Molecular Geometry & Bonding
what the shape decides
Figure

phosphorus pentafluoride: 3 environments

The atoms of phosphorus pentafluoride sorted into orbits under its own symmetry group — two atoms are in the same orbit when an operation of the group carries one onto the other, which makes them indistinguishable by any measurement. There are 2 of F, 1 of P, and a spectrum that resolves environments counts those rather than atoms.
phosphorus pentafluoride: 3 environments. The atoms of phosphorus pentafluoride sorted into orbits under its own symmetry group — two atoms are in the same orbit when an operation of the group carries one onto the other, which makes them indistinguishable by any measurement. There are 2 of F, 1 of P, and a spectrum that resolves environments counts those rather than atoms.

One of the figures on spectra: How many bands there can be, where they sit, and what an absent one proves.

Two 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

A spectrum counts environments, not atoms

Phosphorus pentafluoride’s atoms sorted into orbits. The two axial fluorines are carried into each other by the horizontal mirror plane and are one orbit; the three equatorial fluorines are carried into each other by the threefold axis and are another; the phosphorus is alone. Three environments in a molecule with six atoms, and the two fluorine environments have different sizes, so they cannot be exchanged by anything.

Benzene, where the count goes the other way. Twelve atoms and two environments: all six carbons are carried into one another by the sixfold axis, and so are all six hydrogens. The orbits are of size six with stabilisers of order four, and six times four is the group’s twenty-four. One carbon signal and one hydrogen signal, whatever the instrument.

Ethene’s six atoms in two orbits. The four hydrogens are one orbit of size four with stabilisers of order two, against a group of order eight. Whether a given pair of them is cis or trans is not a distinction the group makes, which is why it is not a distinction a spectrum makes either.

A ratio that squares what it measures

The equivalent atoms of ammonia, which is what a distortion destroys. The three hydrogens are one orbit of the point group, and stretching one of them is exactly the operation that breaks the orbit and lets a forbidden derivative appear.

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