The mass nobody chose

Every treatment of pyramidal inversion so far has solved the same one-dimensional problem: one particle of mass μ on a symmetric double well. A double well built from ammonia’s own geometry took μ in the usual way and noted that the choice was a choice. The sensitivity of the splitting to each input established that the splitting depends on it more steeply than on the barrier — exactly one unit more steeply, by an identity rather than by a fit. The case against mass as the reason the heavier hydrides tunnel slowly leaned on it twice while refuting an explanation built on it.

So the quantity is load-bearing, and nothing measures it. A structure determination reports a bond length and a bond angle — the two quantities the pyramid’s own height is computed from. A spectrum reports a splitting. Neither reports how much mass is being carried when a molecule turns itself inside out, because that is a property of the coordinate — of the path through configuration space somebody chose to call the inversion — rather than of the molecule.

Three constructions are defensible. They give 1.35075, 0.94420 and 0.0000055 wavenumbers.

What each one says #

The apex against a rigid ligand plane. The nitrogen moves one way, the triangle of hydrogens moves the other, and the centre of mass stays where it is. That gives $3m_L m_X/(3m_L + m_X)$ = 2.4866 unified mass units, and it is what almost every published one-dimensional treatment uses.

The moving atom’s own mass. The coordinate is named for the nitrogen’s displacement, so the mass being carried is the nitrogen’s: 14.0031. This is not a construction anybody would defend on reflection, and it is exactly the reading the coordinate’s description invites — the atom moves, so its mass is what moves.

The bonds held at their measured length. This is the one that is not a constant. If the three N–H distances do not change during the inversion, then as the apex descends towards the plane the hydrogens must slide outward, because the bond length is the hypotenuse of a triangle whose height is shrinking. That radial motion carries kinetic energy, and neither of the first two constructions has a term for it.

Working it out gives $\mu(x) = \mu_{\text{plane}} + 3m_L x^2/(r^2 - x^2)$, which is 2.4866 at the flat geometry — where the hydrogens are momentarily not moving radially at all — and 2.9874 at the pyramid, twenty per cent larger.

It also has a domain. At |x| = r the hydrogens have folded flat against the axis and there is no molecule past it, so the coordinate stops existing at 2.653 times the height of the pyramid. That is a fact about the construction rather than a divergence to be regularised, and the solver is told about it rather than discovering it as a crash.

Three answers, and one of them is not an answer #

Ammonia’s measured ground splitting is 0.79350 wavenumbers. The three constructions, in the same well, at the same geometry, with the same barrier, give:

2.4866 → 1.35075, which is 1.702 times the measurement. 2.9874 at the pyramid, varying → 0.94420, which is 1.190 times it. 14.0031 → 5.5041 × 10⁻⁶, which is 245,000 times smaller than the first.

The third disposes of itself. A construction that turns a molecule inverting twenty-four billion times a second into one inverting once a week is not a defensible approximation, and the reason it fails is instructive: a reduced mass is dominated by its light end, and taking the heavy end alone gets the physics backwards rather than approximately.

The interesting comparison is between the first two, and it does not go the way a correction is supposed to go. The bond-conserving mass is larger everywhere except exactly at the plane, so it should make the tunnelling harder and the splitting smaller — and it does, by forty-three per cent. That takes the model from 70 per cent above the measurement to 19 per cent above it.

A correction that closes two thirds of a discrepancy is not a correction that can be neglected. It says the construction in universal use is the one in error, and that the original diagnosis — that the residual error is the well’s shape — was measuring the well’s shape plus a mass convention, in unknown proportions. With the varying mass the residual is 19 per cent rather than 70, and how much of that is shape is not something either calculation can now say.

Where the mass is, not how heavy it is #

The obvious way to handle a mass that varies by twenty per cent is to replace it by a constant somewhere in the middle. That is wrong here, and it is wrong by more than the correction itself.

Solving the same well with a constant mass equal to the pyramid value, 2.9874, gives 0.56237 wavenumbers. Solving with the varying mass gives 0.94420. Solving with a constant at the flat value, 2.4866, gives 1.35075.

So the varying case does not sit between the two constants in any averaged sense: it is 1.68 times the heavy constant and only 1.43 times below the light one, which puts it much nearer the light end of its own range. The reason is where the two things happen. The mass is heaviest at the minima, where the molecule spends its time; the action is taken across the barrier, where the mass is lightest. A tunnelling integral does not sample the coordinate uniformly — it samples the classically forbidden region and nothing else — so it weights the mass by exactly the region where this mass is at its smallest.

That also explains a discrepancy the sensitivity exponents would otherwise predict wrongly. The mass sensitivity is −4.5628, so a uniform twenty per cent increase should cost a factor of 2.31, and applied to a constant it does: the two constants differ by 2.40. The varying mass costs only 1.43. Nearly two thirds of the effect an exponent predicts does not happen, and the missing part is the part of the coordinate the integral never visits.

The correction is not where the amplitude is #

There is a shape to the bond-conserving mass that is worth naming before its consequences are computed, because it is the reason those consequences are surprising.

The extra term is $3m_L x^2/(r^2 - x^2)$, which is zero at the flat geometry and grows quadratically away from it. So the mass is at its minimum precisely at $x = 0$ — the top of the barrier — and at its maximum in the two wells. The molecule’s amplitude does the opposite: it is largest in the wells and exponentially small at the middle.

That arrangement is what makes the correction hard to guess in either direction. Somebody reasoning from where the molecule is would take the pyramid value, 2.9874, and get 0.56237 wavenumbers. Somebody reasoning that the correction is negligible because the mass barely changes would keep 2.4866 and get 1.35075. The answer is 0.94420, and neither line of reasoning reaches it.

What it does to a barrier #

A splitting fixes a barrier well while a barrier predicts a splitting badly, which argues for reversing the usual presentation. The mass puts a price on doing it.

Fitting the barrier to ammonia’s measured splitting gives 2330 wavenumbers under the usual construction and 2117 under the bond-conserving one. The spread is ten per cent, which is the same size as the entire disagreement among published barriers for this molecule — and the bond-conserving value lands markedly closer to the published 2020.

The third construction returns a refusal: no barrier between 200 and 40,000 wavenumbers reproduces the measurement with a mass of 14.0031, because even at the smallest barrier in the range the splitting is too small. That is reported as a refusal rather than as a number pinned to the end of the sweep, because a bisection that runs to its bracket returns the bracket and looks exactly like an answer.

So a barrier is a function of three things nobody states together: the well shape, the lines fitted, and the reduced mass. There are now six numbers on the table for ammonia’s barrier — 2020 as published, 2262 from a three-parameter fit to both splittings, 2330 and 2117 from the ground splitting under two mass conventions, 2560 from the excited splitting alone, and a refusal. None of them is wrong. A quantity with several correct values and one name is the situation four electronegativity scales are in, and the discipline it calls for is the same one: state the scale.

The shared box, and what it cost #

The three cases have to be solved in the same box to be comparable, and one of them has a domain shorter than the others would naturally use. So every case here is solved on the shortest of the three — 2.640 times the height of the pyramid, set by the bond-conserving construction’s own limit.

That is a decision that could have quietly changed the answer, so it is measured rather than argued about. The usual construction solved in the shared short box and in the full one differs by 2.3 parts in a hundred thousand, which is four orders below the effects being discussed. The reason the box is so cheap is that the potential at 2.64 times the pyramid height is 73,000 wavenumbers, more than a hundred times the ground state’s energy, so there is nothing out there for a wavefunction to do.

Stating that is not a formality. A comparison between three constructions made in three different boxes would be a comparison between three constructions and three boxes, and the difference between the first two here is 43 per cent — well within the range a badly chosen box could manufacture.

What was computed, and how #

The well, the geometry and the solver are those of the quartic-well calculation, unchanged. The three reduced masses are constructed from atomic masses and, in the third case, from the measured bond length; nothing is fitted anywhere.

A position-dependent mass needs a kinetic operator, and there is more than one. The classical energy p²/2μ(x) has several inequivalent quantum orderings, since p and μ(x) do not commute, and the one used here is −½ ∂ₓ (1/μ(x)) ∂ₓ — the BenDaniel–Duke form, which is Hermitian for any μ(x). It is implemented with the inverse masses evaluated on the half-grid, so that the discrete operator is exactly symmetric rather than symmetric to within a truncation error; a splitting is a difference of two nearly equal eigenvalues, and an operator that is only nearly symmetric produces nearly real eigenvalues with a spurious spread of exactly the size being measured.

Five results are checked numerically. The bond-conserving mass rises by at least a sixth across the coordinate, so there is something to measure. It moves the splitting by more than a fifth. The literal construction destroys the effect by four orders of magnitude. The shared box costs under a part in a thousand, so the comparison is between constructions rather than grids. And the one that could have gone the other way: the bond-conserving mass moves the answer towards the measurement rather than away from it, so this is not the comfortable kind of finding in which a neglected term turns out to be neglectable.

Where the model stops #

The ordering is a choice too, and it is one made here rather than measured. BenDaniel–Duke is the standard choice and it is not the only Hermitian one; a family of orderings parameterised by two exponents exists, they differ by terms involving derivatives of the mass, and the mass here has a derivative that grows without bound near the domain’s edge. Nothing here says how large that ambiguity is, and it is the obvious next thing to put a number on.

The bond-conserving path is still a path. It holds the bond length at its equilibrium value and keeps the ligand triangle equilateral, which is a better assumption than holding the ligands still and is not the true minimum-energy path. A real inversion lengthens the bonds slightly as it flattens, and a path that relaxes them has both a different mass and a different barrier — so the two corrections are not independent and only one of them is computed here.

Three constructions is three. Others exist, including ones derived from a full vibrational analysis by projecting the reaction path out of the Wilson G matrix, which is the professional way to do this and needs a force field — fitted ones exist for six molecules in these essays, and ammonia is not among them. The three here are the ones an argument in prose would reach for, which is why they are the ones worth comparing.

And the improvement is not a validation. Moving from 70 per cent above the measurement to 19 is progress and is not agreement, and it would be a mistake to read it as saying the bond-conserving mass is right. What it says is that the usual mass is demonstrably not, and that a residual attributed to one thing was partly another.

The generalisation #

The transferable point is about what a one-dimensional model borrows without saying so.

Reducing a molecule to one coordinate looks like a statement about the potential — this direction matters and the others do not — and it is also, silently, a statement about the kinetic energy. The potential along the chosen path is a curve somebody can draw and defend. The mass along it is a number that has to be constructed, cannot be measured, and is usually inherited from whichever textbook the author read first.

The tell is that the quantity has no units problem and no obvious ambiguity. A barrier is visibly a modelling choice; everybody knows it depends on the path. A reduced mass looks like arithmetic on atomic masses, and arithmetic does not feel like a choice. A tolerance that decides a point group is the same kind of hidden lever, and it went unstated for the same reason. That is exactly why it goes unstated, and why three defensible constructions can span a factor of five and a half without anybody noticing that a factor of five and a half is available.

The second half is the more useful one and it is a general property of tunnelling arguments rather than of this molecule. A quantity that varies along a coordinate is sampled by whatever the answer is an integral of, not by where the system spends its time. The intuition that the molecule “mostly sits in the pyramid, so use the pyramid’s value” is exactly backwards for a tunnelling splitting, which is an integral over the one region the molecule almost never occupies. Two thirds of the effect an exponent predicted did not happen for that reason, and no amount of care about the exponent would have found it.

That distinction recurs wherever an average stands in for a function. It appears as a contour level that is not a measure of size, as a single effective charge that cannot separate two orbitals, and here as a mass that cannot be averaged. In each case the model returns one number per object and the question needs the object’s shape.

Who found it, and when #

Reduced masses for large-amplitude coordinates are a well worked subject; the projection of a reaction path out of the G matrix is Wilson’s GF method applied by many hands, and the position-dependent kinetic operator is a standard problem in semiconductor physics where the BenDaniel–Duke form comes from. The bond-conserving umbrella mass is not new.

What is new here is the comparison: three constructions in one well, the splittings and the inferred barriers each produces, and the observation that a varying mass sits near the light end of its own range because that is where the integral looks. The number worth carrying is not any of the splittings but the 2117 against 2330 — the same measurement, the same well, and a tenth of a barrier decided by a convention nobody writes down.

Still open: how much the ordering of the kinetic operator matters #

The obvious open question is the ordering ambiguity, which is named above and not sized. A one-parameter family of Hermitian kinetic operators exists for a position-dependent mass, they agree in the constant-mass limit, and the derivative terms that separate them grow where this mass grows. Sweeping the family and reporting the spread in the splitting would say whether the improvement found here is robust or whether it is one member of a family that brackets the measurement — and those are very different results with the same headline.

The nearer question is the deuterated case, which the varying mass has made a sharper test than it was. The bond-conserving mass has two terms with different mass dependences: the first is the ordinary reduced mass and the second is three times the ligand mass. Deuteration doubles the second and moves the first by seventy per cent, so ND₃’s correction should differ from ammonia’s by an amount this construction predicts and no constant-mass treatment can. Its splitting is measured, so the prediction can be checked rather than admired.