The worst case was the loosest constant

The census that counted what a symmetry label decides under three displacements ended with a caveat it could not price. Every concentration it reported — ammonia’s bond angle at 0.883 under the least-cost displacement, boron trifluoride’s at 0.573, water’s at one — is a statement about the shapes of the normal modes, and the shapes were computed from force fields fitted to observed frequencies. The constants that decide a shape are the interaction constants, which couple a bond to its neighbouring angle, and those are exactly the constants a fit to frequencies pins least.

So the question it left is whether a label’s verdict belongs to the molecule or to the least-determined number in its fit. It can be asked directly: hold that number at other values, refit everything else to every measurement the molecule has, and read the verdict again on each field that reproduces the measurements as well as the published one does.

For water the verdict belongs to the molecule. For ammonia it mostly belongs to the fit, and for boron trifluoride it belongs to nothing at all.

A family of fields, walked #

A general valence force field for ammonia has five constants: the N–H stretch, the H–N–H bend, the coupling between two stretches, the coupling between two bends, and the coupling between a stretch and a bend. Ammonia’s spectrum has four distinct frequencies, and its fully deuterated form four more, so the published field is fitted to eight numbers with five. That is overdetermined, and the fit has a residual — 0.463 per cent, root mean square over every line.

It has a residual for a reason that has nothing to do with the looseness this essay is about. The model is harmonic and ammonia is not: its umbrella vibration runs over a barrier low enough that the molecule turns inside out through it, and no quadratic potential reproduces both isotopologues’ umbrella frequencies at once. A residual of half a per cent is what a five-constant field can do, and a field with the stretch–bend constant held away from its fitted value is either as good as that, or measurably worse.

The walk holds the stretch–bend constant at a value, refits the other four to all eight frequencies, and records the result. Nineteen held values from −1.1 to +0.2 mdyn per radian, plus the points where the fit crosses its own noise level, make twenty-two fields. Water’s family was walked the same way in the essay that first asked what a spectrum fixes, with one isotopologue, where every member fits exactly; here each is fitted to two.

“As good as” needs a number. The standard one takes the published fit’s own misfit as the noise: a held field is indistinguishable from the best when its sum of squared misfits exceeds the best one’s by no more than one noise variance, which is the best sum scaled by one plus one over the degrees of freedom. Ammonia has eight distinct frequencies and five constants, so three degrees of freedom, and the limit is a residual of 0.535 per cent against the best 0.463. The criterion charges all of the misfit to noise, when most of ammonia’s is the harmonic model failing; what that choice does to the band is taken up at the end.

Ammonia’s band runs from −1.000 to −0.634 mdyn per radian. The published −0.831 sits inside it, a little off centre. Water’s, by the same rule, runs from 0.239 to 0.256 — twenty-two times narrower — because D₂O’s three frequencies do what they are supposed to do and pin the constant. Held at zero, water’s fit gets worse by a factor of 463 in its sum of squares; ammonia’s by a factor of 5.4.

The band is not a small wobble in the constants either. Across it ammonia’s stretching constant runs from 5.85 to 6.52 mdyn per ångström, its bending constant from 1.04 to 0.74, the bend–bend coupling changes sign, and the stretch–stretch coupling falls to a sixth of its value. Every one of those fields reproduces all eight frequencies to within the fit’s own half per cent.

Which direction is loose #

The census’s caveat named the interaction constant as the least-determined number, and it is the natural thing to hold. It is not quite what is loose.

The fit’s curvature says how fast the misfit grows along each direction in the space of the five constants, and its eigenvectors are the directions themselves. The loosest is ninety-nine times weaker than the next — the frequencies of NH₃ and ND₃ see one combination of constants a hundred times less well than any other — and its largest component is the bond-stretching constant, at 0.791, with the stretch–bend constant second at 0.427 and the bending constant pulling the other way at −0.349. The looseness is a trade: stiffen the bonds, soften the angles, make the coupling less negative, and every frequency stays where it was to within the harmonic model’s error. It is the same shape as a Hückel parameter that never finds a value — a valley in the misfit that runs along a combination nobody fitted for.

Holding the interaction constant and refitting the rest walks along exactly that trade. The walk’s tangent at the published field has an overlap of 1.000 with the loosest direction, to three decimals, and 0.008 or less with every other. So holding one constant was a good handle on the looseness — the only direction the fit leaves free is the one the walk goes along — but the caveat’s phrase, the least-determined number, was not quite right. It is a direction, and the number it moves most is the one nobody would have called uncertain.

Water’s loosest direction, computed the same way, is 99 per cent its stretch–bend constant and sixty times weaker than the next. There the caveat’s phrase was exactly right, and D₂O fixes it anyway.

The verdicts, across the band #

The E species is where ammonia’s bend and its antisymmetric stretch share a symmetry, so it is where a label has to say which of two vibrations a distortion goes into — the ambiguity that pricing a distortion by its species found in seventy per cent of all vibrations. Across the band:

• the angle displaced at least cost reads from 0.840 to 0.923, against the published 0.883;
• the bond displaced at least cost reads from 0.890 to 0.957, against 0.926;
• the angle under the B row reads from 0.505 to 0.579, against 0.516.

The last of those is the number the measurement at half the distance from a coin called a coin to two decimals, on the commonest distortion there is. It turns out to sit near the bottom of its own range. The curve has a sharp minimum at a stretch–bend constant of about −0.78, where the B-row angle divides its E share almost exactly evenly between the two vibrations — the concentration is the larger of two shares, so a curve through an even split has a corner, not a smooth bottom. The published field lies 0.05 mdyn per radian from that corner. Any other field in the band would have reported a slightly less dramatic coin.

The two clean displacements move more and matter more. The least-cost angle’s 0.840 at one edge of the band is under the census’s own settled threshold of nine tenths; its 0.923 at the other is over it. So whether ammonia’s bond angle joins the thirteen coordinates the census counted as settled under the least-cost displacement depends on which of the fields the spectrum cannot distinguish is used. At the published field it was not settled, and at a third of the band’s fields it would have been.

The $A_1$ species moves further. The least-cost bond reads from 0.809 to 0.937 across the band and the B-row angle from 0.517 to 0.708 — nearly a coin at one edge, clearly off it at the other. That is the species where the umbrella lives, and the umbrella is the vibration the harmonic model fits worst, so it is the one where a field’s freedom to trade the bond constant against the angle constant has most room to change a mode’s shape.

The pattern across both species is the same and it is not subtle: the least-cost displacements climb towards one as the stretch–bend constant rises toward zero, because a smaller coupling makes the modes purer, and they fall as it grows more negative. The published field is where a simplex happened to stop, in a valley whose sides rise by less than a tenth of a percentage point across a third of a unit.

Water, pinned #

Water is the control, and it behaves as a control should. Walked from −0.1 to 0.6 its readings move a great deal — the B-row angle from 0.80 to 0.94, the B-row bond from 0.84 to 0.66 — so its mode shapes are just as sensitive to the stretch–bend constant as ammonia’s. What differs is the fit. Inside water’s band, 0.239 to 0.256, the B-row angle reads 0.874 to 0.878 and the least-cost bond 0.996 to 0.997. Every verdict the census reported for water is a property of water.

The difference is not in the molecules’ mode shapes and not in the displacements. It is in the ratio of measurements to constants and in how well the model fits them. Water’s four constants meet six distinct frequencies and fit them to 0.007 per cent; a small misfit makes a narrow band. Ammonia’s five meet eight and fit them to 0.46 per cent, sixty times worse, and a large misfit is a large noise estimate. The band is wide not because the isotopologue is uninformative but because the model’s own error swamps what it says. A better model of ammonia — one that treated the umbrella anharmonically — would narrow the band without a single new measurement.

Boron trifluoride, with no bottom at all #

Boron trifluoride is different in kind. Its fit has one isotopologue, four distinct frequencies and six constants, and its published residual is 1.4 × 10⁻⁹ — zero, to the arithmetic. An earlier essay found that boron trifluoride’s field has directions in it no frequency can see, and those were directions of the Hessian: combinations of constants that change the matrix of second derivatives not at all, because the three in-plane angles sum to 360 degrees. Walking along one of those changes nothing, including every mode shape, and so changes no concentration.

This is a second kind of freedom and it does change the shapes. Boron trifluoride’s E′ block — the bend and the antisymmetric stretch — holds three independent constants: a stretching combination, a bending combination and their coupling. It has two frequencies to fit them to. So there is a curve of E′ blocks, all different, all putting the two frequencies exactly where they are observed.

The curve runs from a stretch–bend constant of −0.360 to 2.149 mdyn per radian. Past either end no pair of diagonal constants can put both E′ frequencies where they are, and the fit falls off the curve — at −0.4 the best it can do misses by 0.9 per cent. Inside, every field fits to the ninth decimal.

Along it the angle displaced at least cost — the coordinate the census named as its worst case — reads anything from 0.505 to 0.862. It is nearly a coin at a stretch–bend constant of 0.70, where the displacement splits its E′ share evenly between the two vibrations; it is 0.86 at the lower end. The published field is at 0.379 and reads 0.573, because that is where Nelder and Mead’s simplex, started at 0.2, came to rest in a valley with no bottom. The bond displaced at least cost is more robust, 0.86 to 1.00, and the B-row angle has a corner of its own at zero.

So the census’s second headline — that under both clean displacements the coordinates a label cannot resolve are boron trifluoride’s, at 0.573 — survives in its first half and not its second. Boron trifluoride’s angle stays below ammonia’s 0.883 along the whole curve, so it remains the worst case wherever the fit had stopped. But the floor it sets is not a number the molecule has: it is anything from a coin to 0.862, and the census’s count of undecided coordinates, three, would have been three on the middle of the curve and zero at either end of it.

The identity that holds everywhere #

One thing does not move, and its not moving is the check that everything else is real.

The earlier census found an exact identity between its two clean displacements: in a species that appears twice with one bond and one angle coordinate, the bond displaced alone reads exactly what the angle displaced at least cost reads, because the inverse of a two-by-two block is its adjugate. The argument used no property of the force field at all, so the identity must hold on every member of every family here, with each member’s modes rebuilt from its own constants.

It does. Across 81 species-member pairs — every walked and in-band field of water, ammonia and boron trifluoride — the largest gap is 1.1 × 10⁻⁸, the finite-difference floor. The concentrations it relates move by a third of their range along the same walks. That is the statement that the modes really were recomputed each time and not reused from the published field, and it is also the cleanest way to say what the looseness does and does not touch: it moves every concentration, and it cannot move a relation between two of them that holds for any field.

The other checks are the dull ones that make the interesting numbers believable. Walked to its published stretch–bend value, each family refits to the published field — every constant to a part in a thousand — so the family passes through the fit the census used rather than beside it. And the concentrations do respond to the held constant, so the walk is not along a direction the modes cannot see, which would have returned one number for every member and proved nothing about looseness.

What would choose a field #

A family of fields the measurements cannot tell apart is an invitation to find a measurement that can. The fields disagree about everything they were not fitted to, and some of those disagreements are large.

For ammonia the decisive one is a partly deuterated molecule. NH₂D’s N–D stretch runs from 2,673 to 2,746 cm⁻¹ across the band — 74 cm⁻¹, about three times what the worst member of the band misses a fitted line by at that frequency. The reason is the direction the loosest trade takes. NH₃ and ND₃ each carry three equivalent bonds, so they only ever see the bond constant in the combinations $f_r + 2f_{rr}$ and $f_r - f_{rr}$, symmetric and antisymmetric. A single N–D bond among two N–H bonds vibrates nearly on its own, and its frequency reads the bond constant much more directly — which is the constant the loose direction moves most.

The ¹⁵N shift of the symmetric stretch is a weaker lever: 8.3 to 12.3 cm⁻¹, a spread of four wavenumbers, about 40 per cent of the shift itself. Both are ordinary gas-phase measurements.

For boron trifluoride the measurement is already in every spectrum anyone has taken. Natural boron is one fifth ¹⁰B, so every sample of boron trifluoride carries ¹⁰BF₃ alongside ¹¹BF₃, and the ¹⁰B shift of the E′ stretch runs from 38.7 to 59.4 cm⁻¹ along the exact curve, with the published field predicting 53.4. The shift of the out-of-plane bend, 29.4 cm⁻¹, is the same on every member, because the out-of-plane block has one constant and one frequency and nothing to trade. So the one isotopologue that would pin the census’s worst case is one boron trifluoride cannot be prepared without.

None of these predictions is compared with a measured value here, and that is deliberate. The fits here are to harmonic frequencies, derived from observed band centres with anharmonic corrections, and the stored data hold them only for the isotopologues already fitted. A measured N–D stretch of NH₂D or ¹⁰B shift of BF₃ would have to be brought to a harmonic value by the same route before it could vote, and that is a data step, not a calculation.

What the picture cannot show #

The model’s own error sets the width of ammonia’s band. The band is where a five-constant harmonic field cannot tell fields apart at its own residual, and most of that residual is the harmonic model failing on a molecule with a low barrier to inversion. A different noise model — the band centres’ measurement uncertainties rather than the fit’s misfit — would make the band far narrower on paper and would be wrong in the other direction, because it would charge the model’s error to the constants. The truthful reading is that ammonia’s field is fixed to the extent its model is right, and this model is right to about half a per cent.

The count of degrees of freedom is the plain one. By the product rule, ND₃ adds one independent equation per symmetry block to what NH₃ already gives — two, not four — so within the harmonic model ammonia has six independent data for five constants, one degree of freedom rather than three, and a band wider than the one drawn. The plain count is used because the observed frequencies do not obey the product rule exactly, and that disobedience is itself part of the misfit. Water is the sharper case: D₂O adds exactly one equation, to the one block with two frequencies, and one equation is what water’s one-constant family needed, which is why it is pinned.

Only one direction was walked. For ammonia and water the walk lies along the loosest direction to three decimals, so it samples the only freedom that matters; the next-loosest direction is a hundred times stiffer. For boron trifluoride the walk is along the one E′ freedom; the Hessian-flat directions that a projection removes change no mode and were not walked.

Methane and sulfur dioxide are not in it. Sulfur dioxide’s three constants meet its three frequencies exactly and it has no family. Methane has two isotopologues and a redundancy, and its refits have run away along the redundancy before; walking it needs the projection first.

The concentrations are the census’s, unchanged, and so is what they are for. They price a static distortion by the vibrations it would excite — the question a structure that fell along a soft coordinate put to a force field — and every such pricing inherits the band measured here. The displacements, the species decomposition and the settled and undecided thresholds are the ones the three-displacement census defined. Only the force field under them has been allowed to vary.

Who fitted what, and why this was always known in principle #

That force constants are underdetermined by frequencies is as old as the valence force field. Wilson’s GF method of 1939 separated the kinetic and potential parts of the problem precisely so that isotopic substitution — which changes the kinetic matrix and nothing else — could supply extra equations, and the product rule of Teller and Redlich, from 1935 — the identity from which the force constants cancel — says exactly how much new information an isotopologue carries about each symmetry block: the product of its frequencies in a block is fixed by the masses alone, so a block of two frequencies gains one new equation from a substitution, not two, and a block of one gains none. Mills and others in the 1960s added Coriolis coupling constants and centrifugal distortion constants to the fits for the same reason; they are measured from rotational structure and see the mode shapes directly.

None of that is new here. What is computed is the size of the effect on one specific set of published verdicts: how wide each band of indistinguishable fields is, which direction it runs along, how far each concentration moves across it, and which isotopologue would close it.

Still open: the Coriolis constants, and the band the model sets #

The obvious open question is the measurement the mode shapes answer to directly. An E vibration of a symmetric top carries a Coriolis constant, ζ, that is a sum of products of the mode’s own displacement amplitudes, and it is measured from the rotational fine structure of the band rather than from its centre. Every field in ammonia’s band predicts its own ζ for each E vibration, and if they differ as much as the concentrations do, ζ would be the measurement that settles which verdict is ammonia’s — with no new isotopologue needed. It is a sum over the modes already computed on every member.

The nearer question is the one the band’s width comes from. Ammonia’s residual is mostly its umbrella, and the inversion’s double well is already computed here. Fitting the four other constants to every frequency except the two umbrella lines — or to the umbrella’s harmonic frequency at the top of the well rather than its observed tunnelling pair — would say how much of the band is the harmonic model’s error on one vibration, and how much would survive a model that got that vibration right.