The rotational spectrum is a moment of inertia

A microwave spectrum of a small molecule is a comb: a series of lines, evenly spaced, marching off towards higher frequency with a spacing that does not change. Everything about it is one number.

That number is the rotational constant,

$$B = \frac{h}{8\pi^{2}cI},$$

and the only molecular quantity in it is I, the moment of inertia — a sum over the atoms of mass times perpendicular distance squared. No orbital, no force constant, no electronegativity and no bond order appears anywhere in the derivation. A rotational spectrum is a measurement of where the mass is.

The tensor, and where its centre is #

The inertia tensor of a set of point masses is a three-by-three symmetric matrix, and diagonalising it gives three principal moments about three mutually perpendicular axes. That is mechanics rather than chemistry and the arithmetic is short.

One detail is not a detail. The moments must be taken about the centre of mass, not about the centroid of the atomic positions, and the two are not the same: for water they are 0.07 ångström apart, which is a four per cent error in the smallest moment and would put every computed constant wrong by that much. This site centres molecules on their unweighted centroid for drawing, so the shift is made explicitly wherever a moment is computed.

The conversion factor is computed rather than quoted: with the moment in unified mass units times ångström squared, B in wavenumbers is 16.85763 divided by I. That number is Planck’s constant over eight π² times the speed of light, in those units, and a reader can check it.

Four kinds of rotor, and a theorem about which #

Three moments admit four patterns, and the classification is the first thing a rotational spectrum is read for.

Linear — one moment zero, two equal. Carbon dioxide, carbonyl sulfide, hydrogen cyanide.

Spherical — all three equal. Methane, sulfur hexafluoride.

Symmetric — two equal, one different: oblate when the odd one is larger, prolate when it is smaller. Ammonia, boron trifluoride, benzene and xenon tetrafluoride are oblate; phosphorus pentafluoride is prolate.

Asymmetric — all three different. Water, sulfur dioxide, ethene, formaldehyde, hydrogen peroxide.

What makes this more than bookkeeping is that the classification is also a consequence of the point group, arrived at by a completely different route.

The inertia tensor is a symmetric second-rank tensor built from the atomic positions, so every operation of the molecule’s group leaves it unchanged. Now consider a threefold rotation about z. It maps the tensor’s (x, y) block onto itself, and the only two-by-two block invariant under a rotation by 120 degrees is a multiple of the identity. So an axis of order three or more forces the two perpendicular moments to be equal, and a molecule with such an axis cannot be an asymmetric top.

Both computations are run for every molecule here — a search over coordinates for the operations, a diagonalisation of a mass-weighted sum for the moments — and they are required to agree. Nothing connects them but the theorem, so a disagreement would be a real fault in one of the two.

The converse direction is the useful one experimentally. A spectrum that is a simple evenly spaced comb came from a molecule with an axis of order three or more, which is a structural conclusion drawn from the shape of a chart. An asymmetric top gives a spectrum with three constants in it and a pattern that is not a comb at all.

The asymmetry parameter measures how far from symmetric a molecule is: −1 at the prolate limit, +1 at the oblate one. Phosphorus pentafluoride comes out at −1.000 and benzene at +1.000, both being genuine symmetric tops; water is −0.432, which is about as asymmetric as a molecule gets, and sulfur dioxide −0.941, which is nearly prolate despite being bent.

Where the lines fall #

For a linear or symmetric top the rigid-rotor levels are

$$E_J = BJ(J+1), \qquad J = 0, 1, 2, \ldots$$

and the selection rule permits ΔJ = ±1, so the transition from J to J+1 sits at 2B(J+1). Successive lines are 2B apart, exactly, and the spacing is the whole spectrum.

Two things about that figure are worth drawing out.

The spacing scales as one over the moment, so it is a measurement of size and mass rather than of anything chemical. Hydrogen cyanide’s constant is 1.477 wavenumbers and carbonyl sulfide’s is 0.203; the ratio is the ratio of their moments and nothing else.

Rigid is doing work. A real molecule stretches as it spins, which pulls the high-J lines closer together by a term in J³. For carbonyl sulfide the centrifugal distortion constant is about 1.3 × 10⁻⁷ wavenumbers against a B of 0.203, so the fortieth line is displaced by about a thousandth of the spacing — small, real, and never absorbed into B here, because absorbing it would move the recovered bond length by an amount nothing in the figure could account for.

The dipole requirement, which is pure symmetry #

A pure rotational transition is driven by the molecule’s permanent dipole moment. No dipole, no absorption — and whether a molecule has one follows from its point group with no reference to bonding at all.

Symmetry forbids a dipole sets the rule out in full: a molecule may be polar only if its group is C₁, Cₛ, Cₙ or Cₙᵥ, and the usual argument from adding up bond dipoles gets the right answer for the easy cases by a route that does not generalise.

The consequence here is stark. Carbon dioxide, methane, benzene, sulfur hexafluoride and xenon tetrafluoride all have perfectly well-defined rotational levels, computed above, and not one of them can show a pure rotational spectrum at any sensitivity. This is an absence of exactly the kind what an absence proves is about: a theorem rather than a limitation, and it is why microwave spectroscopy is a technique for polar molecules and Raman rotational spectroscopy — which has a different selection rule, needing only an anisotropic polarisability — exists to cover the rest.

Two checks on the tensor that come from outside it #

A quantity computed once has been computed once. Two independent relations are available here, and both hold.

The perpendicular axis rule. For any planar body the moment about the axis perpendicular to the plane is the sum of the two in-plane moments. It follows from the definition in two lines — the perpendicular distance from an in-plane axis is one coordinate, and from the out-of-plane axis it is both — and it uses nothing about how the tensor was built or diagonalised. Benzene’s 88.00 and 88.00 sum to 176.00, which is what comes out; boron trifluoride’s 48.90 and 48.90 sum to 97.81.

The group’s verdict on the classification. Computed above, and worth repeating as a check rather than as a result: the operations are found by a search over coordinates and the moments by a diagonalisation of a mass-weighted sum, and only a theorem connects them.

That last point is worth dwelling on, because “asymmetric top” sounds like a statement about a molecule being unsymmetrical and is not. Ethene has eight symmetry operations, a centre of inversion, three mirror planes and three twofold axes, and it is an asymmetric top. The classification depends on one thing only: whether an axis of order three or more exists.

What the constants come out at #

Some numbers, because the range is larger than intuition suggests and the reason is instructive.

Hydrogen cyanide’s rotational constant is 1.477 wavenumbers. Carbonyl sulfide’s is 0.203. Sulfur hexafluoride’s is 0.091. Benzene’s is 0.192 about its sixfold axis. Water’s largest is 14.6 wavenumbers — an order of magnitude above everything else here, because it is three atoms of which two are hydrogen.

The spread is a factor of a hundred and sixty, and every bit of it is mass and size. A rotational constant is a measurement of a molecule’s bulk, and the reason microwave spectroscopy is confined to small molecules is arithmetic rather than technical: a large molecule’s lines are so closely spaced, and so numerous, that they merge into an unresolvable forest.

Why this is the best structural measurement in the subject #

Rotational spectroscopy determines geometries better than almost anything else available, and the reasons are all visible in the arithmetic above.

The observable is a frequency, and frequencies are the most precisely measurable quantities in physics. Rotational constants are routinely quoted to eight significant figures.

The model has no adjustable parameters. B equals a constant over a moment, and the moment is a sum over masses and positions. There is no force field to fit, no basis set to choose and no exchange functional to argue about. Compare the force field is not in the spectrum, where four constants cannot be determined by three frequencies.

The masses are known exactly. Nuclear masses are measured to ten figures or better, so the only unknowns in the moment are the positions.

What that buys is the subject of the next essay: with one constant and one unknown length a bond length comes straight out, and with two isotopologues two bond lengths do.

The other selection rule, and what it reaches #

A pure rotational transition needs a permanent dipole, which excludes half the molecules in this collection. There is a second experiment with a different rule, and its coverage is exactly complementary.

Rotational Raman scattering is driven by the anisotropy of the polarisability rather than by a dipole. A molecule whose polarisability is the same in every direction — a spherical top — has none, and everything else does. So methane and sulfur hexafluoride are invisible to both experiments, while carbon dioxide, benzene and xenon tetrafluoride, which have no dipole, all have anisotropic polarisabilities and give rotational Raman spectra.

The selection rule is ΔJ = ±2 rather than ±1, because the polarisability is a second-rank tensor where the dipole is a vector, so the lines come at 4B(J + 3/2) and the spacing is 4B rather than 2B. That is the same arithmetic doubled and it gives the same moment of inertia.

Between the two experiments, every molecule with any anisotropy at all is reachable, and the only permanent exclusions are the spherical tops. Carbon dioxide’s bond length, which the microwave method cannot touch, comes from exactly this route.

The molecules with no dipole are not left with nothing: carbon dioxide and benzene have no pure rotational spectrum at all, and both have rotational Raman spectra, because an anisotropic polarisability does the job a dipole cannot. Methane and sulfur hexafluoride have neither, being spherical tops with isotropic polarisabilities — which is a symmetry statement about the molecule and not a limitation of the instrument.

What this treatment leaves out #

Vibration. A real molecule’s rotational constant depends on which vibrational state it is in, because the average moment changes as the molecule vibrates. The constant a spectrum gives is the ground-state one, and it is not the constant of the equilibrium geometry. That gap is small and systematic, and it is the main limit on the accuracy of a structure.

Asymmetric tops in detail. Their energy levels have no closed form and are found by diagonalising a matrix for each J. Everything above is computed for them — the moments, the constants, the classification — and the line positions are not, because the figures here draw evenly spaced series and an asymmetric top does not have one.

Intensities and populations. How strong a line is depends on how many molecules are in the lower state, which is a thermal question, and on the transition moment. Neither is computed. The lines drawn here mark positions.

Nuclear spin statistics. For a molecule with identical nuclei, some rotational levels are missing entirely — the reason ortho and para hydrogen exist — and that is a symmetry effect the level counts here do not include.

The measurement is far better than the model that reads it #

A line’s position being a multiple of one number is a simple statement, and the consequence that follows from it is the reason this whole anchor exists — because of all the quantities in chemistry, that number is the one measured best.

Rotational transitions fall in the microwave region, and a microwave frequency is the most precisely measurable quantity in physics. A frequency can be counted against a standard rather than compared against a scale, so a rotational line’s position is known to eight, nine or ten significant figures without heroic effort.

The rotational constant inherits that precision, and so does the moment of inertia it is a conversion constant divided by.

Compare that with what else is known about a molecule. A bond energy is known to three figures at best; a vibrational frequency to five; a computed geometry to two or three. A moment of inertia is known to nine, which is four to six orders of magnitude better than anything else in the subject.

That imbalance is what shapes everything downstream, and it sets the character of structure determination from rotational spectra.

The measurement is never the limitation. No result in rotational spectroscopy is limited by how well the lines were located, and improving the instrument improves nothing that matters.

The interpretation is the limitation, always. Turning a moment of inertia into a bond length requires a model of what the molecule was doing while it rotated, and every model available is wrong at the level of a part in a thousand — because the molecule vibrates, and the moment measured is an average over that vibration rather than a value at any geometry.

So that work is not a series of attempts to squeeze more out of a difficult measurement. It is a series of attempts to read an extraordinarily good measurement with a model that is a million times cruder than it is, and its hard problems are the places where the model runs out first: a structure that needs more parameters than the spectrum supplies, a constant the spectrum cannot see, a correction that was invented, and an axis that goes the wrong way.

That ordering is worth carrying, because it inverts the usual relationship between theory and experiment. Elsewhere in chemistry a computed quantity is compared against a measurement of comparable accuracy and the agreement is the result. Here the measurement is exact for every practical purpose, and what is being tested is always the model — which is why the honest output of a rotational study is a structure with a subscript on it saying which model produced it.

The precision has one further consequence that is easy to overlook and is the reason the technique identifies molecules at all. A rotational constant is a fingerprint: nine significant figures is far more than enough to distinguish one molecule from every other molecule, and from every isotopic variant of itself.

That is why radio astronomy can say which molecules are present in an interstellar cloud from a handful of lines, with no sample, no chemistry and no possibility of a second measurement. The lines are matched against a laboratory catalogue, the match is either exact to the eighth figure or it is not a match, and there is no room for the coincidences that plague identification by any other kind of spectrum.

So the same precision that makes the interpretation the weak link makes the identification unassailable — and the two uses of one measurement sit at opposite ends of what it is good for. Asking which molecule it is exploits the precision fully; asking what shape it has throws almost all of it away.

Who worked it out #

The rigid rotor is nineteenth-century mechanics; its quantisation is 1920s quantum mechanics. What made it a chemical technique was the microwave hardware built for radar during the Second World War and released to laboratories afterwards, which is why molecular microwave spectroscopy begins abruptly in 1945–46. Within a few years the structures of dozens of small molecules had been determined to a precision diffraction could not approach.

Carbonyl sulfide is the standard worked example, and it was among the first: its constants were measured in the late 1940s and its structure derived from them by the argument the next essay computes.

The same argument run backwards #

Everything here runs forwards, from a structure to a spectrum. A bond length out of a spectrum runs it backwards — two measured line spacings, two bond lengths, and an inversion that is checked by round-tripping a known structure through it to twelve figures before being trusted with a real measurement.