The forty-five that are fixed

Ten directions no frequency can see counted the directions in methane’s space of force constants that leave every frequency exactly where it was. There are ten of them, out of fifty-five, and the count is a theorem rather than a measurement.

It ended by naming what to do next, and the sentence is worth repeating because it is a request for a table rather than for a number:

The flat space is a subspace of the fifty-five-dimensional space of constants, so its orthogonal complement is a set of forty-five combinations that are fixed by the spectrum — and those combinations are computable, being the singular vectors belonging to the non-zero singular values.

They are, and here they are. Two things come out of the same eigen-decomposition, and the first is more useful than the second.

How much of each constant is fixed #

Project the axis belonging to one force constant onto the determined subspace and take the squared length. One means the constant is fixed on its own; anything less is the fraction of it that is, and the shortfall is the part that can be traded against other constants for nothing at all.

For methane:

Every stretching constant is fixed exactly and not one bending constant is. That is a sharper statement than the field is underdetermined, and it is a different statement: a reader of the flat-direction count would reasonably conclude that all fifty-five numbers are somewhat uncertain, and forty-five of them are not uncertain at all.

Boron trifluoride is the same shape with worse numbers. Its three stretching constants are at one, and its three bending constants are at 0.3000 — under a third of each bend constant is fixed by everything the spectrum knows.

Why the stretches escape is worth a sentence, because it makes the pattern predictable rather than lucky. A flat direction has to produce no Cartesian displacement, and the only combination of these coordinates that produces none is the redundancy among the angles — there is no redundancy among the bonds, because four bond lengths are four independent facts about where four atoms are. Every flat direction therefore has an angle in it somewhere, and a constant with no angle in it cannot be part of one. The division between the fixed and the free is not about how strongly a constant affects the spectrum; it is about which coordinates it names. A weakly determined constant in the ordinary sense would show as a small singular value, and none of these is small: the split here is exactly at zero.

The fractions sum to the rank, 45 for methane and 21 for boron trifluoride. That is a trace identity rather than a result, and it is here because it is the arithmetic check that the projection is a projection.

Which combinations, rather than how much #

A fraction says how much of a constant is fixed and not which combination of a set is. Restricting the projection to one kind of constant and diagonalising the block answers the second question: an eigenvalue of one is a combination the spectrum fixes exactly, a zero is one it cannot see at all, and anything between is partly fixed because the rest of it lives outside the block.

Methane’s blocks:

The bending block is the interesting row and it is the one a spectroscopist would most like to have. It contains no ones and no zeros. There is no combination of methane’s six bend constants that the spectrum fixes, and none that it is entirely blind to; every one of them is partly determined and partly traded against couplings elsewhere. An uncertainty attached to a single quoted bend constant cannot express that, because the quantity that is fixed is not in the block.

The eigenvalue of 0.714 belongs to the totally symmetric combination — the sum of all six angles’ constants — and the five at 0.500 belong to the rest. That is the opposite of what the redundancy argument would suggest at a glance: the combination lying along the redundancy is the best determined of the six, not the worst. It is determined because the redundancy’s own direction in the space of constants is r rᵀ, which is a single direction, while the block has six; and what the block is short of is not that one direction but the part of every direction that leaks into the stretch–bend constants.

Boron trifluoride’s bending block is 0.500, 0.200, 0.200 — same shape, less of it.

What a flat direction is made of #

The four zeros in methane’s stretch–bend block are worth writing out, because a null vector printed as a list of fifty-five numbers tells nobody anything and this one has a shape.

Each is the sum of one bond’s couplings to every angle at the same centre, with equal weight and the same sign: the coupling of the C–H₁ stretch to H₁–C–H₂, to H₁–C–H₃, to H₁–C–H₄, to H₂–C–H₃, to H₂–C–H₄ and to H₃–C–H₄, added up. One such direction per bond, four bonds, four directions.

That it has this shape is not an accident and it is the same fact the flat-direction count starts from. The redundancy among methane’s coordinates is the totally symmetric combination of its six angles, which describes no displacement of any atom. Coupling a bond to a combination of angles that does not move anything cannot move anything either — so the whole set of couplings from one bond to that combination is invisible, and there is one of them per bond.

The remaining six flat directions are of the same kind one level up: they mix bending constants with bending–bending constants, which is exactly the direction a fit was found wandering along when boron trifluoride’s field was refitted from four starting points.

The constraint, written down #

That refit is worth restating, because the projection turns it from a warning into a repair.

Boron trifluoride’s field fitted from four starting points that differ only in where the bend and bend–bend constants began returns bend constants of 0.4193, 0.6622, −0.2191 and 0.2878 mdyn per ångström. One is negative. All four reproduce every observed frequency of the molecule and of its isotopologue, with residuals agreeing to three figures.

The recorded repair was one linear constraint per redundancy, and the constraint is now available: project the fitted constants onto the determined subspace and throw the rest away.

A spread of 0.8813 becomes a spread of 0.0027 — a factor of three hundred and twenty-six — and no frequency moves by more than five parts in a thousand million.

The projected field is the smallest one reproducing the spectrum, it is what every starting point projects to, and it removes exactly the part no measurement of this molecule could reach. That last clause is what makes it the natural choice rather than merely a choice: nothing measured is discarded.

The projected bend and bend–bend constants come out equal and opposite, at ±0.2559, and that is the flat direction showing itself in the answer: their sum is the direction the spectrum cannot see, so the projection sets it to zero and what survives is their difference.

What the projection removes, and it is not small #

The removed part is 0.46 per cent of the length of methane’s whole force field and 0.21 per cent of boron trifluoride’s, which sounds like rounding.

It is not, and the reason is worth stating carefully. A small vector in a fifty-five-dimensional space can be a large change to any one of its components, and it is: the projection moves ten of methane’s constants by more than a tenth of a mdyn per ångström, and it moves the bend constant itself from 0.4941 to 0.3530, a change of twenty-nine per cent in the number that gets quoted.

A quantity can be small as a fraction of a field and decisive as a fraction of a constant, and which of the two a reader is being shown is exactly what a table of fitted numbers does not say. The norm of the removed part is the wrong statistic to have quoted and it is the one that would have been quoted, because it is the one a least-squares fit reports. This is the same difficulty a susceptibility curve’s four parameters have and a substitution structure’s bond lengths have: the fitted object is fine and the individual numbers taken out of it are not.

What this does not fix #

The projection removes the freedom between the force constants and the Hessian. There is a second freedom above it, and it survives.

A spectrum fixes eigenvalues, not a matrix. Two Hessians related by a transformation that preserves the symmetry blocks and the masses have the same frequencies, so even a force field with no flat space at all is not determined by frequencies alone. That is the reason a fit needs isotopologues: changing the masses changes which matrix gives which eigenvalues, and enough isotopologues pin it down.

It shows in the arithmetic here. The four projected fields agree to 0.0027 in the bend constant but only to about 0.1 in the worst of their twenty-eight components, and the residue is this second freedom rather than the first. Projection is a complete answer to one of the two problems and no answer at all to the other, which is worth knowing before anybody projects a field and calls it determined.

What is quoted, and what is computed #

The observed frequencies are quoted, for the molecule and for one isotopologue in each case, and they are the only quoted numbers here. Every force constant is fitted to them and every statement about which combination is determined is computed from the geometry.

The determined subspace is found by building the map from force constants to the Cartesian Hessian one basis element at a time — fifty-five Hessians for methane, twenty-eight for boron trifluoride — and diagonalising the Gram matrix of the result. A zero eigenvalue is a flat direction and the rest span the complement; the threshold is relative to the largest eigenvalue, because the matrix carries the square of whatever units the coordinates are in.

The two halves are checked together rather than separately: perturbing the field along the best-determined combination moves a frequency by 57 per cent and perturbing it along a flat one moves nothing by more than seven parts in ten thousand million. Either check alone would pass on a bug that ignored the perturbation entirely.

What this cannot say #

Nothing here is about which force field is physically right. The projected field is the unique one the spectrum determines; it is not the one a chemist should transfer to another molecule. Transferability is a claim that a constant means the same thing in two structures, and the projection is defined by this molecule’s geometry — so a projected constant is comparable between two fits of the same molecule and not between two molecules.

The flat space is a property of the coordinate set. Methane’s is ten-dimensional because its ten internal coordinates are one redundant, and a non-redundant coordinate set chosen by hand would have none. What a non-redundant set buys is uniqueness; what it costs is the symmetry of the description, since no non-redundant set of methane’s angles is invariant under its group.

The projection assumes the fit converged to something reproducing the spectrum. It removes the component of a field along a direction the spectrum cannot see, and says nothing whatever about a field that does not fit. Two fields with different residuals project to different places, and the four rows above are comparable only because their residuals agree to three figures — which was checked before they were compared, and is the reason the residual is in the table the fit produces rather than left implicit.

And two molecules is not a survey. Methane and boron trifluoride are the two here with both a redundancy and a fitted field. The identity is exact so it does not need a survey, but the sizes — a bend constant a third determined, a projection that moves it by twenty-nine per cent — are two data points, and both molecules are small, symmetric and built around one centre. A molecule with two centres carrying angles would have a larger flat space and probably a worse ratio, and the collection’s larger structures have no fitted fields to try it on.

Why the table sorts by stretch and bend #

The pattern in the table — every stretching constant fixed, no bending constant fixed — looks like a chemical fact and is a geometric one, and it follows from where the redundancy lives.

The redundant combination for methane is a relation among angles: the six H–C–H angles cannot all open together, because their sum is constrained. It has no component on any of the four bond stretches at all.

The invisible directions are all built from that combination paired with something else, so every one of them touches at least one angle coordinate. A force constant connecting two stretches — a bond to itself, or a bond to another bond — has neither of its indices among the angles, so no invisible direction can move it.

That is the whole of the sorting. Stretch–stretch constants are determined because the redundancy does not reach them; every constant with an angle in it is exposed, because every invisible direction has an angle in it.

The rule generalises without further work. Locate the redundancy, list the coordinates it has non-zero components on, and any constant whose two indices both avoid that list is determined. A molecule whose redundancy involved a stretch as well as angles — a ring closure, where a bond length and the angles around it are related — would have some stretching constants exposed too, and the table would sort differently.

So the useful output of the whole exercise is not a list of forty-five numbers but a rule for producing the list: a constant is determined exactly when the redundancy does not touch either of its coordinates, which can be evaluated from the geometry before any spectrum is measured or any fit is run.

What was checked #

The determined fractions sum to the rank, exactly, for both molecules. A projection whose trace is not its rank is not a projection, and this is the check that the eigen-decomposition was read correctly.

At least one constant is visibly short of being fixed on its own, which is the check that fails if a bug returned every fraction as one — the failure mode that would make the whole essay say nothing while looking tidy.

The best-determined combination moves the spectrum and a flat one does not, checked as a conjunction.

Water is the refusal. It has no redundancy, no flat space and six constants every one of which is fixed on its own — so a routine that reported a flat direction for it would be reporting an artefact of its own threshold.

Still open: transferability, and a second isotopologue #

The obvious open question is transferability, which the projection now makes answerable in a different currency. A constant transferred between molecules is transferred under whatever convention its source used, and the projection is a convention — a very specific one. Computing the projected fields of methane, silane and germane and comparing those down the group would say whether the chemical trend survives being written in the only convention that is not arbitrary, and it is the first version of that comparison in which the numbers being compared are uniquely defined.

The nearer question is the second freedom. The projection leaves about 0.1 of spread in the worst component, and attributes it to the eigenvalue problem rather than to the coordinate set — but that is an attribution rather than a measurement. Refitting with two isotopologues instead of one, and watching whether the projected spread falls, would test it directly, and boron trifluoride has the data: the fitted field uses one substitution where two are published.