The rotor that stretches

The rotational spectrum is a moment of inertia turns a microwave spectrum into a structure in three steps: every line sits at $2B(J+1)$, so the spacing is $2B$; $B$ is $h/8\pi^2cI$; and $I$ for a diatomic is $\mu r^2$. Three lines of arithmetic, one bond length.

The first of those three steps assumes the molecule is rigid, and no molecule is. A rotating object is pulled outward by its own rotation, the bond stretches — against a force constant the frequency is not the bond strength is careful to distinguish from a bond energy — the moment of inertia grows and the lines crowd together instead of staying evenly spaced.

This essay is about how much, and the answer is not a fitting parameter.

The relation #

Balance the centrifugal force against a harmonic bond and the correction to each line comes out as

$$\tilde\nu(J) = 2B(J+1) - 4D(J+1)^3, \qquad D = \frac{4B^3}{\omega^2}$$

with everything in wavenumbers. The cube is what makes it grow so fast up the series of lines, and the second expression is the interesting one: the distortion constant is fixed by the rotational constant and the vibrational frequency, both of which are measured, neither of which is a microwave measurement of a distortion.

So this is a prediction about a microwave spectrum made from an infrared one. The two experiments use different apparatus, sit in different parts of the spectrum, and are usually done by different people.

The agreement is best for hydrogen fluoride — $2.149 \times 10^{-3}$ against $2.151 \times 10^{-3}$, one part in a thousand — and worst for hydrogen chloride at four per cent. Both of those are hydrides, so the difference is not a mass effect; it is anharmonicity, and the sign is right for it, as the last section explains.

What a rigid reading does to a bond length #

The distortion is small: for carbon monoxide $D$ is $6 \times 10^{-6}$ cm⁻¹ against a $B$ of $1.92$, a ratio of three parts in a million. The reason it matters is the cube.

At $J = 40$ the correction is $4D(J+1)^3 = 4 \times 6.1 \times 10^{-6} \times 68921$, which is $1.7$ cm⁻¹ against a line position of $154$ — one per cent. And the spacing between neighbouring lines, which is what a rotational constant is read from, has shrunk by rather more than that.

Read the spectrum as a rigid rotor and the bond length comes out differently depending on which part of the spectrum is read:

The molecule did not change between those readings. What changed is which part of its own spectrum was used, and the drift is one-way: every step up in J gives a longer bond. A random error would not do that, which is the signature that separates a systematic error from a measurement one.

And the repair, which is one more parameter #

Fit both constants at once — $B$ and $D$ together, which is still linear and still one solve — and the whole series of lines is described exactly. The fit returns the $B$ and the $D$ the lines were generated from, to twelve decimal places, with no residual left at all.

That is what a spectroscopist does, and it is why published rotational constants come with a $D$ beside them. What is worth noticing is the shape of the situation: a one-parameter model fitted to a limited range gives an excellent fit and a biased parameter, and the bias is invisible from inside the fit. The residuals of the rigid fit at low $J$ are tiny; it is only when the range is extended that the model’s inadequacy becomes visible, and by then the parameter has already been quoted.

The force field is not in the spectrum is the same lesson from the vibrational side: more constants than the data can determine there, and here fewer constants than the data requires. Both are about the relation between what is measured and what is fitted, and neither is about the molecule.

Which bond length is the bond length #

There are three and they are all different, and the distinction is not pedantry.

$r_e$ is the distance at the minimum of the potential. It is what a calculation computes and what nothing measures.

$r_0$ is what a rotational spectrum gives after the distortion has been taken out properly: the effective distance in the vibrational ground state. It is longer than $r_e$, because a molecule in its lowest vibrational state is not sitting at the minimum — it is spread over a range of the potential, and the potential is steeper on the inside than the outside, so the average is displaced outward.

And the value a rigid fit returns is longer still, by the amounts in the table, and depends on the experiment.

The gap between the first two is anharmonicity and it is where the four per cent disagreement in the table above comes from. The relation $D = 4B^3/\omega^2$ assumes a harmonic bond; a real bond is softer than harmonic at long range, so it stretches slightly more under rotation than the harmonic estimate says, and the fitted $D$ comes out slightly larger than the predicted one — which is what happens for all four molecules here.

Four out of four in the same direction, on a prediction that could have erred either way, is the check that the residual disagreement is anharmonicity rather than noise.

What this does to the structure determination #

A bond length out of a spectrum determines two bond lengths in a linear triatomic from two isotopologues, by inverting a pair of moments of inertia. That argument used $B$ values and it is worth asking what centrifugal distortion does to it.

The good news is that the effect is largely common. Isotopic substitution changes the reduced mass and barely changes the potential, so the two isotopologues have distortion constants in a fixed ratio, and a structure determined from the difference of two moments is less affected than either moment alone.

The bad news is that it is not entirely common, and the residual is exactly the size of the disagreement between structures determined from different isotopic pairs — which is the thing that determination reported as its own uncertainty. A spectrum that changes when only a mass does is the vibrational half of the same isotopic argument. So the caution belongs where that essay put it: the numbers agree to about a thousandth of an ångström, and a claim at the ten-thousandth level would be a claim about the corrections rather than about the structure.

The size of the effect, in one sentence per molecule #

The distortion constants above span three orders of magnitude and the reason is worth reading off the relation rather than looking up.

$D = 4B^3/\omega^2$ is large when the molecule rotates fast — a big $B$, which means a small moment of inertia — and small when it vibrates fast, which means a stiff bond. Hydrogen fluoride has both a very large $B$ and a very large $\omega$, and the cube beats the square: $D$ comes out at $2 \times 10^{-3}$, the largest here. Nitrogen has a small $B$ and a stiff bond and comes out at $6 \times 10^{-6}$, the smallest.

The practical consequence is that the light hydrides are where a rigid reading fails first. For hydrogen fluoride the correction at $J = 10$ is already three per cent of the line position; for nitrogen it is three parts in ten thousand. So the molecules whose rotational spectra are easiest to observe — light, fast-rotating, widely spaced lines — are exactly the ones where the rigid model is worst, which is a nice inversion of the usual relation between how easy a measurement is and how well a model describes it.

Why the effect was worth predicting rather than fitting #

There is a reasonable objection to this whole exercise: the distortion constant is measured to six figures by the same spectrum it corrects, so what is a prediction worth that agrees to only two?

Three things, and they are the reasons this site prefers a computed number to a fitted one wherever it can get away with it.

It says the two experiments are about the same object. A microwave spectrum measures how a molecule turns and an infrared spectrum measures how it vibrates, and the relation between their leading corrections says the bond doing the stretching under rotation is the bond that vibrates. That could have come out false — for a molecule with a low-lying excited state, or a floppy internal coordinate, it does — and where it does, the disagreement is the finding.

It bounds what a fit can hide. A two-parameter fit will absorb almost anything into its second parameter, including an instrumental artefact. Knowing what $D$ should be to a few per cent means a fitted $D$ that comes out at twice the prediction is evidence of something rather than a number.

And the residual disagreement is itself a measurement. All four molecules here have a fitted $D$ larger than the predicted one, by between one part in a thousand and four per cent, and the direction is fixed by the bond being softer than harmonic at long range. A prediction that agreed exactly would have said nothing about anharmonicity; one that disagrees in a consistent direction measures it.

Where the correction has been used as a measurement #

The distortion constant is a nuisance parameter in a structure determination and a signal somewhere else, and the difference is worth a paragraph.

$D = 4B^3/\omega^2$ can be read in either direction. Used forwards it corrects a rotational spectrum. Used backwards it gives $\omega$ — the stretching frequency — from a microwave spectrum alone, without any infrared measurement at all.

That is a real technique and it matters for molecules whose vibrational spectrum is hard to get: a transient species in a discharge, a weakly bound complex, a molecule present at concentrations a microwave spectrometer can see and an infrared one cannot. The rotational spectrum is often far easier to observe, its lines are sharp, and the distortion constant falls out of the same fit that gives the structure.

The accuracy is what the table in this essay measures. Inverting the relation for carbon monoxide’s fitted $D$ gives an $\omega$ about seven-tenths of a per cent low, and for hydrogen chloride about two per cent — good enough to identify a bond, not good enough to compete with a direct measurement. The error is the anharmonicity, in the same direction every time, so it can be corrected for if the molecule is similar enough to one that has been measured both ways.

Which is the ordinary situation for a relation between two experiments: it is a check when both are available and a measurement when only one is.

Reading the relation backwards gives a force constant #

The relation between the distortion constant and the stretching frequency is used here as a prediction — take the frequency, predict the distortion — and it is used in practice in the other direction, because for a large class of molecules the frequency is the harder of the two to obtain.

The species in question are the ones that exist only in a discharge, a flame or a molecular beam: radicals, molecular ions, and short-lived intermediates. A microwave spectrum of such a species is difficult and is routinely obtained; an infrared spectrum of it is much harder, because the sample is dilute and the vibrational transition is at a wavelength where the background is worse.

But the microwave spectrum already contains the vibrational information. The lines are not evenly spaced, the departure is the distortion, and the distortion follows from the stretching frequency by the relation checked above. Measure the departure and the frequency comes out, from a spectrum taken in a region where the molecule was easy to observe.

The accuracy is limited by the relation rather than by the measurement, which is the standing condition of rotational structure work. The relation is good to a few per cent across the four molecules here, so a frequency inferred this way carries a few per cent of uncertainty — which is far worse than a direct measurement and far better than nothing, and is often the only number available.

There is a second and subtler use of the same arithmetic. A molecule with several stretching modes has a distortion constant that is a combination of them, so measuring the distortion constrains the force field without determining it — one more equation for a force field that has more constants than the molecule has frequencies. On a molecule where every frequency is known, that equation is a check; on one where some are not, it is a constraint that costs nothing extra to obtain.

Which is the general shape of what a distortion constant is worth. It is a vibrational quantity hiding in a rotational spectrum, it is measured to far better precision than it can be interpreted, and its value is greatest exactly where the direct measurement is unavailable.

Where the model stops #

Harmonic. The relation between $D$ and $\omega$ assumes a bond that obeys Hooke’s law, and the four per cent discrepancies are the measure of how much that assumption costs.

Diatomic. A polyatomic molecule has several distortion constants — and the moment that is the sum of the other two is the planarity relation those constants spoil — — five for an asymmetric top — and they mix with the rotational constants in ways one line of algebra does not cover.

No vibration–rotation coupling, which is what the isotope shift is arithmetic would have to include to go beyond a harmonic ratio. A real spectrum’s $B$ depends on the vibrational state, with $B_v = B_e - \alpha_e(v + \tfrac12)$, and the constants quoted here are ground-state ones. That correction is of the same order as the distortion and is a separate effect from it.

The line positions here are computed, not measured, in the same spirit as how many frequencies, not how many modes’s counting: the lines are generated with the measured constants and read back rigidly, which isolates one effect exactly. A real spectrum has all of the corrections above in it at once.

A final note on magnitudes, since three parts in a million sounds like something to ignore. The reason it is not is that a rotational spectrum is one of the most precise measurements in chemistry: line positions are routinely known to seven or eight figures, and bond lengths derived from them are quoted to five. A correction of one part in a million is enormous against a measurement of that precision.

That is the general position for this kind of work. The corrections that matter are not the ones that are large; they are the ones that are large compared with the precision of what they correct. A rigid-rotor bond length quoted to five decimal places from lines around $J = 20$ is wrong in the fourth, which is a statement about the model rather than about the spectrometer.

What changes when the constant depends on J #

The simplest uses of a microwave spectrum get a moment of inertia out of it, invert two moments to get two bond lengths, and use the sum rule on three moments to decide whether a molecule is flat. Every one of them treats the rotational constant as a number the spectrum has.

Distortion makes it a number the spectrum has at a given J. The drift is seventeen parts per thousand across forty lines of carbon monoxide, and its size can be predicted from a completely different experiment — the stretching frequency — agreeing within a few per cent across four molecules and three orders of magnitude. The effect was never hidden: it is in every microwave spectrum that reaches high J, and it goes unnoticed only because the lowest lines return the equilibrium constant so well.

The remaining correction of the same size is the vibrational dependence — $B_v$ rather than $B_e$ — the one that separates a bond length in a state from a bond length in a potential.