water: 4 valence bands

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.

Three 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.

What a photoelectron spectrum measures

Methane’s valence photoelectron spectrum: two bands, at 12.7 and 23.0 electronvolts, each carrying the symmetry species the reduction below produces. The species and the three-to-one intensity ratio are computed from four hydrogen positions with no energy calculation of any kind; the two energies are measured. A band assigned to a species the valence basis does not span would stop this figure being drawn, which is what separates an assignment from a caption.

Ammonia’s valence bands with their species. The valence basis — nitrogen’s s and p functions and one s on each hydrogen — spans 3a₁ ⊕ 2e in C₃ᵥ, and every band shown carries one of those labels. A band assigned to a species the reduction does not produce would stop this figure being drawn, which is what makes this an assignment rather than a caption.

Water's lone pairs are not a pair

Water’s four valence photoelectron bands with the symmetry species of each, beside the species its valence basis spans — oxygen’s s and p functions and one s on each hydrogen, reduced in C₂ᵥ. Four bands, four different labels, and no two of them the same. A band carrying a species the reduction does not produce would stop this figure from being drawn.

Ammonia for comparison, with its species computed the same way. Three valence bands, one of them doubly degenerate — and the degeneracy is available here because C₃ᵥ has a two-dimensional representation, which C₂ᵥ does not. Where water can have no degenerate orbital at all, ammonia must have a degenerate pair of bonding orbitals.

Fewer bands than electrons

Ammonia’s three valence bands with the species each belongs to. Four occupied valence orbitals — one of them a degenerate pair — falling into three species, and the measured ionisation energies beside them. The species are computed from the group; the energies are measured and are marked as measured.

Two shells side by side, which is the comparison the essay’s count rests on. The number of bands is the number of symmetry species the valence functions span, and the number of electrons is twice the number of occupied orbitals — the two are different counts of the same set, and only the first is what a spectrum resolves.

Methane’s two bands. The lower-energy one carries six electrons and the upper two, so the intensity ratio should be about three to one — which it is, and which is the observation that fixed the assignment long before anybody could compute the energies.