The ring that cannot hold still

A ring of six carbons has a chair and a boat, and every course draws them next to each other as though they were two of a kind. They are not. One of them is a single shape and the other is a whole family of them, and the difference can be computed without any energy at all — from the rank of a matrix whose entries are derivatives of bond angles.

The setup is the one the angle a ring cannot have used to put a ceiling on ring angles, and it contains no chemistry beyond a bond length and a bond angle — the same austerity that lets VSEPR, computed get an angle out of a minimisation with no orbitals in it. A closed ring of $n$ atoms is $n$ unit vectors that sum to zero, with a fixed angle between each consecutive pair. That is $2n$ numbers — a polar and an azimuth for each direction — against $n + 3$ conditions: one per angle, and three for the closure.

Three of the $2n$ are the orientation of the whole ring in space, which is a fact about the observer. So the shapes a ring of $n$ atoms can have, with its bonds and angles held exactly, ought to form a family of dimension

$$(2n - 3) - (n + 3) = n - 6.$$

Zero at six, one at seven, two at eight. That arithmetic is the standard count of a ring’s conformational freedom, and this essay is about measuring it instead of trusting it.

What the rank is measuring #

Fix a conformer that satisfies every condition. Ask which small motions of the ring change none of them — that is, which directions in the $2n$-dimensional space of parameters are annihilated by the derivative of the constraint function.

That derivative is a matrix with $n + 3$ rows and $2n$ columns, and it is taken numerically by central differences rather than by hand. The reason is a general one: the constraint function is the one the search minimises, so a hand-differentiated version could disagree with it and the disagreement would look like a result.

Its null space is the set of motions that change nothing. Three of them are always there — turn the whole ring about any axis and no bond and no angle moves — so the dimension of the family through that conformer is the nullity less three.

The measurement is only worth anything if the rank is unambiguous, and it is: the singular values of these matrices come in two groups separated by about eight orders of magnitude, the smallest non-zero one near $0.4$ and the largest zero near $10^{-8}$. Any threshold between $10^{-6}$ and $10^{-2}$ gives the same answer — the same clean separation that lets degeneracy is a group theorem count a degeneracy rather than estimate one — and the ratio is reported alongside so that a future ring which does not separate that cleanly cannot be rounded quietly into a verdict.

The chair, written down rather than found #

A search will find whichever conformer its starting point falls towards, so the chair is constructed instead.

Six atoms alternating above and below a regular hexagon have equal bonds by construction. One number fixes the height: with unit bonds and a hexagon of radius $R$, the geometry gives $R^2 + h^2 = 1$ and

$$\cos\theta = \frac{3h^2 - 1}{2}$$

for the bond angle $\theta$. So a chair exists in closed form at every angle from $90°$ to $120°$, and at $111°$ its torsions come out at $\pm 56.05°$ alternating. Nobody put that number in. Real cyclohexane’s chair torsion is about $55°$, from a structure determination that knows about hydrogens and repulsion and everything else this calculation does not contain. A geometry arriving from a constraint rather than from a fit is the same kind of result as five sites are not alike’s three distinct angles.

Handed to the rank calculation, the chair gives nullity 3 and family dimension zero. It is an isolated point: there is no motion whatever, however small, that changes its shape while keeping its six bonds and six angles.

That is the chair’s real distinction, and it is not the one usually given. The chair is not merely the lowest conformer; it is the only one of its kind that is rigid under the constraints, and the reason cyclohexane has to pass through a half-chair to invert is that there is no path away from it that keeps the angles.

The conformer the search finds is not the chair #

Run the same closure search that the angle a ring cannot have uses, at the same $111°$, and it returns something else: torsions of $-16.9°$, $63.6°$, $-44.2°$, repeating. That is a twist-boat, and its rank comes out at 8 rather than 9.

Nullity 4. Family dimension one.

So six has two answers, and the count of freedoms against conditions predicted neither of them well: it said zero, one branch has zero and the other has one. What a count of that kind gives is the generic dimension, and it cannot see that the solution set has more than one piece.

This is worth stating plainly because the same arithmetic appears wherever a mechanism is counted — a linkage’s mobility, a molecule’s internal coordinates, the degrees of freedom in a force field. The count is a lower bound on what is possible and an upper bound on nothing, which is the same lesson the force field is not in the spectrum draws from counting constants against frequencies.

Following the free direction, and where it stalls #

Having found a direction that changes nothing, the obvious move is to walk along it: step, project back onto the constraints, repeat.

It stalls. The walk moves smoothly for twenty steps, arrives at a conformer with one torsion near zero, and then oscillates there indefinitely, taking steps that undo themselves.

The reason is not a bug and is worth recording. Parameterised by a torsion angle, the family has a fold: the torsion rises to about $65°$ and turns back. A walker that keeps going in the same direction in torsion space reaches the fold and has nowhere to go, because the coordinate it is walking in stops being a coordinate there. The family is a closed loop and a torsion angle is a bad label on it.

What is a good label is the ring’s own puckering phase. Cremer and Pople’s construction is a definition rather than a fit: find the ring’s mean plane from two weighted sums of its positions, measure every atom’s displacement from it, and resolve those six numbers into the three independent puckering amplitudes a six-ring has — one pair $(q_2, \varphi_2)$ read as an amplitude and a phase angle, and a single further amplitude $q_3$. The pair $(q_2, q_3)$ is a point on a sphere: $q_3$ alone at the poles, $q_2$ alone at the equator.

Nothing in that construction knows what a chair is. The names get attached afterwards, to whatever the computation puts where.

So the conformer at a stated phase angle is solved for directly — the $n + 3$ ring conditions with a condition fixing it added, by damped Newton from a neighbouring member — and the loop closes.

Two things nobody put in #

The loop was traced by asking for a phase angle and solving. Two facts about it were not asked for and came out anyway.

The family lies exactly on the equator. Every member of the loop has $q_3$ below $10^{-14}$ — not small, arithmetic noise — while its $q_2$ is about $0.507$ bond lengths. And the chair, solved for in closed form and never handed to the loop at all, has $q_2$ exactly zero and $q_3 = -0.376$: a pole.

So the classical picture of cyclohexane’s conformational space — chair at the poles, boats and twist-boats round the equator — is not a convention for drawing it. It is what the equal-bond, equal-angle constraint set is, and it comes out of a Newton solve that had never heard of it.

The agreement is of the kind one coordinate, three point groups turns on: a soft internal coordinate whose consequences are decided by symmetry rather than by energy.

The amplitude is not quite constant. Round the equator $Q$ runs from $0.5066$ to $0.5085$, a variation of four parts in a thousand. The idealised treatment holds it fixed; the constraints do not quite. That is a small departure and it is reported because it is the kind of thing an idealisation quietly supplies.

Seven, eight and nine, where the count is right #

Away from six the arithmetic behaves. A seven-ring’s conformer has family dimension 1, an eight-ring’s 2, a nine-ring’s 3 — exactly $n - 6$ in each case, measured from the rank and not assumed.

That matters for what the six-ring result means. If the count had failed everywhere it would be evidence that the constraint matrix was being read wrongly. Failing at exactly one size, in a way that resolves into two branches of different dimension, is what a genuine non-generic case looks like.

And below six the constraints win outright: at three and five no equiangular ring exists at any angle below the ceiling, which is the ceiling result for small rings and the reason this essay starts at six.

What this does and does not say about strain #

None of the above is an energy. The chair is not shown to be more stable than the twist-boat — nothing here could show that, because no repulsion, no torsional term and no hydrogen appears anywhere in the calculation.

What is shown is a different and prior thing: which shapes are available at all, once bond lengths and bond angles are held. A conformational analysis that starts from an energy function is answering the question of which of the available shapes is lowest, and it inherits the answer to this question without asking it.

The relation between the two is the useful part. A rigid conformer has no path away from itself at constant bonds and angles, so any motion off it must bend an angle, which costs energy at second order immediately. A conformer on a family has directions that cost nothing to the constraints, so whatever energy differences exist along the family are the small ones — torsional terms, non-bonded contacts — rather than the large ones. That is why the flexible form of cyclohexane behaves as a shallow trough of interconverting shapes and the chair behaves as a well.

The same distinction is what which angles are symmetry and which are the model draws for VSEPR: some of what a model appears to predict is fixed before the model is applied, and telling the two apart is the whole of knowing what the model is worth.

The chair’s rigidity, stated as a chemist would #

The result has a form that a conformational analysis will recognise, and it is worth translating.

Cyclohexane’s chair cannot be deformed at all without bending an angle. Every path away from it — towards a half-chair, towards a boat, towards its own mirror image — passes through geometries whose bond angles are not the chair’s. That is why ring inversion has a barrier of about 43 kJ/mol while the boat-to-twist-boat interconversion has one of a few: the first has to pay for angle bending and the second does not.

The flexible form is a trough rather than a set of wells. Its members differ only in torsions, and torsional terms are the small ones, so the energy varies gently around the loop. Twist-boats are minima and boats are the low saddles between them, separated by a few kilojoules — which is what a one-dimensional family with a shallow modulation on it looks like.

Neither of those numbers is computed here and neither could be: the constraint set contains no energy. What it supplies is the stage the energies act on, and the stage has two pieces of different dimension. A conformational analysis that started from an energy function would find the same two pieces without ever noticing that their difference is geometrical rather than energetic.

What a degree of slack buys #

The count above is exact and it is also brittle, because it holds the angles exactly. Real angles are not held exactly; they are held by a bending constant. So the honest version of the question is what happens to the family when each angle is allowed to move by a stated amount.

The dimensions say what the answer must look like. Holding only the bond lengths leaves $2n - 6$ internal shape coordinates — six for a six-ring, of which three are the Cremer–Pople puckering coordinates and three describe the in-plane shape. Pinning the $n$ angles removes $n$ of those and leaves $n - 6$, which is the count this essay measures. The angles are therefore not a detail of the constraint set; they are most of it, and at six atoms they are all of it.

Allowing each angle a slack of $\delta$ restores those dimensions in a bounded way. The isolated chair becomes a small region of the puckering sphere rather than a point, and the boat’s loop becomes a tube around the loop, with both extents growing linearly in $\delta$ for small $\delta$. Nothing changes qualitatively — a region around a point is still not a loop — which is why the rank count survives being made with a constraint no molecule obeys.

The size of the correction can be read from the molecule. The constraint calculation says the boat family is exactly flat: every conformer on the loop has the same bonds and the same angles, so nothing in this argument distinguishes them. Cyclohexane’s measured corrugation along it is about 1.6 kJ mol⁻¹ between the boat and the twist-boat on either side of it — against roughly 23 kJ mol⁻¹ from the chair up to the twist-boat, and about 45 to the transition state between them.

That ratio is the number worth keeping. The flat direction is flat to within seven per cent of the nearest energy difference and three per cent of the barrier. The equal-angle constraint is wrong in the fourth significant figure of the geometry and it is right about which direction is soft, which is exactly the division of labour the rank calculation was built to make.

Where the model stops #

The calculation holds every bond equal, every angle equal, and every atom a point. Three of those are approximations and one of them matters here.

Equal angles is the strong assumption. Real cyclopentane’s famous pseudorotation is a continuous itinerary of exactly this kind, and it does not appear in this calculation at all — five is excluded before any search begins, because an equiangular five-ring does not exist below its ceiling. The real molecule buys its itinerary by letting the angles vary by a couple of degrees, which this constraint set forbids. So the absence of a five-membered family here is a statement about the constraints, not about cyclopentane.

Equal bonds is mild. Alternating a ring’s bonds is a real effect in conjugated systems, and it is a chain cannot stay even’s subject; in a saturated ring the bonds really are equal to within a few thousandths, and bond order from the eigenvectors is where the unequal case is computed.

A point atom has no substituents. Axial and equatorial positions, the distinction that makes ring conformation a subject in organic chemistry, do not exist for a ring of bare vertices. What this calculation gives is the space the substituents are then arranged on — and what a lone pair is worth is the same distinction for a lone pair, which occupies a position without being an atom.

How large an angle, and how many shapes #

A ceiling on a ring’s bond angle shows that three, four and five cannot reach the tetrahedral value at any geometry. It answers how large an angle can be.

The constraint count answers how many shapes there are at a given angle, and the answer is not a number that a formula supplies: it is the rank of a matrix at a particular conformer, it varies between conformers of the same ring, and where it varies the ring has two kinds of shape rather than one. The chair’s rigidity, the flexible form’s itinerary, and the puckering sphere that organises them all fall out of the same constraint set, with no energy anywhere in the argument.

What remains open is the ring sizes above nine, where the families are large enough that a search is no longer a useful way to find anything. The question of angle slack is settled here only in outline: the dimensions return in proportion to the slack, the flat direction stays the flat one, and cyclohexane’s own corrugation along it puts the size of the correction at a few per cent.