Five sites are not five of a kind

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.

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.

Five sites are not alike

The five-point minimum with the two axial sites marked apart from the three equatorial ones. Their neighbour angles differ, which is what makes them different places rather than five of a kind.

Two models that disagree about the shape

Three arrangements of five ligands, with each ligand’s charge from the σ model and the Coulomb repulsion of the same five directions. The one with a spread of zero is the one with the highest repulsion.

The Berry path with both quantities scaled to their own range. One model sees a coordinate that is nearly flat; the other sees the ligands’ charges being redistributed by half.

Where the two kinds of ligand come from: five sites minimised on a sphere are not five of a kind. Two of them end up axial and three equatorial, related to each other by no operation and to their own kind by all of them — which is the same partition the molecule’s own symmetry operations produce, arrived at from a repulsion rather than from a group.