The steps were a square root of a mass

There are four molecules in this argument and eight measured numbers, and until now it has used four of them.

Putting the two partly deuterated molecules between the pure ones needed a term neither symmetric molecule has, found it, and produced a prediction: the logarithm of the splitting does not fall in equal steps as hydrogens are replaced one at a time. The steps come out 1.164, 0.951 and 0.801 — the first half again the last. That essay’s closing paragraph named the obvious follow-up, which is that each of these molecules has a second measured inversion interval, and a second step pattern read off the same well would say whether the concavity belongs to the potential’s shape or to the tunnelling arithmetic.

Neither, as it turns out, in the sense that question meant. The concavity is in the variable.

The well has to be fitted to two lines, not one #

Before the second pattern can be read, the well has to be able to hold two lines at once.

A quartic with a stated barrier has one shape parameter, so fitting it to a molecule’s ground doublet uses the parameter up and whatever it then says about that molecule’s excited doublet is a prediction — a prediction that comes out nine per cent high. Read a second step pattern off that and it is measuring the shape error as much as the mass. That is the trap the sweep of the kinetic operator’s orderings had to avoid one level down, and for the same reason: a quantity read off a fit is a statement about the fit until the fit has been shown to be tight.

So the well used here is the two-parameter Gaussian-hump form — the shape Manning wrote down and Swalen and Ibers refitted, and the one the published barrier of 2020 wavenumbers comes out of — with both of its shape parameters set against both of NH₃’s measured lines, 0.7935 and 35.81 wavenumbers, and its minimum pinned at the measured geometry. It reproduces both to a part in a thousand, which is the refusal this figure runs: a series read off a well that has not reproduced what it was fitted to is measuring its own failure. The barrier that comes out is 2098 wavenumbers, and then nothing further is fitted — the other three molecules get the same well with only their masses replaced.

The excited line’s steps are 0.969, 0.828 and 0.716. The spread between largest and smallest is 1.35, against the ground line’s 1.45. So the second pattern exists, it is concave the same way, and it is slightly flatter.

And the step-by-step ratio of the two is nearly constant: 0.832, 0.871, 0.894. Three numbers within four per cent of each other where the steps they are ratios of vary by forty per cent. That is not two patterns that happen to look alike. It is one pattern and a scale factor, and a scale factor has a cause that neither pattern’s own shape can name.

The variable in which both are straight #

A tunnelling splitting falls as the exponential of the action, and the action is an integral of the square root of twice the mass times the barrier’s height above the state. The mass appears under a square root, so the logarithm of a splitting is linear in $\sqrt{\mu}$ and not in μ — and it is certainly not linear in a count of substituted atoms, because replacing one hydrogen with deuterium in a three-ligand molecule does not change the reduced mass by a third.

Plotted that way both series straighten. The worst departure from a straight line is 0.0126 in the logarithm for the ground doublet and 0.0084 for the excited one — about one per cent in the splitting.

Plotted against the substitution count the same fits leave 0.100 and 0.0676. The residual is eight times smaller in the right variable, on both lines.

So the unequal steps are not a finding about ammonia’s potential. They are the concavity of a square root: the reduced mass rises from 5446 to 9519 electron masses across the series in steps that shrink, its square root rises from 73.8 to 97.6 in steps that shrink further, and the logarithm of the splitting follows that curve to one per cent. Nothing in the shape of the well enters, which is why the same curvature survives a change of well and survives holding the mass constant along the coordinate.

And the scale factor is now named. The two fitted slopes are −0.1228 and −0.1058 per unit $\sqrt{\mu}$, whose ratio is 0.8614 — the number the three step ratios are within four per cent of. The excited state sits higher in the well, its turning points are further apart and the barrier it tunnels through is thinner, so it is less sensitive to the mass by exactly that factor. One number describes the whole second pattern, and it is a property of where the state sits rather than of the molecules.

What one per cent is, and what makes it interesting #

The residual that survives is small and it is not nothing, and there is a candidate for it that the same four solves can test.

One per cent in the logarithm is the amount by which the action’s linearity in $\sqrt{\mu}$ fails, and it has to fail somewhere, because the mass here is a function of the coordinate rather than a constant. The bond-conserving construction has the ligands sliding outward as the apex descends, so the mass rises from its planar value by a fifth across the range the molecule occupies, and the action is an integral of a position-dependent $\sqrt{\mu}$ rather than $\sqrt{\mu}$ times an integral. Scaling every nuclear mass at once scales that function everywhere and the integral does follow a square root; substituting deuterium for one ligand at a time does not, because it changes the two terms of the construction by different amounts.

So the prediction is that a constant mass should straighten the series further, and it does — dramatically. Run with the planar reduced mass held fixed along the coordinate, the ground line’s worst residual against $\sqrt{\mu}$ falls to 0.0014, a factor of nine better than the position-dependent case and seventy-six times better than the same data against the substitution count. The excited line improves less, to a factor of five, which is consistent: a state higher in the well spends more of its time where the mass function is steepest.

And the residual reverses sign between the two lines, which is the detail that says it is structure rather than noise. The ground line’s four residuals run positive, negative, negative, positive — convex against $\sqrt{\mu}$. The excited line’s run negative, positive, positive, negative — concave, by about two thirds as much. A numerical error would not know which doublet it was in. A curvature that comes from where a state sits relative to a varying mass would, and that is what is left after the variable has been chosen correctly.

That makes the one per cent the interesting quantity rather than the error bar: it is a measurement, on published intervals, of how much the position dependence of the umbrella mass matters.

The line nothing was fitted to #

The second use of the second line is a test, and it is the only test in this argument that the missing correction cannot pass by construction.

The correction is a barrier shift, isotope-dependent because the frozen transverse modes have different zero-point energies in different isotopologues, and its size is set entirely by the two ground doublets: the barrier fitted to NH₃’s is 2116.9 and the one fitted to ND₃’s is 2019.9, so the shift is ninety-seven wavenumbers. Every number in the three essays that sized it came from those two measurements.

With one barrier for both molecules the ground isotope ratio comes out 19.00 against a measured 14.94 — twenty-seven per cent high, which is the defect the correction exists to remove. With the shift it is 14.94, exactly, because that is what the shift was fitted to.

The excited isotope ratio was never touched. With one barrier it is 12.44 against a measured 10.69, sixteen per cent high. With the shift it is 10.13 — five per cent low.

That is the strongest single piece of evidence this argument has produced for the mechanism, and it is also a bound on it. A shift fitted entirely to one pair of measurements improves an untouched quantity by a factor of three, which no arbitrary parameter does; and it passes through the right answer rather than arriving at it, which says the shift is not the whole of what the model is missing.

Where the over-shoot comes from, and it is not the mass #

The over-shoot has a diagnosis, and it is available from the same four solves.

Fit a quartic separately to each molecule’s own ground line and ask what each says about its own excited line. NH₃ gives 39.02 against the measured 35.81, nine per cent too large. ND₃ gives 3.852 against 3.35, fifteen per cent too large.

Both are too large, and that settles what the residual is. An isotope-dependent term raises one molecule’s effective barrier more than the other’s, so it can only move the two errors in opposite directions or move one much more than the other. It cannot make both too large by nine and fifteen per cent. What can is a well whose barrier is the wrong shape at the energy the excited state sits at, and every well in this argument is either a quartic or a Gaussian hump — two-parameter guesses at a surface with no reason to be either.

So the excited line separates the model’s error into two parts that the ground line alone could not. There is an isotope-dependent part, worth ninety-seven wavenumbers of barrier and confirmed threefold on a quantity nothing fitted, and an isotope-independent part, worth nine to fifteen per cent on the excited doublet and belonging to the well’s shape. The first is the transverse zero point. The second is not, and no amount of the first will fix it.

That division is worth more than either half. Every essay of this argument that swept a parameter was asking the same question in a less answerable form: is the seventy per cent over-prediction in the mass, in the ordering, in the well, in the reduction. Sweeps bound a contribution and cannot separate one, because moving a parameter moves everything the parameter touches. A second measured line on each of two molecules separates them by their isotope signature instead — one part changes between the molecules and the other does not — and that is a distinction no sweep of one molecule’s parameters can draw.

The two doublets of one molecule #

There is one more reading available, and it is the cleanest quantity in the essay because it cancels the most.

The ratio of a molecule’s two doublets is a ratio inside one molecule, so the barrier, the geometry and most of the well’s shape error cancel out of it. Measured, it is 45.1 for NH₃ and 63.1 for ND₃. Computed from the two-parameter well it is 45.1 for NH₃ — by construction, since both of its lines were fitted — and 67.6 for ND₃, seven per cent high.

The four-molecule series runs 45.1, 54.9, 62.1, 67.6, rising steadily. The mechanism is that a heavier molecule sits lower in its well relative to the barrier, so its ground state has a wider barrier to get through while its excited state does not, and the two doublets separate. That makes the ratio a reading of how far up the barrier the ground state sits — and the two mixed values, 54.9 and 62.1, are predictions against measurements that exist and are not quoted here.

The steps in that series are 9.7, 7.2 and 5.5, which is the same concavity again and for the same reason: the ratio is a difference of two logarithms, both linear in $\sqrt{\mu}$ with different slopes, so the ratio’s logarithm is linear in $\sqrt{\mu}$ with the difference of the slopes. Everything in this essay is one straight line in one variable, seen from four angles — which is what it means for a finding to be about a variable rather than about a molecule. The seven per cent by which the ND₃ ratio comes out high is then the same shape error as the nine and fifteen per cent above, measured in the one quantity where the barrier has cancelled, and it is smaller than either of them because the cancellation is partial.

What the four solves assume #

The masses are the bond-conserving construction’s. Three constructions are defensible and they differ by forty-three per cent in the splitting, so every absolute number here carries that. The straightness in $\sqrt{\mu}$ does not: run with a constant mass the series is straighter still, the factor between the two variables goes from eight to seventy-six on the ground line, and the slope ratio is 0.853 against 0.861. That is what makes the finding about the variable rather than about the construction.

The two mixed molecules’ splittings are not quoted. Both are measured. Nothing above compares against them, and the reason is the one this argument has used since the mixed masses arrived: a prediction is worth more than a comparison against a number chosen to be compared against. The step pattern, the straightness and the slope ratio are all statements about the four together.

The excited doublet is the first excited one and the well holds no more. Above the second doublet the states are near the top of the barrier and pairing stops meaning anything, so there is no third pattern to read. The number of states below the barrier is reported by the solver and is four for every molecule here — which is also why the ground state’s position in the well matters as much as it does: it sits a quarter of the way up the barrier rather than at the bottom, and the excited one sits most of the way up.

And the two-parameter well is still a two-parameter well. Fitting both of NH₃’s lines removes the shape error for NH₃, at NH₃’s two energies, and nothing guarantees it removes it at ND₃’s two energies — which sit at different fractions of the barrier because the mass is different. That is very likely part of the seven per cent in the doublet ratio, and it is not separable from the rest without a third measured line.

A series is straight in something #

The habit: when a series against an integer is not straight, the integer is usually not the variable.

Four isotopologues indexed by how many hydrogens have been replaced is the natural way to lay them out, and it is the way that makes the series curve. The curve then invites an explanation in terms of the physics — the potential’s shape, the tunnelling arithmetic, the asymmetry of the mixed molecules — and each of those is a real effect that is worth about a tenth of what the curve shows. The variable the exponent is actually linear in was available from the formula the whole argument is built on, and putting it on the axis left a residual small enough to be interesting in its own right.

The corollary is about what a straightening buys, which is more than tidiness. Straight in $\sqrt{\mu}$, the two lines differ by one number, and that number has a physical reading — how far up the barrier the state sits. Bent against a count, the same two lines are six numbers with no relation visible between them, and the relation is the finding.

And it buys a residual worth looking at. A one per cent departure from a straight line is a measurement; a forty per cent variation in the step sizes is a plot of the wrong variable. The first can be attributed — here, to the position dependence of the mass, by the control that holds the mass constant and watches the residual fall by an order of magnitude — and the second can only be described. The same exchange happened one level down when a factor of two in the required stiffening turned into one and a third per cent in the barrier: the useful move was not making the sweep narrower but finding the quantity the sweep did not reach.

Who measured the lines, and when #

The four inversion intervals used here — NH₃’s ground and first excited doublets and ND₃’s — are microwave and far-infrared measurements of long standing, and the ground-state interval of NH₃ is among the most accurately measured intervals in molecular spectroscopy. The Gaussian-hump well is Manning’s, 1935, in the form Swalen and Ibers refitted in 1962. That a tunnelling splitting is exponential in $\sqrt{\mu}$ is as old as the semiclassical treatment of a double well.

What is computed here is the second step pattern, the straightness of both patterns in $\sqrt{\mu}$ against their curvature in the substitution count, the slope ratio that relates them, and the excited doublet used as a free test of a correction fitted to the ground one.

The number worth carrying is eight — the factor by which the residual falls when the series is drawn against the square root of a mass rather than against a count of atoms.

Still open: the third line, and the shape the over-shoot wants #

The obvious open question is what shape of well would close the nine and fifteen per cent. The residual is isotope-independent, which means it is a statement about the potential and not about the reduction, and it has a direction: both excited doublets come out too large, so the barrier is too narrow at the excited state’s energy relative to the ground state’s. A well with a third shape parameter could be fitted to three of the four measured lines and asked to predict the fourth, which is the only way to test a shape rather than accommodate it — and the three-parameter family has a member, the Gaussian hump with its harmonic force constant released, that this argument has so far kept pinned to the measured geometry.

The nearer question is the doublet ratio of the mixed molecules. Both are measured and neither is quoted above, and the prediction is 54.9 and 62.1 against the pure molecules’ 45.1 and 63.1 — a series that rises steeply at first and flattens. A ratio inside one molecule cancels the barrier and most of the shape error, so it is the one quantity in this argument where a comparison against a mixed isotopologue would be testing the mass construction and nothing else. That is two numbers, already published, and the reason to be careful about them is that the prediction for ND₃ is already seven per cent high.