The coordinate an isotope reports

Three numbers is not a structure ended badly for the fitting approach and said what the field does instead.

Formaldehyde has three structural parameters and two independent rotational constants, because a planar molecule’s third moment is the sum of the other two. Fitting is therefore two equations in three unknowns, and the solutions are a curve: a family of structures whose C=O lengths span more than 0.2 Å and whose HCH angles span more than forty degrees, every one of them reproducing both measured constants to the last digit the solver carries. A second isotopologue picks out one member, and the measured constants themselves fail the planarity relation that every member satisfies exactly — so the fit is fitting an excluded model.

The honest continuation was named there: Kraitchman’s equations get one atom’s coordinates from the change in the moments when that atom alone is substituted, and the reason they work better than a global fit is that the vibrational contributions to the two sets of moments are similar and largely cancel in the difference.

This essay computes both halves of that sentence. The first half is exactly right and is worth seeing. The second half is true, is measurable, and does not do what the sentence implies.

The change of variable that makes it possible #

The three principal moments are second moments about axes. The planar moments are second moments about planes:

$$P_a = \sum_i m_i a_i^2 = \tfrac{1}{2}(-I_a + I_b + I_c),$$

and the other two by permutation. The change of variable is linear and takes two lines, and it is the whole trick: substituting one atom changes its mass and therefore changes $\sum m a^2$ by a term in that atom’s own $a$ alone. No $I$ has that property, because every $I$ mixes two coordinates.

From there Kraitchman’s result follows:

$$a^2 = \frac{\Delta P_a}{\mu}\left(1 + \frac{\Delta P_b}{I_a - I_b}\right)\left(1 + \frac{\Delta P_c}{I_a - I_c}\right)$$

with $\mu = \Delta m\,M/(M + \Delta m)$ the reduced mass of the substitution, and the two partners by cyclic permutation. The result is in the parent’s principal axis system, which is the part that looks like magic and is not: the substituted species has its own axes, tilted and shifted, and the correcting factors do the transformation implicitly.

It is an identity, and that is the check #

For a rigid structure the equations are exact. Not accurate — exact, in the sense that a triangle’s angles sum to two right angles is exact — and testing that is the only check available that does not use the same formulae twice.

Formaldehyde’s four atoms are each substituted in turn: carbon to ¹³C, oxygen to ¹⁸O, and each hydrogen to deuterium, which breaks the molecule’s symmetry and makes it a genuinely asymmetric top with no two moments alike. The recovered coordinates agree with the true ones to better than a part in ten million, for every atom and every axis.

That agreement is what makes everything after it a statement about the method rather than about an implementation.

Every sign is lost #

The first cost is visible in the figure and is not a numerical matter at all.

Formaldehyde’s true coordinates in its own axes are: carbon at a = −0.6023, oxygen at +0.6010, the two hydrogens at a = −1.1831 and b = ±0.9348. What comes back is 0.6023, 0.6010, 1.1831, 0.9348 — six non-negative numbers where there were six signed ones.

For a molecule with a known connectivity the signs are usually obvious: an oxygen and a carbon on opposite sides of the centre of mass have to have opposite signs, and two equivalent hydrogens have to be mirror images. But usually is doing real work. Any structure and its mirror image give identical moments for every isotopologue, so a substitution structure cannot be chiral without an external decision. And a molecule with two chemically distinct atoms at nearly the same distance from a principal plane leaves a genuine ambiguity that no further substitution resolves.

Costain’s case, which arrives without being invited #

The second cost is sharper and it is the one the method is remembered for.

Every atom of formaldehyde lies in the molecular plane, so every out-of-plane coordinate is exactly zero. Feed the equations exact moments and they return zero, as they must. Feed them moments carrying a fractional error of one part in a hundred thousand — which is far below the error a real vibrational average introduces — and every one of the four $c^2$ values comes back negative.

That is not a loss of precision. A small coordinate does not come back as a small number with a large uncertainty; it comes back as an impossible number. The literature’s remedy is Costain’s rule — set such a coordinate to zero and accept the error — and the reason the remedy is a rule rather than a calculation is that there is nothing in the equations to interpolate between a real answer and an imaginary one.

Note what makes it happen. $\Delta P_c$ for a planar molecule is exactly zero, so the leading term is a difference of two numbers that are equal; the fractional error in that difference is the absolute error in either divided by nothing. Any coordinate small compared with the molecule has the same problem in weaker form, which is why the method is trusted for atoms far from the axes and not for atoms near them. The constant a spectrum cannot see is the same shape of failure one level up: a quantity that is present in the model and absent from the measurement.

The cancellation, isolated and measured #

Now the claim the method is defended with.

A measured rotational constant is not a rigid molecule’s; it is an average over the zero-point motion, and the difference is a few tenths of a per cent — enormously larger than the precision of the measurement. That is why a structure fitted directly to the constants is not the equilibrium structure. The defence of the substitution method is that the vibrational contributions to two isotopologues of one molecule are nearly the same, so they cancel in a difference.

The model of that error here has the two parts the real thing has. Each moment is inflated by its own fraction, ε times a fixed weight, because the zero-point contribution is not the same about every axis and a uniform inflation would be a dilation of the molecule rather than an error in it. And the substituted species gets almost the same inflation — the same fractions times 1.05 — because deuterating a molecule changes its zero-point amplitudes a little and not much.

Isolating the cancellation is then a matter of running the same calculation with the two errors independent and taking the ratio.

A factor of 13.2, and it is real. It is also close to what the model puts in: the mismatch between the two species’ errors is five per cent, and one over five per cent is twenty, so the method recovers most of what a perfectly correlated error would give.

And it is not enough #

The last column of that table is the one that decides.

A structure fitted straight to the moments inherits about half of ε, because a moment goes as the square of a length. At ε = 10⁻³ that is 0.0405 per cent. The substitution structure’s worst coordinate at the same ε is 0.1016 per cent — 2.51 times worse.

The reason is the amplification the defence leaves out. Kraitchman’s equations are driven by $\Delta P$, the change in a planar moment, and for a light substitution that change is a small fraction of the moment. An absolute error of εP in each of two moments is therefore a relative error of order εP/ΔP in their difference, and here that ratio is around thirty. Thirteen of cancellation against thirty of amplification leaves a factor of two and a half the wrong way.

The remedy is in the numbers too, and it is the one the field actually uses. Sorting the four substitutions by how well their coordinates survive puts oxygen first — ¹⁶O to ¹⁸O, a mass change of 2.004 — and carbon last, at 1.003. The atom whose coordinate survives best is the one whose mass changed most, because a larger Δm makes a larger ΔP and reduces the amplification. Substitution structures are quoted for heavy atoms with large isotope shifts and treated with suspicion for hydrogens, and this is the arithmetic behind that practice.

The other route inverts the moments of two isotopologues directly, which is what a bond length out of a spectrum does for a linear molecule. It needs fewer substitutions and gives a length rather than a set of coordinates, and its conditioning is the same problem in a different basis.

What this does to the fitted family #

The family of fitted structures is not repaired by any of this, and it is worth being clear about which problem each method solves.

The family exists because two measurements cannot fix three parameters. The substitution method escapes it not by adding information about the molecule but by adding measurements: one isotopologue per atom, each contributing three numbers. Four atoms substituted gives twelve moments where the parent had two independent ones, and twelve is comfortably more than three.

What it does not escape is the model. Every moment used is a rigid molecule’s, and the measured ones are not — which is the refusal the fitted family ended on, and it applies to the substitution structure exactly as it applied to the fitted family. A structure built out of vibrationally averaged moments by exact equations for a rigid rotor is not an equilibrium structure, and the amount by which it is not is what has been measured here.

A measured length is not an equilibrium length, because the vibrational average depends on the isotope. Every moment here is an average over a zero-point distribution, which is exactly why the correction above has to exist.

What twelve numbers buy #

It is worth counting the information, because the substitution method is often described as though it added something qualitatively new and it does not.

Formaldehyde’s parent gives three moments, of which two are independent. Each of four singly substituted species gives three more, of which two are independent because each is also planar. That is ten independent numbers for six coordinates — four atoms in a plane, less three for the centre-of-mass conditions and one for the orientation — so the problem is comfortably overdetermined where the parent’s alone was underdetermined.

What the equations do with that surplus is refuse to use it. Each atom is located from its own substitution and from nothing else, so the ten numbers are consumed four at a time with no cross-checking at all. That is the method’s defining feature — each atom stands alone, so an error in one substitution does not spread — and it is also why the centre-of-mass conditions come out only approximately satisfied on real data, which is the standard diagnostic that something has gone wrong.

What a measurement of these moments actually consists of is a pattern of asymmetric-top levels fitted to three constants per isotopic species. Every number here is downstream of such a fit, and the fit’s own uncertainty is smaller than the correction being argued about.

What this cannot say #

The error model is a model. A real vibrational correction depends on the force field, on which mode is excited, and on the anharmonicity, and it is not a fixed fraction of each moment. What the model has right is the structure — a systematic error, correlated between isotopologues, with a small mismatch — and the factor of thirteen is a property of the mismatch chosen, not a measurement of formaldehyde.

One molecule, four substitutions. A molecule with an atom close to the centre of mass has an even worse amplification and a molecule with heavy atoms far out has a much better one; nothing here surveys that range.

No measured constants for the substituted species. Real singly deuterated formaldehyde has been measured; nothing here uses those numbers, because the point is the method rather than the molecule. The top that reports all three is where the measured asymmetric-rotor constants do get used.

No symmetric tops. Kraitchman’s equations need three distinct moments, and a top with two equal ones is refused outright rather than answered. That is a real limitation of the method and not of this implementation: a symmetric top has no unique a and b, and a routine returning numbers there would be reporting its own arithmetic.

And no measured moments are used. The parent structure is the accepted one and every moment is computed from it, so what is being tested is the method rather than formaldehyde’s geometry.

What is done with an imaginary coordinate #

An out-of-plane coordinate coming back imaginary is a failure with a standard handling, and the handling is worth stating because it builds an assumption into the answer without announcing it.

The equations return the square of a coordinate. For an atom genuinely in a principal plane the true value is zero, the computed square is a small number of either sign depending on which way the measurement’s error fell, and half the time it is negative — at which point taking its square root gives an imaginary coordinate.

Nobody reports an imaginary coordinate. The universal practice is to set it to zero, which is very nearly always the right answer, because an atom whose square coordinate comes out at the noise level is an atom in the plane.

The awkwardness is what happens when it is not. An atom sitting genuinely a little out of the plane — a hydrogen on a nearly-planar amine, a substituent tilted by a few degrees — has a small but real coordinate whose square is comparable to the error, and the same procedure applies: the value comes back negative as often as not, it is set to zero, and the atom is placed exactly in a plane it is not in.

So the method’s failure mode is not a wrong number. It is a structural claim — that the atom is planar — produced by a convention for handling a negative square, and the claim is made in exactly the cases where planarity was the question.

That is worth putting beside the inertial defect, which answers the same question and does not have this difficulty. The defect uses the moments directly rather than the difference between two sets of them, so a small departure from planarity shows as a small negative number rather than as an imaginary coordinate that gets rounded away.

Two methods, one question, and only one of them has a convention that answers it in advance.

What was checked #

The equations return the true coordinates for a rigid structure, for every atom and every axis, to a part in ten million — the identity everything else rests on.

The true structure has coordinates of both signs and every returned coordinate is non-negative, both halves checked together, so that the claim about lost signs is not vacuous.

A symmetric top is refused rather than answered, which is the tripwire that the correcting factors are not being divided by zero silently.

Correlated error is worth several times independent error, at four fractional errors, with the factor stable across two orders of magnitude.

Every out-of-plane coordinate is imaginary at every non-zero error, which is Costain’s case appearing without being provoked.

And the refusal: the cancellation does not make the method more accurate than a direct fit here, by a factor of 2.51 the wrong way — with the atom whose mass changed most having the best-surviving coordinate, which is the arithmetic that says when the method is worth using.

Still open: double substitution, and a computed correction #

The obvious open question is the double substitution. Substituting two atoms at once gives a difference of coordinates rather than each separately, and the differences have a different amplification — a two-atom substitution changes ΔP by more than either one alone, which is exactly the quantity the amplification divides by. Whether a structure built from pairs is better than one built from singles is a question with an arithmetic answer, and it is the construction behind the mass-dependent methods the field now prefers. A spectrum counts environments, not atoms is the reason a pair substitution is often the only one available: two equivalent atoms cannot be substituted one at a time in a real sample.

The nearer question is the one the error model here is standing in for. A vibrational correction to a moment is computable from a force field, and force fields exist for several molecules of exactly the kind used here. Computing the real correction for formaldehyde and its four singly substituted species — rather than modelling it as a fixed fraction with a mismatch — would replace the one invented number here with a calculated one, and would say whether the factor of thirteen is generous or mean.