Deuterium cannot tell the masses apart

A one-dimensional model of ammonia’s inversion needs a reduced mass, and the mass nobody chose found that the construction everybody uses is not the one a rigid-bond picture implies. Holding the three N–H bonds at their measured length makes the mass a function of position, and in one well at one barrier it moved the ground splitting forty-three per cent towards the measurement. Sweeping the orderings of the kinetic operator then showed that the choice of ordering could not have produced that, since the five in use span under half a per cent.

The mass comparison ended by proposing the test that should settle whether the improvement means anything. The bond-conserving mass is the usual reduced mass plus a term proportional to the ligand mass alone, so deuterating the hydrogens changes its two parts by different factors, and the prediction for ND₃ differs from any constant-mass prediction in a way that can be computed. ND₃’s splitting is measured. The prediction can be checked rather than admired.

It has been checked, four ways. The isotope cannot choose between the two masses, and the reason it cannot is more useful than a verdict would have been.

What deuteration does to the mass #

The bond-conserving mass is μ(x) = μ_plane + 3m_L x²/(r² − x²). The first term is the familiar reduced mass of the apex against a rigid plane, and deuteration raises it from 2.4866 to 4.2210 unified mass units, by seventy per cent. The second term is three times the ligand mass times a function of geometry, and deuteration doubles it. Because the second term grows faster, it is a larger share of ND₃’s mass than of NH₃’s.

At the pyramid, NH₃’s mass is 20.1 per cent above its planar value and ND₃’s is 23.7 per cent above. That is the prediction no constant mass contains: a constant mass rescales by a fixed factor on deuteration, while this one changes shape. If the bond-conserving construction is right and the usual one wrong, the difference should show up in how the splitting responds to deuteration, and that response is the most accurately measured isotope effect in molecular spectroscopy — 0.79350 wavenumbers for NH₃ against 0.05310 for ND₃, a ratio of 14.94.

The difference in shape is not large, and it is worth saying before the tests how large an effect it could produce. The three and a half points between 20.1 and 23.7 per cent act on the region the tunnelling integral samples, which the mass comparison found weights the mass towards its light end. So the construction’s distinctive isotope prediction is a correction of a few per cent to a quantity the model has to get right to many tens.

One potential should need one barrier #

The cleanest form of the test does not predict anything. A potential energy surface does not depend on the masses of the nuclei — one curve, two isotopes and two average bond lengths is the ordinary consequence — so a model of NH₃ and a model of ND₃ must share their barrier. Fitting the barrier to each molecule’s measured ground splitting separately, under a stated mass and a stated well, gives two numbers that ought to be one. How far apart they fall is a test that a fit to either molecule alone can never make, because each fit can always be satisfied.

In the quartic well the usual mass needs 2330 wavenumbers for NH₃ and 2220 for ND₃, so ND₃’s is 4.70 per cent lower. The bond-conserving mass needs 2117 and 2020, 4.58 per cent lower. The construction that moved each barrier by two hundred wavenumbers moved the gap between them by an eighth of a point.

In a well whose shape is fitted rather than assumed — a parabola with a Gaussian hump, whose two shape parameters take both of each molecule’s measured splittings, the well that first separated a barrier from what a splitting measures — the usual mass needs 2262 and 2185, 3.42 per cent apart, and the bond-conserving mass 2098 and 2009, 4.25 per cent apart. Changing the shape of the well moved the gap by more than a point. Changing the mass in that better well moved it by eight tenths of a point in the wrong direction.

So on the one test that involves no prediction at all, the bond-conserving mass is indistinguishable from the usual one in the quartic and worse than it in the fitted well. Neither comes near a single barrier.

The ratio, predicted six ways #

The more familiar form of the test predicts ND₃ from NH₃. Fit the well to NH₃, replace the mass, and compare the splitting that comes out with ND₃’s measured one, or equivalently compare the predicted ratio with 14.94.

In the quartic fitted to NH₃’s ground splitting, the usual mass predicts a ratio of 19.26 and the bond-conserving mass 19.00: ND₃’s splitting comes out at 0.04121 and 0.04176 against 0.05310. In the well shaped to both of NH₃’s lines, the ratios are 17.63 and 18.49. Every one of those overshoots the measurement, by between eighteen and twenty-nine per cent. The bond-conserving mass is nearer in the quartic, by a quarter of a unit, and further in the fitted well, by nine tenths.

That is the answer to the question as posed. The two constructions differ from each other by less than either differs from the measurement, and which of them is nearer depends on the shape of the well. A test in which the ranking reverses when an ingredient the test is not about is changed has no power to rank. It is the same flaw as a test that cannot come out the other way, met from the other side: that test’s outcome was fixed by its construction, and this one’s by an ingredient it was not meant to depend on.

The ratio also says something about why. All six predictions overshoot, which means the model makes the heavier molecule tunnel too slowly relative to the lighter one whatever mass it is given. A one-dimensional splitting is an exponential in an action that grows as the square root of the mass, so an isotope effect that is too strong is a model whose action is too large or whose barrier is too wide — properties of the potential and of the path, which the mass construction does not touch.

A 0.04 per cent agreement that NH₃ refutes #

One comparison looks much better than all the others, and it deserves a section because it is the kind of result that gets quoted.

At the published barrier, with nothing fitted to either molecule, the bond-conserving mass gives ND₃ a splitting of 0.05308 wavenumbers against a measured 0.05310 — agreement to four hundredths of a per cent. Taken alone that is a striking confirmation of a construction with no free parameters.

It is not a confirmation. In the same well, at the same barrier, with the same construction, NH₃ comes out at 0.94420 against 0.79350, nineteen per cent high. The usual mass is seventy and sixty-one per cent high for the two molecules, so it misses both badly and by similar amounts, which is at least consistent. The bond-conserving mass misses one by nineteen per cent and the other by nothing, and a model that is wrong for NH₃ by nineteen per cent cannot be right for ND₃ by construction; it is right for ND₃ because ND₃’s number happens to fall where the error in the barrier and the error in the ratio cancel.

This is the ordinary hazard of checking a model against one member of a family. A spectrum that changes when only a mass does is the reason isotopologues are measured at all: a second mass on the same potential is a second equation, and a model that satisfies one equation by accident is exposed by the other. Here the second equation was measured decades ago, and the four-figure agreement survives only if NH₃ is not looked at.

Two lines carried across #

The well shaped to NH₃’s two measured lines makes the strongest prediction the model can make, since both of its shape parameters are spent on one molecule and the other molecule’s two lines are then predictions.

Under the usual mass it predicts 0.0450 and 3.10 wavenumbers for ND₃’s ground and first excited splittings; with the bonds held fixed, 0.0429 and 2.90. ND₃ measures 0.0531 and 3.35. Both constructions are low on both lines, by fifteen and eight per cent under the usual mass and by nineteen and thirteen with the bonds fixed. The measured excited ratio is 10.69 and the two predictions are 11.57 and 12.34.

The pattern is the same as in the ground ratio, and it is consistent across the two lines, which is what makes it a statement about the model rather than about one measurement. Deuteration suppresses tunnelling too much in every version of this one-dimensional description. The bond-conserving construction suppresses it slightly more, because its extra term grows with the ligand mass and puts more mass exactly where the heavier molecule already has more.

Nothing in the kinetic energy moves the gap #

Having found the mass construction powerless on the isotope test, the natural remaining suspect in the kinetic energy is the ordering, which was sized on NH₃ alone.

Under the five orderings in use, with the bonds held fixed, ND₃’s fitted barrier is lower than NH₃’s by 4.58, 4.61, 4.64, 4.60 and 4.60 per cent. The orderings move each barrier by a few wavenumbers and move the gap by six hundredths of a point. The mass construction moved it by an eighth of a point in the quartic and eight tenths in the fitted well; the shape of the well moved it by more than a point.

So the kinetic energy of this model — the mass and its ordering, together — has been swept as fully as it can be, and the gap between the isotopologues does not close under any of it. Whatever makes the two molecules need different barriers is in the potential or in what the reduction to one coordinate leaves out.

Where the gap comes from, and why it should exist #

That conclusion has a physical reading, and it is the reason the gap is less embarrassing than it first looks.

A potential energy surface is mass-independent. The potential in a one-dimensional model is not a surface. It is the energy along one path with every other coordinate relaxed and silently carried along, and in any honest reduction those other coordinates bring their own zero-point energy with them. Ammonia has five other vibrations, and their frequencies change as the molecule flattens; their zero-point energy is therefore a function of the inversion coordinate, it adds to the effective barrier, and it scales with the masses of the hydrogens. The atoms are not at the points is the reminder that this energy is not small: it is thousands of wavenumbers in total, and a few per cent of it varying along the path would be tens of wavenumbers.

So a one-dimensional model should need a different barrier for each isotopologue, and the difference it needs is a measurement of how the other vibrations’ zero-point energy changes between the pyramid and the plane. The sign found here — ND₃ needs the lower barrier — is what that account gives if the other vibrations stiffen as the molecule flattens, since the lighter molecule would then carry more extra energy over the top. Whether they do, and by how much, is a calculation over a force field this model does not have, and it is stated here as the reading the evidence points at rather than as a result.

What that reading changes is the status of the test. The mass comparison proposed deuteration as a test of the kinetic energy. It is dominated instead by a mass dependence in the potential that every one-dimensional treatment absorbs and none states — a gap of three and a half to five per cent in the barrier, against changes of an eighth of a point and eight tenths of a point from the two masses it was meant to separate.

How the comparison was run #

Both isotopologues use the same geometry — the pyramid height from NH₃’s measured bond length and angle, since an equilibrium structure does not depend on the nuclear masses — and every solve for either molecule, under either mass, is made in the same box, cut at 0.995 of the bond length where the bond-conserving coordinate ends. That is the choice the mass comparison measured as costing under a part in a thousand, and making it for both molecules means no difference between them comes from the grid.

The quartic’s barrier is found by bisection on the logarithm of the barrier until the computed ground splitting matches the measured one, with the bracket checked before the search so that a target outside it is refused rather than pinned to an endpoint. The shaped well’s two parameters are found by a damped Newton iteration on the logarithms of both parameters and both residuals, and it converges for both molecules under both masses. The measured splittings are 0.79350 and 35.81 wavenumbers for NH₃ and 0.05310 and 3.35 for ND₃, from their microwave inversion spectra for the ground doublets and their umbrella-band spectra for the excited ones.

The checks are these. The shaped well reproduces both lines of each isotopologue under both masses. Every construction, well and ordering needs an ND₃ barrier at least three per cent below NH₃’s. Every prediction of the isotope ratio from a fit to NH₃ overshoots the measurement by more than fifteen per cent. The two masses differ from each other by less than the smaller miss, and the sign of that difference is opposite in the two wells. And the refusal: at the published barrier the bond-conserving mass must land on ND₃ to within a per cent while missing NH₃ by more than fifteen, which is what makes the single-molecule agreement a demonstrated coincidence rather than a suspected one.

What two isotopologues cannot settle #

Two molecules is one isotope effect. NH₂D and NHD₂ would add two more ratios on the same potential, with symmetric-top rotational structure complicating the inversion spectra, and a mass construction that depends on the ligand mass would predict their splittings as a function of the number of deuterium atoms. Whether the overshoot grows linearly with that number would separate a mass effect from a zero-point effect more sharply than the two end members can.

The two wells are two shapes. A quartic and a Gaussian-on-a-parabola are the simplest one- and two-parameter wells, and the finding that shape moves the gap more than mass does is a finding about those two. A well fitted to four lines — both isotopologues’ ground and excited splittings at once — would have enough freedom to close the gap, and would then be a fit rather than a test.

And the zero-point reading is a reading. It explains the sign and the rough size of the gap and it is not computed. The mass comparison’s rigid bonds are part of the same omission, since a real inversion path lengthens the bonds as it flattens and the bond-length change is exactly the kind of coordinate whose zero-point energy varies along the path.

A test is only as good as what it holds fixed #

The transferable point is about what makes a test discriminate between two models, and it is not that the models make different predictions.

The two masses do make different predictions for ND₃, and the difference is computable and was computed. What the test lacked is that everything else in the prediction was held fixed while that difference was measured. The prediction also depends on the well’s shape and on a mass dependence hidden in the one-dimensional potential, and both of those move the answer by more than the difference between the models does. When that happens the models’ ranking changes with the unexamined ingredient, which is the precise sense in which a test has no power: not that it gives the wrong answer, but that its answer is set by something else.

The check that exposes it is cheap and it is the one run here. Before reading a test’s verdict, vary an ingredient the test is not about and see whether the verdict survives. Here changing from a quartic to a fitted well reversed which mass looked better, and the reversal is the finding. The same check is what a control that outranked the mechanism supplied for a correlation, and what a second isotopologue supplies for a force field: something varied that the argument did not mean to vary, and the argument either survives it or does not.

The second point is about agreement with a single number. The best match in this essay — 0.05308 against 0.05310 — is also its clearest coincidence. A model with no free parameters that lands on one measurement to four significant figures invites belief, and the defence against that is not scepticism about the number but a second measurement on the same potential. The isotope shift is arithmetic for exactly this reason: a mass changed on a fixed potential produces relations that a model has to satisfy all of, and satisfying one of them is cheap.

Who measured it #

The inversion spectra of NH₃ and ND₃ are among the best measured in molecular physics, from the microwave work that founded the subject onward, and the one-dimensional double-well treatment of them goes back to Dennison and Uhlenbeck in 1932 and Manning in 1935. That the effective inversion potential of an isotopologue absorbs the zero-point energy of the other modes is a standard point in the reaction-path treatment of large-amplitude motion.

The comparison of two mass constructions, two wells and five orderings against the isotope effect, and the finding that it cannot separate them, are computed here. The mass comparison deserves the credit for proposing a test that could be run and for saying in advance that it would be sharper than a constant mass allows, which is what made its failure to discriminate a result rather than a disappointment.

Still open: the zero-point energy along the path #

The obvious open question is the one the gap points at. Computing the other five vibrations’ frequencies at several points along the inversion path, for both isotopologues, and adding their zero-point energy to the one-dimensional potential would turn the gap between the two fitted barriers from a symptom into a prediction. If the vibrationally adiabatic barriers for NH₃ and ND₃ differ by the four per cent the fits need, the model’s isotope problem is solved without touching the mass; if they differ by much less, something else in the reduction is missing. It needs a force field for ammonia along a path, which this model does not yet have.

The nearer question is the partly deuterated molecules. NH₂D and NHD₂ are measured, they sit on the same potential, and they break the equivalence of the three ligands, so the bond-conserving mass is no longer a single function of one coordinate but depends on which ligands are heavy. Whether the overshoot in the isotope ratio grows in equal steps with the number of deuterium atoms is a cheaper test than the force field, and a step pattern that is not equal would say the error is in the path rather than in the potential.