Strong outside, weak inside

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.

Hypervalency does not stop at three centres

The bond orders along each chain. The three-centre case has two equal bonds of 0.707 — not the one half the electron count suggests. Every longer chain alternates, strong at the ends and weak in the middle, and the spread grows with the chain.

The electron population on each centre. The odd-numbered sites carry more than one and the even-numbered ones carry exactly one, at every chain length. The three-centre case is the half-charge on each end that every account of hypervalency quotes, and it is the first entry in a series.

Measured iodine–iodine distances against the computed bond orders. Ordinary I₂ has one bond at 2.67 Å. The three-centre anion has two equal bonds at 2.90, far longer than a full bond because each is worth about seven tenths of one. The five-centre anion is 2.81, 3.17, 3.17, 2.81 — short outside, long inside.

One scale, from two centres to a cage

The hypervalent chains, whose pairs are fewer than their bonds. Their bond orders are known to alternate strong–weak–weak–strong; the localisation of the same systems says the corresponding thing about the orbitals, which is that each pair is not quite a two-centre bond and reaches into the next bond by a measurable amount.

The answer a search is most likely to give

Where the multicentre argument starts: a bond spread over more centres than two, computed rather than assumed. Everything about the multicentre picture survives; what does not survive is the idea that a molecule has one of them.