The second molecule with a blind spot

Methane’s forty-five determined combinations were written out as a table — every stretching constant fixed on its own, no bending constant fixed at all, and ten directions in a fifty-five-dimensional space that no spectrum can touch.

One molecule is a case and not a pattern. There is exactly one other here with a redundant coordinate set.

Why there are only two #

A redundancy is a combination of internal coordinates that describes no displacement of any atom, and it appears when a molecule has more coordinates than it has ways to vibrate. Of the six molecules with force fields here, four do not — and a force field is already underdetermined by its own spectrum before any redundancy enters.

Water and sulfur dioxide have two bonds and one angle against three vibrations; carbon dioxide the same. Ammonia has three bonds and three angles against six vibrations — exactly enough, with nothing spare, because a pyramidal centre’s three angles are independent.

Methane has four bonds and six angles against nine vibrations, so one coordinate is spare: the six angles at a tetrahedral centre are five coordinates’ worth. Boron trifluoride has three bonds, three angles and an out-of-plane coordinate against six vibrations, so one is spare there too: the three angles at a planar centre sum to a full turn.

One redundancy each, and two quite different geometries. That is the comparison, and it is the only one available.

The two geometries differ in the way that matters to this question rather than in an incidental way. A tetrahedral centre’s redundancy is a relation among six angles, all of them equivalent by symmetry; a planar centre’s is a relation among three, also all equivalent. So neither molecule’s redundancy singles out a coordinate, and both spread their one relation evenly over the angles it names — which means any difference between the two is a difference of counting rather than of which coordinate got unlucky. That is what makes the comparison clean, and it is why the numbers below come out as integers over integers rather than as whatever a lopsided constraint would give.

A quarter and a fifth #

Methane’s flat space is ten-dimensional in fifty-five constants. Boron trifluoride’s is seven-dimensional in twenty-eight.

Eighteen per cent against twenty-five. The smaller molecule has the larger blind spot, which is the reverse of what a count of coordinates suggests — and the reverse of what most readers would guess, since a larger molecule is the one with more to be uncertain about.

The arithmetic behind it is worth following because it explains why “one redundancy” is not a useful measure of anything. A redundancy is one linear relation among the coordinates. A force constant is a second derivative with respect to a pair of coordinates, so what a single relation costs is a relation among every product that involves the redundant combination — which for a space of n coordinates is n relations, one for each partner, with the self-pair counted once.

Methane has ten coordinates and loses ten directions; boron trifluoride has seven and loses seven. Both exactly. So the flat space’s dimension is the number of coordinates and not the number of redundancies, and the share it takes is therefore n over n(n+1)/2, which falls as the molecule grows.

That is a general statement and it inverts the intuition completely: a redundant coordinate set costs proportionally less the bigger the molecule, because the number of constants grows quadratically and the damage grows linearly.

Eight kinds, and three the tetrahedron has no version of #

Methane’s constants come in five kinds: stretch, stretch–stretch, stretch–bend, bend and bend–bend. Boron trifluoride’s come in eight, because it has an out-of-plane coordinate — a wag of the boron through the plane of the fluorines — and that coordinate brings a constant of its own and couplings to everything else.

The pattern found there holds on the new molecule. Every stretching constant is fixed on its own, all three of them. No bending constant is fixed on its own; the three bends between them supply 0.90 of a determined combination out of three, and the three bend–bend constants supply 2.10 out of three. The nine stretch–bend constants give six fixed and three free.

And the out-of-plane coordinate is untouched. Its own constant is fixed on its own, and so are all three of its couplings to the stretches. Two of its three couplings to the bends are fixed and one is free.

The reason is exact rather than empirical. The redundancy is the statement that the three angles at a planar centre sum to a full turn — it is a relation among those three coordinates — so the products it spoils are the products involving one of them. A wag is not one of those angles, so the wag’s own constant is not in the relation at all, and its couplings to the three bonds are not either.

So a molecule’s blind spot is not spread evenly across its force field. It sits on the coordinates the redundancy names, and a coordinate outside the relation is measured exactly as well as it would be if the molecule had no redundancy.

Methane’s own blocks, for the comparison #

That table is worth putting beside the new one rather than referred to, because the shared pattern is the whole of what two molecules can establish.

Methane’s four bonds give four stretching constants, all fixed on its own; its six stretch–stretch couplings likewise. Its bending constants are not fixed and its bend–bend couplings are not, and the stretch–bend block is partly determined.

So the two molecules agree about every kind they share, and they agree in a way that is not a near-agreement: a stretching constant is fixed exactly on its own in both, meaning its block’s projection has an eigenvalue of one at that constant, and a bending constant is fixed in neither, meaning no eigenvalue reaches one anywhere in that block.

The exactness is what makes the agreement worth reporting. Two molecules whose stretching constants came out 95 and 92 per cent determined would be two numbers; two molecules whose stretching constants are determined exactly are two instances of a statement that might be a theorem. The section below says what the theorem would be.

Two molecules is not a rule #

The shared pattern — stretches fixed alone, bends not — holds on a tetrahedral centre and a planar one, which are the two most different redundant geometries available. That is worth something and it is not a rule.

What would make it one is an argument, and the argument is available in outline. A redundancy among angles is a relation among bending coordinates, so the products it spoils all involve a bend; a stretching constant is a product of two stretches and is in no such product. On that reasoning every molecule whose redundancy is among its angles has all its stretching constants determined, which covers every planar three-coordinate centre and every tetrahedral one — a large share of chemistry.

It does not cover a molecule whose redundancy involves a bond length, and rings are where those live: a ring’s bond lengths and angles are not independent, because the ring has to close. Nothing here has a ring, and a ring is where the argument would be tested rather than repeated.

What the fractional traces mean, and where they come from #

The bending blocks are the interesting ones and their numbers are easy to misread.

Boron trifluoride’s three bending constants have a determined trace of 0.90. That is not nine tenths of one constant. It is the rank of the projection of the determined subspace onto the three-dimensional space those constants span — so 0.90 of three directions are reached, and none of the three coordinate directions is reached whole.

The three bend–bend couplings have a trace of 2.10, which is 70 per cent of three. Add the two: 3.00 of six, so the spectrum determines exactly half of the six-dimensional space the bends and their couplings span between them.

That exact half is not an accident and it is the redundancy speaking. The angle-sum relation is one linear condition among three bending coordinates, and what it removes from the space of their pairwise products is one direction per coordinate — three of the six — leaving three.

The same counting on methane’s six bending coordinates predicts six removed from twenty-one, leaving fifteen. Computed, methane’s bend block has a trace of 3.214 of six and its bend–bend block 11.786 of fifteen: fifteen exactly, of twenty-one. Two molecules, two different redundancies, and both land on the integer the counting gives with the two blocks’ fractional parts summing away.

So a bending block’s fractional trace is a count in disguise. The determined share of the whole bending sector is 1 − n/(n(n+1)/2), which is 1 − 2/(n+1): a half for three bending coordinates and five sevenths for six. The blind spot is proportionally worse the fewer coordinates the redundancy involves, which is the same inversion the whole-field comparison showed, one level down.

And it explains why the split between the two blocks is fractional when their sum is not. The removed directions are combinations of a bend with each coordinate in turn — one of which is itself — so they lie partly in the bend block and partly in the bend–bend one, in a ratio the geometry fixes and neither block alone reports. Quoting either block’s trace on its own is quoting half of a count.

And the two halves are tied together exactly, which is what makes the counting checkable. Write $n$ for the number of bending coordinates, so the bend block has dimension $n$ and the bend–bend block $n(n-1)/2$. The redundancy removes $n$ directions from the two blocks between them, and it removes them from the $n(n+1)/2$ directions the two blocks span, leaving $n(n-1)/2$ — the bend–bend block’s own dimension. Subtract the two statements and the bend block’s determined trace has to equal the bend–bend block’s removed trace, whatever the geometry does with the split.

Measured, they do, on both molecules and to four figures: boron trifluoride’s bending block is determined to 0.9000 and its bend–bend block loses 0.9000; methane’s bending block is determined to 3.2143 and its bend–bend block loses 3.2143. Neither number is a round one and neither was put in by hand — both come out of diagonalising a fifty-five- or twenty-eight-dimensional matrix — so the agreement is a test of the whole construction rather than a restatement of it. A projection that had lost or double-counted a direction would break it.

What was computed, and how #

The determined subspace is that essay’s construction, unchanged: the second derivative of every frequency with respect to every force constant, assembled into a matrix whose null space is the directions no spectrum can see, and diagonalised. A block is the projection of that subspace onto the constants of one kind, and its trace is how many combinations of that kind the spectrum determines — a rank rather than a count, so it can be fractional.

Ten things are checked. That both molecules with a redundant coordinate set are in the census. That every block’s determined count lies between zero and the block’s size, which is what a projection’s trace has to do and is the refusal that would catch a broken projection. That every stretching constant is fixed on its own in each molecule, and that no bending constant is. That the planar molecule’s constants fall into more kinds than the tetrahedral one’s. That the out-of-plane coordinate’s own constant is fixed on its own. And that a larger share of the smaller molecule’s field is invisible.

The fractional traces are the part worth being careful about. A block’s trace of 0.90 out of three does not mean nine tenths of one constant is known; it means the projection of the determined subspace onto the three bending constants has rank nine tenths, which is a statement about a subspace and not about any constant. A paper can quote a constant only when its block’s fixed count includes it, which is the stronger condition and the one the table reports separately.

Where this stops #

Two molecules, and they were not chosen. They are the only two in the collection with a redundant coordinate set, so the comparison is a complete census of what is available rather than a sample of anything. Whether the pattern is a fact about angle redundancies or about these two molecules is a question a ring would settle and nothing here has one.

The coordinate sets are choices. A different set of internal coordinates gives a different force field and a different redundancy — using five of methane’s six angles removes the redundancy and breaks the symmetry, which is worse — so the flat space is a property of the coordinates as much as of the molecule. What is not a choice is that some redundancy exists for a symmetric set, since the count of coordinates exceeds the count of motions however they are labelled.

And a determined constant is not a transferable one. The projection makes a force field unique and it does so by choosing the smallest field reproducing the spectrum, which is a convention. A constant a chemist wants to carry to another molecule is a constant under whatever convention its source used, and the projection is a very specific one. Being determined and being comparable are different properties and only the first is measured here.

The generalisation #

The habit is to find the second instance before generalising from the first, and to expect it to differ rather than to confirm.

That table was a complete answer for one molecule and it invited being read as a statement about force fields. Running the second available case produced the same qualitative pattern and three different numbers, one of them inverted from the obvious expectation — the smaller molecule’s blind spot is the larger — and a whole class of coordinate the first case had no version of. None of that is visible from one molecule and none of it required new data.

The corollary is arithmetic and is the transferable part. A constraint among k of a system’s n degrees of freedom costs, in the space of their pairwise couplings, a number of directions equal to n rather than to the number of constraints; and the space of pairwise couplings grows as n². So a constraint’s relative cost falls as the system grows, which is the opposite of the usual intuition about underdetermination and is worth carrying anywhere a model’s parameters are second derivatives.

Who found it, and when #

The redundancy of a symmetric internal-coordinate set is Wilson, Decius and Cross, 1955, and the projection onto the determined subspace is the standard repair. The force fields are fitted here to quoted frequencies. What is computed here is boron trifluoride’s block census, its comparison with methane’s, and the identification of the out-of-plane coordinate as the one the redundancy cannot reach.

The number worth carrying is not seven. It is that one redundancy costs ten directions in one molecule and seven in another, and that both numbers are the count of coordinates rather than the count of redundancies.

Still open: a ring, and a second isotopologue #

The obvious open question is the molecule whose redundancy is not among its angles. A ring closes, so its bond lengths and its angles are not independent, and the relation involves a stretching coordinate — which would put a stretching constant into the flat space for the first time and test the argument that predicts the pattern here. Cyclopropane is the smallest case and its frequencies are measured; what it needs is a force field, which is a fit rather than a new idea, and a fit to one molecule’s spectrum has its own difficulty.

The nearer question is the one raised there about the projection’s own residual. It leaves about a tenth of spread in the worst component across four starting points, and attributes that to the eigenvalue problem rather than to the coordinate set — an attribution rather than a measurement. Refitting with two isotopologues rather than one is the obvious test, and methane already has both in the library. What that would actually establish is worth thinking about before running it: a redundancy is a combination of constants that displaces no atom, and a mass cannot see a displacement that is not there.