A ceiling that rises where the measurements fall

Take the two standard explanations of the gem-dimethyl effect — that a substituent bends the chain’s angles towards closure, and that it depopulates the extended rotamers — and the first has the wrong sign while the second has a ceiling. The measured 250-fold acceleration of a five-membered ring closure sits above that ceiling, so neither account survives.

It closed by noticing something that looked like a problem for the second account and turning out to be the opposite. The ceiling rises with ring size, because closing a bigger ring means freezing more rotations, while the measured accelerations fall — 250-fold for a five-ring, tenfold for a six. Working out where the two cross, it said, would turn the ceiling from a limitation into a prediction.

That is what is done here, and the prediction is unusually clean: the ceiling has a closed form.

The ceiling, written down #

Turn the substituent’s penalty on an extended rotamer up without limit. Every conformation with a rotor left anti becomes infinitely costly, so the substituted chain is confined to the all-gauche states: there are $2^r$ of them for $r$ rotors, and exactly two — all $g^+$ or all $g^-$ — curl the chain into a ring rather than a helix going nowhere.

So the closable fraction of the substituted population tends to $2/2^r$, which is a number and not a limit that has to be searched for.

The unsubstituted chain’s closable fraction is the Boltzmann one: with $x = e^{-g/kT}$ for the gauche energy $g$, the two closable states carry weight $2x^r$ out of $(1 + 2x)^r$ over all three states per rotor. The ratio is therefore

$$\text{ceiling} = \left(\frac{1 + 2x}{2x}\right)^{r}$$

At the measured butane gauche energy of 3.8 kJ/mol and room temperature, $x = 0.215908$ and the base is 3.3255. The ceiling is 10.99 at two rotations, 36.46 at three, 120.88 at four and 400.81 at five.

The 36.456 found numerically reported for three rotations was found by driving the penalty up numerically until the ratio stopped moving. It agrees with the closed form to a part in a thousand, which is the check that both are computing the same thing.

Two things about that expression are worth noticing before it is used. It has no substituent in it — the substituent’s size, its shape and the size of the penalty it imposes have all gone, because the ceiling is what happens when the penalty is large enough that its value stops mattering. And it depends on the temperature only through the gauche energy, so a rate measured at a different temperature is being compared against a different ceiling.

Where it crosses #

A closure to a ring of $n$ atoms has to freeze about $n - 2$ internal rotations. So:

The two curves cross between five and six, and they cross the wrong way round for anybody hoping for a single explanation. The account is refuted where the effect is large and comfortable where it is small.

That is a stronger statement than one ceiling could support. One measurement above one ceiling is a failure. This has a failure and a success on two neighbouring ring sizes, which is a pattern — and a pattern says that whatever is missing from the account is missing specifically from the five-membered case.

What that says about the missing term #

Something makes a five-membered closure faster than the rotamer statistics allow, and does not need to make a six-membered one faster at all.

Five-membered rings are already awkward for a different reason: a ring of five cannot have the tetrahedral angle at a planar geometry and puckers to get near it, and the puckering itself has no single answer. Its measured strain is 26 kJ/mol against cyclohexane’s zero. So the five-ring’s transition state is under angular strain in a way the six-ring’s is not, and a substituent that shifts the preferred angle has something to relieve there and nothing to relieve in the larger ring.

That is the first account — the one found to have the wrong sign — reappearing with a condition attached. It predicts a slowing for an unstrained closure, which is why it failed; where the closure is already strained, the sign of the effect need not be the same, because the ring’s own preferred angle is not the tetrahedral one.

This essay does not compute that, and says so plainly under what this cannot say. What it establishes is where the missing term has to act: on the five-ring and not the six, which is a much narrower target than a single failed ceiling offered.

The objection to the ceiling, and what it comes to #

The other thing left recorded was an objection to both accounts. A gem-dimethyl group makes the chain more crowded, and it makes the closed ring more crowded too — so part of whatever it buys on the open side it should give back on the closed side, and neither account has a term for that.

The first half of the answer needs no model at all and is a counting fact.

A transannular contact is a pair of ring atoms four or more bonds apart. A pair one bond apart is the bond; two apart is the bond angle; three apart is the torsion. A ring of $n$ atoms has a pair four apart only when $n \geq 8$.

Both measured accelerations are on rings of five and six. The term is exactly zero for both of them, for a reason with no parameter in it, and the objection therefore cannot be part of the explanation whatever its size would be somewhere else.

The second half is a calculation, for the rings where the term does exist. Modelling the substituted carbon as one larger united atom and recomputing the contacts of the least crowded conformer, one position at a time:

It always costs, at every position of every ring — so the objection is real for medium rings. That is the same term a double bond cannot be put into a small ring against: a constraint that is vacuous below a ring size and decisive above it, with the boundary set by counting rather than by a parameter. And the position matters between four and sixty-seven times more than the substituent’s presence does, which is a fact about medium rings rather than about substituents and is the same fact the conformers of a nine-membered ring already showed.

Why the acceleration is measured on small rings #

The two halves fit together into a small practical conclusion.

A gem-dimethyl group accelerates a closure by removing extended conformations, and the removal is worth more the more rotations there are to remove — so the effect should be largest for the biggest rings. It is not measured that way, and this essay says why: from eight atoms onwards the substituent starts paying a transannular price in the ring it is trying to form, and the price is between 0.16 and 48 kJ/mol depending on where it sits.

So the acceleration is measured on five- and six-membered rings because those are the sizes where the mechanism has something to gain and nothing to give back. The ceiling rises past them and the penalty rises with it.

What a kineticist should take from it #

A ceiling is a test, not an excuse. A model with a largest possible effect can be refuted by a single measurement larger than it, and the refutation needs no fitting — which makes it a much cheaper instrument than a fit, and a much sharper one.

The rotor count is the whole of the sensitivity. Everything depends on it exponentially and on nothing else exponentially, so the useful thing to know about a closure is how many rotations it actually freezes. Two closures to the same ring size through different tethers have different ceilings.

And the transannular objection has a size threshold. Below eight atoms it is not a small term, it is no term at all. Above eight it is between a fifth of a kilojoule and forty-eight, depending entirely on where the substituent sits, which means it cannot be estimated from the substituent alone. The same ring measured at its several conformers spans more than the substituent does.

What is quoted, and what is computed #

Three things are quoted, and all three are measurements: the butane gauche energy of 3.8 kJ/mol, the two gem-dimethyl accelerations of 250-fold and tenfold, and the united-atom contact parameters. The last of these is the one number the transannular conclusions are sensitive to, and it is used for distances rather than for a total energy — the same caution recorded where it was introduced.

Everything else is computed: the closed-form ceiling, the numerical limit it is checked against, the rotor counts, the ring conformers, the transannular pairs and every contact energy.

The ceiling is computed twice, once as a limit of the Boltzmann sum and once by driving the penalty to a large value and reading the ratio. Neither is the other read back to itself: the first is algebra on the state count and the second is an enumeration over $3^r$ states.

What this cannot say #

The rotor count is an estimate. Taking $n - 2$ rotations for a ring of $n$ is a reasonable count of what has to be frozen and it is not a calculation; a real closure has a tether whose rotations are not all equivalent. The ceiling depends on $r$ exponentially, so an error of one rotor is a factor of 3.3 — which is smaller than the factor of 6.9 by which the five-ring’s measurement exceeds its ceiling, but not by much.

The gauche energy is butane’s. A substituted chain’s gauche interactions are not butane’s, and a larger one would raise $x$ and lower the ceiling. That moves the conclusion in the direction it already points.

There is no transition state anywhere. Everything here is about the population of closable conformations, which is a ground-state statistic. A rate is about a barrier, and what the strain accounts do and do not reach is the boundary every strain argument here works inside.

And the substituent is one number. Modelling a gem-dimethyl carbon as a larger united atom with the same well depth is the crudest possible description; a real one has two methyl groups that rotate and can be pointed away from the ring, which is exactly the freedom the position dependence above is a crude version of. A substituent that can move out of the way is a substituent whose cost is an average over an ensemble, and averaging over an ensemble is what changes the answer wherever it has been looked for.

What was checked #

The ceiling agrees with its closed form at every rotor count from two to five, to a part in a thousand — two computations of one number, one algebraic and one an enumeration.

And it rises with the number of rotations frozen, which is the property that makes it a prediction rather than a bound.

There are measurements on more than one ring size to test it against, which is what makes a crossing possible to see at all.

The account is refused by one measured ring and sufficient for another, which is the finding.

A ring of five, six or seven has no pair four bonds apart, so the transannular term is exactly zero — checked by counting the pairs of a computed conformer rather than by trusting the arithmetic.

And rings large enough to have one do, with a bulkier position costing something at every placement, and the placement mattering by more than three times what the substituent does.

The refusal is a substituent no larger than what it replaces, which must cost exactly nothing wherever it is put — the check that the enlargement is doing the work rather than the recomputation.

The angular account has a ceiling too, and where its sign changes #

The rotamer account was rescued from being a limitation by having its ceiling written down. The angular account deserves the same treatment, and it does not survive it.

Its relief is $-k\delta\,(2\Delta + \delta)$, where $\Delta = \theta_{\text{ring}} - \theta_0$ is how far the ring’s interior angle sits from the tetrahedral preference and $\delta$ is the shrink the substituent imposes. Read as a function of $\delta$ that is a downward parabola, so it has a maximum, and the maximum is at $\delta = -\Delta$ with value

$$\text{relief}_{\max} = k\,\Delta^2 .$$

The shrink that extracts the most from the mechanism is exactly the ring’s own mismatch — no more, no less. Pushing harder overshoots and the relief falls again.

Putting the same constant into that gives the largest acceleration the mechanism can produce at any shrink whatever. For a five-ring, $\Delta = -1.47°$ and the ceiling is 0.083 kJ mol⁻¹, a factor of 1.03. The measurement is 250.

So the refutation at five is not about the sign after all, or not only. No value of the shrink reaches the measurement, because the ring’s angle is already within a degree and a half of where the chain wants it and there is nothing to relieve. A mechanism whose whole content is the removal of a mismatch cannot produce more than the mismatch was worth.

The six-ring’s ceiling is larger — $\Delta = 10.53°$ gives 4.24 kJ mol⁻¹, a factor of 5.5 against a measured 10 — and it is unreachable for a different reason. Attaining it needs $\delta = -10.53°$, a substituent that opens the ring’s angle by ten degrees, which is the opposite of the premise the account is built on. With a shrink of any positive size the six-ring’s relief is negative, as computed for the angular account.

That also locates the sign change the previous section asked for. The relief vanishes at $2\Delta + \delta = 0$, so the boundary sits at $\theta_{\text{ring}} = \theta_0 - \delta/2$, and with the quoted six-degree shrink that is 106.47° — a ring of

$$n = \frac{360°}{70.53° + \delta/2} = 4.90 .$$

Below 4.90 the mechanism helps and above it hinders, and both rings the effect has ever been measured on are above it. Nor is that an artefact of the quoted shrink: the boundary reaches five only for a shrink under 2.94°, and rises no higher than 5.10 even as the shrink goes to zero. Every shrink large enough to matter puts the boundary below five.

So the two failed accounts are not one working account with a boundary in it. The angular mechanism’s useful side is entirely at three and four atoms, where the effect is not measured and where the closure is dominated by something else; on the rings where it is measured the mechanism is small, wrongly signed, and capped a factor of two hundred below the observation. That leaves the rotamer account as the only one of the two with anything left, which is why the effort goes into its rotor count rather than into this one.

Still open: the three- and four-membered rings #

The angular account with its condition attached is the section above: its sign changes at 4.90 rings, both measured cases are on the hindering side of that, and its own ceiling caps the five-ring two hundred-fold below the observation. What that leaves undone is the three- and four-membered case, where the mechanism is large and helpful and where nobody has measured a gem-dialkyl acceleration to compare it against — a measurement rather than a calculation, and the one that would test the half of the account nothing here has been able to refuse.

The nearer question is about the rotor count the ceiling is exponential in. The count of $n - 2$ was taken as read here, and the exponential dependence makes it the largest uncertainty in the whole comparison: one rotor is a factor of 3.3, which is half the gap the five-ring’s measurement has to clear. Counting the rotations a specific closure actually freezes — a lactonisation has an ester tether whose own rotations are hindered differently from a carbon chain’s — would replace an estimate with an integer, and it is the only step that could make the refusal at five airtight rather than probable.