An ordering worth half a per cent

The mass nobody chose found that ammonia’s inversion splitting, computed in one double well from one barrier and one geometry, falls from 1.35075 wavenumbers to 0.94420 when the reduced mass is built by holding the bonds at their measured length instead of holding the ligand plane rigid. That is forty-three per cent, towards the measured 0.7935, and it was presented with a caveat that it did not size.

A mass that depends on position does not commute with the momentum. The classical kinetic energy p²/2μ(x) therefore has no unique quantum form, and the calculation used one of several — the BenDaniel–Duke ordering, −½ ∂ₓ (1/μ) ∂ₓ — because it is the standard one. If a different ordering moved the splitting by as much as the mass construction did, the forty-three per cent would be a property of a convention stacked on a convention, and the improvement would say nothing about the mass.

The orderings can be swept, and the answer is not close. The five in use span 0.47 per cent between them.

A potential of eight wavenumbers #

Von Roos wrote every Hermitian ordering of that kinetic energy as

$$T = \tfrac14\left(\mu^{\alpha} p\, \mu^{\beta} p\, \mu^{\gamma} + \mu^{\gamma} p\, \mu^{\beta} p\, \mu^{\alpha}\right), \qquad \alpha + \beta + \gamma = -1,$$

and each ordering that has been argued for since is a point in that plane. Expanding the products and collecting terms turns every one of them into the BenDaniel–Duke operator plus an ordinary potential:

$$T = -\tfrac12\,\partial_x \frac{1}{\mu}\,\partial_x + s\,\frac{\mu'^2}{\mu^3} + c\,\frac{\mu''}{\mu^2}, \qquad s = \tfrac12(\alpha + \gamma + \alpha\gamma),\quad c = -\tfrac14(\alpha + \gamma).$$

That rearrangement is the whole of the method, and it changes the question from one about operators into one about potentials. An ordering is a small extra potential added to the well, and the question is how large it is and where it sits.

For the bond-conserving mass, μ(x) = μ_plane + 3m_H x²/(r² − x²), both derivatives are in closed form, so every ordering’s potential can be written down and drawn.

The largest of them inside the region the molecule occupies is about eight wavenumbers, on a barrier of 2020. Gora–Williams adds 8.04 at the top of the barrier and 6.67 at the minima; Zhu–Kroemer adds 8.04 at the top and 8.15 at the minima; Li–Kuhn and Mustafa–Mazharimousavi add roughly half as much. BenDaniel–Duke adds nothing, by construction, since both its coefficients vanish.

What a tunnelling splitting responds to is not the size of an added potential but the difference it makes between the barrier region and the wells, because a constant added everywhere shifts both members of a doublet together. Gora–Williams raises the barrier top by 1.37 wavenumbers more than it raises the minima, which makes the barrier effectively taller. Zhu–Kroemer raises the minima by a tenth of a wavenumber more than the top, which makes it effectively lower. The two are opposite in sign before anything is solved, and the curves show why: one bows up in the middle and the other is nearly flat until the edges.

Five solves #

Solved in the same well, the same box and on the same grid as the mass comparison, the five orderings give 0.94420, 0.94036, 0.94477, 0.94228 and 0.94338 wavenumbers. Relative to BenDaniel–Duke that is 0, −0.407, +0.060, −0.204 and −0.087 per cent. The spread across all five is 0.469 per cent.

The mass construction moved the same splitting by 43.1 per cent. The ordering is ninety-two times smaller, and it points in both directions, so nothing about the choice among these five could have manufactured the improvement. The sign pattern is the one the added potentials predicted: the two orderings that make the barrier effectively taller lower the splitting, and the one that lowers it raises the splitting slightly.

The excited doublet behaves the same way at a smaller relative size, from 44.303 wavenumbers under BenDaniel–Duke to 44.166 under Gora–Williams, a change of 0.31 per cent. An excited state sits higher in the well, spends more of its amplitude over the barrier and is less sensitive to a few wavenumbers added to it, which is the ordinary behaviour of a splitting that depends on a barrier less steeply the nearer the barrier top a state sits.

Two integrals for the whole family #

Because every ordering adds a combination of the same two functions of position, first-order perturbation theory has a particularly compact form here. The shift of a level is the expectation of the added potential in the unperturbed state, so the shift of the splitting is

$$\delta\Delta \approx s\,a + c\,b,$$

where a and b are the differences between the two members of the BenDaniel–Duke doublet in their expectation of μ′²/μ³ and of μ″/μ². Those are two numbers, 0.0353 and 0.0553 wavenumbers, and they describe the entire plane of orderings at once. It is the same economy a single detuned level showed on a much smaller problem: when a perturbation is a fixed function with a coefficient in front of it, first order needs the function’s expectation once and the coefficients forever.

Fifteen orderings with both exponents on a grid from −1 to 0 were solved exactly and compared with that formula. The changes run from −0.41 to +1.06 per cent, and the worst disagreement between exact and first order is 0.005 per cent of the splitting — a two-hundredth of the largest change on the grid, and an eighth of the smallest one that is not zero. First order is not an approximation to the answer across the whole region where orderings have been proposed. It is the answer, to the precision anybody could want.

That has a practical consequence and a conceptual one. Practically, sizing the ordering ambiguity for any position-dependent mass does not need a family of solves; it needs two integrals over one doublet. Conceptually, it says the ambiguity lives in a two-dimensional space rather than in von Roos’s three exponents, since only s and c enter — and two different orderings with the same s and c are the same operator.

How far out an ordering has to be #

The family is unbounded. Any real α and γ define a Hermitian ordering, so somewhere in it is an ordering giving any splitting at all, and the statement that “the family brackets the measurement” is true and empty. The question with content is how far out the bracketing ordering has to sit.

Drawn in the two coefficients that actually enter, the family has a shape. Real exponents exist only where (α + γ)² ≥ 4αγ, which in these coordinates is the region under the parabola s = 2c² − 2c, and everything proposed sits within a quarter of a unit of the origin. The lines are first-order estimates of where the splitting would reach the measurement and where it would return to the constant-mass value; both run through the admissible region, well away from the cluster.

The exact positions come from following two rays and solving for the crossing. Along α = −γ = t only the μ′² term acts, and the splitting falls, reaching the measured 0.7935 at t = 3.04 — an ordering with exponents +3.04, −1 and −3.04. Along α = γ = −t the splitting rises, and it undoes the bond-conserving mass entirely, back to 1.35075, at t = 4.63, where β is +8.26.

Every ordering in use has exponents of magnitude one or less, and inside that strip neither ray moves the splitting by one per cent. The ordering that would make the model agree with experiment has exponents three times larger than any that has been argued for, and the one that would erase the mass correction five times larger. Out there first order is no longer exact. At the ordering that lands on 0.7935 it predicts 0.781, and at the one that lands on 1.35075 it predicts 1.287 — misses of two and five per cent, which is what the terms second order in the added potential should do once its coefficients are no longer small.

Both crossings were checked for an artefact that only a large exponent can produce. The μ′² term diverges at the edge of the coordinate’s domain, where the ligands fold flat, with the sign of αγ; along the first ray that sign is negative, so the ordering digs a well at the edge that could in principle hold a state of its own and be mistaken for a member of the doublet. It does not: at both crossings the ordered potential’s floor beyond the wells stays above the fourth level, at 4,751 and 5,344 wavenumbers against 1,490 and 1,704.

Why the answer is small #

A ninety-fold gap between two effects wants a mechanism rather than a coincidence of numbers, and there is one. It is visible in the formula before anything is computed.

The ordering’s potential is s·μ′²/μ³ + c·μ″/μ². In atomic units the Planck constant is one and so it does not appear, but restored it multiplies both terms as ħ², and both terms carry one reciprocal power of the mass once the derivatives are counted. So the ordering is a quantum correction: a term of order ħ²/μ that vanishes in the classical limit. The mass construction is not. It changes the mass inside the tunnelling action, ∫√(2μ(V − E)) dx, which is the leading term of the splitting’s logarithm and grows with the mass rather than shrinking.

That argument makes a prediction that can fail. Multiplying every nuclear mass by a factor λ while holding the potential fixed is the same as dividing ħ² by λ, so the ordering’s potential must fall as exactly 1/λ, its effect on the splitting must shrink, and the mass construction’s effect must grow.

It does all three. Gora–Williams’s contribution to the effective barrier is 2.746, 1.373, 0.687 and 0.343 wavenumbers at λ = 0.5, 1, 2 and 4 — exactly inversely proportional, to nine figures. The spread among the five orderings falls from 0.63 per cent to 0.47, 0.34 and 0.25, and its ratio from one doubling to the next approaches $\sqrt{2}$, which is what a potential falling as 1/λ acting on a splitting whose sensitivity grows as $\sqrt{\lambda}$ should give. The mass construction’s effect rises from 34 per cent to 43, 54 and 70. The ratio between the two runs 55, 92, 159, 285.

So the gap is not ammonia being lucky. Any molecule whose large-amplitude coordinate carries a smoothly varying mass will find the same ordering of effects, and a heavier one will find it more strongly — which does not make mass the reason heavier molecules invert slowly, since phosphine’s rigidity was never mostly its mass. It also says where the ordering does matter: where the mass changes over a distance comparable to the wavelength, so that μ′/μ is large. That is the setting BenDaniel and Duke were writing for — an electron crossing an abrupt interface between two semiconductors, where the effective mass jumps — and it is exactly the setting a molecular coordinate is not.

What was solved, and how #

The well, the geometry, the box and the grid are those of the mass comparison: a quartic with its minima at the pyramid height computed from ammonia’s measured bond length and angle, a barrier of 2020 wavenumbers, and a 2,400-point grid out to 0.995 of the bond length, where the coordinate ends. The kinetic operator is BenDaniel–Duke on a half-grid of inverse masses, so the discrete operator is exactly symmetric, and each ordering enters only as a potential added to the diagonal.

The rearrangement into BenDaniel–Duke plus a potential was not taken on trust. The symmetrised product form was applied directly to a smooth test function by nested finite differences, for each of the five orderings at four positions across the well, and its difference from BenDaniel–Duke applied the same way reproduces s·μ′²/μ³ + c·μ″/μ² times the function to within a thousandth. The derivatives of the mass are closed forms, and that route used none of them.

The checks are these. BenDaniel–Duke reproduces the bond-conserving splitting of the mass comparison. The five orderings span under one per cent, and the mass construction’s effect is more than fifty times their spread. First order reproduces all fifteen grid points to within 0.02 per cent of the splitting. Both rays reach their targets with no state bound at the domain’s edge, and only at an exponent more than twice any in use. The ordering’s potential scales as exactly 1/λ, the spread falls with λ, and the mass effect rises. And the refusal: with a constant mass all five orderings are the same operator, so their splittings must agree to the last bit the solver carries — and they do, which is what says the ordering potentials are not manufacturing a difference out of rounding.

What sweeping a family cannot settle #

Von Roos’s family is not every possible kinetic operator. It is the two-parameter set of symmetrised products, and other Hermitian forms exist. The five orderings here are the ones in use, and the result is a statement about them and about the region of the family near them.

Choosing an ordering is itself a symptom. A one-dimensional kinetic operator is what remains after the other coordinates have been removed, and a proper reduction — the Laplacian written in curvilinear coordinates, with the Wilson G matrix supplying the mass and its determinant supplying an extra potential — produces one definite operator rather than a family to choose from. That operator need not sit in von Roos’s plane at all, since its extra potential depends on the geometry of the full configuration space and not only on μ(x) — and a reduction that carries the other coordinates along also carries their zero-point motion into the potential. It is not computed here, and it is the honest way to remove the ambiguity rather than bound it.

The bound is on this well. Every number here is for a quartic with a barrier of 2020. A different shape changes where the doublet’s amplitude sits and therefore the two integrals, though not the scaling argument, which is about orders in ħ rather than about ammonia.

And the ordering was the last free choice in the kinetic energy, not the last thing wrong with the model. The bond-conserving mass left the splitting nineteen per cent above the measurement, and the orderings cover under one per cent of that. What is left is the potential and the path, which the account of what a barrier measures already showed carries most of the discrepancy.

A choice is worth what it moves #

The transferable point is about how to size an unresolved modelling choice, and it has two halves.

The first is that a choice with a continuum of options is sized by how far into the continuum the answer has to go, not by whether the continuum contains it. An unbounded family contains everything, so “some ordering reproduces the measurement” was guaranteed before any solve and carries no information. What carries information is that the one that does needs exponents three times larger than anything proposed. The same move turns a sweep of a tolerance or a threshold into a finding: not whether the answer changes somewhere, but where, against where anybody would put the setting.

The second is that the size of a choice often has a reason that can be read off its form before it is computed. The ordering terms carry ħ² and a reciprocal mass; the mass construction lives in the action. That predicted which would win and how the gap would move with mass, and a prediction of that kind is worth more than the ratio at one mass, because it transfers. Sweeping λ was the right test for the same reason a sweep of the quantum defect was the wrong one for a Stark shell: a sweep only tests an approximation if it moves the parameter the approximation is controlled by. It is the same kind of reading that separated the harmonic and anharmonic terms of a vibrationally averaged moment: identify the order of each term in the parameter that controls it, and the numbers stop being a surprise.

There is a third, smaller point about first-order theory. When every option is a fixed set of functions with coefficients in front of them, first order needs the functions’ expectations once, and a whole family of calculations collapses into a linear map. It is worth checking for that structure before running a sweep, since here it replaced fifteen solves with two integrals and was exact across the whole region that mattered.

Where the orderings come from #

The ambiguity of a position-dependent kinetic energy was met in semiconductor physics. BenDaniel and Duke proposed their form in 1966 for electrons at heterojunctions; Gora and Williams used another in 1969; Zhu and Kroemer argued for theirs in 1983, the same year von Roos wrote down the general family; Li and Kuhn’s appeared in 1993, and Mustafa and Mazharimousavi’s in 2007. None of the arguments among them is settled, which is exactly why a sweep is the right thing to report rather than a choice.

The bond-conserving umbrella mass and the observation that its ordering ambiguity is two orders below its effect are computed here. The mass comparison deserves the credit for naming the ordering as the question and for saying that a family bracketing the measurement would be a different result with the same headline — which is the sentence that made the sweep worth running to the end of each ray.

Still open: whether deuterium agrees, and the operator nobody chose #

The obvious open question is the deuterated molecule, which the bond-conserving construction said it would pass differently from any constant mass. Its second term is three times the ligand mass, so deuteration doubles it while raising the usual reduced mass by seventy per cent, and ND₃’s splitting is measured. With the ordering now known to be negligible, the isotope test is a test of the mass and nothing else in the kinetic energy.

The nearer question is the operator a proper reduction would supply. The ordering family is what a one-dimensional model has to choose among because it was not derived; writing the full kinetic energy in curvilinear coordinates and reducing it onto the umbrella path gives a definite operator with its own extra potential. Whether that operator lands inside the cluster of orderings in use — which the scaling argument suggests it must, since its extra terms are of the same order in ħ — is a calculation over a force field that ammonia does not yet have here.