A barrier is not what a splitting measures

Ammonia is drawn as a pyramid, and a pyramid has a handedness in the same way a tetrahedral carbon does, and for the same reason a group theorist would give: three ligands and a lone pair, in one of two arrangements that are mirror images. Nothing about the drawing says the two arrangements exchange. They do, about twenty-four billion times a second, and the number that says so is one of the most accurately measured quantities in molecular spectroscopy — the ground-state inversion doublet of ammonia is split by 0.79350 wavenumbers, which is a line at 23.79 gigahertz and is the transition the first maser was built on.

The standard account of why is a barrier, quoted as 2020 wavenumbers, or twenty-four kilojoules a mole. The nitrogen has to pass through the plane of the three hydrogens, that costs energy, and the molecule tunnels through the cost rather than climbing over it. Every textbook statement of the matter is that sentence with the number in it.

Put the number into the simplest well ammonia’s own measured geometry allows, and ask what it predicts. It predicts 1.3508 wavenumbers, which is seventy per cent too large, and it predicts the next doublet up at 68.37 against a measured 35.81, which is ninety-one per cent too large. Everything else about the same calculation is right to a few per cent.

The coordinate, and what it is not #

The umbrella coordinate is the signed perpendicular distance of the nitrogen from the plane of its three hydrogens. It is zero at the flat geometry, positive on one side and negative on the other, and it is the one direction in which the molecule’s two arrangements are connected.

That distance is not a number anybody measures. What a structure determination reports is a bond length and a bond angle — 1.0124 ångström and 106.67 degrees for ammonia, the angle being the quantity the repulsion account of shape exists to predict — and the height follows from them: three ligands at that angle sit on a circle whose radius is the circumradius of their own triangle, and Pythagoras does the rest. It comes to 0.38163 ångström, and it is computed here rather than quoted, because a collection that quoted it would have no way of noticing if the three numbers disagreed.

The well itself is the shortest expression that has minima at ±0.38163 ångström and a stated height in between: A(x² − x₀²)², with A fixed by the barrier. It has two numbers in it and both are measured or quoted. Nothing else about ammonia enters.

Solving it needs a mass, and the umbrella coordinate does not obviously have one. The choice made here is the usual one — the apex moving against a rigid triangle of hydrogens with the centre of mass held still, which is 2.4866 unified mass units. That the choice is a choice, and an expensive one, is a calculation of its own.

What the model gets right #

Before the failure, the agreements, because they are what make the failure specific.

The lowest state comes out at 586.81 wavenumbers above the bottom of the well. The harmonic frequency at the minimum — the curvature of the well divided by the reduced mass, square-rooted — is 1226.6 wavenumbers, against an umbrella fundamental observed near 950 and a harmonic value fitted potentials put in the range of eleven to twelve hundred. Four states lie below the barrier, which is what a spectrum of ammonia shows. The molecule is not a harmonic oscillator sitting in one of two wells and it is not a free rotor passing over the top; it is a system with a handful of doublets below a barrier and singlets above it, and the two-parameter well reproduces that whole structure.

It also reproduces something less often stated. The ground state sits 29.05 per cent of the way up its own barrier. A drawing of a pyramid implies a molecule resting at the bottom of a well, and it is not resting there — the same correction that puts water’s atoms away from the points its structure names — nearly a third of the barrier is spent before the tunnelling question is even asked, which is why an argument about inversion that quotes a barrier without quoting a zero-point energy has left out one of its two terms.

The two states are the ordinary consequence of a symmetric potential. Every stationary state of a well that is unchanged by reflection is even or odd about the middle, so the lowest two are the nodeless combination and the one-node combination of what would be a state in each well. They have almost the same shape, they have almost the same energy, and the entire difference between them is what happens in the middle, where both are small.

Neither stationary state is a pyramid #

This is worth a section rather than a clause, because it is where the picture and the physics come apart.

A pyramid is a molecule with its nitrogen on one side. Neither of the two lowest states is that: both have equal amplitude on both sides, by symmetry, exactly. The states that are pyramids are the sum and the difference, and those are not stationary — a system prepared in one of them oscillates into the other at a frequency equal to the splitting, and back again.

So the drawing of ammonia is a superposition of two states rather than a state, and the splitting is the rate at which the drawing stops being true. At the measured 0.79350 wavenumbers the exchange takes 21.0 picoseconds. The molecule is a pyramid for about as long as a bond takes to vibrate twenty times, which is the regime in which a point group stops being the right description of a molecule and a permutation group starts being one.

That is also why the arrangement has no optical activity to speak of. A pyramidal nitrogen with three different substituents is a stereocentre in exactly the sense a carbon is, and it cannot be resolved, because the two forms interconvert faster than anything can separate them. The interesting cases in synthetic chemistry are the ones where something — a ring, a heavy neighbour, a charge — pushes the splitting down far enough that the molecule stays put, and how far down is the question that decides whether such an amine can ever be resolved.

The failure, and it is only in one place #

The computed ground splitting is 1.3508 against a measured 0.79350, a factor of 1.7023. The computed excited splitting is 68.372 against a measured 35.81, a factor of 1.9093.

Two things are worth reading off that pair. Both are too large, so this is not a level being misplaced; and they are too large by nearly the same factor, so whatever is wrong is wrong in a way that affects the whole barrier rather than one energy.

The repair anybody would try first is to raise the barrier, and it half works. A barrier of 2329.9 wavenumbers reproduces the ground splitting exactly. It then puts the excited splitting at 47.30 against 35.81 — still 32 per cent out. And running the fit the other way, against the excited splitting alone, asks for 2559.8, which is 9.9 per cent higher again.

One height cannot satisfy two lines, and that is the first sign that the height is not the free parameter the problem has. A well with one number in it can be made to pass through one measurement, and it passes through the second wherever its shape puts it.

What a splitting is actually an exponential of #

The quantity that decides a tunnelling splitting is not a height. It is the action under the barrier,

$$S \;=\; \int \sqrt{2\mu\,(V(x) - E)}\;\mathrm{d}x$$

taken between the two turning points at the energy the state actually has, and the splitting falls off as its exponential. Three things go into it: the mass being carried, the distance it is carried over, and how far the potential rises above the state on the way. A height is one point of the third.

The comparison that makes the point is a second well with one more parameter. A harmonic term with a Gaussian hump on it — the form Manning wrote down in 1935 and Swalen and Ibers refitted in 1962, which is where the quoted 2020 comes from — has three constants, of which one is spent making the minimum sit at the measured geometry. That leaves two, and there are two measured splittings.

Fitted to both, it reproduces both exactly, and the barrier that falls out of it is 2262.0 wavenumbers. That number is a prediction rather than a fit: nothing in the fitting aimed at it, both free parameters having gone to the splittings. It lands twelve per cent above the published 2020.

So the two wells agree about the height to about a tenth and disagree about the splitting by a factor of 1.70. The height is not the quantity.

The area, where the width is the same #

The two wells’ barriers are almost exactly as wide as each other at the energy that matters. Measured at the ground state’s own energy, the half-widths are 0.2591 and 0.2607 ångström — six parts in a thousand apart. A picture of the two barriers side by side does not show a fatter one and a thinner one.

Their actions are 5.6872 and 6.1229, which differ by 0.4357. Exponentiate that and it is a factor of 1.546, against a measured discrepancy in the splitting of 1.7023. So the area accounts for ninety per cent of the whole error, and the remaining ten per cent is the prefactor — the frequency with which the state attempts the barrier, which the two wells also disagree about slightly because their curvatures at the minimum differ, 1226.6 wavenumbers against 1178.5.

The lesson is in where the extra area comes from. It is not the height and it is not the width; it is the shape of the shoulders between them. The quartic falls away from its maximum faster than the Gaussian-on-a-parabola does, so it spends less of its width near the top, and a difference distributed thinly across a whole interval is invisible in a drawing of two curves and decisive in an integral of them.

What is quoted, and what is computed #

Two numbers are quoted and everything else is computed. The bond length and the bond angle come from ammonia’s microwave spectrum. The barrier of 2020 wavenumbers is Swalen and Ibers’s, and it is worth being precise about what kind of number it is: not a measurement, but a parameter of a potential they fitted to an inversion spectrum — the same distinction that has to be made about an oxidation state that sits outside the range of every charge computed from the same wavefunction. The two splittings, 0.79350 and 35.81, are measurements.

The height of the pyramid is computed from the length and the angle. The well is built from that height and the barrier. The Schrödinger equation in one coordinate is discretised by finite differences and the lowest eigenvalues found by Sturm-sequence bisection, which counts how many eigenvalues lie below a trial energy and brackets each one on its own. That choice is not decoration. Every question about inversion asked here sweeps a parameter and reads a splitting off each solve, so a sweep is hundreds of solves; and the two numbers being subtracted agree to five figures, which is exactly the case where computing each to the last representable bit beats computing the whole spectrum to a tolerance.

The solver is checked against a potential whose answer is known in closed form — a harmonic well, on the same grid the figures use, whose levels must be evenly spaced and must come out at half-integer multiples of its own frequency. The check is stated as a fraction of the level spacing rather than of the level, because a constant offset in every level is precisely what a discretised second derivative produces and precisely what a difference of two levels does not care about. It is also required to refuse: the harmonic well’s levels must come out evenly spaced rather than paired, since a solver that reported doublets there would report them anywhere.

And the splitting is required to be converged in the grid. Halving and doubling the point count moves it by 1.3 parts in ten thousand, which is four orders below the discrepancy this essay is about.

The bug that would not have shown itself #

One property of the numerical method is worth recording, because it is the failure this whole calculation is least able to notice.

The wavefunctions are obtained by inverse iteration at each eigenvalue. The natural starting vector is something smooth — a sine on the grid — and a sine on a grid symmetric about zero is even or odd about the middle by construction. So is every state of a symmetric well. A start with a definite parity has no overlap at all with half the spectrum, and the iteration then converges to the wrong member of a doublet while returning a perfectly normalised, perfectly smooth, entirely wrong function.

Nothing about the result looks wrong. The two members of a doublet differ by a part in four hundred in energy and by everything in shape, so a figure drawn from it shows two identical curves where it should show one with a node. The repair is a start with no parity and a Rayleigh quotient checked against the eigenvalue that was asked for, which costs one pass over the vector and cannot be fooled: it returns the energy of whatever came back.

That is the hardest shape of defect to see, because its symptom is an absence — the same shape as a plate of orbitals drawn at one contour value, in which a 3d simply has no surface — and every comparison asks whether something is right rather than whether it is there. The discipline that catches it is the one the vibrational census uses: compute the same quantity by a second route that shares nothing with the first, and require them to agree.

Where the model stops #

One coordinate. A real umbrella motion is coupled to the three bond stretches and to the molecule’s rotation, and neither is here. The bonds are held at their measured length and the hydrogen triangle stays equilateral, which is an assumption about the inversion path rather than about the potential along it, and what that assumption costs is part of the question of which reduced mass to use.

The barrier is not a measurement. It is a fitted parameter, and different fits give different values because they fit different potentials to different lines. Beside the 2020 published, this calculation alone yields three more — 2329.9 from the ground splitting in a quartic, 2559.8 from the excited splitting in the same quartic, 2262.0 from a three-parameter fit to both. Quoting one of them without saying which well it came from is quoting a number without its units.

The reduced mass is a choice, made here in the usual way and not defended. It enters the action under a square root, so a twenty per cent error in it is a ten per cent error in an exponent of about six, and a quantity with several defensible constructions is exactly the hazard four measures of an orbital’s size spanning a factor of two and a half was written about, which is a factor of nearly two in the answer — the same size as the discrepancy this essay is about. That calculation is worth doing on its own.

And a splitting is not a rate. The exchange time quoted above is the period of a coherent oscillation of an isolated molecule, which is not what happens in a gas at pressure or in a solvent. It is the right quantity for a spectroscopic line and the wrong one for a chemical lifetime.

The generalisation #

The transferable point is about which parameter of a model is the one a measurement constrains, and it is not always the one the model is named after.

A double well has a height, a width, a curvature at its minimum and a shape between them. Three of those are easy to state, easy to compute and easy to quote; the fourth is a function rather than a number, and it is the one the splitting is sensitive to. So the quantity that ends up in every textbook is the one that is quotable, and the quantity that decides the answer is the one that is not.

The tell is available before any comparison with a measurement. The threshold at which a convention becomes load-bearing is the same one a tolerance sweep on a point group had to draw. A quantity that appears inside an exponential in the answer is a quantity whose uncertainty is multiplied rather than added, and a quantity that appears as an integral over a region is one that no single value of the integrand can summarise. The action here is both. A model whose output depends on a parameter that way needs its parameter reported as a curve or a range, not as a number, and this is the same discipline the counterpoise correction arrived at from the other end when it found a published answer holding over a window seven tenths of a per cent wide.

There is a second half, and it is the more useful one. The failure was located by the model getting everything else right. A well that had misplaced the zero-point energy, or the number of states below the barrier, or the harmonic frequency, would have been a bad well and the splitting would have been one symptom among several. This one is right about all three and wrong about the splittings by a factor, which is what says the defect is in a property the other quantities do not depend on — and there is only one such property here.

Who found it, and when #

The inversion of ammonia was identified as tunnelling by Hund in 1927 and the doublet was measured by Cleeton and Williams in 1934, which was the first observation of a microwave absorption line in any gas. Manning’s potential is from 1935 and Swalen and Ibers’s refit from 1962. Townes and Gordon’s maser, in 1954, ran on this transition.

None of that is this collection’s. What is its own is the arithmetic: a two-parameter well built from the two structural numbers and the one quoted barrier, solved, and compared against the two measured splittings — and the observation that the disagreement lives almost entirely in an area that two nearly identical-looking curves differ by.

Still open: how steeply a splitting depends on a barrier #

The obvious open question is the sensitivity, and it has a number already. The splitting depends on the barrier as a power with an exponent of −3.56, which says two things that pull in opposite directions: a barrier known to ten per cent predicts a splitting to thirty-six, and a splitting wrong by a factor of two fixes a barrier to a fifth. The precise instrument is the measurement and the derived quantity is the published number, which is the reverse of the order in which the two are usually cited. Putting the exponent on an axis and reading it both ways explains why the literature carries several barriers for one molecule without anybody having made an error.

The nearer question is the isotope. The same potential with a different reduced mass is a prediction with nothing fitted, since an equilibrium structure does not depend on nuclear mass — and ND₃’s splitting is measured. If the over-prediction found here is a property of the well’s shape it should survive the substitution; if it is somehow a property of the mass, it should not — and a worked case where changing an isotope changes a length without changing a potential is there to read it against. That is one more solve of a well already built, and it is the cheapest test of this diagnosis available.