Two moments about two different lines

Splitting the zero-point correction by axis shows that one dimensionless coefficient cannot serve all three principal axes, and points at the isotope. The substitution method works because the zero-point error nearly cancels between a molecule and its deuterated form; asked axis by axis, that cancellation ought to be three numbers rather than one, and one of water’s three corrections has the opposite sign to the other two.

The three numbers are here, and they are large: 64.6 per cent, −20.9 and −20.1 for water, and −162.7, 85.4 and −17.4 for ammonia. Against the five per cent of a correction invented rather than computed that is an order of magnitude, and it would be the whole of this essay if the arithmetic were the whole of the question.

It is not. Before those three numbers can be compared with each other, something has to be true that nobody checks: that the parent’s a axis and the daughter’s a axis are the same line.

The frame turns #

A principal axis is an eigenvector of the inertia tensor, and the inertia tensor is built from the masses. Change a mass and the tensor changes; change the tensor and its eigenvectors move. Nothing in that sentence is subtle, and it is not usually said, because the quantities a rotational spectrum reports are three numbers with three letters attached and the letters look like names of directions rather than of ranks.

They are names of ranks. A is the smallest moment, C the largest, and which physical line each of them is depends on the masses that went into the tensor.

For water and one deuterium the turn is 21.122°. That is not a perturbation of an axis; on the page it is a visibly different pair of lines through the same three nuclei. The third axis does not move by so much as a rounding error, and the reason is worth separating from the reason a symmetry usually gives: water is planar, the normal to a plane is always a principal axis of a planar body whatever the masses on it are, and no substitution within the plane can tilt it. It is fixed by the geometry rather than by the point group — the point group has already fallen from C₂ᵥ to Cₛ.

Ammonia’s turn is 9.388°, and there one axis is genuinely held by the surviving mirror plane. Both molecules therefore keep exactly one axis and turn the other two, and they keep them for different reasons.

What the comparison is actually comparing #

The clean way to see the size of the error this introduces is not to compare angles but to compare moments, and the daughter has two sets: its own three, and the three it would have about the parent’s axes.

Monodeuterated water’s smallest moment is 0.72911 u Ų. Its moment about the parent’s smallest axis is 0.87431 u Ų, which is 19.9 per cent larger. Neither number is wrong and they are not the same number, and a comparison that quietly takes the first while meaning the second has an error in it about a fifth the size of the quantity, before any physics has been done at all.

Methane is worse — 23.5 per cent — for a reason the last section of this essay is about. Boron trifluoride is exactly zero, for a reason the section after that is about. Ammonia is 2.69 per cent, which is small enough to be mistaken for the effect being looked for: the zero-point corrections themselves are one to two per cent of a moment, so an artefact of the frame is here the same size as the physics, on the one molecule where it is small enough to be missed.

There is no need to take the two frames on trust. Write the daughter’s tensor in the parent’s principal frame and look at it.

The matrix is not diagonal. Its largest off-diagonal entry is 0.37587 u Ų — about half the parent’s smallest moment — and the angle that diagonalises the leading two-by-two block comes out at 21.122°, which is the turn, recovered from the tensor without ever computing an eigenvector or comparing two directions. The off-diagonal entry is the rotation. It is not a residual, not a fitted quantity and not a small parameter: it is a number the tensor has, in a frame somebody chose, and it says that the choice was wrong for this molecule.

Two of the nine entries are exactly zero and stay exactly zero, and those are the ones coupling the in-plane axes to the normal. That is the planarity again, and it is the reason the turn is a rotation in a plane rather than a general one — which is what makes a single angle enough to describe it.

The three numbers, now that they can be read #

With the frame understood, the per-axis mismatches can be read for what they are.

Water’s worst axis is its smallest, at 64.6 per cent, and that is the axis whose correction is negative — the one that shrinks rather than grows. Its correction is the smallest of the three in magnitude, at −0.46 per cent against 1.88 and 1.07, so a mismatch expressed as a fraction of it is being divided by the smallest available denominator. That much is arithmetic and would be unremarkable.

What is not arithmetic is ammonia and methane, where the mismatch exceeds a hundred per cent. A mismatch of more than a hundred per cent means the correction changed sign: ammonia’s parent has +1.4714 per cent on its two degenerate axes and its daughter has −0.9231 on one of them. Methane is the same shape, +1.6785 becoming −0.8790.

A near-cancellation between two quantities of opposite sign is not a near-cancellation. It is a sum, and the substitution method’s whole justification — that the systematic error is common to both species and subtracts away — has no purchase on a case where the error is positive in one species and negative in the other. Two of these four substitutions are that case.

The mechanism is the same one that produced the sign in the first place. A moment about an axis is a sum over atoms of their distances from it, some of which a given vibration increases and some of which it decreases; whether the sum comes out positive depends on which atoms are where relative to the axis. Move the axis — which is exactly what these substitutions do — and the balance can go the other way. The sign change and the frame rotation are one effect, not two.

The substitution that changes nothing #

Boron trifluoride is in this set because it does something none of the others can.

Its boron sits at the centre of mass and on every symmetry element the molecule has. Changing ¹¹B to ¹⁰B therefore changes no moment of inertia at all — the three differences come back as exactly zero, 0.000e+0, not as small numbers — because a mass at the centre of mass contributes nothing to any moment about any axis through it. The frame does not turn, because there is nothing to turn it.

That is the identity behind a warning every account of the substitution method gives: an atom at or near the centre of mass cannot be located by substituting it, because Kraitchman’s equations invert a change in the moments and there is no change to invert. Here it is not a warning. It is a column of zeroes, and it says that a coordinate reported for such an atom is a coordinate computed from arithmetic noise.

And now the surprise, which is the reason boron trifluoride is worth the column rather than merely worth the caution.

The rigid moments do not change and the spectrum does. ¹⁰BF₃’s rotational constants differ from ¹¹BF₃’s by 310 kHz on the two degenerate axes and 58 kHz on the third. Every one of those hertz is zero-point motion: the two isotopologues have identical equilibrium geometries and identical rigid moments, and different vibrational amplitudes, so their vibrationally averaged moments differ where their equilibrium ones do not. A microwave transition is measured to about a kilohertz. Three hundred and ten kilohertz is not near the noise; it is three hundred linewidths.

Methane’s is larger still. CH₃D’s threefold axis passes through the deuterium, so its moment about that axis is methane’s moment, identical to twelve figures — and its rotational constant about it differs from methane’s by 4,038 MHz, four gigahertz of pure zero-point difference on a rigid moment that did not move.

So the case where the rigid model predicts exactly nothing is the case where everything measured is the thing the rigid model omits. That is a better test of a zero-point correction than any of the cases where the correction is a small part of a large difference, because here it is the whole of the difference and there is no large number to hide inside. It is the same shape as an overlap that symmetry forbids: a prediction of exactly zero is a prediction that a residual measures one thing and one thing only.

Where there is no frame at all #

Methane’s parent is a spherical top, and a spherical top has no principal axes.

Its three moments are equal, so its inertia tensor is a multiple of the identity, and every orthogonal triad diagonalises it. The angles a diagonaliser reports between methane’s axes and CH₃D’s — 54.74°, 45.00°, 35.26° — are recognisable on sight as the tetrahedral angles of a coordinate triad aligned with the molecule’s own bonds. They are properties of the routine, not of the pair of molecules, and a different diagonaliser would return different ones.

The failure this makes possible is sharper than an undefined angle. Read CH₃D’s moments in methane’s arbitrary triad and they come out as three equal numbers, 3.9216 u Ų — the trace shared out equally, which is what any wrong frame gives for a molecule with a threefold axis. In its own frame CH₃D is 3.1757, 4.2945, 4.2945: a symmetric top. In the parent’s frame it is reported as a spherical top, which it is not, and the report is internally consistent, dimensionally correct and wrong.

The same shape appears in a permutation that collapsed a group to half its size with every element correct: the answer is not noisy, it is a different answer to a different question, and nothing in it looks wrong.

What was computed, and how #

The moments are eigenvalues of the inertia tensor built from the molecule’s own coordinates and a mass list, with the masses taken from the standard table and substituted by index. The axes are the corresponding eigenvectors, sorted by moment, which is the convention that gives the letters a, b, c their meaning.

The turn is computed twice, by two routes that share nothing. The first pairs each daughter axis with the parent axis it has the largest absolute dot product with and takes the arc cosine. The second expresses the daughter’s tensor in the parent’s frame — a similarity transform by the matrix of the parent’s eigenvectors — and reads the angle that would diagonalise the leading block off the off-diagonal entry. For water both give 21.122°, and the agreement is the check: the first route is a comparison of directions and the second never computes the daughter’s eigenvectors at all.

The zero-point corrections are the standard harmonic ones: a sum over modes of half the second difference of a principal moment with respect to that mode’s mass-weighted coordinate, times the mode’s mean-square amplitude, at a symmetric displacement of 0.05 in mass-weighted units so the first-order term cancels. Degenerate moments are averaged within their degenerate set at a relative tolerance of 10⁻⁵, for a practical reason: ammonia’s two equal moments differ in the seventh figure because the geometry is built from angles.

The rotational constants are $B = h/8\pi^2 I$ in the usual units, evaluated once on the equilibrium moment and once on the moment inflated by its computed correction. The difference between the two isotopologues is then taken in each case, which is why the rigid column can be exactly zero while the other is not — the two are not a value and an approximation to it, they are two different expressions that happen to agree when the molecule is not vibrating.

The degeneracy detection is a relative test on the parent’s moments and it is tested in the direction that would fail if it were removed: methane must be reported as degenerate, rather than merely reported with a small number. A version of this that checked the angle was small would pass for boron trifluoride and fail for methane, and would be checking the wrong thing in both cases.

Where the model stops #

Everything here is harmonic, and the corrections it compares are harmonic corrections. The mean bond length in a real ground vibrational state exceeds the equilibrium one for a reason a quadratic potential has no term for, and that effect is common to both isotopologues in the same qualitative way the harmonic one is. Whether it cancels better or worse between them is not answerable in this model, and it is the obvious next thing to ask.

The frame rotation, by contrast, is not a model result at all. It follows from the definition of a principal axis and a table of masses, and it would be the same number computed at any level of theory, from any geometry, with any force field or none. That is worth separating clearly: this essay contains one exact geometrical statement and one approximate physical one, and the exact one is the one that undermines the method.

Four molecules is four molecules, and three of them have a symmetry axis. The two hydrogens of water are related by its twofold axis, so both single substitutions give the same daughter and the census has four rows rather than eleven. A molecule with several inequivalent hydrogens would give several different turns, and whether the turn correlates with anything — the distance of the substituted atom from the centre of mass, most obviously — is not answerable from four cases.

And the numbers about the parent’s axes are exact statements about a hypothetical. No spectrometer measures a molecule’s moment about another molecule’s axis. What that column bounds is the size of the error in treating the two as comparable, which is what a per-axis analysis does whether or not it says so.

The generalisation #

Two things here are not about moments of inertia.

A quantity indexed by rank is not a quantity indexed by identity. The three principal moments are ordered by size and named by position in that order, so a change that reorders them, or that moves the lines they belong to, silently changes what each name refers to. Nothing in the notation records it. The same failure appears in a set of orbital energies compared between two systems, where the third-lowest level of one atom is not the same orbital as the third-lowest of another, and in a shell whose extra degeneracy no group predicts, where two levels sharing a number is a fact about a potential rather than about a label. Sorted output is a convenience that destroys an identity, and the destroyed identity is usually the one being reasoned about.

And a degenerate spectrum has a basis nobody chose. When two eigenvalues coincide, the eigenvectors are not determined and whatever a routine returns is its own arbitrary choice inside that subspace. Reading structure off those vectors is reading the routine. The safe form is the one used here: detect the degeneracy and refuse the question, rather than answer it with the number that comes back. The same discipline is what makes a degenerate pair of vibrational modes reportable at all, where the individual shapes are arbitrary and only their span is not.

Who found it, and when #

Kraitchman’s equations are from 1953 and the caution about atoms near the centre of mass is as old as their use; Costain’s discussion of substitution structures in 1958 is where the practical rules come from. That the principal axes rotate under substitution is not a discovery — it is why the equations are written in terms of coordinates in the parent’s principal-axis system rather than the daughter’s, and any careful account says so.

What is easy to lose, and what this essay exists to state as a number, is that the rotation is large. Twenty-one degrees is not a correction. The literature’s habit of writing the moments as $I_a$, $I_b$, $I_c$ for both species encourages exactly the comparison this essay refuses, and it is easy to make in good faith, having computed the corrections per axis and lined them up by index.

The zero-point isotope shift in a molecule whose rigid moments are identical is standard in high-resolution work, where it appears as the difference between $B_0$ and $B_e$ for isotopologues of the same equilibrium structure. Computing it here from a force field, and finding it at three hundred linewidths for boron trifluoride, is a demonstration rather than a discovery.

Still open: the anharmonic term, and boron trifluoride’s shift #

The obvious open question is the anharmonic term, which is now the only contribution to a vibrationally averaged moment with no expression here and the one every result above is exposed to. It is not a small correction on top of the harmonic one: for a bond it goes as the mean displacement rather than the mean square, which is first order in the anharmonicity and enters a reciprocal moment with the opposite sign. A diatomic is where that can be computed exactly, and Morse potentials can be solved from measured constants.

The nearer question is the one boron trifluoride poses, which is only pointed at here. If the entire isotope shift there is zero-point, then measuring it measures the zero-point correction directly, with no structural difference to subtract — and the ratio of the measured shift to the computed one is a test of a harmonic force field of a kind no frequency fit provides, because a force field fitted to frequencies is not fixed by them. Two isotopologues of one planar molecule would give it, and the number to beat is 310 kHz.