One carbon, three lithiums, and no way to make it two

One of the figures on hückel systems: Adjacency matrices diagonalised: levels, coefficients, bond orders, and the shell closures that decide which rings are stable.

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.

Four centres, and the pair that will not localise

The level pattern of the tetramer’s framework, solved as a graph. Eight atoms, eight electrons, four levels filled — and eighteen contacts between them. The shortfall is the count, and everything below is the same statement made about the orbitals rather than about the number.

Two separate ethenes and the tetramer, localised by the same sweep. The ethenes give orbitals on exactly two atoms with a participation number of 2.000; the tetramer’s sit on four. The difference is the molecule, not the method.

The four localised orbitals, with the share of each pair on each of the eight framework atoms. Each sits on one carbon and three metals, and the three per cent on the fourth metal is what a genuinely four-centre orbital leaves behind.

One scale, from two centres to a cage

The tetramer’s four capped faces, which is the picture the tetramer localisation produces: one carbon and the three lithiums of its face, four times over. The count of four atoms comes from this picture; the participation number of 2.54 says that most of each pair is on the carbon, which the picture does not say and cannot.

The answer a search is most likely to give

What one of the descriptions looks like — the picture a paper would print. Several of the fourteen would be described in the same words, which is why the disagreement between them has not been noticed.