Which rings close a shell

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

Six 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.

Where two-centre bonding stops

Rings of four, six, eight and ten carbons, each filled and asked whether its shell closed. Every orbital in every row is spread over the whole ring — none of them belongs to a bond — and the level pattern is what decides the answer.

The two-dimensional version: every ring filled and asked, with the level pattern of one orbital and then degenerate pairs shown to the left of each row. The shell-closure logic behind Wade’s rules is this pattern in three dimensions rather than two.

Aromaticity as a computed shell closure

Every ring from three to eight, filled with its own pi electrons and asked whether its highest occupied shell came out full. The dots to the left of each row are the level pattern — one orbital, then degenerate pairs — and the pattern is where the rule comes from. The function producing this does not know the phrase “4n+2”; it fills and reports.

The ring with a twist in it

Every twisted ring from three to eight, filled with its own number of π electrons and asked whether the highest occupied shell is full. Four and eight close; three, five, six and seven do not. The flat version of this table closes at six and nowhere else in the same range.

The same ring, three charges

Six rings from three-membered to eight-membered, each filled with the same six π electrons. Only the ones whose highest occupied shell comes out full are closed, and the closure is computed rather than looked up.

The ring's levels are its group's characters

The levels of six rings as the eigensolver returns them, with the shell closures marked. This is the picture the calculation gives; the argument above says it could have been drawn from the group without running the calculation at all.

A stabilisation is measured from somewhere

The shell closures themselves, which neither reference affects: each ring filled with its own electrons and asked whether the highest occupied shell came out full. This is the part of the story that is a property of the level pattern and not of any subtraction, and it is the part that survives changing the reference.