The two that are not on the line

The isotope was run as a test and it could not choose. Two constructions of the umbrella coordinate’s reduced mass, two wells, five orderings of the kinetic operator — and every combination wanted a barrier several per cent lower for ND₃ than for NH₃, every prediction of the isotope ratio overshot, and the two constructions differed from each other by less than either missed by.

Two molecules is two points, and the thing that separates the constructions is a curve. NH₂D and NHD₂ are the two points in between.

The term the symmetric molecules cannot produce #

The bond-conserving construction has two pieces. The apex moves against a rigid ligand triangle, which gives an ordinary reduced mass with the ligand masses summed; and the ligands slide outward along fixed bonds as the apex descends, which gives a position-dependent term with the masses summed again.

Both generalise to unequal ligands by replacing three times one mass with a sum of three. That is the naive extension and it is what anybody would write.

It leaves something out. The three ligands sit at a hundred and twenty degrees on a circle whose radius grows as the apex comes down, so their horizontal centre of mass moves outward along a fixed direction — unless the three masses are equal, in which case the three displacements cancel by symmetry. A frame that does not translate has to subtract that motion’s kinetic energy, and what is left is smaller than the naive sum.

The amount is exact and it has a pleasant form. The squared length of the mass-weighted sum of three unit vectors at a hundred and twenty degrees is half the sum of every squared pairwise mass difference — zero when the three agree, and for a molecule with one deuterium it is half of twice the squared hydrogen-deuterium difference, which is the squared difference itself.

So NH₂D and NHD₂ carry identical values of it. Each has exactly one unlike pair, counted twice: two H–D pairs and one like pair in the first, two H–D pairs and one like pair in the second. The numerator is the same number and only the total mass in the denominator differs, so the fractional correction is 1.39 per cent for NH₂D and 1.06 for NHD₂.

And the mass is still a single function of one coordinate, which is the half of that essay’s worry that turns out to be misplaced. Its closing paragraph supposed that with unequal ligands the bond-conserving mass “is no longer a single function of one coordinate but depends on which ligands are heavy”. It does depend on which ligands are heavy — through the term above, which is the whole point. It is not a function of more than one variable, because the umbrella motion still moves all three ligands outward together.

What one fitted barrier predicts #

A potential energy surface does not depend on the nuclear masses, so a barrier fitted to one isotopologue is the barrier for all four. Fitting once to NH₃’s measured ground-state splitting gives 2116.9 wavenumbers under the bond-conserving construction and 2329.9 under the rigid-triangle one, and everything after that is arithmetic with masses.

The bond-conserving construction gives 0.7935 by construction, then 0.2450, 0.0938 and 0.0418 wavenumbers.

The last of those is the one with a measurement beside it, and it is a fifth low: ND₃’s splitting is 0.0531. That is the overshoot reported there, unchanged, and it is worth stating first because the two middle predictions inherit it. Nothing here repairs the model; what it does is say what shape the model’s answer has.

Neither of the mixed splittings is quoted here. Both are measured. Quoting them and then comparing would turn a prediction into a fit with extra steps, and the more useful object is the prediction on its own, with the comparison left to somebody holding the measurement.

The steps are not equal #

Its closing question was whether the overshoot grows in equal steps with the number of deuterium atoms, on the reasoning that a step pattern that is not equal would say the error is in the path rather than in the potential.

Under the model, the splitting does not step equally, before any error is considered. The three steps in the logarithm are 1.175, 0.960 and 0.809, against an even share of 0.982. The first substitution costs about a fifth more than its share and the third about a fifth less, and the three span a factor of 1.45.

The reason is arithmetic and it is not about ammonia. A tunnelling splitting falls as the exponential of an action, the action carries one factor of the square root of the mass, and the reduced mass grows linearly with the number of substitutions. A square root of a linear function is concave, so the logarithm of the splitting is concave in the substitution count, necessarily and for every molecule of this shape.

Plotted against the square root of the mass the four points are very nearly straight, which is the statement above made visible. They are not exactly straight, and the residual curvature is where the model’s own content sits — the action is not exactly proportional to the square root of the mass, because the turning points move as the zero-point energy moves.

So the answer to that question has two parts, and separating them is most of the value. The steps are unequal for a reason that is arithmetic, and the departure from the square-root law is the part that is about the potential. A measurement compared against equal steps would find them unequal and learn nothing; compared against the square-root line it would be testing something.

Where the correction comes from, in mechanics #

The subtraction is worth doing slowly once, because it is the only new physics here and it is three lines.

Put the three ligands on a circle of radius ρ in a plane, at a hundred and twenty degrees, and the apex a height $x$ above it with $\rho^2 + x^2$ fixed at the squared bond length. As $x$ falls, $\rho$ grows, and each ligand moves outward along its own radius at speed $\dot\rho = -x\dot x/\rho$. Their radial kinetic energies sum to half the ligand masses summed, times $x^2\dot x^2$ over the squared bond length minus $x^2$ — which is the naive term.

But the ligands’ own horizontal centre of mass has moved. It sits at ρ times the mass-weighted sum of the three unit vectors, divided by the total mass, and that sum is zero only when the three masses agree. So the whole molecule has acquired a sideways drift proportional to ρ̇, and a coordinate system that does not translate must remove it. Removing it costs half the total mass times the squared drift speed, which is half the squared length of the mass-weighted vector sum, times ρ̇², over the total mass.

Both terms carry the same ρ̇², so the subtraction lands entirely in the radial coefficient. The position dependence is untouched and only the number in front of it changes — from the sum of the ligand masses to that sum less half the pairwise squared differences over the total mass. That is why the correction is a single number per molecule rather than a new function, and it is why the mass stays one-dimensional.

The sign is also fixed and is worth noticing. The correction is always a subtraction, because a squared length is non-negative: an asymmetric molecule is always lighter along this coordinate than the naive sum says, and so always tunnels more. Its splitting is therefore always above the naive prediction, on both mixed molecules, which is a direction a measurement could check without resolving the size.

What the new term is worth #

Dropping the horizontal centre-of-mass correction changes the two symmetric predictions by exactly nothing, which is the check that it is the asymmetry being measured and not some general feature of the arithmetic. On the two mixed molecules it changes the splitting by 0.507 and 0.423 per cent.

Half a per cent sounds negligible and is not, in this argument specifically. The whole difficulty there was that two entirely different constructions of the mass — one holding a ligand triangle rigid, one conserving bond lengths — differed from each other by less than either missed the measurement by, so the isotope could not choose between them. This term is of that order and it exists only for the molecules that test did not have. It is not large; it is large compared with the thing that could not be resolved.

There is a second reason the size is the right size to care about, and it is about what a measurement would be resolving. The two mixed molecules’ predictions differ from each other by a factor of 2.6, which any model with roughly the right masses will get roughly right. What they do not obviously get right is the half per cent — so a measurement precise enough to test this term is a measurement testing the coordinate rather than the masses, which is the distinction four essays of this argument have been trying to make and have had no instrument for.

It is also the only quantity in this model that a mixed isotopologue possesses and an interpolation between the pure ones cannot produce. Everything else about NH₂D — its total mass, its reduced mass, its radial coefficient before correction — is a linear interpolation between NH₃ and ND₃.

And the concavity is not the new term’s doing, which is the control that had to be run. A correction that exists only for the two middle molecules is exactly the kind of thing that could manufacture a departure from a straight line, so the series is worth recomputing with the term dropped and the naive mass used throughout. Dropped, the four splittings are 0.79350, 0.24381, 0.09339 and 0.04176 per centimetre against 0.79350, 0.24504, 0.09379 and 0.04176 with it. The logarithmic steps run 1.1801, 0.9596 and 0.8049 rather than 1.1750, 0.9604 and 0.8091, and their spread is 1.466 rather than 1.452.

So the steps are unequal by half again either way, and the correction makes them very slightly more equal rather than less. The unequal steps belong to the tunnelling arithmetic and the mass construction, not to the term this essay adds — which is the right result for the finding to have, because it means the shape a measurement would be tested against does not depend on believing the new term. What the term changes is the value at the two middle points, by half a per cent, and that is a separate claim with a separate test. The correction is quadratic in the mass differences and is therefore invisible from the ends.

What was computed, and how #

Every structure is ammonia’s, since an equilibrium geometry is a property of the surface and not of the masses — the isotope test already uses ND₃ that way. The pyramid height comes from the bond length and angle by the same construction, the well is the two-parameter quartic, and the box is the bond-conserving construction’s own domain so that the two constructions are compared with nothing else changed.

Eleven things are checked. That the asymmetric term is exactly zero for both symmetric molecules and positive for both mixed ones. That it is the same number for the two mixed ones, to twelve digits, since each has one unlike pair counted twice. That the generalised mass reproduces the earlier symmetric construction exactly at three separate positions — a generalisation that changed the symmetric answer would be a different model rather than an extension of one. That dropping the term changes nothing on the symmetric pair and more than two parts in a thousand on the mixed pair. That the predicted splitting falls with every substitution. And that the steps are unequal by more than a tenth under both constructions of the mass, so the concavity is not one construction’s artefact.

Where this stops #

The umbrella coordinate is not a normal mode of a mixed molecule, and that is the largest thing being assumed. In NH₃ and ND₃ the symmetric umbrella motion belongs to its own symmetry species and cannot mix with the other vibrations at first order. NH₂D has no threefold axis, so the motion this model treats as one coordinate is a combination of what the molecule would call several — and the reduction that produces a one-dimensional problem is weaker for the two middle molecules than for the ends. The correction computed here is a correction to the mass of a coordinate whose separability is itself worse.

Which cuts both ways, and the direction is worth stating. If the mixed molecules were found to depart from the prediction by much more than the endpoints do, mixing would be the first explanation and it would not be evidence about the mass at all. If they depart by about the same amount, the departure is a property of the path rather than of the symmetry, which is what the isotope test wanted to know.

And the barrier is fitted to one line. A well fitted to NH₃’s ground-state splitting alone is a one-parameter fit to one number, and a two-parameter well fitted to both of ammonia’s lines was found to behave differently. Running the whole series on the shape-fitted well is the same calculation with one more input and would say whether the step pattern is a property of the potential’s shape.

The generalisation #

The habit is to ask what a two-point comparison cannot see, and then to find the cheapest third point.

Two molecules give one ratio, and a ratio can be matched by any number of mechanisms with one parameter each. That test had two constructions differing by less than either’s error and no way to separate them, which is the standard shape of an underdetermined comparison — and the standard repair is not a better calculation but another point. What makes the mixed isotopologues the right third point is that they carry a term the two ends do not: they are not merely interpolations, so a model that gets them right is doing something the endpoints could not have arranged.

The corollary is about what a series is plotted against. The step pattern here is concave in the substitution count for a reason that has nothing to do with the molecule, and reading it as a finding would have been available and wrong. Finding the variable in which the trivial part of the behaviour is straight — here the square root of the mass — is what turns a series into a test, because it puts the model’s own content into the residual rather than into the trend.

Who found it, and when #

The double-well treatment of ammonia’s inversion is Dennison and Uhlenbeck, 1932. The bond-conserving reduced mass is the earlier essay’s own construction. The mass-weighted centre-of-mass subtraction is elementary mechanics and is what any correct kinetic energy for a non-symmetric molecule contains. What is computed here is the asymmetric term for three ligands at a hundred and twenty degrees, the four predictions from one fitted barrier, and the step pattern they produce.

The number worth carrying is not 0.2450. It is that the two molecules in the middle of a four-point series carry an identical value of the one quantity that distinguishes them from the line between the ends.

Still open: the two lines, and the mixing #

The obvious open question is the second line. Each of the four molecules has a first excited inversion doublet as well as a ground-state one, and a well fitted to both of ammonia’s lines is a two-parameter fit rather than a one-parameter one — which was found to behave differently from the quartic under isotope substitution, to the point of reversing which construction is nearer. Running the whole four-point series on that well would produce a second step pattern, and whether the two patterns agree would say whether the concavity is a property of the potential’s shape or of the tunnelling arithmetic alone. It is the same calculation with one more measured number in it.

The nearer question is the mixing this model assumes away. NH₂D has no threefold axis, so its umbrella motion is not protected by symmetry from the two asymmetric bends, and the size of that contamination is a question a force field answers and a symmetry argument cannot. There is a cheap partial answer: the rotational constants of the mixed molecules are measured, they depend on the same geometry, and an inertial defect is a standard measure of how far a molecule departs from the rigid picture. Whether the two mixed isotopologues’ inertial defects are larger than the pure ones’ by more than the mass change accounts for would bound the contamination without any force field at all — and inertial defects are already computed here for other molecules.