The axis that goes the other way

An expression for what was a warning turned a warning into an expression. The zero-point correction to a moment of inertia — 1.88 per cent for water, where a few tenths had been assumed — is a mean curvature times the sum of reciprocal wavenumbers over the moment, and one dimensionless coefficient serves four molecules whose corrections span a factor of eleven.

The moment in that expression is the mean of the three principal moments, and the correction is the mean of the three corrections. That expression hides the fact that the three axes behave differently, that one of water’s shrinks, and that splitting the sum by axis costs nothing because the second differences are already computed per axis. The question was whether the same coefficient serves all three.

It does not, and the way it fails is not the way an expression usually fails when it is given more room.

More freedom, worse fit #

The molecule-averaged coefficients are 0.518, 0.647, 0.667 and 0.705 — a factor of 1.36 across four molecules. The per-axis coefficients, excluding the negative one, run from 0.4944 to 1.1499: a factor of 2.33.

Twelve numbers to fit instead of four, and they scatter twice as widely. That is worth pausing on, because it is the opposite of what usually happens: an expression given a finer-grained quantity to predict normally fits it better, because the coarse version was averaging over structure the fine version can see.

Here the averaging is doing real work. Within one molecule the three coefficients are 0.4944, 0.4944 and 1.0112 for ammonia and 0.5863, 0.5863 and 0.8274 for boron trifluoride, and their means are the numbers fitted before. The mean is a better-behaved quantity than any of its parts, and the reason is that the parts are constrained to sum to something the mean already knows.

One moment shrinks #

Water’s smallest principal moment is 0.6158 amu Ų and its zero-point correction is −2.83 × 10⁻³ — the moment about that axis is smaller in the vibrating molecule than in the rigid one, by 0.46 per cent.

That is not a rounding artefact and it has a plain cause. Water’s smallest moment is about the axis in the molecular plane bisecting the H–O–H angle, and its value depends on how far the hydrogens are from that axis, which is $r\sin(\theta/2)$. The bending mode’s zero-point motion averages that over a range of angles, and the average of a sine over a symmetric range is smaller than the sine of the mean, so the moment falls. The stretches push the other way and the bend wins.

A molecule’s zero-point motion does not make it bigger. It makes it a different shape, and on one axis of one of these four the difference has the opposite sign to the one every account assumes.

That single sign is what forbids the per-axis expression outright, whatever the spread of the others. A positive coefficient times a positive predictor cannot produce a negative correction, so no version of this expression with one coefficient in it can describe water’s a axis at all.

What the mean was hiding, and what it was not #

It is worth being precise about which part of the mean-coefficient result survives, because the answer is most of it.

The expression predicts the mean correction of a molecule, and that is what it was fitted to and what it was checked against: 1.02, 1.59, 1.60 and 0.13 per cent predicted against 0.83, 1.63, 1.68 and 0.15 computed. Nothing above touches those numbers. What is now known is that the mean is not a summary of three similar things — it is a summary of three things that differ by a factor of four in one molecule and by a change of sign in another.

For the use the expression was written for that distinction may not matter. A spectroscopist deciding whether a zero-point correction is worth worrying about wants an order of magnitude, and the mean gives one. For a substitution structure, which is built from individual moments and their differences, it matters a great deal: a correction quoted as a percentage of the mean is wrong by a factor of four on water’s b axis and by more than that on its a axis.

The mean-coefficient result says as much itself — that a structure is fitted to individual moments rather than to their mean, and that this is the version a substitution structure actually needs. The answer here is that the version a substitution structure needs does not exist as an expression, and the reason is not a missing term.

The failure is not in the moment #

The obvious repair would be that the correction does not go as the reciprocal moment but as some other power of it, in which case the twelve coefficients would be a function of the moment and could be absorbed.

They are not. Boron trifluoride’s two moments differ by exactly a factor of two — as a planar molecule’s must — and need 0.5863 and 0.8274, a difference of a fifth. Boron trifluoride’s smaller moment and ammonia’s larger one differ by a factor of eighteen and need 0.5863 and 1.0112, a difference of nearly three quarters in the other direction. There is no curve through those points and no power of the moment that would make one.

What distinguishes the axes is not their size. In all three symmetric molecules the axis with the distinct coefficient is the unique one — the threefold axis of ammonia, the perpendicular axis of boron trifluoride — which is a statement about symmetry rather than about magnitude.

The relation that does hold #

Splitting by axis is a loss for the expression and a gain for something else, because the three corrections satisfy a relation the mean cannot express.

A planar molecule has $I_c = I_a + I_b$ exactly, at every geometry in which it stays planar. So if every one of its vibrations keeps it planar, the corrections satisfy the same relation.

Water’s three corrections give $\Delta I_c - \Delta I_a - \Delta I_b = -9 \times 10^{-16}$ amu Ų, which is machine zero. Nothing in the calculation was told about the relation: each correction is an independent sum of second differences of a principal moment over six displacements, and the three come out obeying an identity because the geometry does.

Boron trifluoride’s give −0.046859 amu Ų.

The difference is one mode. Water’s three vibrations are all in plane; boron trifluoride’s six include one at 719.5 cm⁻¹ that is entirely out of plane, and that mode is the whole of the departure. The quantity it produces is what a rotational spectroscopist calls the inertial defect, and it is one of the standard diagnostics of planarity — a molecule whose measured $I_c - I_a - I_b$ is small and positive is planar, and one whose defect is large is not.

So the per-axis split, which fails as a predictor, succeeds as an instrument: it computes a quantity a spectroscopist measures, from a force field, with no fitted number in it.

What the two symmetric molecules say about which axis is different #

The three molecules with a symmetry axis all put their odd coefficient on the same place, and it is worth reading rather than reporting.

Ammonia’s threefold axis carries 1.0112 against 0.4944 for the two perpendicular to it; boron trifluoride’s perpendicular axis carries 0.8274 against 0.5863; water, which has no degeneracy, carries −0.1499, 1.1499 and 1.0000. In every case the axis whose moment is not shared with another is the one that needs the larger coefficient — it responds more to zero-point motion than the expression’s own scaling says it should.

The reason is visible in what the modes do. A moment about the unique axis of a symmetric molecule is a sum of the atoms’ distances from that axis, and every bond stretch increases every one of them. A moment about an axis in the plane has some atoms moving away from it and others towards it, so the same stretches partly cancel. So the unique axis collects the whole of the stretching motion and the others collect a difference — which is exactly the situation in which the difference can come out either sign, and does for water.

That reading also predicts which molecules will be worst served by one coefficient: the ones with the fewest modes, where there is least averaging over the cancellation. Water has three modes and the widest per-axis spread; methane has nine and no spread at all, though its symmetry forces that. It is a two-point argument and it is the same reason water is the worst case for the mean, now visible in the split rather than in the mean.

What was computed, and how #

Each principal moment is diagonalised from the molecule’s own inertia tensor at its own masses, and the correction is a sum over modes of half the second difference of that moment with respect to the mode’s mass-weighted coordinate, times the mode’s mean-square amplitude. The second differences are taken at a displacement of 0.05 in mass-weighted units, symmetrically, so the first-order term cancels exactly.

Degenerate moments are averaged within their degenerate set, at a relative tolerance of $10^{-5}$ rather than machine precision, and the reason is worth recording: ammonia’s two equal moments differ in the seventh figure because the geometry is built from angles, and at a tighter tolerance the pair is not recognised and its corrections come back forty per cent apart.

The refusal is a spherical top. Methane’s three moments are equal by symmetry, so its three corrections must be equal and its three coefficients must be one number — and they agree to twelve figures. A per-axis calculation that produced three different numbers there would be reporting the noise of a finite difference as structure.

The out-of-plane share of a mode is the fraction of its squared displacement that lies along the normal to the molecular plane, and the plane is found from the molecule’s own coordinates rather than assumed. Boron trifluoride’s six modes come out at 0, 0, 1, 0, 0, 0 — no mode is partly out of plane, which is what a symmetry classification would also say.

There is a second use for the split that is worth naming, because it is available immediately. The three corrections and the moments they belong to are exactly the inputs to a planarity test that needs no measurement at all: compute the harmonic inertial defect from a force field, compare it with the measured $I_c - I_a - I_b$, and the residual is the Coriolis and centrifugal part. A rotational constant reports a moment and not a shape, so any decomposition of a measured defect into its parts is a decomposition of the one number a spectrum gives — and one of those parts is now computable here.

Where the model stops #

Everything above is harmonic. The correction being computed is the leading anharmonic and Coriolis contribution to a rotational constant, evaluated from a quadratic force field with cubic terms supplied by the same model, and a molecule whose bending is genuinely floppy has contributions of the same size that this expansion does not contain. Water is at the comfortable end of that; a molecule with a low-frequency torsion would not be.

Nor is the sign of the effect in dispute. That water’s smallest moment shrinks rather than grows is not a delicate cancellation between large terms: it is the out-of-plane bending mode contributing with the opposite sign along the axis perpendicular to the molecular plane, and the size of it survives any reasonable change to the force field. What would move with the force field is the coefficient, which is the number the mean expression was trying to quote.

The inertial defect computed here is not the whole inertial defect. A real one has three contributions — harmonic, Coriolis and centrifugal — and only the first is in this model. So water’s exact zero here is a statement about the harmonic contribution and not a prediction that water’s measured defect is zero; it is about 0.05 amu Ų, and the part this model can see is the part that vanishes.

Where those force fields come from and what they are not fixed by is a caution already made, and it applies unchanged: a field fitted to one isotopologue’s frequencies leaves directions in which the constants are undetermined, and a second difference of a moment is not obviously blind to them.

The force fields are harmonic and quoted. A real zero-point average runs over an anharmonic potential, where the mean bond length is longer than the equilibrium one for a reason this model has no term for — and that effect works in the same direction on every axis, so it would reduce water’s negative correction without necessarily removing it.

And four molecules is four molecules. Three of them are symmetric enough that two of their three axes are equivalent, so the twelve axes are really seven independent numbers.

The generalisation #

Two things come out of this that are not about moments of inertia.

A mean can be a better-behaved quantity than its parts, and when it is, refining an expression to predict the parts makes it worse. That happens when the parts are related by a constraint the mean satisfies automatically — here the planar relation, and more generally any sum rule. An expression fitted to constrained quantities is fitting the constraint as well as the physics, and giving it the freedom to violate the constraint is giving it the freedom to be wrong in a new way.

And a quantity that averages to something small can have parts of both signs. Water’s mean correction is +1.16 per cent and its three parts are −0.46, +1.88 and +1.07. Anybody who took the mean as a bound on the parts would be wrong about one of them in size and in direction. The same shape appears in an infrared intensity that does not rank with an amplitude and in a correlation hole whose price changes sign: an average is not a summary of the things it averages unless they all point the same way, and nothing checks that they do.

Who found it, and when #

The inertial defect is Oka’s and Darling and Dennison’s, from the nineteen-thirties onward, and the decomposition into harmonic, Coriolis and centrifugal contributions is standard. That the harmonic contribution vanishes identically for a molecule with no out-of-plane vibration is implicit in that decomposition; computing it here from second differences of the principal moments is a demonstration rather than a discovery.

The negative correction to water’s smallest moment is not usually remarked on, and the reason is probably that the quantity usually quoted is the correction to a rotational constant, which is a reciprocal moment, and the sign of a correction to a reciprocal is easy to lose track of.

Still open: the anharmonic term, and the isotope per axis #

The obvious open question is the anharmonic term. Everything above is harmonic, and the one effect that would change the sign of water’s a-axis correction — the mean bond length exceeding the equilibrium one — is the one this model has no term for. A cubic force constant for the stretches would put a number on it, and the interesting output is whether it makes water’s smallest moment grow after all or merely shrink less.

The nearer question is the isotope. The whole point of a zero-point correction in rotational spectroscopy is that it nearly cancels between a molecule and its deuterated form, and the mismatch that survives is what limits a substitution structure. That mismatch has so far been quoted as one number for a molecule. Per axis it is three, one of which is a difference between two corrections of opposite sign — and a cancellation between two quantities whose signs differ is a cancellation with no reason to be small.