The constant a spectrum cannot see

The rotational spectrum is a moment of inertia is the starting point, and its claim is that every line in a microwave spectrum sits at a multiple of one number, and that number is a conversion constant divided by a sum of mass times distance squared. That is exactly right, and the phrase to notice in it is one number.

A molecule has three moments of inertia. For a symmetric top two of them are equal, so there are two distinct constants — and only one of them is the one the spectrum is a multiple of.

Three moments, two constants #

The moments are computed from the coordinates and the masses, with no bonding argument anywhere. For ammonia they come out at 1.710, 1.710 and 2.659 atomic mass units times ångström squared: two equal and one different, which is what makes it a symmetric top.

The two distinct rotational constants follow — 9.8601 wavenumbers for the two equal moments and 6.3406 for the unique one — and both describe real motions. The molecule turning end over end has one energy scale and the molecule spinning about its threefold axis has another, and the two differ by thirty-six per cent of the larger.

The consequence is easiest to see on the molecule with the most lines to hide it in, where nothing about the spectrum looks impoverished.

And it is worth seeing the levels themselves, which do depend on both constants, beside the transitions between them, which do not.

The classification into spherical, symmetric and asymmetric tops is a statement about the moments, and it is decided by the point group: an axis of order three or higher forces two of the three moments to be equal. So the very symmetry that makes ammonia a symmetric top is what creates the pair of constants, and it is also — as the rest of this essay shows — what makes one of them unobservable.

Where the second constant lives in the levels #

The rigid rotational energy of a symmetric top is

$$E(J, K) = B\,J(J+1) + (A - B)\,K^{2}$$

where J counts the total angular momentum and K counts how much of it is about the unique axis. Both constants are there, and the second one is not hiding.

Looking at those levels, the second constant seems entirely accessible: it is the spacing between the K = 0 and K = 1 levels of any J, and there are as many determinations of it as there are values of J.

And where it goes #

A pure rotational transition has ΔJ = +1 and ΔK = 0. The second of those is the reason for everything that follows and it is worth being precise about where it comes from: a molecule absorbs a photon through its dipole moment, a symmetric top’s dipole lies along its unique axis, and a dipole along an axis exerts no torque about that axis. So the component of angular momentum about it cannot change.

Subtract two levels with the same K:

$$E(J+1, K) - E(J, K) = B\,(J+1)(J+2) - B\,J(J+1) = 2B(J+1)$$

The K² term appears in both levels with the same coefficient and cancels. Not approximately — identically, for every J and every K.

The molecule’s energy levels contain both constants and its spectrum contains one. What was lost is not resolution, and no better spectrometer recovers it.

Three tops, and two of them are worse off #

Two of the three are worse off than ammonia, and for a reason that belongs to a different essay in this collection.

Benzene and boron trifluoride have no dipole moment at all — both are centrosymmetric or planar-symmetric enough that their point group forbids one — so neither has a pure rotational spectrum in the first place. For them both constants are invisible to the technique, and the reason is a stronger version of the same argument: no dipole, no torque, no absorption at any J.

So benzene’s 0.1916 and 0.0958 wavenumbers are computed here from an arrangement of twelve atoms and are not obtainable from a microwave spectrum of benzene at all, because benzene has no microwave spectrum.

How much is lost, in the units that matter #

It is worth quantifying the loss rather than leaving it as a fact about a parameter, because a structure is what the measurement is usually for.

Ammonia has two structural unknowns: an N–H bond length and an H–N–H angle. Its two moments of inertia are two functions of those two unknowns, so the pair of constants would determine the structure outright. One constant is one equation, and one equation in two unknowns has a one-parameter family of solutions: a whole curve of (length, angle) pairs all of which reproduce the measured B exactly.

So the blindness costs a structure. What is recovered instead, by measuring the isotopologue ND₃, is a second equation on the same two unknowns — and that works precisely because the two isotopologues have the same geometry and different masses, which is the trick a bond length out of a spectrum is built on.

The irony worth noticing is that the second measurement recovers what the second constant of the first molecule would have given, at the cost of preparing an isotopically substituted sample — and it is available only because the electronic structure does not know about neutrons.

What is actually being measured, then #

The useful way to state the situation is in terms of what the experiment determines rather than what it fails to determine.

A pure rotational spectrum of a symmetric top determines one number. That number is a sum of mass times distance squared about an axis perpendicular to the unique one, and for a molecule with several unknown bond lengths and angles it is one equation.

A bond length out of a spectrum is the essay about what to do with one equation and two unknowns: substitute an isotope, get a second equation on the same structure, and solve. That strategy is available here too and it is the only one available, because the second constant of the same molecule is not a second equation — it is not obtainable.

The degeneracy is real and is not the point #

A reader who knows the subject will object at this stage that the K levels are not merely invisible in the spectrum — they are populated, they affect the intensities, and the pattern of intensities carries something.

That is true and it does not recover the constant. The population of a level depends on its energy through a Boltzmann factor, so the relative intensities of the K components of a line do depend on A. What they depend on is the ratio of (A − B)K² to kT, which at room temperature and a constant of a few wavenumbers is a small number, and which is entangled with the nuclear spin statistics that also weight the K components.

So the information is there in principle, buried in intensities, and it is not where a rotational spectrum’s precision lives. A line position can be measured to a part in ten million; a relative intensity to a few per cent. A parameter that only appears in the second is not measured in the sense the first sets.

The distinction is the same one what an absence proves draws between a band that is forbidden and a band that is merely faint. Here it is a parameter that is forbidden from the positions and merely faint in the intensities.

The constant comes back when the model fails #

There is one route to A, and it is worth its own section because of what it says about models.

A real molecule is not rigid: it stretches as it spins, and the rotor that stretches computes the size of that effect from the vibrational frequency with no fitting. The correction to the energy is a term in J²(J+1)², and for a symmetric top there is a second correction of the form

$$-D_{JK}\,J(J+1)K^{2}$$

which depends on both J and K. Subtract two levels with the same K and this term does not cancel: it leaves a residue proportional to (J+1)K².

So K reappears in the line positions, the lines split into K components, and the splitting is measurable. It is small — a few ten-thousandths of a wavenumber at moderate J — but it is there, and fitting it gives a constant that carries information about the unique axis.

A linear molecule’s lines are evenly spaced at 2B(J+1) and have no second constant to hide, because a linear molecule has only one. The symmetric top’s blindness is the price of having two constants and a selection rule that only ever compares levels with the same value of the second one.

The structure of that statement is worth pausing on. The rigid rotor is the model that makes the spectrum simple and the constant invisible. Correcting the model makes the spectrum more complicated and the constant visible. The quantity is measurable only through the failure of the approximation that concealed it, which is a pattern rather than an accident: an exact cancellation is a property of an idealisation, and the way to see past it is to find the term the idealisation dropped.

The general shape: a selection rule is a filter on information #

This is a case of something broader, and the pieces of it appear in two other places.

A selection rule decides which transitions occur, and it is usually presented as a rule about intensities: this transition is allowed, that one is forbidden. Selection rules are one theorem is the essay about where they all come from — an integral over all space vanishes unless the integrand is totally symmetric.

The consequence nobody draws from that is the one here. A selection rule is also a filter on what the spectrum can say about the molecule, because a quantity that only appears in the difference between two levels connected by a forbidden transition appears in no observable frequency. The rule ΔK = 0 does not merely remove some lines; it removes an entire parameter.

What an absence proves is the neighbouring statement about what a missing band means. This is the complementary one: what a rule costs, in parameters rather than in bands.

The three cases, ranked by how much is visible #

The pattern across the three molecules in the table is worth setting out, because it makes the point that this is a hierarchy rather than a single failure.

An asymmetric top — water, sulfur dioxide, formaldehyde — has three distinct constants and a spectrum in which all three appear. Its levels have no closed form, its lines are not evenly spaced, and the complexity is exactly what carries the information.

A symmetric top with a dipole — ammonia, methyl fluoride — has two constants and a spectrum reporting one. The levels are simple and the loss is one parameter.

A symmetric top without a dipole — benzene, boron trifluoride — has two constants and no pure rotational spectrum at all. The loss is total.

A spherical top — methane — has one constant and no dipole, so again no spectrum, and here nothing is lost by the blindness because there was only ever one number to find.

Read down that list and the amount recoverable falls as the symmetry rises, which inverts the usual relation between symmetry and tractability. High symmetry makes the levels easy to write down and makes them carry less.

What the molecule is doing, meanwhile #

It is worth stating what the invisible constant physically is, because the temptation is to treat an unmeasurable parameter as a fiction.

A of 6.3406 wavenumbers is the energy scale for ammonia spinning about its threefold axis. The molecule really does that; the states with K ≠ 0 really are the ones where it is doing so; the energy really is (A − B)K² above the K = 0 level of the same J. Nothing about the physics is uncertain and nothing about it is conventional.

What is missing is a transition that changes K, because the dipole cannot drive one. The states exist, the energies exist, and no photon connects two of them that differ in K alone.

That is a good general picture of what spectroscopy is: not a window onto a molecule’s energies but a list of the differences between the pairs a particular operator connects. Everything else is inferred, and what can be inferred depends on which operator was used — which is why the same molecule looks different in the infrared, the Raman and the microwave, and why two structures give two spectra is a workable identification method at all.

Two constants, one axis, and a naming trap #

A last practical note, because the literature’s notation makes this easy to get backwards.

By convention A, B and C label the three rotational constants in decreasing order, so A belongs to the smallest moment. Ammonia’s smallest moments are the two equal ones, so by that convention its pair are A = A = 9.8601 and C = 6.3406, and the invisible one is C. Benzene is the same shape of case.

The essay above has called the unique axis’s constant A throughout, which is the convention used for a prolate top, where the unique axis genuinely does have the smallest moment. Both usages are current and both are defensible, and a reader comparing a table with a formula has to check which is meant before subtracting anything.

What is not ambiguous is the physics: there are two constants, the one belonging to the unique axis cancels out of every rigid transition, and which letter it is written with does not affect that.

Where the missing constant is available instead #

The blindness is a blindness of the pure rotational spectrum, and it is worth saying that the constant is obtainable — because a quantity that no measurement could reach would be a different and more serious situation than one that a different measurement reaches easily.

Two routes are standard.

The rotational structure of a vibrational band. An infrared band of a symmetric top has rotational fine structure, and for a band whose transition moment lies perpendicular to the symmetry axis the selection rules allow the quantum number about that axis to change. The term that cancelled out of every pure rotational transition does not cancel out of those, and the spacing of the resulting sub-bands gives the constant directly.

Transitions the distortion makes weakly allowed. A real molecule is not a rigid symmetric top; centrifugal distortion mixes states slightly, and transitions that the rigid selection rules forbid acquire small intensities. Those forbidden lines do depend on the missing constant, and measuring a few of them recovers it — at the price of hunting for lines orders of magnitude weaker than the allowed ones.

So the constant is not unmeasurable. It is unmeasurable in the spectrum where every other rotational constant is easy, which is a more specific and more interesting statement.

The practical consequence is a change of instrument rather than a limitation. A structure determination on a symmetric top that needs the missing constant is a determination that needs an infrared measurement as well as a microwave one — and the two are done in different laboratories, at different precisions, and are combined into one structure with the accuracy of the worse of them.

That is the shape of the difficulty rotational spectroscopy keeps meeting. The quantity a measurement cannot see does not stop being a property of the molecule, and recovering it costs the precision that made the original measurement worth making.

What is left #

The whole treatment here is rigid, so the escape route above is described rather than computed: the centrifugal terms are given their form and their size is not derived, because deriving $D_{JK}$ for a symmetric top needs the vibrational–rotational coupling and the calculation here has only the rigid moments and the harmonic frequencies separately.

The asymmetric case is not treated either, and it is the one where the situation is different in a way worth naming. A molecule with three distinct moments — water, for instance — has levels that do not have a closed form at all, and every one of its three constants shows up in the pattern of lines. So the blindness here is specific to the symmetric top, and it is caused by the very symmetry that makes its levels simple enough to write down.

That is a trade that turns up from both sides: symmetry makes a problem tractable and it makes some of the problem’s content unobservable, and the two are the same fact.