Molecular Geometry & Bonding
what the shape decides
Figure

The angles each arrangement gives

Every distinct angle subtended at the centre, for each number of sites, measured off the minimised arrangement. Five is the case with three distinct angles, which is the arithmetic signature of sites that are not all equivalent.
The angles each arrangement gives. Every distinct angle subtended at the centre, for each number of sites, measured off the minimised arrangement. Five is the case with three distinct angles, which is the arithmetic signature of sites that are not all equivalent.

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

VSEPR, computed

Every distinct angle each arrangement produces, measured off the minimised configuration rather than quoted. The rows are the familiar shapes and none of them was assumed.

Five sites are not alike

The distinct angles each arrangement produces. Every row has one or two — except five, which has three. That is not a quirk of the drawing; it is the signature of positions that are not all the same.

The two counts either side of the boundary, with their energies. Phosphorus pentafluoride’s axial and equatorial fluorines sit at measurably different bond lengths and the arrangement is exactly D₃ₕ; the octahedral case has one kind of site and no such distinction to make. The structure is not in doubt — what is in doubt is whether a room-temperature spectrum can see the difference.

Four, five and six sites with their distinct angles side by side. Four gives one angle and six gives two, and every site in each is equivalent to every other; five gives three angles and its sites are not all alike. The anomaly is in the middle of the sequence rather than at either end of it.

The shapes above six coordination

The same table as the low-coordination one, with two columns added: each arrangement’s computed repulsion energy beside the published Thomson minimum. The rightmost column is the one that grows, and the growth is what makes these shapes hard to name.

The sites are not the same size

The distinct angles of the five-, six- and seven-site minima, with each arrangement’s energy checked against the published Thomson value. Six has one kind of site and is immune to the whole argument above; seven has three kinds and a very flat valley between arrangements, so every term in the competition matters and none of the answers is pinned by symmetry.

Two systems a model cannot tell apart

What VSEPR does compute, and computes exactly: the angles a repulsion minimum has for each count of pairs. The counts are right and the model’s resolution is the counts.

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