The exponent that runs both ways

A double well built from ammonia’s own geometry, with the published inversion barrier put into it, gives a ground splitting of 1.3508 wavenumbers against a measured 0.79350. The discrepancy has a location: the two wells being compared agree about the barrier’s height to twelve per cent and about the area under it to eight, and eight per cent in an area of about six is a factor of one and a half in an exponential.

That leaves a quantity unmeasured, and it is the one every argument in this subject leans on without stating. How hard does a splitting depend on a barrier? The answer is a single number and it is worth having, because it governs both directions of a traffic that runs both ways: barriers are quoted and splittings predicted from them, and splittings are measured and barriers inferred from them — the same two-way traffic between a fitted parameter and the measurement it was fitted to that keeps turning out to point the wrong way. The same derivative does both jobs and it is not equally good at them.

There is no power law, and the reason is worth having #

Sweeping the barrier from a third of the published value to three times it, with the geometry and the reduced mass held where the measurements put them, gives a curve that is bent.

The phrase “tunnelling is exponential in the barrier” appears in every account of this and is not what the arithmetic does; neither does the natural repair, “it is a power law in the barrier”. The local logarithmic slope is −2.5325 at 1212 wavenumbers, −3.5628 at 2020, −4.5692 at 3030 and −6.1504 at 5050. It more than doubles across a factor of four in the barrier, and a single straight line fitted through the whole sweep leaves a worst residual of 0.90 in the logarithm, which is a factor of two and a half — larger than the entire discrepancy in ammonia’s splitting.

What the splitting really is exponential in is the action, and the action’s own dependence on the barrier is not simple. The height enters it under a square root, which would give an exponent proportional to the square root of the barrier; but raising the barrier also raises the state inside it, which widens the region the integral is taken over, and the two effects do not cancel at a fixed rate. So a sensitivity exists at every barrier and no single number describes the family.

That is a limitation and it is also the answer to a question. The exponent at ammonia’s own barrier is −3.5628, and that is the number every argument about ammonia needs. Quoted with the barrier it was measured at, it is a measurement; quoted alone, it is a tangent presented as a curve.

What ten per cent on a barrier is worth #

A barrier quoted to ten per cent — which is a good quotation, and better than the spread between published values — predicts a splitting to thirty-six per cent. A splitting known to a factor of two — which is a terrible measurement, and worse than anything a microwave spectrometer would produce — fixes a barrier to nineteen per cent.

Those two sentences are the same derivative and they are not the same statement. In one direction the uncertainty is multiplied by 3.56 and in the other it is divided by it, so the arrangement in which a barrier is the primary datum and a splitting the derived one is the arrangement that wastes information.

It is also the arrangement in which the numbers are usually presented. A barrier is what gets a sentence in a textbook, because it is an energy in familiar units and it sounds like a property of a molecule — the same appeal an oxidation state has, and for the same reason. A splitting is what a spectrometer measures, to five figures, in an afternoon. The barrier is a fitted parameter of somebody’s potential and the splitting is the observation it was fitted to — and the derived quantity is the one that gets quoted. That is the same reversal a symmetry measure suffers when a verdict is quoted and the tolerance that produced it is not.

Why one molecule has several barriers #

This explains something otherwise puzzling about the literature, and it does so without anybody having made a mistake.

Fitting the two-parameter well to ammonia’s ground splitting alone asks for a barrier of 2329.9 wavenumbers. Fitting the same well to its first excited splitting alone asks for 2559.8. Fitting a three-parameter well to both at once returns 2262.0, and the published value is 2020.

Four numbers, one molecule, no errors. Each is the barrier of a particular well fitted to a particular subset of the data by a particular procedure, and because a splitting responds to the barrier as a steep power, procedures that differ mildly produce barriers that differ visibly. A collection of published barriers spread over fifteen per cent is not a controversy; it is the shape of the inference. Four measures of an orbital’s size spanning a factor of two and a half is the same finding about a different quantity: several correct answers, one word.

The practical consequence follows directly and is worth stating in the form somebody could use. A quoted barrier is only interpretable alongside the well it was fitted in and the lines it was fitted to. Substituting it into a different well is the operation performed deliberately on ammonia’s quartic well, to see what it would do, and it produced an answer seventy per cent out while every other property of the same calculation came within a few per cent.

The three sensitivities are one number #

There are three inputs — the barrier, the reduced mass and the height of the pyramid — and the obvious expectation is three independent slopes to measure. There is one, and the other two follow from it exactly.

The mass against the barrier. Multiply the reduced mass by λ and divide the well’s coupling by the same λ. The kinetic term picks up a factor of 1/λ and so does the potential term, so the whole operator is the original one divided by λ — which leaves every eigenfunction untouched and divides every eigenvalue by λ. The barrier is the coupling times a fixed power of the geometry, so it is divided by λ too. Therefore

$$\Delta(\lambda\mu,\; H/\lambda) = \Delta(\mu, H)/\lambda$$

and differentiating at λ = 1 gives the mass exponent as exactly one below the barrier’s. Not approximately, not in the deep-barrier limit: exactly, and for any potential whatever.

The geometry against the mass. Substitute x = αy in the same equation. The second derivative brings down 1/α², the quartic’s minimum moves to x₀/α, and its coupling picks up α⁴ — and the barrier, being the coupling times the fourth power of the minimum, is unchanged by that pair of moves. So the spectrum at a given barrier is a function of μα² and x₀/α jointly, and the geometry exponent is exactly twice the mass’s.

Both are three lines of algebra and neither is checked by being believed. Computed at four barriers, the difference between the mass and barrier exponents comes out at −1.000000 and the ratio of the geometry to the mass exponent at 2.000000, to six decimal places in each case.

Two things follow, and one of them decides what is worth measuring next. The model has one free sensitivity rather than three, so measuring the barrier’s fixes the others — and the mass is always the steeper of the two by exactly one unit, which makes an uncertain mass strictly worse than an equally uncertain barrier. And the geometry’s exponent is twice the mass’s: at ammonia’s barrier it is −9.1256, which is 2.56 times the barrier’s. A ten per cent error in the height of the pyramid is worth two and a half times a ten per cent error in the barrier, and the height is the one quantity of the three that nobody quotes at all, because it is derived from a length and an angle rather than reported — and the angle is exactly the quantity the repulsion account of shape spends its whole effort predicting to within a degree or two.

The check the barrier cannot supply #

There is an obvious worry about all of this, and the quartic-well calculation named it: the well was solved with one particular construction of the reduced mass, chosen because it is the usual one and not because anything defended it. If the over-prediction were an artefact of that choice rather than of the well’s shape, everything above would be about the wrong thing.

The test is free, because an equilibrium structure does not depend on nuclear mass. The same well, with hydrogen replaced by deuterium and nothing else touched, is a prediction with nothing fitted, and ND₃’s ground splitting is measured at 0.0531 wavenumbers.

The computed value is 0.08569, which is 1.614 times the measurement. Ammonia’s factor is 1.702. The reduced mass has gone from 2.4866 to 4.2210 unified mass units — a change of seventy per cent — the splitting itself has fallen by a factor of 15.8, and the ratio between computation and measurement has moved by five per cent.

That is what settles it. An error that survives a seventy per cent change in one of the model’s three inputs, while the answer it is an error in falls by more than an order of magnitude, is not an error in that input. It is a property of the potential’s shape, exactly as the original diagnosis said, and the isotope substitution is the cheapest way of saying so.

The excited splitting behaves less well — 7.400 computed against 3.35 measured, a factor of 2.21 against ammonia’s 1.91 — and that is worth recording rather than hiding. The excited state sits higher in the well, where the two candidate potentials differ more, so a shape error should show up more strongly there. It does, in the right direction, by an amount nothing predicted.

Raising the barrier is the wrong repair, quantified #

With the exponent in hand it is possible to say precisely how much of the discrepancy a barrier adjustment can absorb, and the answer is: the whole of one line and none of the disagreement between two.

Raising the barrier from 2020 to 2329.9 — a fifteen per cent increase, which is what a factor of 1.70 divided by an exponent of 3.56 comes to, and the agreement between the fitted value and that arithmetic is a check on both — puts the ground splitting exactly on its measurement. It puts the excited splitting at 47.30 against 35.81.

So the residual, after the best possible one-parameter repair, is 32 per cent on a quantity that started 91 per cent out. Two thirds of the excited splitting’s error was the barrier being too low and the last third is the well being the wrong shape, and no barrier removes the last third because the two lines want the barrier moved by different amounts in the same direction.

This is the same shape of finding the counterpoise correction reached about a threshold: a result that rests on one adjustable number has no redundancy in it, and the way to find out is to ask the model for two things at once.

What was computed, and how #

The well, the geometry and the solver are those of the quartic-well calculation, unchanged, so this extends that measurement rather than a neighbouring one. The sweep is forty-five barriers spaced logarithmically over a factor of eight, each a fresh solve of the same tridiagonal problem, and the exponent is a central difference in the logarithms at one per cent either side.

The barriers implied by each splitting are found by bisection on the logarithm of the barrier rather than on the barrier itself, because the splitting falls by orders across the bracket and a linear bisection spends its first twenty steps in a region where the answer is numerically zero. The bracket is checked before the search rather than after: a target outside it returns a refusal, since a bisection that runs to its endpoint returns the endpoint and looks exactly like an answer.

The isotope calculation changes one number. The reduced mass of the umbrella coordinate under the construction used throughout is $3m_L m_X/(3m_L + m_X)$, and substituting deuterium’s mass for hydrogen’s is the whole of it; the potential, the geometry, the grid and the box are identical.

Four results are checked numerically, and the third is the one that could have overturned the argument. The sweep is a straight line on logarithmic axes to within a stated tolerance, so calling the exponent an exponent is licensed. The exponent read at two different barriers differs, so it is reported as a local slope rather than as a law. The two isotopologues’ over-prediction factors agree to within a tenth, which is what makes the diagnosis a diagnosis — had ND₃ come back at 1.05 or at 3.0 the shape explanation would have been refuted. And the barrier implied by the ground splitting agrees with the one predicted from the exponent and the discrepancy, which is two routes to one number that share nothing but the model.

Where the model stops #

The exponent is local. It is a property of a quartic well at ammonia’s geometry near ammonia’s barrier, and it moves by nine per cent across a fifteen per cent change in that barrier. It is not a transferable constant and no argument here treats it as one; what transfers is that it is large, negative, and of order three or four for any well with a state well below its barrier.

One shape. Every number above is computed in the two-parameter quartic, including the barriers that other procedures are being compared against. A three-parameter well has its own exponent and its own implied barriers, and the comparison between published values remains a comparison between procedures rather than a measurement of anything.

The isotope test moves one input and not two. ND₃’s equilibrium structure is taken as ammonia’s, which is right for the electronic potential and slightly wrong for anything derived from a vibrationally averaged structure — the same distinction that gives one bond two lengths depending on which isotope was measured, and which the zero-point motion of a bond angle makes unavoidable. The effect is far below the factors being discussed and it is not zero.

And the excited splitting is the weaker measurement. Ammonia’s 35.81 and ND₃’s 3.35 are quoted to four and three figures against ground-state values known to five, and the excited state is also where the one-coordinate approximation is worst, since a state higher in the well samples more of the bond stretching this model has no term for — the coupling a normal-mode analysis would have to supply.

The generalisation #

The transferable point is about which direction of inference a steep dependence helps.

The instinct on finding that an answer depends on a parameter as a steep power is that the parameter is badly constrained and the model is fragile. Half of that is right. The prediction is fragile: anything computed forwards from the parameter inherits its uncertainty multiplied. The inference is the opposite — the observation pins the parameter far better than the parameter would ever have pinned the observation, and a steep dependence is precisely the property that makes a measurement informative.

So the question worth asking of any such model is which end of it carries the measurement. Where the measured quantity is the sensitive one, a steep exponent is a gift, and the right way to report the result is as a parameter with a tight bound. Where the measured quantity is the parameter, it is a hazard, and the right way to report the result is as a range rather than a value.

Ammonia is the first kind and is presented as the second. Its splitting is measured to five figures and its barrier is derived; the derived quantity is the one that appears in every account, is quoted to three figures without a well attached, and disagrees between sources by more than the measurement it came from disagrees with anything. Reversing the presentation costs nothing and would make the fifteen per cent spread among published barriers legible instead of embarrassing.

The second half is about testing a diagnosis by an input it should not depend on. The quartic-well calculation concluded that the error was in the potential’s shape. That conclusion makes a prediction — the error should survive a change of mass — and the prediction is cheap, since one of the model’s three inputs can be changed by seventy per cent by writing a different element symbol. A diagnosis that could not have been refuted by that substitution would have been worth much less, and a verdict inside its own error bar is the standing example of what happens when the refuting test is not run.

Who found it, and when #

The exponential dependence of a tunnelling rate on the action is Gamow’s, from 1928, and the WKB treatment of a symmetric double well is standard. The inversion spectra of ammonia and ND₃ are measured quantities of long standing.

The sweep, the exponent, the four barriers and the isotope check are new arithmetic on a well built from ammonia’s own geometry. The observation worth carrying is not any of the numbers but the asymmetry: one derivative, two directions, and the literature’s convention pointing the wrong way down it.

Still open: which reduced mass #

The obvious continuation is the molecule that does not invert. Phosphine’s barrier is quoted near 12,300 wavenumbers, six times ammonia’s, and the account in circulation is that phosphorus is heavier so it tunnels less. Three things about phosphine differ from ammonia and the mass is one of them; the pyramid is twice as tall and the barrier six times higher. Each can be substituted into ammonia’s well on its own, which is the only way to find out what each is worth, and the exponent above says in advance that the answers will not be close to each other.

The nearer question is the reduced mass, which every calculation so far has taken in the usual way and none has defended. It enters the action under a square root, so a twenty per cent error in it moves an exponent of six by ten per cent and the answer by a factor of nearly two — which is the size of the whole discrepancy in ammonia’s splitting. There is more than one defensible construction of it, and at least one of them is not a constant at all.