The Coriolis constant was an isotope shift

Boron trifluoride’s E′ block has three force constants and two frequencies. The exact fits form a closed loop, along which the B–F stretching constant runs from 3.9 to 12.1 mdyn per ångström while every field reproduces the spectrum of ¹¹BF₃ to a thousandth of a wavenumber. A ¹⁰B spectrum cuts the loop to two fields — the one the data came from, at 7.425, and a partner at 3.966 — and no third isotopologue that keeps the molecule’s symmetry can separate them, because the two fields give the same frequencies at every symmetric choice of masses.

That essay named the next candidate. A Coriolis constant measures how the two partners of a degenerate vibration are coupled when a symmetric top rotates, and it is computed from the mode shapes, not from their frequencies. Two fields that share eigenvalues need not share eigenvectors — an earlier refit had already shown how differently two fields can move a molecule while agreeing on its frequencies — so ζ looked like the quantity that could decide.

It cannot, and the reason turns out to be a single line connecting two measurements usually treated as unrelated.

Every value, twice #

The axial Coriolis constant of an E′ pair is a sum over atoms of the turning sense of each atom’s motion in one partner into its motion in the other, mass-weighted, with the partners oriented by the threefold rotation so that the sideways translation comes out at exactly +1. It is the same construction that read ammonia’s two E vibrations, and here it is evaluated at two hundred and one fields spaced around the loop.

Around the loop $\zeta_3$ takes every value it can, from −1 to +1. At one point of the loop the stretching pair circulates exactly like a rigid sideways translation; at another it circulates exactly against it. So before any isotopic data, a measured $\zeta_3$ is enormously informative: it is a continuous coordinate around the whole loop.

But a closed curve meets any level twice. The fitted field’s $\zeta_3$ of 0.7910 is reached at exactly two points of the loop, at B–F constants of 7.425 and 3.966 — and those are the fitted field and its ¹⁰B partner. The Coriolis constant does not merely fail to separate the two fields. It picks out the same pair the isotope does.

A planar top has one Coriolis number #

The first thing to settle was whether two constants were in play. For ammonia, $\zeta_3$ and $\zeta_4$ obey a sum rule, $\zeta_3 + \zeta_4 = I_z/2I_x - 1$, fixed by the geometry and the masses. A planar molecule has every atom in the plane perpendicular to the axis, so by the perpendicular-axis theorem $I_z = 2I_x$, and the rule becomes $\zeta_3$ + $\zeta_4$ = 0.

On every one of the two hundred and one fields the sum is zero to $10^{-15}$. The rotations about axes in the plane are not E′ motions in a planar molecule — they belong to E″ — so the E′ space contains only the translation and the two vibrations, and with the translation at +1 the total is one, as it was for ammonia. $\zeta_4$ is −0.791 under the fitted field, and the E′ block carries one axial Coriolis number, not two.

The constant is the boron’s share of the motion #

Why one number should coincide under two fields three mdyn per ångström apart becomes clear when ζ is read atom by atom. Boron sits on the axis. Its sideways motion can only circulate as a translation does, so its term in ζ is its whole share of the pair’s motion, with the sense of the rotation — exactly as ammonia’s nitrogen behaved. The fluorines carry the rest.

Around the entire loop, $\zeta_3 = 2s/(1 - m_{\mathrm B}/M) - 1$, where s is the boron’s share and $m_{\mathrm B}/M$ its fraction of the molecule’s mass, to three parts in a thousand million. The line runs from −1, where the boron does not move in the stretch at all, to +1, where it carries everything a vibration can give it — 0.838, since the rest of its sideways freedom belongs to the translation. The fluorines’ contribution is therefore not free either: the identity says that once the boron’s share is known, how the three fluorines circulate is known too.

And the boron’s share has a second meaning. Make the boron heavier by a small amount and each squared frequency falls in proportion to how much of the mode the boron carries:

$$s_k = -\frac{\partial \ln \lambda_k}{\partial \ln m_{\mathrm{B}}},$$

which is a sensitivity of the spectrum to the boron mass — the quantity a boron isotope shift measures, in the limit of a small shift, and the same kind of mass derivative from which an isotope’s substitution reads an atom’s position. Differencing the frequencies around ¹¹B reproduces the share to a part in a million under both fields. So $\zeta_3$ is a boron isotope shift written in another form.

That closes the argument. The essay on the third mass showed that the fitted field and its partner give the same E′ frequencies at every boron mass, because the sum of the squared frequencies is linear in the reciprocal masses and their product obeys Teller and Redlich’s rule. Two curves that are equal at every mass have the same slope at every mass. The slope is the boron’s share; the share is ζ.

The same cut, from either side #

The two measurements cut the loop identically, and the picture shows how closely they track each other everywhere else too: the ¹⁰B mismatch and the ζ mismatch rise and fall together around the loop, which is what two readings of one function of the boron mass — one over a finite step, one at a point — would be expected to do. A measured $\zeta_3$ can stand in for a ¹⁰B spectrum. It cannot add to one.

That inverts the usual expectation. Coriolis constants are read from the rotational fine structure of a band and isotope shifts from its centre, and the two are normally reported in different tables as independent constraints on a force field. For boron trifluoride’s E′ block they are one constraint measured twice. Where a fit uses both, it has not gained an equation; it has gained a consistency test on the harmonic model, since the two must agree if the model is right.

What one number fixes, and what it leaves #

A measurement that cuts a closed loop to two points has still told the loop something definite, and it is worth asking what the two surviving fields have in common rather than only how they differ. The table answers it in one line. The fitted field and its partner differ in the B–F stretching constant by 3.46 mdyn per ångström and in the bending constant by a factor of four — and their stretch–bend interaction constants agree to a part in ten million, at 0.3786.

That is not a property of this pair. Around the entire loop, the interaction constant is a straight line in $\zeta_3$,

$$f_{r\theta} = 3.1748 - 3.5349\,\zeta_3 \ \text{mdyn/rad},$$

to a ten-thousandth everywhere the loop has been sampled. Every level of $\zeta_3$ meets the loop at two fields, and each pair shares its interaction constant: at $\zeta_3$ = 0 both have 3.175, at $\zeta_3$ = −0.9 both have 6.356. A measured $\zeta_3$ fixes the interaction constant exactly and leaves the two diagonal constants two-fold. The ¹⁰B shift, being the same quantity, does the same.

That changes what the two-fold ambiguity is about. It was described as a choice between two force fields, which made it sound as though everything about the E′ block were in doubt. It is narrower than that. The coupling between a B–F stretch and its neighbouring F–B–F bend is settled by the data already in hand; what remains open is only how the remaining stiffness is shared between the bond and the angle. And that is the question the symmetry-breaking substitution addresses: one heavy fluorine puts one bond in a different environment from the other two, which weighs the bond’s stiffness separately from the angle’s.

It is also why Coriolis constants have been used in fitting the force fields of small symmetric molecules: a degenerate block’s interaction constant is the parameter frequencies alone leave free, and ζ fixes it. What this loop adds is that, in a planar molecule, a boron isotope shift fixes the same parameter to the same precision, and the two cannot be added together to fix anything more.

The rate of exchange #

If $\zeta_3$ and the ¹⁰B shift are two readings of one quantity, each can be priced in the other’s units, and that is the practical content of the identity.

The loop passes through every value of $\zeta_3$ twice, once on each half, and both halves give the same ¹⁰B shift at the same $\zeta_3$, for both bands — to within the sampling of the loop, a few hundredths of a wavenumber. So the shift is a function of $\zeta_3$ alone, and the function has a simple shape. At $\zeta_3$ = +1 the boron carries all of its share in the stretch: the stretch’s band moves by 59.4 cm⁻¹ on going to ¹⁰B and the bend’s not at all. At $\zeta_3$ = −1 the boron is entirely in the bend, which then moves by 19.6 cm⁻¹ while the stretch stands still. Between them the boron’s motion is traded from one vibration to the other, and $\zeta_3$ is the share of the trade the stretch has taken. Under the fitted field the stretch shifts by 53.3 cm⁻¹ and the bend by 1.9.

Near the fitted field $\zeta_3$ changes by 0.034 for every wavenumber of the stretch’s shift, and by 0.108 for every wavenumber of the bend’s. A Coriolis constant measured to ±0.005 is therefore worth a ¹⁰B shift of the stretch known to ±0.15 cm⁻¹, or of the bend to ±0.05. That is the comparison an experiment would make: whichever of the two is easier to measure to the equivalent precision is the one worth taking, and taking both buys a check on the model rather than a second equation. For boron trifluoride specifically the shift is the easier one, since natural boron is a fifth ¹⁰B and every spectrum of the compound already contains the ¹⁰B bands.

And the couplings across the axis #

The axial constant is not the only Coriolis coupling. Rotation about an axis in the plane couples each E′ pair to the out-of-plane wag, and those couplings depend on the shapes too — through the wag’s own shape, which is fixed, since its block has one constant and one frequency, and through the E′ pair’s.

They agree as well: 0.209 for the bend and 1.791 for the stretch under both fields, to $10^{-7}$. And they are not new numbers. Each E′ pair’s coupling to the wag is 1 + its axial ζ, at every point of the loop, so the couplings across the axis are the axial constant again. Every Coriolis constant of the molecule’s vibrations is a function of the one number the boron isotope shift already fixes.

How each claim is held #

The orientation and the sum rule come from the same checks that ammonia’s constants used: the translation pair gives +1 exactly under the same orientation rule, and $\zeta_3$ + $\zeta_4$ vanishes to $10^{-12}$ at every point of the loop.

The identity is checked as an identity, at every sampled field, between the eigenvector sum and the line through the boron’s share, and separately the share is checked against the spectrum: a central difference of the squared frequencies in the reciprocal boron mass reproduces it to $10^{-6}$ under both fields. The two routes have nothing in common but the force field.

The coincidence is checked where it matters: the two fields’ stretching constants differ by more than three mdyn per ångström, and their axial constants for both isotopologues and their couplings to the wag agree to $10^{-7}$. The cut is checked as a cut: the fitted $\zeta_3$ is met at exactly two points of the loop, and each lies within 0.02 mdyn per ångström of one of the two fields.

The refusal is a field off the loop. Moving the fitted field’s bending constant by a tenth must change $\zeta_3$ by more than a thousandth; a quantity every field shared would be no measurement of anything, and the identity would be trivially true.

Where the argument stops #

The loop is sampled, not solved in closed form. Every identity here is checked at two hundred and one of the 1,402 fields the loop was traced through, evenly spaced around it, and at the two fields themselves after they were polished to the ninth decimal; the linearity of the interaction constant in $\zeta_3$ is a measured statement about those points, not a derived one, and why the line should be straight is not shown here.

The ¹⁰B data are synthetic. As in the essays this one continues, the ¹⁰BF₃ frequencies are what the fitted field predicts, so the question throughout is identifiability — what a measurement of a given kind could ever decide — and not agreement with a table. A real ¹⁰B shift carries anharmonic corrections, and so does a real ζ, and neither is corrected here.

Every constant is harmonic and first-order. The identity is a property of the quadratic field at equilibrium. A measured ζ includes higher-order contributions and, for boron trifluoride’s $\nu_3$ and $\nu_4$, the effects of nearby levels; the identity says the harmonic parts of the two measurements are one quantity, not that the measured numbers will agree to the last digit.

The identity is a planar molecule’s. It needs the on-axis atom’s share to be the only thing the E′ block leaves free, which holds because the in-plane rotations sit in a different species. In ammonia the rotation pair is in the E species with the vibrations, and its two Coriolis constants did carry information about a field its frequencies could not fix; whether some isotopologue’s spectrum would have carried the same information there is not computed.

Two measurements that were one #

The habit this leaves is a question to ask of any proposed new constraint: is it a derivative of something already measured? A Coriolis constant felt like a different kind of data because it comes from a different part of the spectrum, and the force field is not in the spectrum in exactly the way that makes new kinds of data valuable. But kind is not independence. The boron’s share of the motion is what ζ counts and what the isotope shift weighs, and two instruments pointed at one quantity give one equation.

Who measured what #

Coriolis splitting of degenerate bands was worked out for symmetric tops in the 1930s, the sum rules for the ζ constants of a species were set out in general by Meal and Polo in 1956, and planar XY₃ molecules were among the first to which they were applied, with $\zeta_3$ + $\zeta_4$ = 0 a textbook result. That isotope shifts and Coriolis constants constrain overlapping parts of a force field has long been known in practice.

What is computed here is $\zeta_3$ around the entire loop of exactly fitting fields, the identity tying it to the boron’s share of the motion and so to the boron mass-sensitivity of the frequencies, the coincidence of every Coriolis constant under the two ¹⁰B-consistent fields, and the finding that a measured $\zeta_3$ cuts the loop at the same two points as a ¹⁰B spectrum.

Still open: the off-axis atom, and a molecule with two chlorines #

The nearer question is the substitution that breaks the symmetry. Replacing one fluorine separates the two fields’ frequencies by about half a wavenumber, and a molecule with one heavy fluorine is no longer a symmetric top, so its E′ pairs split and its Coriolis couplings become a matrix of first-order constants between non-degenerate modes. Whether any of those separates the two fields by more than the half wavenumber the frequencies manage — or whether each is again a derivative of a mass the frequencies already weigh — is a calculation on the same loop.

The obvious question is boron trichloride. Its natural chlorine isotopes give mixed species — two chlorine-35 atoms with one chlorine-37, and the reverse — in every sample, so the symmetry-breaking substitution boron trifluoride lacks is already in the spectrum. The same construction — a loop, its ¹⁰B cut, and the mixed species under both fields — would say whether a real molecule’s natural isotopic mixture decides its E′ block.