The count that was never written down

One number decides which way it breaks ran into a quiet problem and said so rather than picking a side: the rotor count that goes into the rotamer ceiling is assigned in more than one way, and the ways disagree. It computed everything with one of them, named the discrepancy, and left it — because settling it inside an argument about temperature would have meant changing the calculation it was extending.

This essay settles what can be settled. The question is small and the answer has two halves that point opposite ways.

What the disagreement is #

The rotamer account of the gem-dimethyl effect says that a substituent accelerates a ring closure by shifting the open chain’s population towards the conformations that can close. The largest acceleration it can produce is a ceiling: however hard the substituent pushes, each internal rotation can at most be driven gauche, so the closable population cannot exceed a fixed fraction and the rate ratio cannot exceed

$$\left(\frac{1 + 2x}{2x}\right)^{r}, \qquad x = e^{-g/kT}$$

with $g$ butane’s gauche energy and $r$ the number of internal rotations the closure freezes. Every comparison of the account with a measurement is a comparison against that number, and the ceiling is what does the refuting. The conformer statistics underneath it are the ones the ring that cannot hold still set up, and the contacts they are competing against are the atoms that meet across a ring.

$r$ is the problem. It is not measured, it is rarely derived, and different treatments assign it in different ways.

One common count is n − 2: three rotors for a five-membered closure, four for a six-membered one. The verdict on the gem-dimethyl comparison used four for the five-ring and two for the six. And an ordinary count of torsions gives a third answer, n − 3 — a chain of n backbone atoms has n − 1 bonds, of which the two terminal ones do not change the backbone’s shape, leaving n − 3 internal torsions for the closure to freeze.

The first and third differ by one and agree about direction. The second is the odd one, and it is odd in a way that cannot be repaired by shifting: it gives the larger ring fewer frozen rotations. Two counts that order their cases oppositely are not two calibrations of one quantity. At most one of them is a count of anything.

Why one rotor is not a detail #

The ceiling is a power in $r$ by construction, so a disagreement about the count is a disagreement about a factor. At butane’s gauche energy and room temperature each rotor multiplies the ceiling by 3.32: one rotor is 3.32, two rotors is 11.0, three is 36.5, four is 121.

So the conventions in dispute are up to two rotors apart on the same ring, which is a factor of eleven in the quantity every verdict is decided against. That is far larger than any uncertainty in the measurements being compared to it, and it is not the kind of uncertainty that averages out — it is a choice, made once, that multiplies.

That is the case for being nervous, and it was made when the discrepancy was first noticed. What it could not do without running the calculation is say whether the nervousness was warranted.

Every verdict survives every convention #

Three accelerations are measured: a 250-fold and an 11,000-fold on γ-butyrolactone formation from a five-membered closure, and a 10-fold on δ-valerolactone formation from a six-membered one.

Under n − 2, the five-ring ceiling is 36.5 and the six-ring ceiling is 121. Under the decided verdict’s counts they are 121 and 11.0. Under n − 3 they are 11.0 and 36.5.

Every one of the five-ring measurements is above every one of the five-ring ceilings. The six-ring measurement is below every one of the six-ring ceilings. Nine calculations, and the verdict column never moves: the account is refuted on the five-membered ring under all three conventions, and unrefuted on the six-membered ring under all three.

This is the half of the answer that matters most, and it is worth stating in the form the argument actually needs. Nothing concluded about the gem-dimethyl effect rests on the unsettled count. The refutation the main argument turns on — that the rotamer account cannot produce a 250-fold acceleration on a five-membered closure, let alone an 11,000-fold one — is safe by margins of 2.1, 6.9 and 22.7 depending on how the rotors are counted, and all three of those are refutations.

A better way to say it #

There is a stronger statement available than “the verdict is the same under all three”, and it is worth having because it does not mention the conventions at all.

The ceiling rises monotonically with the rotor count, so for each measured acceleration there is a least rotor count that would reach it. That number can be compared against the most rotors the chain could possibly have — and the honest ceiling on that is not a convention at all, it is the number of bonds in the chain.

The 250-fold acceleration on the five-membered closure needs five rotors. The 11,000-fold needs eight. A five-atom chain has four bonds, so it cannot supply five rotations under any convention, and it cannot supply eight under any imaginable one.

That is the refutation without a margin in it. The account does not fall a little short on the five-membered ring; it would need more internal rotations than the molecule contains. And because the comparison is made against bonds rather than against a convention’s count, it is not merely that all three conventions agree — it is that all three conventions, and every fourth one anybody could propose, are below the requirement.

The six-membered case comes out the other way by the same arithmetic: its 10-fold acceleration needs two rotors, and the most any convention gives a six-membered closure is four. Reachable, and therefore unrefuted, and therefore not explained.

This is the form the conclusion should have been quoted in from the start. It is the same result and it depends on strictly less.

And every margin moves #

The other half is the one actually worth worrying about, and it is not reassuring.

The six-membered ring’s survival is quoted with a margin, and the margin is whatever the convention says it is. Under the decided verdict’s count the ceiling is 11.0 against a measured 10, which clears the measurement by 9.9 per cent. Under n − 3 it is 36.5, which clears it by 265 per cent. Under n − 2 it is 121, a factor of 12.1.

Those are three very different statements about the same case. The first says the account is only just adequate on the six-membered ring and would be refuted by a modestly larger measurement or a modestly different gauche energy. The third says the account has an order of magnitude in hand. Anything that leaned on how narrowly the six-ring survived was leaning on a choice nobody had made.

And the convention the decided verdict happened to use is the tightest of the three. That is the arrangement that would have been dangerous: a verdict decided by ten per cent, on the most pessimistic of three unstated counts, where two other defensible counts give the same answer with far more room. It came out safe, and it came out safe by luck rather than by anyone having checked.

So which one is right #

This essay does not settle that, and it is worth being explicit about why rather than leaving it as an omission.

Deciding which count is correct means deciding what “the closure freezes this rotation” means for a specific reaction, at a specific transition state, in a specific solvent — and that is a question about a mechanism rather than about a geometry. The n − 3 count is the one that follows from counting internal torsions of a chain, which is why it is included here; it is not obviously the right one either, because the transition state for a lactonisation is not the closed ring and does not necessarily freeze every torsion the product does. That gap between a product’s geometry and a transition state’s is the same one the explanation with the wrong sign turned on, and it is not closed by anything computed here.

What can be said is that the question does not need answering for any verdict, and does need answering before any margin on it is quoted. Those are different obligations and the second one is now discharged by stating the range instead of a number.

What was computed, and how #

The ceiling is the closed form above, evaluated at butane’s gauche energy — the one quoted quantity in the calculation — and at 298 K. The measured accelerations are the three standard ones and are also quoted. Everything else is arithmetic: three ring sizes crossed with three conventions, each giving a rotor count, each rotor count giving a ceiling, each ceiling compared against the measurement for that ring.

A second check covers the supply argument: that every measured acceleration is reachable by some rotor count, so the search reports a refusal rather than running off the end; that the five-membered cases need more rotors than the chain has bonds; and that the six-membered case does not. The first of those is the guard — a search that silently returned its cap would have made every case look refuted by supply, including the one that is not.

The convention check covers four things, and two of them are about the conventions rather than the chemistry: that the three counts really do disagree in direction as well as in value, so that the comparison is not three names for one number; that every case’s verdict is the same under all three; that at least one case is refuted whichever is used, so the sweep is not vacuously agreeing that everything survives; and that the margins spread by more than a factor of ten, so that the second half of the finding is not an artefact of rounding.

That third check is the one worth pointing at. A sweep in which every case survives under every convention would have produced the same headline — the verdicts are convention-independent — and would have established nothing, because a comparison that never rejects anything cannot be evidence that a choice does not matter. The refutation on the five-membered ring is what gives the agreement content.

Where the model stops #

The gauche energy is quoted and held fixed, and the ceiling depends on it as strongly as it depends on the rotor count. A sweep in temperature found the verdicts stable over the range a solution-phase closure is run in; a sweep in the gauche energy would be the same shape of check and is not done here.

The ceiling is a ceiling, so an unrefuted case is not an explained one. The six-membered ring’s 10-fold acceleration sits under every ceiling computed here, and that means the rotamer account could produce it, not that it does. A verdict inside its own error bar is where how little an unrefuted case is worth gets established, and nothing here improves it.

And three conventions is not all of them. These are the two in use and the one an ordinary torsion count gives; somebody could defend others, and the finding as stated is that these three agree rather than that every possible count would. The property that makes the agreement robust is visible in the figure, though: the ceilings for a ring size span a factor of eleven and the nearest one is still a factor of two from the measurement it has to clear, so a fourth convention would have to differ from all three by more than any of them differ from each other to change a verdict.

The generalisation #

The transferable point is about what an unsettled convention threatens.

The instinct on finding one is that everything computed with it is suspect, and the instinct on finding that the conclusions do not move is that it did not matter. Both are wrong here in the same way: they treat “the conclusion” as one thing. A verdict — this account is refuted, that one is not — survived. A margin — by how much — did not, and margins are conclusions too, quoted in the same sentences and read with the same confidence.

So the useful discipline is not to settle every convention before using it. It is to know which of the things being reported are robust to it, and that is answered by recomputing under the alternatives rather than by arguing about which is right. The recomputation here is nine evaluations of a closed form and it is the cheapest calculation in the whole argument; the argument about which count is correct would have taken far longer and would have answered a question nothing depended on.

How a convention gets chosen twice #

It is worth a paragraph on how two conventions come to sit side by side in one line of argument, because the mechanism is ordinary and it will happen again.

Neither was adopted as a convention. A sweep over ring sizes needs a rotor count per ring size, and n − 2 is the shortest expression that gives sensible numbers across the sizes swept. A verdict on two specific rings needs a rotor count for those two, and takes the values its own argument is framed around. Each is a local choice, correct-looking in its own context, and neither is compared against the other until a calculation needs both at once — here, the temperature sweep.

That is the shape to watch for: not a disagreement anybody had, but a parameter that two calculations each supplied for themselves. A formula derived twice is easy to compare; a number chosen twice is not, because each copy looks like a detail of its own calculation, and the only thing that brings the two into view is a third calculation that happens to need both.

The remedy that follows is not to pick a value. It is to state the count as a named convention with its reasoning attached, so the next calculation that needs one chooses from a short list rather than inventing an entry.

Who found it, and when #

The rotamer or Thorpe–Ingold account of the gem-dimethyl effect is old and the ceiling argument is a standard one. The discrepancy is a modelling one rather than a chemical one, noticed by a calculation that was computing something else and needed both counts at once. The sweep across conventions is new here.

It is worth saying plainly that this is a finding about a model’s bookkeeping rather than a discovery about chemistry, and that it matters because the alternative — picking one count quietly and saying nothing — would have left every margin already quoted unqualified.

Still open: the gauche energy, and a derived rotor count #

The obvious open question is the gauche energy, which is the other quoted input and the only one not yet swept. The ceiling depends on it through $x = e^{-g/kT}$, the calculation here uses one value for it, and butane’s gauche energy is reported over a range in the literature rather than as a point. Sweeping it would say the same thing this essay says about the rotor count: which verdicts are safe and which margins are not. The supply argument above suggests the answer for the five-membered ring is that nothing is at risk — a factor of eleven in the ceiling did not reach it, and a plausible range in one energy is a smaller lever than that — but an estimate that can be wrong by two is the standing reminder that a lever’s size is worth measuring rather than assuming.

The nearer question is whether the rotor count can be settled from conformer statistics rather than from a mechanism. A closure freezes the torsions that differ between the open chain’s conformer family and the closed ring’s, and both can be computed — the conformer enumeration is what the ceiling is built on, and the ring’s own family is what the transannular contacts are counted over. Counting the torsions that actually lose their freedom, rather than assigning a number by ring size, would produce a fourth count that is derived instead of chosen, and would say which of these three it is nearest to.