More coordinates than motions

A vibrational assignment is written in internal coordinates — this bond stretching, that angle bending — and it has been shown that the percentages in such an assignment are a property of the coordinates rather than of the mode. Four defensible conventions for sharing one mode out among its coordinates gave boron trifluoride’s 1454 cm⁻¹ band as 36.7, 49.9, 97.9 or 94.6 per cent a B–F stretch.

That essay ended by naming the next difficulty and declining it: a molecule with a ring has more internal coordinates than degrees of freedom, so the coordinates are not independent and the sharing-out has a null space in it. This essay takes that up, and the first thing it finds is that a ring is not needed. Methane has the problem.

Ten coordinates, nine motions #

The valence coordinate set is built from the structure: one stretch per bond, one bend per pair of bonds at a common atom, one out-of-plane coordinate at a planar three-coordinate centre. For methane that is four stretches and six angles — the six pairs of four bonds — which is ten. A five-atom non-linear molecule has $3 \times 5 - 6 = 9$ vibrations.

The comparison worth pausing on is ammonia against boron trifluoride. Both have three bonds to a central atom and three angles between them; both have six vibrations. Ammonia’s coordinate set is exactly six and is not redundant. Boron trifluoride’s is seven — the three angles at a planar centre are joined by an out-of-plane coordinate, because without one nothing in the set describes the atom leaving the plane — and it is redundant by one.

The reason is that a flat centre’s three angles sum to $2\pi$ and a pyramidal centre’s do not sum to anything in particular. So the redundancy is a fact about the geometry, arrived at from the geometry, and neither the count of atoms nor the count of bonds predicts it.

What the redundancy is #

Formally it is the left null space of the matrix $\mathbf{B}$ that turns a Cartesian displacement into a change of the internal coordinates: a vector $\mathbf{r}$ with $\mathbf{B}^\mathsf{T}\mathbf{r} = 0$. A combination of coordinates changing by these amounts corresponds to no displacement of any atom whatever.

It is found here as a null eigenvector of $\mathbf{B}\mathbf{B}^\mathsf{T}$, which is a symmetric matrix the size of the coordinate set, so the same eigensolver that finds normal modes does the work.

Boron trifluoride’s is the same shape and shorter: nothing in the stretches, nothing in the out-of-plane coordinate, and $1/\sqrt{3}$ in each of the three angles. Formaldehyde’s is more interesting, because formaldehyde is not equilateral.

That difference is small and it is not noise. It is the same constraint expressed in a coordinate system that weights the angles unequally, and it is worth seeing because it makes the point that a redundancy is a vector rather than a fact: which combination is redundant depends on how the coordinates were scaled, even though that one is redundant does not.

The direction a fit can run along for nothing #

The consequence is in the force field. The Cartesian Hessian at the reference geometry is $\mathbf{B}^\mathsf{T}\mathbf{F}\mathbf{B}$, exactly — the second term in the chain rule carries a factor of the displacement from the reference and vanishes there. So adding any multiple of $\mathbf{r}\mathbf{r}^\mathsf{T}$ to the force constant matrix changes the Hessian by

$$\mathbf{B}^\mathsf{T}(\lambda \mathbf{r}\mathbf{r}^\mathsf{T})\mathbf{B} = \lambda (\mathbf{B}^\mathsf{T}\mathbf{r})(\mathbf{B}^\mathsf{T}\mathbf{r})^\mathsf{T} = 0.$$

Not approximately zero. Zero.

The control is what makes the first half worth stating. A calculation that ignored the added term altogether would also report no change, and would be wrong; the second nudge is the same size, along a direction that is not a redundancy, and it wrecks the spectrum.

So a redundant coordinate set gives a least-squares fit an entire direction along which the residual is flat. And a least-squares search, offered a flat direction, will walk along it: an unconstrained methane fit came back with the bending constant and the bend–bend constant both at −97,596 millidynes per ångström, equal to four significant figures and opposite in effect, reproducing all eighteen observed frequencies to half a per cent. The molecule was fine. The numbers were not numbers.

The repair this site made was to drop one of the two constants, which costs 0.022 per cent of residual and returns a C–H constant of 5.47 millidynes per ångström, the value the molecule is supposed to have. That is the right repair and it is worth being clear about what it is: a constraint chosen by hand, because the data cannot choose one.

The shape of the difficulty, and where else it appears #

A force field determined only up to a direction is a shape of difficulty that turns up all over the place, and it is worth naming.

The force field is not in the spectrum established the general version: a molecule with more force constants than observed frequencies has a family of fields reproducing every one of them. Three numbers is not a structure is the same statement for a rotational spectrum, and one spectrum, a line of models is it for a Hückel parameterisation.

What is different here is the source. In every one of those the shortfall is a shortfall of data — more parameters than measurements, so measure more. Here the number of measurements is irrelevant. Methane has eighteen observed frequencies across two isotopologues against four constants, which is generous by any standard, and the flat direction survives all of them, because it is not a direction in which the predictions fail to separate: it is a direction in which they are identically equal, forced by the geometry of the coordinate set. No amount of data closes it.

Why not simply drop a coordinate #

The obvious alternative is to remove the redundancy at the source — throw away methane’s tenth angle, keep nine coordinates for nine motions, and have no null space.

It breaks the molecule. The force field here is keyed by what the coordinates are made of, so that symmetry-equivalent coordinates automatically get the same constant, and dropping one of six equivalent angles makes them inequivalent. The computed mode species then stop matching the ones the character table predicts, which is a check that would fail. That agreement is what a mode’s symmetry species being computable from the coordinates alone rests on, and it is not worth spending to tidy a null space away.

That is not a defect of the site’s implementation. It is the general reason redundant coordinate sets are used: a redundant symmetric set and a non-redundant unsymmetric one are the two options, and the second gives up an exact statement to avoid an inconvenience.

The standard treatment in the literature keeps the symmetric set and projects the redundancy out. The projection is a choice — the same word the percentage conventions earned, and this is the same difficulty one level down: there, four conventions shared a mode out among coordinates; here, a convention decides which of a family of force fields is the force field.

The opposite fault #

Carbon dioxide breaks the pattern in the other direction, and the way it does is worth the section.

So carbon dioxide’s three valence coordinates span two directions, and the molecule has four vibrations. Its coordinate set is not redundant; it is short, by two, and the two missing directions are the two components of the bend. That is why a linear molecule needs a different construction — a pair of coordinates that describe the displacement of the central atom from the line, rather than an angle — and it is the reason force-field calculations treat linear molecules in a finite subgroup rather than in their own group.

There is a subtlety here that is easy to miss. The same bending coordinate is perfectly usable for describing the mode: displace the atoms along the bending eigenvector by a finite amount and the angle does change, at second order, and a composition analysis reports carbon dioxide’s bend as a hundred per cent bend. So one measurement of “how much of this mode is that coordinate” says the coordinate is the whole of it and another says the coordinate does not exist. Both are right. The first uses a finite displacement and the second uses a derivative, and a stationary quantity is exactly the case where the two come apart.

Carbon dioxide’s four vibrations include two that are the same bend in perpendicular planes, and neither of them has a valence coordinate at all. The angle that would describe them is stationary at a hundred and eighty degrees — moving either oxygen sideways changes it only at second order — so it has no first derivative with respect to any displacement, and the two directions in coordinate space those modes would occupy are simply absent. A coordinate set can be short as well as long, and the same rank calculation reports both.

What a redundancy is not #

Three things it is easy to confuse it with, and none of them is what is computed here.

Not a zero mode. The six motions that deform nothing — three translations and three rotations — are directions in Cartesian space that leave every internal coordinate alone. A redundancy is a direction in coordinate space that corresponds to no Cartesian displacement. The two are null spaces of the same matrix read from opposite sides, and the counts are unrelated: methane has six zero modes and one redundancy.

Not a degeneracy. Methane’s t₂ and e modes are degenerate because the group requires it, and the arbitrariness there is a choice of basis inside a set of modes with the same frequency. The redundancy has nothing to do with frequency and is not a set of modes at all.

Not a symmetry-forbidden constant. A force field can have constants that symmetry sets to zero, and those are visible in the pattern of the matrix. The flat direction is a combination of constants that symmetry permits, that are not zero, and that no measurement of this kind can separate.

Where the model stops #

The Hessian is harmonic. Everything above is exact at the reference geometry and only there, since it is the vanishing of the second chain-rule term that makes $\mathbf{H} = \mathbf{B}^\mathsf{T}\mathbf{F}\mathbf{B}$ exact. Away from equilibrium the redundancy is still a redundancy of the coordinates and is no longer a flat direction of the energy.

The count is for one geometry. A molecule that changes shape can change how many redundancies it has, and a fluxional one has no single answer — which is one more reason a rigid point group is the wrong group for it.

Nothing here is about a real potential. The force constants are fitted to observed frequencies by a search whose limits are described elsewhere, within a harmonic model with no anharmonicity in it, so the flat direction is a flat direction of that model. What a real potential energy surface does along the same combination of coordinates is a different question, and it is not flat: the redundancy is exact only to second order.

And the redundancy is a fact about a chosen set. A different set of coordinates — symmetry coordinates, or a set built from a ring’s own topology — has its own count. What does not change is the number of vibrations, which is why the comparison in the first figure is between a coordinate count and that.

The redundancy can be written down #

The excess is described here as a combination of coordinates that describes no displacement, which is what it is. It is also, in each of these three molecules, an identity anybody can write down from the geometry — and doing so explains why the excess is one in all three rather than some other number.

Boron trifluoride. The three fluorines lie in a plane around the boron, so the three F–B–F angles sum to 360°. That sum cannot change: however the molecule distorts within its plane, the three angle changes add to zero. One relation among three coordinates, and one redundancy.

Formaldehyde. The same statement at its planar carbon. Two H–C–O angles and one H–C–H angle sum to 360°, so again their changes sum to zero, and again the excess is one.

Methane. The six H–C–H angles around a tetrahedral centre are similarly constrained: to first order in the displacements their changes sum to zero, because the four bond directions cannot all move apart from one another. One relation among six coordinates, one redundancy.

So the count is not a coincidence of three molecules. Each has exactly one centre with a closure condition on its angles, and each has exactly one redundant coordinate, and the correspondence is one relation per constrained centre.

That also predicts where the excess will be larger. A molecule with two planar centres — ethene, with its two carbons — has two such relations and should carry two redundancies, and a molecule with none has none: a linear or an open-chain arrangement with no centre surrounded by three or more neighbours has independent coordinates and no excess at all.

And it identifies the redundant direction explicitly rather than as an output of a rank calculation. It is the vector with equal components on the angles around one centre and nothing anywhere else — the all angles opening together motion, which no set of atoms can perform because the angles around a centre have a fixed total.

Which makes the flat direction concrete. Adding force constant along it means stiffening a molecule against a motion it cannot make, and the frequencies do not move because there is nothing there to stiffen. The five parts in a hundred million measured here is arithmetic noise around an exact zero, and the exact zero follows from a sentence about the sum of some angles.

One qualification belongs with the methane case and not with the other two. The relation among the three angles at a planar centre is exact at any distortion — the three angles sum to 360° whatever the molecule does, as long as it stays planar. The relation among a tetrahedral centre’s six angles holds only to first order in the displacements, because the six angles of a distorted tetrahedron do not have a fixed sum. So the redundancy is exact for the planar molecules and approximate for methane, and it is exact enough there because a vibrational analysis is itself a first-order theory: the Hessian is a second derivative at one point, and a relation that holds to first order in the displacements is all that a harmonic treatment can notice.

What was checked #

The counts, for six molecules: the coordinate count, the vibration count, the redundancy as the difference, and the rank as the smaller of the two.

The shape of the redundancy. For both planar three-coordinate centres it has no bond stretching in it at all, to $10^{-12}$ of its length; for methane it is exactly $1/\sqrt{6}$ in each of six angles, to $10^{-8}$.

The flat direction, with its control. Two molecules, the frequencies unmoved to better than a millionth along the redundancy and moved by tens of per cent along a coordinate.

And the refusal, which is carbon dioxide. Its bending row is identically zero and its rank is two against four degrees of freedom, so a function that reported its shortfall as a redundancy would be reporting the same number for the opposite fault. The check requires the rank to be below the degrees of freedom there and above them nowhere.

Still open: redundancy in rings, and what a projection picks #

The redundancy is one number for these molecules and it is not one number for a ring, where the count grows with the number of independent cycles. Benzene’s valence set has more coordinates than methane’s has in total, and its null space is several-dimensional; whether the null vectors of a ring correspond to anything recognisable — one per cycle, say, or one per symmetry species — is a question the same calculation could answer directly and has not been asked.

The nearer question is what a projection does. The literature’s repair is to project the force constant matrix onto the space orthogonal to the redundancy, which picks one point on the flat line, and the point it picks depends on the metric the projection uses. That is a third convention on top of the two already found, and the honest thing to measure is how far apart the force fields it produces are — not whether the frequencies differ, since they cannot, but whether the constants a paper quotes would.