Molecular Geometry & Bonding
what the shape decides
Figure

The density of states of a chain of 2000

The 2000 levels of a linear chain, binned into 34 intervals across the band, with the closed-form density drawn through them. The density piles up at both edges because that is where the level spacing turns over, and nothing periodic was assumed to get it.
The density of states of a chain of 2000. The 2000 levels of a linear chain, binned into 34 intervals across the band, with the closed-form density drawn through them. The density piles up at both edges because that is where the level spacing turns over, and nothing periodic was assumed to get it.

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.

Four 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.

In the essays

A density of states is not a spectrum

The density of states of a chain of two thousand: how many levels lie in each interval of energy, binned from the computed eigenvalues, with the closed form drawn through them. Everything in this figure is a property of one set of levels. Nothing in it says which of those levels an electron could be moved between, or how strongly.

The same density of states at twice the length, which changes nothing about it. That is the first thing a density of states is not: it is not a property of a particular sample, and doubling the number of atoms leaves the shape where it was. A spectrum of a particular sample is not like that.

And the same chain binned more coarsely. The divergences at the edges survive the coarsening and the shape between them is unchanged, so nothing in the picture depends on the binning — but the height of every bar does, which is worth noticing before any of it is read as an intensity.

One defect is a level, many are a band

The host band’s own density of states, for scale: the levels the impurities are being pulled out of, and the edges they close on. Everything in this essay happens below the left-hand edge of this figure.

A band becomes a bell curve

A chain’s own density of states, computed by diagonalisation. Adding second and third neighbours to it keeps everything a function of one variable — the wavevector, though this collection declines to use the word — so no matter how many neighbours are added the spectrum stays a one-dimensional object, and its shape is free to go anywhere.

Two bands, if the chain is short enough

What the density of states of a pure chain looks like when nothing is alloyed into it — the shape every one of the alloy’s sub-bands is a squeezed copy of. The two spikes at the edges are the reason a small contrast changes the shape so much: there are many states near the edges to push about.

Every figure · Every orbital, by what it encloses · All essays