The gap against size: uniform against δ = 0.15

One of the figures on extended structures: Chains and rings taken far enough to behave like solids — bands, gaps, fillings, defects and ends — every one of them a finite matrix diagonalised, with no lattice anywhere in the argument.

Seven 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.

The width of a band is a count of neighbours

Two families of chains, one uniform and one with its bonds alternating by fifteen per cent, with the gap above the filled levels plotted against length. The lower family is heading for zero and the upper family settles at four times the alternation. Every chain here has essentially the same mean coordination, so the second moment is the same across the whole figure and would have predicted neither curve.

Where a molecule stops being one

The two sequences on one pair of logarithmic axes: a uniform chain’s gap and an alternating chain’s, at fifteen per cent alternation. The lower one is a straight line heading down with no end — the same slope of −1 the rates figure above fitted — and the upper one flattens onto four times the alternation. A single computed point on either curve would have said nothing about which of the two behaviours it belonged to, and quoting one is the commonest way a cluster calculation is misread.

The gap is not the band width

Two families of chains with the same coordination throughout, and therefore essentially the same band width. Their gaps are entirely different: the uniform chains’ falls without limit and the alternating chains’ settles at four times the alternation. Nothing about the width predicted either curve.

The same pair of sequences at a smaller alternation. The alternating gap settles at four times the distortion — a smaller number than before — and the width of the band has not moved at all, because the width is a count of neighbours and the gap is not. Two quantities, one of which depends on the alternation and one of which does not.

And at three times the alternation. The gap has tripled with it while the width is unchanged, so the ratio between them is not a property of the material — which is the whole objection to reading one off the other.

A half-filled band is not always a metal

Two ways a gap can open in this model — not at all, or by the structure distorting. A third way exists and is absent from both curves: the electrons declining to move past one another. That third gap is the same size whatever the chain does, because it is set by an atomic quantity rather than by a structural one, and no sequence computed here would reveal it.

A band gap is not a bond energy

The HOMO–LUMO gap of two families of chains against their length. One family’s gap heads to zero and the other’s settles on a finite value. Both are “the gap” and only one of them has a band gap to converge on, which is the first way the four quantities part company.

The gap at a small alternation, against the length. Both quantities the essay is about live on this figure: the gap is the separation between two adjacent levels somewhere in the middle of a column, and the bond energy is roughly the drop from the atomic level to the bottom of the column, doubled. One of them settles as the chain grows and the other is proportional to it.

The same measurement at six times the alternation. The gap settles at four times the alternation as before, so the number it settles on has moved by a factor of six and nothing about the binding has moved at all — which is as sharp a demonstration as the model affords that the two quantities are not related by a constant.

The third way to be an insulator

The first kind, as a limit: an alternating chain’s gap survives as the chain grows and a uniform chain’s does not. This is the whole of the band-theory criterion, and it is a statement about two sequences.

The distortion the ends decide

The gap of an alternating chain at the distortion used throughout, against length. It settles at four times δ and is the same for both phases, so it plays no part in the comparison above — which is unusual among Peierls arguments and is the reason the end energy comes out as a clean constant.