Molecular Geometry & Bonding
what the shape decides
Figure

The projectors, as matrices

The projection operator onto each representation benzene's 6 pz functions actually span. Every one of them squares to itself to better than 10⁻¹², its trace is the dimension of the subspace it projects onto, and the whole set adds up to the identity — so nothing in the space is missed and nothing is counted twice. Blue is positive and amber negative; the area of each square is the size of the entry.
The projectors, as matrices. The projection operator onto each representation benzene's 6 pz functions actually span. Every one of them squares to itself to better than 10⁻¹², its trace is the dimension of the subspace it projects onto, and the whole set adds up to the identity — so nothing in the space is missed and nothing is counted twice. Blue is positive and amber negative; the area of each square is the size of the entry.

One of the figures on representations: Character tables generated from a molecule's own operations, bases reduced in them, and what a descent in symmetry does to a level.

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

The projector is unique, the basis is not

The four projectors that are not zero for benzene’s π functions, drawn as their matrices. Each square’s area is the size of the entry and its colour is the sign. The one-dimensional projectors are simple — every entry ±1/6 — and the two-dimensional ones are not; every one of them squares to itself, and the four add to the identity.

The two results, drawn as coefficients round the ring: circle area is the size of the coefficient and colour is its sign. Both are E1g functions; both are outputs of the same operator; they differ by a rotation within the subspace that nothing in the symmetry chooses.

The same construction on a smaller group. Boron trifluoride’s three fluorine σ functions span a₁′ ⊕ e′ in D₃ₕ, so one projector returns a line and the other a plane — and the two matrices are built by the same sum over operations, from the same characters, with nothing chosen anywhere in the arithmetic.

How far, and along which coordinate

The projectors themselves, applied to a different space: benzene’s six π orbitals resolved into symmetry-adapted combinations. The same operators, the same group, and a different vector space — which is the whole reason projection is worth building once.

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