Molecular Geometry & Bonding
what the shape decides
Concept

Closure — where it appears

The requirement that combining two members of a set gives another member, which is what makes a set of symmetry operations a group. Generating a molecule's operations by closing its elements under multiplication is how a point group is found from coordinates rather than recognised.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

The twist the ring forces on the bond. For each ring size, the largest torsion about the double bond that the ring will close on — 180° being a flat trans arrangement and 90° being a π bond broken outright. The six-ring will not close at any torsion tested. The eight-ring reaches 139.3°, against 136° measured in trans-cyclooctene by diffraction: a model with bond lengths and bond angles in it and no energy anywhere agrees with the crystal to a few degrees.

The double bond a ring cannot hold

A trans double bond needs a ring of nine carbons to sit flat, and the eight-ring will hold one twisted by 40.7 degrees — against 136 degrees measured in trans-cyclooctene. The same eight-ring threshold, applied by counting ring sizes, sorts nine bridgehead alkenes correctly with no bridgehead anywhere in the argument.

shape · Strain One distortion, resolved into the species of the group it left. An arbitrary displacement of benzene, projected onto each symmetry species of D6h. The weights add to 1, which is the check that the projectors resolve the whole of it: the largest is E2g at 35.0 per cent, and the totally symmetric part — 3.4 per cent here — is the part that changes every distance and no symmetry at all.

How far, and along which coordinate

A continuous symmetry measure returns one number: how far a structure is from a shape. Projecting the same displacement onto the twelve symmetry species of benzene's group turns it into a list that sums to 1.000000000000, says which coordinates the structure left along, and shows that the totally symmetric part — 3.4 per cent of this one — moves every atom by 0.0200 ångström and lowers the symmetry measure by nothing at all.

symmetry · Point group The same neighbours, and a fifth of the binding between them. five structures in which every site has 4 neighbours. Their second moments are identical — 4 for every one, which is the coordination and is what the band width is read from. Their bindings per site are not: they run from 1.28 to 1.64, and the least bound is the one with the most four-step walks.

Two structures with the same neighbours

Five structures in which every atom has exactly four neighbours. Their second moments are 4.000 to nine decimal places, because that identity is the coordination and nothing else. Their bindings per site run from 1.2756 to 1.6363 — a spread of twenty-two per cent — and two of them that agree on the second, third and fourth moments together still differ in the third decimal place.

solids · Cohesion Ten answers, and none of them is another one turned round. Every localised description the search found for a twelve-vertex cage, placed by the value of the functional it maximises. There are 10 of them, spanning 0.02, and the two closest differ by 0 — far more than the 10⁻¹³ the sweep converges to, so they are different maxima rather than one maximum reached to different precision. Each has its own multiset of participation numbers, and a symmetry of the cage permutes sites without changing that multiset, so no two of these are related by one. The controls above find one answer each.

How many descriptions a cage has

A localisation is a maximisation, and running it once reports the maximum it reached rather than the maximum there is. Run to exhaustion on a twelve-vertex borane it finds ten answers and then three batches of twelve starts in a row that find nothing new — and none of the ten is another one seen from a different side, because a symmetry of the cage cannot change a multiset of participation numbers and all ten multisets differ.

beyond · Multicentre The 6-ring at 111.5°: the alternating form and what a search finds. The alternating ring — every atom displaced above or below the plane in turn — with its dihedral angles and its two strain terms, beside the lowest-torsion member a search constrained only by bond angles and closure returns. Both satisfy every geometric constraint; only one of them is staggered.

The explanation with the wrong sign

Two methyl groups on one carbon make a ring easier to close, by up to eleven thousand-fold, and the textbook reason is that they compress the ring's internal angle. Computed from a standard bending term, that compression helps a three-ring and a four-ring and hinders every ring from five up — predicting a slowing of 0.754-fold for the five-ring measured to speed up 250-fold. The account with the right sign is about rotations rather than angles, and it has a ceiling of 36.5 that the measurement is already above.

shape · Strain

Named alongside it

The objects these essays reach for when they reach for this one.

Model limitBond angleConformationConventionDistortionEquivalent atomsGraphLocal minimumMinimisationTorsionApproximationBand width

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