The double bond a ring cannot hold

A double bond does something no single bond does: it fixes six atoms in a plane. Its own two carbons, and the four attached to them, all lie in one plane, and the reason is that the π bond is an overlap of two p orbitals whose axes have to be parallel. Twisting about the bond turns the overlap off, which is why cis and trans alkenes are separable compounds rather than conformers of one another.

In an open chain that constraint costs nothing. In a ring it is a constraint on the ring, and for a trans arrangement it is a severe one: the two chains leave the double bond in opposite directions and have to come back round to meet each other.

The question, made precise #

Take a ring of $n$ carbons with one double bond in it. Every bond length is known — 1.33 Å for the double bond, 1.50 for the two bonds flanking it, 1.54 for the rest — and every bond angle is known: 120° at the two sp² carbons, about 111° at the others. The torsion about the double bond is not free; it is 0° for a cis arrangement and 180° for a trans one.

Everything else is free. The remaining torsions can take any values. Does such a ring exist?

This is the same kind of question as the largest angle a ring can have, and it is answered the same way: write the ring as its $n$ bond directions rather than its $n$ positions, so the conditions become closure, one angle condition per vertex, and one torsion condition. Then search, and read the residual. A ring that exists drives the residual to the floor of the search; one that does not leaves it stuck.

Two numbers, and both are one too large #

Cyclohexene exists and is entirely ordinary; the calculation says a cis double bond needs seven carbons. Trans-cyclooctene has been an isolable compound since 1953; the calculation says a flat trans double bond needs nine.

Both thresholds are one ring too conservative, and that is not a failure — it is the measurement. The model allows no angle strain at all: every sp³ vertex is held at 111° and every sp² one at 120°, and a ring either closes on those numbers or does not. Real rings do neither. They close by giving somewhere, and what they give is precisely what makes them strained.

So the compounds living in the gap between the model’s threshold and the observed one are exactly the ones that pay for the privilege. Cyclohexene’s sp² angles open to about 123.5° rather than 120. And trans-cyclooctene is one of the most familiar strained molecules in organic chemistry — strained enough that its strain energy has been measured, and strained enough that its two mirror images do not interconvert, so it can be resolved into enantiomers despite having no stereocentre.

What the eight-ring will hold #

The model can say how much a ring gives, if the giving is put in the one place the calculation can handle exactly. Instead of asking whether the ring closes on a flat trans bond, pin the torsion at some angle $\tau$ and ask for the largest $\tau$ the ring will close on.

139.3° against 136°. The calculation contains bond lengths, bond angles and geometry. It contains no force constants, no energies, no π overlap and no strain. It agrees with a crystal structure to within three degrees.

That is worth being precise about, because it is easy to over-read. The model does not predict that trans-cyclooctene exists — nothing here says whether the twist it demands is affordable. What it says is that if the compound exists at ordinary bond lengths and angles, its double bond is twisted by about forty degrees, and it is. The energy question and the geometry question have been separated, and only the second has been answered.

The seven-ring is the useful contrast: its limit is 93.8°, a twist of 86°, which is essentially no π bond at all. Trans-cycloheptene has been observed — at low temperature, as a transient, trapped rather than bottled. The model says the geometry it would need is one in which the double bond has stopped being a double bond, and the chemistry agrees.

Why trans is so much harder than cis #

The gap between the two thresholds — seven for cis, nine for trans — is worth an explanation rather than a table entry, because it is not obvious that the trans case should be harder at all. Both fix the same six atoms in a plane; the only difference is which side of the double bond each chain leaves from.

That difference is everything. In the cis arrangement the two chains leave the double bond on the same side, pointing roughly towards each other, and the ring closes across a short gap. In the trans arrangement they leave on opposite sides, so the chain has to travel out, round and back — and the shortest path back passes through the region the double bond’s own plane occupies.

Counting degrees of freedom gives part of it. A ring of $n$ bonds written as directions has $2n - 3$ free parameters once the overall orientation is divided out, and the conditions are three for closure, $n$ for the bond angles, and one for the torsion: $n + 4$ in all. Solutions exist generically when $2n - 3 \geq n + 4$, that is from $n = 7$. That count is blind to the cis-trans distinction, and it gets the cis threshold exactly right and the trans one wrong by two.

The counting cannot see the difference because it is a generic statement, and the same limitation appeared when the rigidity of a six-ring was counted rather than constructed: a count of freedoms against conditions gives the dimension of the solution set where one exists, and says nothing about whether it does. The trans case is a region of the parameter space where the solutions have gone away, and only a search finds that.

Bredt’s rule is the same question #

In 1924 Julius Bredt stated that a double bond cannot be placed at the bridgehead of a bicyclic system. For forty years the exceptions to it were exceptions. Then in 1967 Wiseman pointed out that it had never been a rule about bridgeheads at all.

A bridgehead carbon belongs to two rings. A double bond at a bridgehead is therefore in both of them, and the geometry it has differs between them: it is cis in the smallest ring it can be cis in and trans in the largest. So the question is only ever the one already answered — how large does a ring have to be to hold a trans double bond?

The counting is arithmetic on three integers. A bicyclo[$a$.$b$.$c$] system has bridges of $a$, $b$ and $c$ atoms between its two bridgeheads, so its three rings have $a+b+2$, $a+c+2$ and $b+c+2$ atoms. What decides is the largest of them.

The threshold is calibrated on one compound and the other eight are predictions. Saying which is which is the whole difference between a fit and a rule: trans-cyclooctene sets the scale for how much twist is affordable, because it is the compound that defines the boundary experimentally, and everything else follows by counting.

The pair that makes the point is at the bottom of the table. Bicyclo[3.3.1]non-1-ene is Wiseman’s own compound: its rings are 6, 6 and 8, so it has an eight-ring and it is isolable. Adamantene has the same bicyclo[3.3.1] fragment — and the extra bridge of the adamantane cage closes that eight-ring off. What is actually available is six. It is not isolable, and a rule that looked only at the bicyclic name would have got it wrong.

A twist that can be seen #

A twisted double bond has a consequence that goes beyond strain, and trans-cyclooctene is the standard illustration of it. The molecule has no stereocentre — no carbon with four different substituents anywhere in it — and it is nevertheless chiral, because the twist itself has a handedness. The two enantiomers were separated in 1963 and they racemise only on heating, since interconverting them means passing the double bond through the ring.

Chirality is a statement about the point group and not about a list of substituents, and this is the cleanest available demonstration: the group of a planar trans-cycloalkene would contain a mirror plane, and the twist removes it. Everything that is chiral here is geometry.

The twist also weakens the bond, and it shows in the two places a weakened π bond always shows: trans-cyclooctene reacts far faster than its cis isomer in additions across the double bond, and its stretching frequency is lower. The second of those is tempting to read as a direct measure of the strain, and a frequency is not a bond strength — the normal mode that looks like a C=C stretch in a ring is mixed with everything else in the ring, and a shift in it cannot be assigned to one interaction.

What the residual is measuring #

The floor is not a judgement call, and that has to be shown rather than claimed. The residuals fall into two groups separated by more than two orders of magnitude — the rings that close sit between $10^{-9}$ and $10^{-5}$, the rings that do not sit between $10^{-2}$ and $10^{-1}$ — so any floor drawn between them gives the same thresholds.

The rings that close are not at zero, and the reason is worth knowing. The search is a coordinate descent on $2n$ spherical angles with a polish on the winner, and its floor is set by how far the polish gets rather than by the geometry. The first version of it stopped at a residual of $6 \times 10^{-4}$ for the nine-ring, which sits above the floor the same arrangement reaches when the descent is allowed to finish properly — and a threshold read off residuals cannot be allowed to depend on how long the search was permitted to run.

Two constraints, and they are not the same #

A ring can fail to hold a double bond for either of two reasons and the calculation keeps them apart.

Angle strain is what the closure bound describes: a small ring cannot keep its bond angles near tetrahedral because a closed curve has to turn through 360°, and a double bond makes it worse by demanding 120° at two of the vertices. This is what stops cyclopropene and cyclobutene from being ordinary compounds, and it applies to cis and trans alike. It is also the constraint that has nothing to do with hybridisation, whatever the labels attached to the carbons afterwards suggest.

Torsional strain is the constraint of this essay: the ring cannot deliver a flat trans arrangement even when the angles are comfortable. It is invisible in small rings because the angle strain has already made them impossible, and it is the only constraint operating from about eight carbons up.

The separation matters for reading the numbers. A cis double bond in a ten-ring is under neither constraint, and the residual says so. A trans double bond in a ten-ring is under neither either, which is why the trans and cis residuals converge from about ten upwards — the ring has become large enough that the double bond is a local detail.

What this cannot say #

Three limits, and each is a place where a real answer would need an energy.

Nothing here is an energy. The model reports which geometries are available, not which are paid for. A twist of 40° costs something — the π bond weakens as the cosine squared of the twist, roughly — and how much can be afforded is a thermodynamic question this calculation cannot ask. That is why the threshold has to be calibrated on a compound.

The angles are held rigid. Real rings relieve strain by opening and closing their angles as well as by twisting, and forbidding that is what makes both thresholds one ring too conservative. Allowing it would need force constants, and force constants are a fitted quantity that would replace a geometric statement with a parametrised one.

A single conformer. The search returns one closed ring per case and reports its residual. A ring of ten or twelve carbons has many closed conformers, and which one a real molecule adopts is decided by transannular contacts between hydrogens that this model does not have.

Above the threshold, and the rings that come free #

From nine carbons up the trans double bond stops constraining anything: the residual sits at the search floor and stays there through twelve, and by then the ring has enough torsional freedom that the double bond is a local detail rather than a global constraint.

What that means chemically is that a macrocyclic trans alkene is unremarkable. Rings of twelve, fourteen or twenty carbons carrying trans double bonds are ordinary compounds — several of them are perfumery materials made on an industrial scale — and their double bonds are flat. The whole subject of anti-Bredt chemistry lives in the narrow window between seven and nine, which is why it took forty years to be understood: the interesting cases are three ring sizes wide.

Eight is where saturated rings are already awkward: the conformers are close in energy and none of them is comfortable, so the ring has little margin to spend before a double bond is introduced. That is why the trans-cyclooctene is the smallest isolable member of the series and why it is strained enough to be a useful reagent.

The threshold leaves a stereochemical fingerprint #

A twisted double bond has a consequence the geometry does not mention and a chemist notices immediately: it is chiral.

A flat trans alkene in a large ring has a mirror plane — the plane of the double bond and its four neighbours. Twist it, and the two chains leave that plane on opposite sides in a definite sense, and the mirror image is a different molecule. Nothing has been substituted, no atom is a stereocentre, and the compound is nonetheless resolvable into two enantiomers.

Racemising it means dragging the chain through the double bond’s plane, which is the same closure problem this essay solves, walked rather than solved once. So the barrier to racemisation ought to fall steeply with ring size in exactly the window the threshold picks out — high at eight, where the closure is only just possible, and low at ten, where it is comfortable.

It does. trans-Cyclooctene racemises with a free energy of activation of roughly 150 kJ mol⁻¹, which is a half-life of years at room temperature and is why its enantiomers were resolved and kept. trans-Cyclononene is at about 85, minutes to hours. trans-Cyclodecene is near 45, which is not a resolvable compound at all.

Two ring sizes carry a factor of more than three in the barrier and something like fifteen orders of magnitude in the rate. That is the threshold this essay computed, showing up in a completely different observable — an optical rotation that persists or does not — with no geometry quoted anywhere in the measurement.

Still open: contacts across the ring #

The obvious next constraint is the one that has been left out of every ring calculation here: the contacts between atoms across the ring. Angle strain and torsional strain are both local — they are properties of neighbouring bonds — and the medium rings, eight to eleven carbons, are strained mainly by something non-local, hydrogens on opposite sides of the ring pressed together. That effect has a minimum around nine or ten and it is the reason those ring sizes are hard to make.

Modelling it needs a contact distance and therefore a radius, which is the same missing quantity the five-coordinate arrangements turned out to need. Both point at the same gap: a model of angles is a model of what is geometrically possible, and everything about what is chemically preferred needs a second length scale that a symmetry argument does not supply.

The claim that survives is the one Wiseman made. Bredt’s rule is not about bridgeheads; the bridgehead is where a trans double bond happens to be forced, and the forcing is a property of the largest ring. Counting three integers and comparing the largest against eight sorts the compounds correctly, and the eight came out of a calculation about rings that had never heard of a bicyclic system.