The moment that is the sum of the other two

A molecule’s inertia tensor is built from its masses and its coordinates and nothing else. Diagonalising it gives three principal moments, conventionally ordered $I_a \le I_b \le I_c$, and those three numbers are what a rotational spectrum measures.

Most of what can be said about them is quantitative and approximate, and the rotational spectrum is a moment of inertia works through what a spectrum reports about them. One statement is exact, costs nothing, and answers a structural question outright.

The identity #

Put a planar molecule in the $xy$ plane. The moment about $z$ is $\sum_i m_i (x_i^2 + y_i^2)$; the moment about $x$ is $\sum_i m_i (y_i^2 + z_i^2)$, which for a planar molecule is $\sum_i m_i y_i^2$; and the moment about $y$ is $\sum_i m_i x_i^2$.

So $I_z = I_x + I_y$, term by term, because each atom’s contribution to the first is the sum of its contributions to the other two. That is Pythagoras summed over the atoms with masses attached, and it holds for any planar arrangement whatever — any number of atoms, any masses, any shape.

The quantity $\Delta = I_c - I_a - I_b$ is called the inertial defect, and for a rigid planar structure it is zero.

Computed here:

The planar entries are zero to the last bit or to within a floating-point rounding of it. The non-planar ones are not close to zero and are not close to each other. There is no threshold to choose, which is what makes the test worth having: the answer is a fact about the structure and not about a tolerance somebody picked.

The classification, checked twice #

The three moments also decide what kind of rotor the molecule is, and that decision can be made twice by independent routes.

From the numbers. All three equal is a spherical top; two equal is a symmetric top, oblate if the odd one is largest and prolate if it is smallest; one zero is linear; all three different is asymmetric.

From the group. A rotation axis of order three or higher forces two of the moments to be equal, because the inertia tensor must commute with every operation of the point group and a threefold axis leaves no room for two distinct perpendicular moments. So the point group forbids certain classifications outright, in the same way degeneracy is a group theorem has the group deciding which electronic degeneracies may exist.

Every row of the table is checked against both, and the check is a useful one because the two can disagree in an informative way. Ammonia’s hydrogens are quoted to four decimal places in the structure it carries, which puts them about $5\times10^{-5}$ Å off a perfect threefold arrangement, so its two perpendicular moments differ by one part in a million. At a tolerance of $10^{-6}$ ammonia is classified asymmetric, which is a statement about the rounding in a published structure and not about ammonia.

The tolerance is therefore one part in ten thousand, set by the coordinates rather than by the arithmetic, and the gap to the nearest genuine case is enormous: the smallest real asymmetry in this collection is ethene’s, at three parts in a hundred.

Why a threefold axis forces two equal moments #

The group half of that check deserves its own paragraph, because it is the same theorem that runs through the whole symmetry field here.

The inertia tensor is a symmetric second-rank tensor built from the structure, so every operation of the point group must leave it unchanged. A tensor invariant under a rotation of order three or more about an axis cannot distinguish any two directions perpendicular to that axis: if it did, rotating by $120°$ would produce a different tensor. So the two perpendicular principal moments are forced equal, and the molecule is a symmetric top.

That is Neumann’s principle in miniature — a property tensor must carry at least the symmetry of the structure — and it is the same argument that makes a cubic point group force all three moments equal, which is why methane and sulfur hexafluoride are spherical tops with $\kappa$ undefined rather than merely near some limit.

The check runs the other way too, and that is where it bites. A structure whose group has a threefold axis and whose computed moments are not equal has a fault in its coordinates, and this collection’s tolerance discussion above is exactly the story of one such near-miss being correctly diagnosed as rounding in a published structure rather than as a discovery.

The linear case is the degenerate limit of the same statement. All the mass lies on one axis, so the moment about that axis is zero, the other two are equal, and the classification is decided by comparing a computed number against zero — with the same requirement that the tolerance be set by the coordinates. Carbon dioxide, hydrogen cyanide and carbonyl sulfide all return exactly zero here, since their structures are built on an axis rather than fitted to one.

What the asymmetry parameter decides #

A symmetric top’s rotational spectrum is a series of lines at even spacing, because its energy levels have a closed form in two quantum numbers. An asymmetric top’s do not, and the departure is measured by

$$\kappa = \frac{2B - A - C}{A - C}$$

which is $-1$ at the prolate limit, $+1$ at the oblate one, and somewhere between otherwise.

The values computed here are worth reading as a set:

• benzene, boron trifluoride, xenon tetrafluoride: $+1.000$ — oblate symmetric tops exactly, by symmetry rather than by accident.
• phosphorus pentafluoride: $-1.000$ — prolate, likewise exactly.
• hydrogen peroxide: $-0.994$
• formaldehyde: $-0.962$
• sulfur dioxide: $-0.941$
• ethene: $-0.917$
• water: $-0.432$

Water is the outlier, and by a long way. Its three moments are $0.62$, $1.16$ and $1.77$ u Ų, roughly in the ratio $1 : 2 : 3$, which is about as far from either symmetric limit as a molecule can get.

That is a fact about a light atom in an awkward place. Water’s mass is nearly all in the oxygen, which sits close to the centre of mass, and its two hydrogens are light and at right angles to each other — so the three moments end up comparable and none of them dominates.

The consequence is entirely practical: water’s rotational spectrum is genuinely hard, with no regular pattern to organise it, while sulfur dioxide — the same bent triatomic shape with heavier atoms — sits at $-0.941$ and is nearly a prolate top whose spectrum is nearly regular.

Substitution moves the parameter without moving an atom #

The asymmetry parameter is built from masses as well as positions, so isotopic substitution changes it — and it changes more than the trivial scaling would suggest.

Water is $\kappa = -0.432$. HDO, with one deuterium, is $-0.688$. D₂O is $-0.541$.

Substituting one hydrogen moves the molecule a quarter of the way toward the prolate limit, and substituting the second brings it partly back. Nothing moved: the same structure, the same bond lengths, the same angle, and a rotational spectrum organised quite differently.

That is the rotational counterpart of what a spectrum that changes when only a mass does computes for the vibrational case, and the mechanism is related without being identical. There, substitution removes symmetry operations and splits degeneracies. Here, it moves the centre of mass and reweights the sums, which changes a continuous parameter rather than a discrete count.

Where the model stops, and what the gap is used for #

The defect is exactly zero for a rigid planar molecule, and no molecule is rigid.

A real planar molecule vibrates, and the out-of-plane bending motions carry it away from the plane for part of every cycle, while the in-plane motions stretch it. The two effects have opposite signs and the out-of-plane one usually wins, so a measured inertial defect for a planar molecule is small and positive — water’s is about $0.05$ u Ų, against the exact zero computed here.

That residue is not a nuisance. It is one of the more useful quantities in rotational spectroscopy, precisely because the large rigid part has cancelled out of it: a measured defect of a few hundredths reports on the low-frequency out-of-plane vibrations, and a measured defect of a whole unit or more says the molecule is not planar at all. The sign matters too — a small positive defect is a planar molecule vibrating, and a negative one of any size is a structural statement.

Only the rigid part is computed here, not the correction. Getting the correction requires the vibrational frequencies and their dependence on the rotational state, which is a coupling between the two motions that the rigid-rotor and harmonic approximations both discard by construction.

So the honest statement of what has been computed is this: the identity is exact for the structures given, the structures are rigid by assumption, and the difference between a computed zero and a measured five hundredths is the size of the assumption.

What this does for a structure determination #

The chain from spectrum to structure runs: measure the line positions, extract the rotational constants, invert the constants for the moments, and fit a structure to the moments. A bond length out of a spectrum runs that chain for carbonyl sulfide and finds the answer depending on which isotopic pair is used, by $0.0074$ Å.

The defect enters that chain at two points.

It is a free consistency check. Three measured constants give a defect with no fitting at all. If the molecule is believed planar and the defect comes out at $-1$, the belief is wrong or the assignment is.

The check is free in a strong sense: it uses no structural model, no force field and no assumption about which atom is where. Every other use of rotational data requires a structure to be posited and fitted, and a fit can absorb a wrong assumption into its parameters without complaining. The defect cannot, because it is a relation among the three measured quantities themselves.

It reduces the number of unknowns. A planar molecule has one fewer independent moment than a general one, so the identity removes a parameter from the fit and the fit is that much better determined. That matters more than it sounds, because the fit is badly conditioned in the ordinary case — a bond length out of a spectrum finds the best-conditioned isotopic pair for carbonyl sulfide to be the oxygen substitution rather than the heavy-atom one that looks like the obvious choice.

It is also how the twisted structure of hydrogen peroxide was established: the defect is zero at both planar conformations and $-1.08$ at the measured one, so a non-zero defect on a four-atom molecule with two obvious planar candidates settles the question directly. One coordinate, three point groups works through what that twist then decides about the molecule’s symmetry.

Run the inversion in the direction spectroscopy uses it — two isotopologues’ moments in, bond lengths out — and the conditioning of the problem decides how good the answer is. The inertial defect is the diagnostic that says whether the moments being inverted describe a planar molecule at all before any structure is extracted from them.

The quantity a real planar molecule gives instead of zero #

The relation is exact for a set of points in a plane, and a real planar molecule does not return zero — it returns a small positive number, reliably, and the number is used as a criterion rather than treated as an error.

The quantity has a name, the inertial defect: the largest moment less the other two. For an idealised planar structure it is zero by the arithmetic here. For a real planar molecule measured by microwave spectroscopy it comes out at a few hundredths to a couple of tenths in the units used here, and it comes out positive.

The sign is the informative part and it follows from what the vibrations do.

Out-of-plane vibrations reduce it. A molecule bending out of its plane has atoms with a coordinate perpendicular to the plane, which adds to the two smaller moments and not to the largest, pushing the defect negative.

In-plane vibrations increase it. Stretching and bending within the plane moves atoms further from the centre in the plane, which adds to the largest moment more than to the others.

A planar molecule has more in-plane modes than out-of-plane ones — for a molecule of $N$ atoms, $2N-3$ against $N-3$ — so the in-plane contribution wins and the defect is positive.

That makes it a test with a direction. A small positive inertial defect means planar; a substantially negative one means the molecule is not. A genuinely non-planar molecule has atoms permanently out of the plane rather than momentarily, the perpendicular contribution dominates, and the defect goes negative by an amount that measures how far out of plane they sit.

The numbers here fit that pattern. Ammonia missing by 0.76 and methane by 3.18 are not near-misses to be explained away — they are the defect being large and of the sign a non-planar structure gives, on molecules that are not planar at all.

So the exact relation and its failure are both useful, and they are useful for different questions. The relation identifies which molecules are planar; the defect measures how much they vibrate, and a table of defects across a series is a table of vibrational amplitudes obtained from a rotational spectrum.

The defect also settles a class of question that is otherwise hard, and it settles it with a sign rather than with a fit. A molecule that might be planar or might be shallowly pyramidal — an amide nitrogen, a substituted aromatic ring, an anion with a lone pair on its central atom — is exactly the case where a diffraction structure struggles, because the departure from planarity is smaller than the atoms’ own motion and the crystal averages over it.

A rotational spectrum answers immediately. The defect is positive and small for the planar case and negative for the pyramidal one, the two are separated by far more than the measurement’s precision, and no fitting is involved — the three moments come out of the line positions and the subtraction is one line of arithmetic.

That is an unusually clean structural test and it is worth its own sentence. A shape question with two answers is settled by the sign of one number, computed from a measurement good to nine figures, on a molecule in the gas phase where nothing is packing it into a shape it would not otherwise take.

The one case it cannot settle is the shallow double well — a molecule that is pyramidal at its minimum and inverts rapidly, so that its average structure is planar. There the defect reports the average, which is what a rotational measurement always reports, and distinguishing a flat molecule from a rapidly inverting one needs the vibrational spectrum’s tunnelling splitting rather than a moment of inertia.

Who found it, and when #

The identity is old enough to have no clear attribution: it is a property of moments of inertia rather than a result about molecules, and it appears in classical mechanics texts as the perpendicular axis theorem.

Its use as a diagnostic belongs to microwave spectroscopy, which began in earnest after 1945 when wartime radar work left behind both the hardware and the people. Darling and Dennison had worked out the vibrational contribution for water by 1940, and Oka and others put the inertial defect on a general quantitative footing through the 1950s and 1960s, working out the vibrational contribution in enough detail that a measured defect could be turned into a statement about a specific out-of-plane mode.

The reason it survived into routine use is the one this essay is about: it is a quantity in which everything large cancels. A moment of inertia is dominated by the heavy atoms and the overall size; the defect is a difference of three such quantities, so what is left is small, and small quantities that are exactly zero in an idealisation are where the interesting physics is.

What an exact relation is for #

The moments of inertia first give a molecule’s shape and then, inverted, its structure. This essay takes the one exact relation among the three and shows what an exact relation is for: not to compute a number, but to be checked.

The pattern is the same as the one running through symmetry arguments generally. A statement that holds exactly can be tested against a measurement, and the size of the discrepancy then measures whatever was left out — the vibrations here, the correlation elsewhere. A statement that holds approximately can only ever be compared with another approximation.