The atoms that meet across a ring

Ring strain splits into two terms, and the second is missing from most accounts. Cyclopentane’s bond angles are a degree and a half from tetrahedral and its angle strain, on a standard bending constant, is four tenths of a kilojoule; its measured strain is twenty-six, and the missing twenty-five and a half are torsional — one ethane barrier for every bond in the ring.

Two terms, then: a function of the bond angles and a function of the dihedrals. Between them they account for cyclopropane, cyclobutane, cyclopentane, cyclohexane and cycloheptane.

From cyclooctane upwards they do not, and this essay is about the reason. It is not that the two terms are inaccurate. It is that a third term exists from eight-membered rings onwards and does not exist below them, and it is a function of a coordinate neither of the other two mentions.

The counting fact #

Take two atoms in a ring and count the bonds between them by the shorter way round.

Separated by one bond, they are bonded: the term is the bond length. By two, they define a bond angle. By three, a dihedral. By four or more, they are not related by any internal coordinate of the ring at all — nothing in a description built from lengths, angles and torsions says anything about how far apart they are.

A cycle of $n$ atoms has a pair separated by four or more bonds exactly when $n \ge 8$, because the largest separation available on a ring of $n$ is $\lfloor n/2 \rfloor$.

That is a fact about cycles rather than about chemistry, and it has a consequence worth stating plainly: the two-term account is complete in form for every ring up to seven. It can be inaccurate there, and it can have the wrong constants, but there is no term it is missing. From eight upwards there is.

The measured strains beneath that figure are the reason to care. Cycloheptane is 26 kilojoules a mole and cyclooctane 40; cyclononane is 53 and cyclodecane 50; and then cyclododecane drops to 17. The medium rings are anomalously strained, and their anomaly begins exactly where the third term begins to exist.

Twelve structures a two-term account cannot tell apart #

The sharpest version of the claim does not need any comparison with measurement at all. It needs one ring, one bond angle, and several structures.

Every conformer of a ring at a fixed bond angle has, by construction, exactly the same angle strain. So an account built from angles and torsions separates such conformers by the torsional term alone, and two conformers whose dihedrals happen to give the same torsional energy are, to that account, the same structure.

Two of those twelve differ in torsional energy by 0.026 kilojoules a mole — three parts in ten thousand — and in their closest cross-ring approach by 0.80 ångström, from 2.331 to 3.131.

Two carbon atoms not bonded to each other are in contact at about 3.4 ångström, which is twice the tabulated van der Waals radius. A separation of 3.13 is a squeeze. A separation of 2.33 is shorter than a carbon–carbon bond by only half an ångström, and no such structure exists.

On the quoted contact potential the two structures’ cross-ring terms are 4.6 and 366.5 kilojoules a mole. The two-term account puts them within a rounding error of each other.

The account’s own favourite #

The conformer with the lowest torsional energy in that family is at 44.5 kilojoules a mole, ten below its nearest rival, and it is the structure the two-term account would pick as the ring’s shape.

Its closest cross-ring approach is 2.639 ångström.

So the term that is missing is not a correction to the ranking; it changes which structure is at the top of it. This is the same shape of failure the tolerance sweep found in a different corner of the subject — a procedure that returns an answer, confidently, from a criterion that cannot see the thing that decides.

Why an angle term cannot stand in for it #

There is an obvious objection: a structure with two atoms at 2.33 ångström is not a real structure, so a search that minimised a proper energy would never produce it, and the third term is doing no work that a careful conformational search would not do anyway.

The objection has the argument backwards, and the reason is worth setting out.

A conformational search minimises whatever energy it is given. Give it the two-term energy and it produces the structure circled in the figure above, with its 2.639-ångström contact, and reports a strain. Nothing in the search notices, because the search is a minimisation over the same coordinates the energy is written in, and none of those coordinates is the distance in question.

That is the general shape of the failure and it is not special to rings. An account expressed in a set of coordinates can only be wrong in ways those coordinates can express, and a term outside them is invisible to the account, to the minimiser and to the person reading the output. The same thing happens when a force field is fitted to a spectrum: the fit reproduces the frequencies it was given and the parameters it chooses are not determined by them.

What makes the ring case sharp is that the missing coordinate can be counted. There is no argument about whether a ten-ring has cross-ring pairs; it has fifteen.

What the medium rings do about it #

Real medium rings solve the problem, and the solutions are the reason their conformational analysis is a subject of its own.

Cyclodecane’s preferred conformation is a boat-chair-boat arrangement in which six of its hydrogens point inwards and are in genuine contact — the transannular strain is real, it is measured, and it is why cyclodecane is 50 kilojoules a mole strained while cyclohexane is 0 and cyclododecane 17. Those inward hydrogens are also why medium rings have unusual reactivity: a reagent approaching the ring meets a face that is already occupied, which is a steric argument of exactly the kind a bond angle cannot support.

By twelve atoms the ring is large enough to adopt a conformation with all its torsions near sixty degrees and no atoms in contact: the ring folds into a shape with four straight runs, and the strain falls to a sixth of cyclodecane’s. That drop is the third term switching off, and neither of the other two terms changes much across it.

The two quantities that have to be compared are best seen together, on one pair of axes, over the range of ring sizes the observation covers.

The conformer the count is taken on is the chair, and it is worth drawing once so that the contacts being counted are visible rather than described.

What the third term is, and why it is quoted #

The distance between two atoms four bonds apart is a coordinate; the energy associated with it is a model, and the model here is quoted rather than computed.

A Lennard-Jones potential between united-atom CH₂ groups, with a well depth of 0.46 kilojoules a mole and a minimum at 4.0 ångström, is what molecular mechanics uses for exactly this term. It is a fit, its parameters were chosen against measured structures, and the numbers it produces at short range — the 366 kilojoules above — are far outside the range it was fitted in.

So the essay’s claims are stated in terms of distances wherever possible and in terms of the potential’s energies only for illustration. The distance between two atoms in a computed conformer is a geometric fact; what it costs is a fitted opinion.

That distinction matters more here than usual, because a term with a twelfth power in it is enormously sensitive to the distance and therefore to the conformer search that produced it.

Where the model stops #

The conformers are found by a closure search, not by minimising an energy. Each is a closed ring with equal bonds and equal angles at a stated value, found from a different starting point. They are structures the ring can adopt, not structures it does adopt, and the point of the family is that the two-term account has to choose between them.

The bond angles are held equal, which is the device that isolates the third term and is also the reason no number here should be read as a strain energy. A real ring trades a little angle strain for a lot of cross-ring room, and the atoms in a real structure are not at the points a model puts them in any case. Real medium rings do not have equal angles: they open the angles at the crowded positions, which is precisely the coupling between the three terms that the fixed-angle family removes in order to isolate the third one.

The potential is a fit and its short-range behaviour is an extrapolation. Nothing here should be read as a strain energy.

Only one ring size was taken apart in detail. The family above is a ten-ring, chosen because it has fifteen cross-ring pairs and enough conformers to compare. The eight- and nine-rings have the same qualitative behaviour with fewer structures to show it on, and a ring’s ability to hold still at all is itself a function of size.

And hydrogens are not present. The contacts that matter in a real medium ring are between hydrogens, which stick out from the carbons and meet each other first; a united-atom treatment folds them into the carbon and gets the distance scale roughly right by construction.

And the twist-boat, which has the same bonds and the same angles and a quite different set of cross-ring distances, is where the count changes without anything a bond diagram records changing at all.

The three terms as a hierarchy #

Stated in order, the account arrived at reads as a hierarchy by separation along the chain, and each term applies to a set of pairs that the previous one does not touch.

One bond apart: the bond length. Fixed, in every structure here.

Two bonds apart: the bond angle. Constrained by the ring size, and the source of the small rings’ strain.

Three bonds apart: the dihedral. The source of cyclopentane’s strain and of cyclohexane’s preference for a chair.

Four or more: the distance, which is not determined by the others and is what this essay is about. Between two molecules the same term is the whole interaction, and it has been computed there — a molecule with no dipole at all attracts another through terms of exactly this kind.

That hierarchy is exactly how a molecular mechanics force field is written, and the fourth line is what such a force field calls the non-bonded term. What is unusual about a ring is that a non-bonded term appears within one molecule, between atoms joined by a chain of bonds, and that the number of such pairs is a property of the ring size rather than of the chemistry.

The torsional coordinate on its own is a threefold barrier turned through a full revolution, and it is the same term used here. Everything in this essay is a count over the minima of a surface built from that term, which is why the count of rotors is the quantity the argument is sensitive to.

What is quoted, and what is computed #

Quoted: the measured strain energies of the eight rings; the ethane barrier of 12.1 kilojoules a mole; the bending constant; the Lennard-Jones parameters; the van der Waals contact distance for carbon.

Computed: every conformer, by a closure search on a stated seed; every dihedral, from the coordinates; every torsional energy, from the dihedrals and the quoted barrier; every cross-ring distance, from the coordinates; and the counting fact, from the definition of a cycle.

What a strain energy is a measurement of #

Every measured number in this essay is a heat of combustion in disguise, and the chain from one to the other is worth spelling out because the third term enters it twice.

A ring’s strain energy is its heat of combustion minus the heat a strain-free chain of the same formula would have, per methylene group. So it is a difference against a reference, exactly like a delocalisation energy — a stabilisation is measured from somewhere, and so is a destabilisation.

The reference is an unstrained long chain, and a long chain has cross-ring pairs of its own in every folded conformation. So the third term is present on both sides of the subtraction, at different magnitudes, and the quoted strain of cyclodecane is the difference of two numbers each of which contains it.

That does not make the measured strains wrong. It means that the quantity a two-term model is being compared against was never a sum of two terms, and that the two-term comparison — where the two terms track the measurements up to seven rings — was working because the term missing from both sides happened to cancel.

What was checked #

No ring below eight has a pair of atoms four bonds apart, and every ring from eight up does. Both halves, because the claim is that the term does not exist rather than that it is small.

Two conformers of one ring at one bond angle have torsional energies within 0.6 kilojoules a mole of each other and closest approaches differing by more than 0.4 ångström. This is the essay’s central claim in the form that can fail, and it fails if the family collapses to a single structure or if the two coordinates turn out to be correlated.

And the conformer with the lowest torsional energy is not the one with the most room in it, which is what makes the missing term a change of ranking rather than a correction to one.

What the average is worth, before the term is computed #

The average over the family is the right next quantity and its size can be bounded now, from the spread the twelve conformers already show, without any third term at all.

At 298 K, $RT$ is 2.48 kJ mol⁻¹. The twelve structures here span 0.6 kJ mol⁻¹ in torsional energy, which is a quarter of that — so their Boltzmann populations are nearly uniform, and the ensemble’s mean energy sits 0.29 kJ mol⁻¹ above the lowest conformer. Widen the spread to a more realistic three kilojoules and the mean rises only to 1.15; widen it to eight and it reaches 1.87 and stops, because by then the high members are depopulated faster than they are added. The correction to an enthalpy is therefore of order one kilojoule and is bounded above by about two, whatever the third term turns out to do to the ordering.

That is small against a measured strain of fifty, and it is the wrong place to look for the discrepancy.

The larger effect is in the free energy rather than the enthalpy, and it is not a correction to the strain at all. A ring with $g$ conformers thermally accessible carries a conformational entropy of $R\ln g$, worth $RT\ln g$ — 1.7 kJ mol⁻¹ for two conformers, 3.4 for four, 6.2 for twelve. That is a real stabilisation of the medium rings and it does not appear in a strain energy, because a strain energy is an enthalpy difference taken from heats of combustion.

Which is worth stating as a caution about strain energies generally. A strain energy is an enthalpy and a ring’s conformational freedom is an entropy, so the number every strain account compares against is deliberately blind to the one quantity that grows fastest with ring size. Cyclodecane is not merely strained by fifty kilojoules; it is strained by fifty and paid back several by having many shapes, and only the first of those is in the table.

Still open: a term for the ring as a whole #

The three terms in this account are all local: a length, an angle, a dihedral, and now a distance between two atoms. What none of them is, is a term that depends on the ring as a whole.

There is such a term, and it has already been touched from the other side. A ring’s conformations form a continuous family — a pseudorotation itinerary — and the barrier between two conformers is not a sum of local terms but a property of the path between them. The strain of a molecule that interconverts rapidly between conformers is a Boltzmann average over that family, and a strain energy quoted for a medium ring is an average over a set of structures that this essay has just shown a two-term account cannot even enumerate correctly.

That average is computable, it needs the third term to be worth computing, and it is what would connect ring geometry to the thermochemistry the measured strains actually come from.