A bond length out of a spectrum

Most calculations in molecular structure run from a structure outwards. This one runs the other way: a measured line spacing goes in and a bond length comes out, and it is worth being clear about why that inversion is possible here and so rarely anywhere else.

A rotational constant is one conversion divided by one moment of inertia, and a moment of inertia is a sum over the atoms of mass times distance squared. The masses are known to ten figures. The conversion is a combination of defined constants. So a measured constant is, arithmetically, a statement about the positions — and inverting it is a matter of solving an equation rather than of fitting a model.

One moment is not enough #

Carbonyl sulfide is linear: O, C and S in a line, two bond lengths to determine. Its moment of inertia is one number. Two unknowns and one equation, and the shortfall cannot be made up by measuring the same molecule more precisely.

What supplies the second equation is an isotopologue. Substituting sulfur-34 for sulfur-32 changes the masses and leaves the bonds exactly alone — the electronic energy surface is independent of nuclear mass, which is the same Born–Oppenheimer argument the isotope shift is arithmetic rests on — so the substituted molecule is a second measurement of the same structure with different weights.

Two moments, two lengths, and the equations are independent because substituting one end of the molecule moves the centre of mass and therefore reweights the two bonds differently. Solving them is a two-dimensional Newton step on an expression short enough to write out:

$$I = \sum_i m_i z_i^{2} - \frac{\left(\sum_i m_i z_i\right)^{2}}{M}.$$

The round trip, before the measurement #

The inversion is checked before it is trusted, by the same round trip used for point groups: build a structure, compute what it would give, forget the structure, recover it, and require the two to agree.

Given the equilibrium bond lengths 1.1543 and 1.5628 ångström, the forward calculation produces two moments; the Newton solve, started from 1.0 and 1.0, returns 1.154300000000 and 1.562800000000 with a residual of 3 × 10⁻¹⁴ in the moments. That is the arithmetic proving itself and nothing more. No measurement has entered, and none of the physics of the next section is being tested.

It is worth having anyway. An inversion that quietly converged to the wrong branch, or a Jacobian with a sign error, would still produce a plausible pair of lengths from real data; the round trip is what separates this arithmetic is right from this arithmetic gives an answer.

What the measurement gives, and why it disagrees with itself #

Four isotopologues of carbonyl sulfide have measured ground-state rotational constants, in megahertz: 6081.49 for ¹⁶O¹²C³²S, 5932.83 for ¹⁶O¹²C³⁴S, 6061.90 for ¹⁶O¹³C³²S and 5704.83 for ¹⁸O¹²C³²S. Any two of them determine a structure, and the three pairs including the parent give

The three answers differ by up to 0.0074 ångström in the C–O bond. The measured constants are good to five or six figures, so that spread is not experimental error and is not arithmetic error. It is the model.

The reason is zero-point motion. A rotational constant is an average over the molecule’s vibrational ground state, and the quantity being averaged is 1/r², not r. Averaging an inverse square and then inverting is not the same operation as averaging the distance, so a ground-state moment corresponds to a slightly shorter effective bond than the equilibrium one — and the correction depends on how much the molecule moves, which depends on its masses. A heavier isotopologue vibrates with a smaller amplitude and therefore reports a slightly different structure.

What comes out of this inversion is called the r₀ structure, and it is not the rₑ structure. None of the three rows above is 1.1543 and 1.5628, which are the equilibrium values, and the disagreement between the rows is a direct measurement of how large the effect is.

Which pair to use is not a free choice #

The three pairs are not equally good even setting the physics aside, and the arithmetic says which is which.

Substituting an atom close to the centre of mass barely changes the moment, so the two equations it produces are nearly the same equation and the solve is ill-conditioned. Substituting an atom far from the centre of mass changes the moment a great deal, and the equations are well separated.

The figure reports the independence of each pair — the sine of the angle between the two rows of the Jacobian. The oxygen substitution comes out best at 1.7 × 10⁻², the carbon substitution next at 6.5 × 10⁻³, and the sulfur substitution worst at 4.0 × 10⁻³.

That ordering is not the one intuition suggests. Substituting the heaviest atom gives the weakest constraint, because carbonyl sulfide’s centre of mass sits near the sulfur: changing the mass of an atom that is nearly at the pivot moves the moment less than changing the mass of the light atom at the end. The pair a chemist would reach for first is the least informative of the three, and the ill-conditioning shows up as sensitivity — a small error in either constant moves the recovered lengths more for that pair than for the others.

The simplest case, where one measurement is enough #

A diatomic has one bond and one moment, so the counting works out and no isotopologue is needed. It is worth doing that case explicitly, because it is where the whole technique’s precision is visible without any of the complications.

For a diatomic, I = μr² with μ the reduced mass, so

$$r = \sqrt{\frac{h}{8\pi^{2}c\,\mu\,B}},$$

and every quantity on the right except B is known to ten figures or better. A rotational constant measured to six figures therefore gives a bond length to six figures, which is a precision nothing else in structural chemistry approaches.

The catch is the same one that appears in the triatomic case and is easier to see here. The measured constant belongs to the vibrational ground state, and the molecule in that state is not sitting at the bottom of its well: it is oscillating, and the moment is averaged over the oscillation. So the r₀ obtained is slightly larger than the equilibrium rₑ, systematically, by an amount that depends on the anharmonicity of the potential and on the mass.

That is the whole of the r₀-against-rₑ problem in one bond, and it is why the carbonyl sulfide inversion above gives three different answers from three isotopic pairs: each pair averages differently.

The condition number, and what it is a warning about #

The independence numbers reported above are not decoration, and it is worth saying what an ill-conditioned pair actually does to an answer.

Two nearly parallel equations mean that a small error in either measurement moves the solution a long way along the direction they nearly share. For the sulfur pair, whose independence is 4.0 × 10⁻³, an error of one part in ten thousand in a rotational constant moves the recovered bond lengths by roughly one part in forty — a factor of two hundred and fifty amplification.

That is still tolerable here, because the constants are known to one part in a million. It stops being tolerable as soon as the measurement is less good or the substitution is smaller, and the general lesson is the one every inverse problem teaches: the quality of an inversion is a property of the pair of measurements, not of either one of them.

The practical rule that follows is worth stating, because it is the opposite of the instinct. Substitute the atom furthest from the centre of mass, not the heaviest. In carbonyl sulfide the centre of mass sits near the sulfur, so changing the sulfur’s mass barely moves the moment, while changing the oxygen’s — a light atom at the far end — moves it a great deal.

Why the substitution method exists #

The standard repair for the r₀ problem is Kraitchman’s method, from 1953, and it is worth describing because it is a genuinely different idea rather than a correction factor.

Rather than solving for all the coordinates at once from a set of moments, Kraitchman’s equations give the coordinates of the substituted atom alone, from the difference between the parent’s moments and the substituted molecule’s. Doing that for every atom in turn — substituting each one and measuring — builds up a structure atom by atom, and the zero-point contributions largely cancel in each difference because the two molecules being compared differ in one nucleus.

The result is called the r_s structure, and it lies much closer to the equilibrium geometry than any r₀ structure does. It needs one isotopologue per atom, which is why microwave structure determination is as much a synthetic problem as a spectroscopic one.

The same shortfall, twice on one site #

The shape of this problem — more parameters than independent measurements, filled in by measuring a related system — occurs twice in the essays around this one and it is the same shape both times.

The force field is not in the spectrum has four force constants and three frequencies, and the second isotopologue supplies the missing equations. This essay has two bond lengths and one moment, and the second isotopologue supplies the missing equation. In both cases the substitution works for the same reason — the property being determined does not depend on nuclear mass — and in both cases the answer that comes out depends slightly on which isotope was chosen, for the same reason: the measurement is of a vibrationally averaged quantity and the averaging is mass-dependent.

The difference is in how large the residual ambiguity is. The force-constant family spans 37 per cent in one constant; the structural ambiguity here is 0.6 per cent in a bond length. A microwave structure is the better measurement by a factor of about sixty, and that is why the technique is the reference standard for small-molecule geometry and vibrational spectroscopy is not.

Three more geometries the same way #

The method generalises, and how far it generalises is a counting problem exactly like the one at the top of this essay: n geometric parameters need n independent moments.

A symmetric top gives two distinct rotational constants per isotopologue rather than one, because two of its three moments are equal and the third is not. Ammonia’s structure — one bond length and one angle — therefore comes out of a single isotopologue, with no substitution needed at all.

An asymmetric top gives three constants per isotopologue, which is a great deal of information: water’s two parameters are heavily overdetermined by one measurement, and the excess is used to test the rigid-rotor model rather than to fix the structure.

A large molecule needs one substitution per atom, and the arithmetic is Kraitchman’s rather than a two-by-two solve. That is why microwave structure determination is as much a synthetic problem as a spectroscopic one: obtaining the isotopologues is the work.

The pattern across all of these is the same as the vibrational one. The measurement is extraordinarily precise and the model has no free parameters, so the limit is always how many independent measurements exist rather than how good any of them is.

What this cannot do #

It needs a linear or symmetric molecule to be this simple. An asymmetric top has three constants per isotopologue rather than one, which is more information, but its energy levels have no closed form and its structure comes from a least-squares fit over many constants rather than from a two-by-two solve.

It needs the molecule to have a dipole. Carbon dioxide’s structure cannot be obtained this way at all, because it has no rotational spectrum — the lines in the figure above are drawn and unobservable. Its bond length comes from Raman rotational spectroscopy or from diffraction instead.

It assumes rigidity. Centrifugal distortion is real and is quoted separately here rather than absorbed into the constant, because absorbing it moves the recovered bond length by an amount that nothing in the figure could then account for. For carbonyl sulfide the distortion constant is about 1.3 × 10⁻⁷ wavenumbers against a B of 0.203.

It says nothing about bonding. A structure is a set of positions. Why the C–O bond in carbonyl sulfide is 1.154 ångström while the one in carbon monoxide is 1.128 is a question for the bonding essays, and this measurement’s virtue is precisely that it answers a geometric question without entering that argument.

Why the isotopes disagree #

Two equations in two unknowns having a solution that depends on which isotope was used is the finding, and the cause is not in the arithmetic — it is in the assumption the two equations share.

The method requires that substituting an isotope changes the masses and leaves the structure alone. That is exactly right about the electronic structure: the potential energy surface is a function of the nuclear positions and of the electrons’ solution to them, and a nucleus’s mass does not appear in it anywhere. Two isotopologues have the same potential, to the accuracy of the Born–Oppenheimer separation, and their equilibrium geometries are identical.

Their average geometries are not, and a rotational constant is an average.

The size of that difference can be measured on the simplest case available. Hydrogen chloride and deuterium chloride have equilibrium bond lengths agreeing to three hundredths of a milliångström — the accuracy of the measurement rather than a real difference — and average separations differing by 4.34 milliångström, a hundred times more. The heavier isotopologue sits lower in an asymmetric well, samples less of it, and comes out shorter.

So the two equations are not two measurements of one structure. They are measurements of two structures that differ by a few thousandths of an ångström, solved as though they were the same — and the residue shows up as a solution that moves when the isotope is changed.

That accounts for the disagreement’s size as well as its existence. The shifts are of order thousandths of an ångström, they are largest when the substituted atom is hydrogen (because the anharmonic effect is largest there), and they are smallest for substitutions among heavy atoms.

Which gives a practical rule for any structure determined from isotopologues. Substitute a heavy atom if there is a choice. Carbon-13 for carbon-12 changes the mass by eight per cent and the average geometry by almost nothing; deuterium for hydrogen changes the mass by a hundred per cent and the average geometry by an amount comparable to the precision being sought.

Who did it first #

Microwave spectroscopy became available to chemists in 1945–46, on hardware built for radar, and the structure of carbonyl sulfide was among the first things done with it. Townes and co-workers measured its isotopologues in the late 1940s; the pair-solving argument above is the one used then and it is still what a first course teaches.

Jacob Kraitchman’s 1953 paper is the substantial theoretical advance, and it is a good example of how a measurement improves: not by better instruments, but by noticing that a difference of two moments is a cleaner quantity than either moment, because the zero-point contributions largely cancel in it. The same instinct is behind the isotopic argument in the force field is not in the spectrum, and behind the Teller–Redlich rule, which is a statement about a ratio for the same reason.

Still open: what an ionisation energy measures #

That is as far as rotation goes. The next questions turn from motion to electrons: what a photoelectron spectrum measures asks what an ionisation energy is a measurement of, and finds an answer that is not an orbital energy however often it is called one, and water’s lone pairs are not a pair uses the same spectrum to take apart a picture almost everybody is taught.