Three numbers is not a structure

A bond length can be pulled out of a rotational spectrum, and for a diatomic it works: one line, one moment of inertia, one distance.

That worked because a diatomic has one structural parameter and a spectrum gives one number. This essay is about the arithmetic when it does not work out so evenly, and formaldehyde is the smallest molecule where it does not.

The counting #

An asymmetric top has three principal moments of inertia and its spectrum reports all three, which an asymmetric top’s spectrum shows: break the symmetry and the constant a symmetric top hides becomes measurable.

Three numbers. And formaldehyde has three structural parameters — the C=O length, the C–H length and the HCH angle — so the count looks exactly right.

It is not, because formaldehyde is planar, and for a planar molecule the moment about the axis perpendicular to the plane is the sum of the other two. That relation is exact for any rigid planar structure, it is the planar sum rule, and it means the third constant carries no information the first two do not.

So: two numbers, three parameters, and a curve of structures rather than a structure.

The curve #

These are not near-misses or structures that fit within experimental error. Each one reproduces both constants to nine decimal places, and the third constant comes out at 1.1385 for every member — not because it was fitted, but because planarity fixes it once the other two are set.

The range is chemically enormous. A C=O length of 1.000 ångström is shorter than any known; 1.300 is longer than a single C–O bond. An HCH angle of 74.6 degrees is impossible and 164.5 is nearly linear. Every one of them fits the spectrum.

What a chemist does about it #

The standard answer is isotopic substitution: make the same molecule with a different mass distribution, measure its spectrum, and get more numbers.

Deuterating both hydrogens gives a molecule with the same structure and different masses, hence different moments, hence two more measured constants.

That second sentence is the finding here.

The prediction that is not a prediction #

The deuterated molecule’s A constant is the same for every structure in the family, to nine decimal places, and the reason is worth following.

A is the moment about the axis through the C and O atoms. The carbon and oxygen sit on that axis and contribute nothing; the two hydrogens contribute $2m_H y^2$, where $y$ is their perpendicular distance from it. So A depends on the hydrogens’ perpendicular distance and on nothing else about the structure.

Fitting A for the light molecule fixed $y^2$. Deuterating changes $m_H$ to $m_D$ and leaves $y$ alone. So

$$\frac{A(\mathrm{D_2CO})}{A(\mathrm{H_2CO})} = \frac{m_H}{m_D} = 0.50038$$

for every member of the family, and 9.4053 × 0.50038 is 4.7063.

A measurement whose value is a mass ratio times another measurement is not a new constraint. It is worth making — it checks the assignment, it checks the masses, and a disagreement would mean something was wrong — but it adds nothing to the structure determination, and a count of “two new numbers per isotopologue” is out by one for this molecule.

The general rule behind that is worth stating: substituting an atom that lies on a principal axis adds nothing about that axis. It is why structural work substitutes off-axis atoms where it can, and why some molecules are much harder to pin down than their parameter counts suggest.

What the second isotopologue does say #

The B constant does vary across the family — from 0.917 to 1.182, and the measured value of 1.1215 picks out the member at a C=O length of 1.2495 ångström, a C–H length of 1.0222 and an HCH angle of 134.6 degrees.

The accepted structure of formaldehyde is 1.2033, 1.1005 and 116.3.

So the procedure works, in the sense that it returns a unique answer, and the answer it returns is wrong by four hundredths of an ångström in one bond and eighteen degrees in the angle.

Why it is wrong #

The fault is not in the arithmetic and it is diagnosable from the measurements alone.

Every structure in the family is planar and rigid, so every one satisfies $I_c = I_a + I_b$ exactly. The measured constants do not: converting them to moments gives an inertial defect of 0.057 u Ų, small but far outside the measurement’s precision.

A rigid planar structure cannot have a non-zero inertial defect. So there is no rigid planar structure that reproduces the measured constants, and every fit of one to them is fitting a model the data has already excluded.

What the constants actually describe is a molecule that is vibrating. The measured rotational constant of a vibrational state is an average over the motion in that state, and averaging a moment of inertia over a bond that is anharmonic gives something that is not the moment of any single geometry — the same phenomenon that gives a molecule three different bond lengths at once depending on which average is taken.

Deuterating the molecule changes the amplitude of that vibration as well as the masses, so the two sets of constants correspond to two slightly different average structures — and fitting one rigid structure to both is fitting a compromise that need not be near either.

What would have to be measured #

The counting suggests its own remedy, and it is the one the field uses.

Three parameters need three independent numbers. The light molecule supplies two. One more independent measurement finishes the job, and it has to be one that depends on the parameters differently from the first two — which the deuterated A does not, and the deuterated B does.

Substituting the oxygen would work better still. Oxygen sits on the a axis, so ¹⁸O leaves A unchanged and changes B and C; substituting one hydrogen at a time — making HDCO, which has no symmetry at all — changes all three and locates that hydrogen’s coordinates directly.

So the practical recipe is not “measure more isotopologues” but measure isotopologues that move the atoms in question off the axes they sit on, and the counting above says which those are before any spectrum is recorded.

The three structures a spectroscopist quotes #

The literature’s response to all of this is not to pretend the difficulty away but to name three different structures, and the names are worth knowing because they are what the numbers in tables mean.

$r_0$ is what comes out of fitting a rigid structure to ground-state constants — the calculation above, with its inconsistencies. It is easy, it is common, and it depends on which isotopologues were used.

$r_s$ is the substitution structure, from Kraitchman’s equations: the change in the moments on substituting one atom locates that atom’s coordinates, and the errors of the vibrational averaging partly cancel in the difference. It is the standard method and it is better, and it needs one isotopologue per atom.

$r_e$ is the equilibrium structure, the minimum of the potential, which is what a calculation computes and what nothing measures directly. Getting to it needs constants for several vibrational states and an extrapolation.

The three differ by hundredths of an ångström, which is the same size as the disagreement above, and a bond length quoted without saying which one it is carries that ambiguity.

Where the principal axes fall is what decides which substitution helps. Formaldehyde’s a axis lies along the C=O bond — the top that reports all three draws it on the molecule — so substituting the hydrogens moves mass perpendicular to that axis and says nothing new about it, while substituting the oxygen moves mass along it and would.

The family is a curve, not a cloud #

One feature of the family is worth remarking on because it makes the underdetermination sharper rather than vaguer.

The solutions are not a fuzzy region of parameter space within some tolerance. They are a one-dimensional curve: pick a C=O length anywhere in the range and there is exactly one C–H length and one angle that go with it, found by Newton’s method in a few steps and exact to the solver’s precision.

That is what two equations in three unknowns gives, and it means the ambiguity has a definite shape. Along the curve the parameters trade against each other in a fixed way — lengthening C=O shortens C–H and opens the angle — because both of the fitted moments depend on all three, and holding two moments fixed leaves one direction free.

A structure quoted from such a fit is therefore not “the structure to within an uncertainty”. It is one point chosen from a curve, and what chose it was usually a starting guess close to the expected answer. That is a different kind of error from an experimental one and it does not shrink when the measurement improves.

The generous case is worth naming beside it. A symmetric top has two independent constants and ammonia has two structural parameters, so its bond length and angle are determined by its spectrum exactly with nothing left over — which is why they are known to five decimal places and formaldehyde’s are not. A spectrum of a symmetric top is where those two constants come from.

How much this generalises #

The counting is easy to do for any molecule and it is discouraging.

A molecule of $N$ atoms has $3N - 6$ structural parameters and three moments, so a spectrum of one isotopologue determines the structure only for a diatomic. Symmetry helps enormously: a symmetric top has two independent moments and often only two or three parameters, and formaldehyde’s C₂ᵥ symmetry is what reduces nine parameters to three.

Planarity hurts, by removing one of the three constants. So the molecules whose structures are hardest to determine are the planar ones with several parameters, and the ones that are easiest are the small symmetric non-planar ones — which is why ammonia’s structure is known to five decimal places and a substituted benzene’s is not.

A table of asymmetric tops tabulates the rotational constants of five molecules against the asymmetry parameter each corresponds to, and the column this essay cares about is not in that table: the count of independent constants, one or two or three. It is decided by the symmetry and never by the spectrometer, so a better instrument does not move it.

Where the model stops #

Everything here is rigid. The whole essay is about the failure of that assumption and it does not repair it; repairing it means computing the vibrational averages, which needs a force field and is what the diatomic calculation does.

Only two isotopologues appear. A real structure determination uses one per atom, and the redundancy that provides is what makes the substitution method work.

The fit is exact rather than least-squares. Two constants and two unknowns give an exact solution at each C=O length, so there are no residuals to look at — which is convenient for the argument and unlike real practice, where the residuals are the first sign of trouble.

And centrifugal distortion is ignored, though it is the next term and is measured routinely; it is a subject of its own and would add parameters to the fit rather than constraints.

The vibrational averaging this essay blames is computed elsewhere on the simplest system that has it: the bond length that depends on the isotope puts two isotopologues on one identical potential curve and finds average bond lengths differing by 4.34 thousandths of an ångström. Nothing about the potential differs; the heavier isotopologue simply sits lower in the same well. That difference, in a polyatomic with several bonds, is what a rigid fit to two isotopologues is trying to absorb, and the 7.1 milliangstrom above is what it fails to absorb.

And what is actually measured, before any of this, is a series of lines whose spacing is the rotational constant — a bond length out of a spectrum draws three of them. Everything in this essay is downstream of extracting one number per molecule from a picture like that, and the extraction is the part that works.

What a structure determination is really doing #

Set out plainly, the procedure has four steps and only the first is a measurement.

Measure the line positions, which is done to a part in ten million and is the reason microwave spectroscopy is the most precise structural method there is.

Fit rotational constants to them, which needs a model of the rotor — rigid, or with distortion terms, or non-rigid — and is where the first choices enter.

Assume a rigid structure and fit its parameters, which is where this essay’s difficulty lives: the count may be short, the model is known to be excluded, and the answer is a point on a curve.

And quote a bond length, which by now is three or four steps from anything measured and carries a superscript naming which of the three structures it is.

None of that is a criticism of the field, which has thought about every one of the steps for eighty years and has the superscripts to prove it. It is a description of the distance between a measurement of extraordinary precision and a number a chemist reads as a distance — and that distance turns up wherever one looks, whether the quantity is a moment from a fitted law or a mode’s percentage composition.

What is quoted, and what is computed #

Quoted: formaldehyde’s and deuterated formaldehyde’s measured rotational constants; the accepted structure; the atomic masses; the names and definitions of the three kinds of structure.

Computed: every member of the family, by Newton’s method on the two constants; every moment of inertia, from the coordinates and masses; every predicted deuterated constant; the inertial defects; and the member the deuterated B picks out.

The two repairs, and which one a published structure used #

Seven structures reproducing the constants exactly is an underdetermination, and there are two standard responses to it. Both are used, they produce different kinds of number, and a published structure rarely says which was applied.

Add measurements. Each isotopologue supplies a fresh set of constants against the same structural parameters — new equations, no new unknowns — and enough of them determine the structure outright. That is the honest repair, it is expensive because each isotopologue has to be synthesised and its spectrum measured, and it is what a careful structural study does.

What it does not do is make the answers agree, and two pairs are enough to show that.

So the honest repair fixes the counting and leaves the model error where it was. Enough isotopologues turn an underdetermined problem into an overdetermined one, and an overdetermined problem with a wrong model does not have a solution at all — which is why the substitution structure exists and why its equations are differences rather than fits.

Assume parameters. Fix the C–H length at a value transferred from a related molecule, or fix an angle at a tetrahedral value, and fit only what is left. That is cheap, it is extremely common, and it converts an underdetermined problem into a determined one by supplying the missing information from somewhere the spectrum knows nothing about.

The second is not illegitimate and it is not always signposted. A structure reported as determined from the rotational constants may have had two of its parameters supplied by assumption, and the quoted uncertainties are then the uncertainties of a fit conditional on those assumptions — which are much smaller than the uncertainties of the parameters themselves.

That matters because the assumption is usually the thing being tested. A study asking whether a C–H bond in some environment is unusual, having fixed the C–H bond at a transferred value, has answered a different question — and the seven structures here span 1.557 to 0.952 ångström in exactly that parameter, which is the range the spectrum leaves open.

So the practical rule for reading such a structure is the familiar one for fitted structures, stated for a new quantity. Ask how many independent constants went in and how many parameters came out. If the second exceeds the first, something was assumed, and the interesting question is what.

There is a third response that is better where it is available, and it is neither of the two. Report the family rather than a member of it. The seven structures here are not seven guesses at one answer; they are a one-parameter set, every member of which reproduces the measurement exactly, and describing the set is a complete statement of what the spectrum determined.

What that costs is a number to quote, which is why it is rarely done. A paper reporting the C=O length lies between 1.000 and 1.300 ångström, correlated with the C–H length as follows has said everything the data support and has produced nothing a table can hold — and a table is what a structural result is expected to be.

What the family requires #

Every member of the family reproduces both fitted constants to nine decimal places, and is planar, so its third moment is the sum of the other two.

The third constant is the same for every member, because it was never free.

The structures in the family are not nearly the same molecule — the C=O length spans more than 0.2 ångström, the C–H more than 0.3, and the angle more than 40 degrees.

Every member predicts the same deuterated A, and it is the light molecule’s A times the mass ratio, to five parts in a thousand.

While the deuterated B does vary, by more than 0.2 wavenumbers, so the second isotopologue settles something.

And the refusal: the measured constants themselves fail the planarity relation that every structure in the family satisfies exactly — so no rigid planar structure reproduces them, and the fit above is fitting an excluded model.

Still open: the substitution method #

The honest open question is the substitution method, because it is what the field actually does and because its trick is worth computing rather than describing.

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. That is error cancellation, in the same sense two wrong numbers make a right difference, and it can be measured here in the same way: compute the vibrational average for a model potential, substitute, and see how much of the error survives the subtraction.