Molecular Geometry & Bonding
what the shape decides
Figure

A 6-ring at 111°: the chair

A closed ring of 6 equal bonds meeting at 111°, drawn from the coordinates the closure conditions produce. Its torsions are 56.0°, -56.0°, 56.0° and repeat; its puckering amplitude is 0.376 bond lengths at a phase of 97°. Turning the ring changes none of those numbers, which is what makes them a description of the shape rather than of the view.
A 6-ring at 111°: the chair. A closed ring of 6 equal bonds meeting at 111°, drawn from the coordinates the closure conditions produce. Its torsions are 56.0°, -56.0°, 56.0° and repeat; its puckering amplitude is 0.376 bond lengths at a phase of 97°. Turning the ring changes none of those numbers, which is what makes them a description of the shape rather than of the view.

One of the figures on where the atoms go: Repulsion minimised on a sphere, the angles that fall out of it, and the arrangements that are not all alike.

Five essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

In the essays

The ring that cannot hold still

The conformer the count cannot reach: a six-ring’s twist-boat at the same bond angle as the chair above. The two are the same ring with the same angles at every atom, and they are separated by a path rather than by a number — which is why a count of independent conditions gets six wrong in both directions while it gets seven, eight and nine right.

The chair at 111°, built from the closed form. Six equal bonds, six equal angles, torsions alternating ±56.05°, and — the line that matters — no free directions at all once the three rotations are removed. Turn the ring and every number under it stays where it is.

The twist-boat the search returns, at the same bond length and the same 111° angle as the chair above. Its torsions repeat with period three rather than alternating, and unlike the chair it has one direction left over — one way to deform that changes no bond and no angle.

The double bond a ring cannot hold

The chair conformer of a saturated six-ring, which closes exactly at the ordinary bond angle and is rigid in a sense a count of freedoms cannot see. Adding a double bond removes two of its torsional degrees of freedom and adds a constraint, which is why the six-ring is the boundary case in every calculation here.

The strain that is not in the angles

The chair itself, drawn from the coordinates the construction produces. Its bonds come out equal to 10⁻¹⁵ without that having been imposed — the construction fixes the angles and the alternating displacement, and the equal bonds follow from the symmetry.

The atoms that meet across a ring

Cyclohexane’s chair, where all three terms are at their minimum together: tetrahedral angles, staggered bonds, and no pair of atoms far enough apart along the chain to approach each other. That coincidence is why it is the reference against which every other ring’s strain is quoted.

And the itinerary a six-ring takes between its conformers. A ring with no transannular pairs can be described completely by its torsions, and this is what that description looks like.

A ceiling that rises where the measurements fall

The six-membered ring at the angle it prefers, which is the closure the account is comfortable with. Its measured strain is zero and its ceiling is a hundred and twenty; the ring the account fails on is the one that cannot reach this angle at all.

Every figure · Every orbital, by what it encloses · All essays