Two Coriolis constants are one number

Walking ammonia’s force field along the direction its frequencies see least found a band of fields the spectrum cannot tell apart. Fitted to the four distinct frequencies of NH₃ and the four of ND₃, a five-constant valence field reproduces all eight to half a per cent for any stretch–bend interaction constant between −1.000 and −0.634 mdyn per radian, and across that band the verdict a symmetry label gives on the bond angle moves from 0.840 to 0.923 — from undecided to settled. The verdict belonged to the fit, not to the molecule.

The essay ended by naming the measurement that sees mode shapes directly, which is what a spectrum’s frequencies alone never fix. A degenerate vibration of a symmetric top carries a Coriolis constant, written ζ, which splits its rotational lines by an amount the band analysis reports alongside the band centre, and ζ is built from nothing but the displacement pattern of the two partners of the pair. Each field in the band predicts its own pair of ζ values for ammonia’s two E vibrations, the bend $\nu_4$ and the stretch $\nu_3$.

Both halves of that proposal are now computed. The first holds better than expected: ζ moves by a quarter to a third across a band in which no frequency moves by one per cent. The second does not hold at all. The two constants are not two measurements.

What ζ counts #

When a molecule rotates, each vibrating atom feels a Coriolis force at right angles to its own motion, and in a degenerate pair that force couples the two partners: a vibration along x is pushed toward y. The strength of the coupling about the symmetry axis is

$$\zeta = \sum_k \big(l^{a}_{x,k}\,l^{b}_{y,k} - l^{a}_{y,k}\,l^{b}_{x,k}\big),$$

a sum over atoms of the turning sense of each atom’s displacement in partner a into its displacement in partner b, with the displacements mass-weighted and normalised. It lies between −1 and +1. A pair in which every atom moves in a circle about the axis when the partners are combined with a quarter-period lag has ζ near +1, like a translation; a pair whose atoms circle the other way has ζ near −1. Its sign depends on how the two partners are labelled, so the label is fixed by the threefold rotation: the pair is oriented so that the rotation by 120° carries the first partner toward the second, as it carries x toward y. With that convention the translation pair gives exactly +1, which is the check that the orientation is the one every table uses.

Twenty-two fields, two constants each #

Every member of the family is the field with its stretch–bend constant held at a value and the other four refitted to all eight frequencies. On each, the two E vibrations’ partners are found as degenerate eigenvectors, oriented by the rotation, and ζ is summed.

On the published field $\zeta_4$ is −0.732 and $\zeta_3$ is +0.509. The bend’s partners turn against the rotation, the stretch’s with it. Across the band the frequencies cannot resolve, $\zeta_4$ runs from −0.807 at one edge to −0.641 at the other, and $\zeta_3$ from 0.584 to 0.418. Both are straight lines in the held constant to within a few parts in a thousand, which makes them easy to invert: $\zeta_4$ changes by 0.453 for every mdyn per radian.

That is a large response, and the size is the point. The band exists because the frequencies barely notice where along it the field sits — that is what being unable to tell the fields apart means — so it is worth setting the two kinds of observable side by side.

Every frequency of either isotopologue moves by between 0.28 and 0.87 per cent across the band. $\zeta_3$ moves by 33 per cent and $\zeta_4$ by 23. The frequencies are sums over the field weighted by the mode shapes, and the walk is precisely the direction in which a change of shape and a change of stiffness cancel in those sums. ζ has no stiffness in it at all. It is the shape alone, and the shape is what moves.

So the lead was right that ζ would decide. What it did not anticipate is what the two constants do together.

Where the turning lives #

The sum over atoms in ζ can be read atom by atom, and the reading says what the field is actually changing. Set an E vibration’s two partners running a quarter of a period apart and every atom traces an ellipse in the plane across the axis. Its term in ζ is the ellipse’s signed area, and divided by the share of the pair’s motion the atom carries in that plane it becomes a signed roundness: +1 for a circle traced in the sense of the rotation, −1 for one traced against it, zero for motion back and forth along a line.

The nitrogen’s term is always exactly its own share of the motion. An atom on the axis cannot do anything sideways except move as a translation does, so its path is a circle in the sense of the rotation on every field — the check holds to a part in a million — and it carries little: 1.5 to 2.8 per cent of the bend’s motion across the band, 11 to 12 per cent of the stretch’s. Everything else is the three hydrogens.

What the walk changes is the hydrogens’ ellipses, and it changes their shape rather than their size. In the bend each hydrogen carries 32.0 per cent of the motion at one edge of the band and 30.9 per cent at the other — almost the same — while its signed roundness goes from −0.855 to −0.721. In the stretch the share goes from 22.6 to 23.7 per cent and the roundness from +0.680 to +0.433. The figure shows the two kinds: a bending hydrogen’s ellipse lies mostly across its own N–H direction and a stretching one’s mostly along it, and walking toward the weaker coupling flattens both, which is why both constants move toward zero together. ζ reads the shape of the motion and nothing else, which is exactly what the frequencies cannot see: a frequency weighs each shape by a stiffness, and along the band a change of shape and a change of stiffness are traded against each other in every frequency without moving any by more than the fit’s own misfit.

That is also why the label’s verdict follows $\zeta_4$ so closely. The census of what a symmetry label decides asks how much of a bond-angle distortion goes into the bending vibration rather than the stretching one — which is a question about the same E mode shapes, asked through a distortion rather than through a rotation.

The sum the field cannot touch #

The third curve in the first figure is the sum $\zeta_3$ + $\zeta_4$, and it is flat. On every one of the twenty-two fields, from −1.1 to +0.2 mdyn per radian, it is −0.22247. For ND₃ on the same fields it is −0.19551, equally flat.

Those numbers are not fitted. They are what the geometry and the masses give for one simple ratio: with $I_z$ the moment of inertia about the threefold axis and $I_x$ the moment about an axis across it,

$$\zeta_3 + \zeta_4 = \frac{I_z}{2I_x} - 1,$$

and for NH₃ at the stored geometry that is −0.222467 against −0.222466 summed from the eigenvectors. The agreement holds to a part in a million on every field and for both isotopologues — and ND₃’s value differs from NH₃’s only because deuterium moves the moments — the same moments an isotope’s substitution reads a position from.

The reason is completeness. Every displacement of four atoms belongs to some symmetry species, and the E-species displacements form a space that contains the sideways translations, the rotations about the two axes perpendicular to the symmetry axis, and the two vibrations — nothing else. The sum of ζ over a complete set of pairs in that space depends on the space and not on how it is divided up, the same way the trace of a matrix does not depend on its eigenvectors. The translation pair takes exactly +1; the rotation pair takes $1 - I_z/2I_x$, which is 0.222 for NH₃; and the whole E space adds to exactly one, for NH₃ and ND₃ alike, although every term inside it moves with the masses. Whatever is left for the two vibrations is fixed before any force constant is chosen.

So a force field cannot move $\zeta_3$ without moving $\zeta_4$ by the same amount the other way. Every field in the family, every field outside it and every field anyone has fitted to ammonia agrees on the sum. The two constants a band analysis reports are one measurement of the field and one check on the geometry. This is the same shape of fact as the Teller–Redlich product rule, in which the frequencies of two isotopologues stand in a ratio no force constant can affect — a quantity built from mode shapes whose total over a symmetry block is fixed by the masses.

The verdict, read off the bend’s constant #

One number about the field is still enough, because the family is one-dimensional. The band was a walk along one direction, and every quantity that changes along it — the verdict on the bond angle included — is a function of where on the walk the field sits. $\zeta_4$ is a coordinate for that position.

The verdict is the concentration the three-displacement census defined: of a unit bond-angle distortion, applied in the way that disturbs the molecule’s kinetic energy least, what share goes into the E vibration the symmetry label names. On the published field it is 0.883. It crosses the census’s own settled threshold of nine tenths at $\zeta_4$ = −0.696. A measured $\zeta_4$ less negative than that puts ammonia’s bond angle among the coordinates a symmetry label decides; one more negative leaves it among those it does not. The published field’s $\zeta_4$ of −0.732 sits 0.036 on the undecided side.

Ammonia’s $\zeta_4$ is not compared with a measured value here, and the reason is not a missing number but what the number would mean. The measured ζ of a band comes from fitting its rotational structure, and ammonia’s $\nu_4$ band is entangled with its inversion doubling and with Coriolis and l-type interactions involving the overtone of the umbrella, so the effective ζ a band analysis reports carries those resonances inside it. It would have to be corrected to the harmonic ζ the field predicts before it could vote — the same data step the essay that proposed isotopologues declined to take for frequencies, and for the same reason.

Two isotopologues, one curve #

ND₃ has its own pair of Coriolis constants, and they move across the band as well: $\zeta_4$ from −0.836 to −0.690. It might look like a second, independent measurement. It is not one either.

Both are functions of the same one-dimensional family, so every field puts the pair on one curve. A measured pair would locate the field once and test the model once: if the two measured values fell off the curve, no member of the family would reproduce both, and the fault would lie with the harmonic valence field itself — with the five-constant form — rather than with any constant in it. That is a better use of the second number than a second vote, because the band’s width is set by the model’s own error, and nothing among the frequencies can separate the model’s error from the field’s looseness.

What a measurement to a given precision would buy #

Because $\zeta_4$ is linear across the band with a slope of 0.453 per mdyn per radian, its precision converts directly. A $\zeta_4$ known to ±0.005 confines the stretch–bend constant to 0.022 mdyn per radian, sixteen times narrower than the band’s 0.367. At ±0.02 the window is 0.088, still four times narrower. Only at ±0.05 does the measurement approach what the frequencies already do. For comparison, the whole distance between the published field and the verdict’s nine-tenths crossing is 0.036 in $\zeta_4$, so a precision of about ±0.01 is what it takes to say on which side of the threshold ammonia’s bond angle falls.

How each number is held #

The orientation is checked on a pair whose answer is known. The sideways translations, put through the same orientation rule and the same sum, give ζ = +1 to the last digit; an orientation that had the rotation’s sense backwards would give −1, and every vibrational ζ would change sign with it. Each vibrational pair is also checked to turn into itself under the rotation by exactly 120°, which is the test that the two partners really are a degenerate pair and not two neighbouring modes.

The sum rule is checked from two routes that share nothing: the eigenvectors of every field, summed, against the moments of inertia of the geometry, on all twenty-two fields and for both isotopologues, to a part in a million. The rotation pair, built from the geometry alone, gives $1 - I_z/2I_x$, and with the translation and the two vibrations the E-species total is one on every field.

ζ is checked to respond. Across the band $\zeta_4$ must move by more than a tenth; a walk along a direction the mode shapes could not see would return one ζ for every field and prove nothing about the field’s looseness.

The refusal is a pair with nothing to couple: ammonia’s two $A_1$ vibrations, both unchanged by the rotation, must give zero, and do, to $10^{-6}$.

Where the field’s own form stops the argument #

Every ζ here is harmonic and first-order. It is the constant the quadratic field predicts at the equilibrium geometry, which is what a fitted valence field is about; the ζ a band fit reports is an effective one, with anharmonic and resonance contributions that for ammonia are not small. The comparison needs those removed, and that is not done here.

The geometry is the stored one, and the sum rule inherits it. $I_z/2I_x - 1$ is a property of the bond length and angle, and an error in either moves the sum. It does not move the split between $\zeta_3$ and $\zeta_4$ along the band, which is what reads the field.

Inversion is outside the model. Ammonia’s umbrella runs over a barrier low enough that the molecule tunnels through it, and the harmonic field describes a molecule held on one side. The E vibrations are less affected than the umbrella, but every E level is in fact a tunnelling doublet, and each component can carry its own effective ζ.

Only the one loose direction is walked. The family is the curve along the direction the frequencies see least, which the walk’s own analysis found to be a hundred times looser than the next. A measured ζ fixes the position along that curve; whether it agrees with the curve is the separate question the two isotopologues together would ask.

Coriolis, Teller, and Meal and Polo #

The splitting of degenerate vibrational bands by rotation was understood in the 1930s, in the work of Teller, Johnston and Dennison on symmetric tops, and sum rules for the ζ constants of a symmetry species — the statement that their total is fixed by geometry and masses — were set out in general by Meal and Polo in 1956. They have long been used in the other direction: a measured ζ and the sum rule together give a second constant without a second band analysis, or a check on a geometry. Ammonia’s general valence force fields, fitted to NH₃ and ND₃, go back to the same period.

What is computed here is $\zeta_3$ and $\zeta_4$ on every member of the band of fields ammonia’s frequencies cannot tell apart, the size of their response against the frequencies’, the verdict on the bond angle as a function of $\zeta_4$ with its crossing at −0.696, and what a stated precision in $\zeta_4$ would buy.

Still open: the effective ζ, and the one-curve test #

The nearest question is the data step. Ammonia’s $\nu_4$ and $\nu_3$ band analyses report effective ζ constants, and turning one into the harmonic $\zeta_4$ this field predicts means removing the anharmonic contribution and the resonance with the umbrella’s overtone. If that can be done to ±0.01, the measurement says on which side of nine tenths ammonia’s bond angle falls, and to ±0.005 it narrows the field sixteen-fold.

The sharper question is the one-curve test. NH₃ and ND₃ each give one usable ζ, and the family predicts the pair lies on one curve. Measured values for both, corrected the same way, would either land on it — locating the field and confirming the five-constant form — or miss it, which would be the first evidence from outside the frequencies that the form, not the constants, is what limits ammonia’s field.