is the action, which comes to 41.93 against ammonia’s 5.69, and exponentiating the difference puts phosphine’s splitting near 2.5 × 10⁻¹⁶ wavenumbers — an exchange time of about a day, against ammonia’s twenty-one picoseconds.

A day is a chemically meaningful answer and it is the model’s rather than the molecule’s. What it says correctly is the shape of the situation: phosphine’s two forms do interconvert, by tunnelling, on a timescale that has nothing to do with the timescale of a molecular vibration, and that is why a resolved phosphine stays resolved and a resolved amine does not.

It also says what would change it. Bringing phosphine’s splitting up to a thousandth of a wavenumber — a line a microwave spectrometer could find — would need its barrier cut to 1,383 wavenumbers, a ninth of the quoted value, with the geometry untouched. Nothing about a phosphorus does that, and the reason to compute the number anyway is that it puts a scale on how far from observable the case is.

What was computed, and how #

The well, the solver and the geometry construction are those of the quartic-well calculation for ammonia, unchanged. Phosphine’s bond length and bond angle are quoted from its microwave spectrum; its barrier is quoted, and it is the least secure number here — published estimates cluster near 150 kilojoules a mole and the spread among them is wider than ammonia’s, since there is no inversion spectrum to fit one to.

That last point deserves to be said plainly rather than left in a caveat. Ammonia’s barrier is fitted to the very splitting being computed; phosphine’s cannot be, because the splitting has never been seen. Two numbers with the same name and different provenances is the defect three rotor conventions in one library turned out to be. So the two quoted barriers are not the same kind of number, and the comparison between them inherits whatever a computed barrier is worth. It happens not to matter here — halving phosphine’s barrier still leaves it costing a factor of three hundred, which is two orders above the mass — and that is checked rather than assumed.

The decomposition is five solves of the same well with different inputs, and the action for each is a six-thousand-panel midpoint integration between the turning points at that solve’s own ground-state energy. The turning points are bisected on the potential rather than read off the grid.

Four results are checked numerically. The mass alone costs less than a factor of three. Either of the other two alone costs more than a factor of a thousand. Phosphine’s own splitting is below anything the calculation can resolve, so it is reported as a bound — the only honest way to report an inability to see, as opposed to a value of zero. And the three costs do not add up to the whole, so the decomposition cannot be read as a factorisation.

Where the model stops #

Phosphine’s barrier is computed, not measured, and everything above inherits that. What survives it is the ordering, since the geometry alone — a quantity taken from a measured structure — already costs four orders of magnitude more than the mass.

The inversion path is assumed. Both molecules are carried through the flat geometry with their bonds held at the measured length and their ligand triangle equilateral. That is a one-dimensional cut through a six-dimensional surface, chosen because it is the symmetric one, and a real inversion relaxes the bond length as it goes — which lowers the barrier by an amount this model has no term for and which is probably larger for phosphine, whose bonds are longer and softer — the same omission a repulsion model that had never been given a bond length was found to be carrying.

The reduced mass is the usual construction, and what that is worth is a question of its own. It matters more here than for ammonia alone, because the two molecules’ ligand masses are the same while their central masses differ, and every construction of the reduced mass weights those two differently. A construction that weighted the central atom more heavily would give the mass a larger share — and the argument here survives it, because the share would have to grow by four orders of magnitude to matter. That a quantity with several defensible constructions can be leaned on this hard without anybody naming which one is in use is the hazard four measures of an orbital’s size was written about.

And two molecules is two. Arsine’s barrier is quoted higher again and its pyramid taller again, so it should extend the pattern; nothing here computes it, and a pattern established on two points is an observation rather than a trend, which is the standing warning a two-ring anomaly that turned out not to be the first of a series left on this collection.

The generalisation #

The transferable point is that an explanation naming one of several inputs has to be checked against the others, and the check is usually cheap.

The mass account of phosphine is not a bad piece of reasoning. Tunnelling really is more difficult for a heavier particle, phosphorus really is heavier, and the conclusion is true. It is an explanation that identifies a real effect pointing the right way and does not ask how large it is — and the size is what the sentence is being used to claim.

The test is one substitution per input, into a model that takes all of them, and it costs whatever one solve costs. What makes it work here is that the model’s inputs are separable in the sense that each can be moved alone, even though their effects are not separable in the sense that the costs would add. Those are different properties and only the first is needed.

There is a sharper version worth naming. The three inputs here are not equally visible. A mass is an atomic property that can be looked up; a barrier is quoted in every textbook; the height of a pyramid above its ligand plane is in no table anywhere, because it is derived from two quantities that are. The input that turned out to matter most is the one with no name and no tabulated value, and it is hard to believe that is unrelated to its absence from the explanation. The same shape appears where the count that decides a hypervalent molecule’s charge had to be constructed before it could be seen to be doing anything.

Who found it, and when #

The pyramidal inversion of ammonia and the rigidity of phosphine are both long established, and the qualitative attribution of the difference to the barrier rather than the mass is standard in the specialist literature — it is the textbook summaries that reach for the mass. The measured structures and the quoted barriers are not this collection’s.

What is its own is the arithmetic: three substitutions into one well, the action for each, and the observation that the smallest of the three effects is the one the short explanation names. The value of doing it is not the conclusion, which was not in doubt among people who had computed it, but the ratio — four orders of magnitude is a much stronger statement than “mostly the barrier”, and it is the kind of statement a comparison can support and a plausibility argument cannot.

Still open: whether the ligands or the centre decide the mass #

The obvious open question is the reduced mass, which every calculation so far has taken in the usual way and none has defended. The argument above leans on it twice — once to say that phosphine’s is only ten per cent larger than ammonia’s, and once to say that the ligands rather than the central atom decide it — and both statements are properties of one construction among several. There is at least one other that is not a constant at all: if the bonds are held at their measured length, then as the apex descends the ligands must slide outward, and their radial motion carries kinetic energy no constant mass accounts for.

The nearer question is what a barrier means when it is computed rather than fitted. Ammonia’s is fitted to a splitting and phosphine’s cannot be, so the two enter this comparison on different footings — and the exponents make that asymmetry expensive, since a barrier is exactly the quantity a splitting determines well and a calculation determines badly. Asking what range of computed barriers is consistent with phosphine having no observable splitting would turn the quoted 12,300 into a bound with a stated confidence, which is what the evidence actually supports.