Molecular Geometry & Bonding
what the shape decides
Concept

Fluxionality — where it appears

The rapid interconversion of a molecule between equivalent structures. What a spectrum then sees is an average over the rearrangement, so the rigid point group of any one structure is the wrong group to use.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Five sites are not five of a kind. The minimised arrangement of five points, with the two axial sites marked apart from the three equatorial ones. Their neighbour angles differ, so the two kinds of position are genuinely different places — which the shape's name does not convey.

Five sites are not alike

Every other common arrangement has one or two distinct angles. Five has three, because two of its positions are on an axis and three are round an equator — and a molecule built that way does something about it.

shape · VSEPR Three counts for each molecule, and the barrier between them. Four molecules, with the number of operations available to each: the point group, which is what a rigid rotation or reflection can do; the feasible group, which adds whatever a crossable barrier makes available; and the whole permutation-inversion group, which is every rearrangement of identical nuclei whether a molecule can reach it or not. The middle column is the one that describes a real spectrum, and it equals the first only when nothing moves.

The group of a molecule that will not hold still

A point group is a group of rotations of space, and it presupposes a structure for them to act on. Ethane has no one structure — its methyl groups turn billions of times a second — and the group that describes its spectrum has thirty-six elements where the point group has twelve, out of two thousand eight hundred and eighty conceivable.

symmetry · Point group The pucker is the flat ring's softest motion by a factor of a hundred. Every internal curvature of the flat pentagonal bipyramid under a 1/r repulsion, on a logarithmic axis: the eleven eigenvalues of the energy's second derivatives once the three rigid rotations are removed. The pair at the far left is the ring's pucker, a degenerate pair; the next motion is a hundred times stiffer with equal bonds. IF₇'s longer ring bonds make the pucker three times stiffer and leave it still thirty-four times softer than anything else.

The ring held flat by its long bonds

Given iodine heptafluoride's own composition — five long bonds and two short — the repulsion model puts the short pair on the axis, as the molecule does, and keeps the equatorial ring flat. But the flatness is the softest thing in the arrangement by a factor of a hundred. With equal bonds, a pucker of the ring costs nothing at all to second order under an inverse-square law and lowers the energy under any steeper one. What holds IF₇'s ring flat is the four per cent by which its ring bonds are longer, and that holds only under laws softer than about the 4.4th power.

shape · VSEPR A few thousandths of the energy in a contact term puckers IF₇'s ring. For each contact law added to the Coulomb repulsion of IF₇'s bond domains, the share of the Coulomb energy the contact term must carry before the flat ring stops being a minimum, on a logarithmic axis. Steep powers need very little: 2.8 per cent at 1/r⁶, 0.16 per cent at 1/r¹². An exponential contact needs 0.09 to 2.5 per cent while its range is short — and from a range of 0.35 Å it needs an infinite amount, because it holds the ring flatter instead. So does 1/r⁴. Fluorine's own contact — the square of its 2p overlap, with Slater's exponent for the atom and for the anion — needs 0.11 and 0.16 per cent. Open circles mark the share needed with equal bond lengths, for comparison.

The fluorines' range decides the ring

IF₇'s flat ring was held by its bond lengths under every repulsion softer than about 1/r^4.4, with the fluorines left out. Put them back as a second set of repelling points and under the same law they change nothing at all — they sit on the same rays, and the model cannot tell where along a ray a point sits. They matter only as a contact repulsion of a different shape. Then a steep power puckers the ring with a few thousandths of the energy. An exponential contact puckers it only if its range is under 0.346 ångström, and a longer one holds it flatter. Fluorine's own orbital gives a range near 0.16. And the ring that puckers does not settle: its envelope and its twist differ by a ten-thousandth of the pucker's depth, so the pucker runs round the ring.

shape · VSEPR Xenon hexafluoride's fluorines move its lone pair off the axis and through a face. At a lone-pair weight of 1.5, the energies of the capped octahedron and the edge-capped octahedron relative to the pentagonal pyramid, against the six fluorines' Born–Mayer contact repulsion as a share of the domain energy; filled dots are the lowest structure found from any nudged start. With no contact the pyramid, lone pair on the axis, is lowest by 0.063. The capped octahedron reaches it at a share of 2.3% and is the lowest from 3 per cent; in between the lowest structure has left the axis and keeps only a mirror plane.

The lone pair goes through a face

Xenon hexafluoride is the other famously floppy seven-domain molecule, and its seventh domain is a lone pair: a domain with no atom at its end, which a contact repulsion between fluorines cannot see. With the domains alone the model puts the lone pair on the axis of a pentagonal bipyramid at every lone-pair weight — a pentagonal pyramid of fluorines. Add the six fluorines' contact repulsion and, past a share of the energy that rises with the lone pair's weight, the lowest structure becomes a distorted octahedron with the lone pair through one face. Every contact law tried does this. On the way the lone pair leaves the axis continuously, and past it the molecule's floppiness is the lone pair hopping from face to face over an edge — not a ring's pucker moving round a ring.

shape · VSEPR

Named alongside it

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

MinimisationModel limitRepulsionVSEPRAxialBond lengthCoordination numberDegeneracyEquatorialPoint groupClosed-shell configurationsConjugacy class

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