Molecular Geometry & Bonding
what the shape decides
Concept

Band filling — where it appears

How many of a band's levels are occupied, fixed for a structure of one orbital per site by the electrons each site brings. It decides whether the structure conducts and which distortion, if any, it is unstable to.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

Five sixths of the bonds, nine tenths of the binding. What a site in the outer layer of an open block keeps, measured three ways: the fraction of its bonds, the fraction the second-moment rule predicts of its binding, and the fraction the calculation gives. The last two agree and the first does not.

A surface is not a count of broken bonds

Cut a crystal and every atom in the new face has lost one of its six neighbours. The standard estimate follows immediately: a surface costs one sixth of the cohesive energy per atom exposed. Computed, it costs a little over half that — the atom keeps 91.2 per cent of its binding while keeping only 83.3 per cent of its bonds, because the bonds that survive get stronger when their competitors are removed.

wrong · Cohesion One band width, three shapes. three densities of states, each computed from a wrapped structure of 4,096 levels and drawn against the band scaled to run from −1 to +1. A chain piles its states into the two edges; a cubic structure piles them into the middle and thins to nothing at the edges.

Where the states pile up

Two bands of the same width can be entirely different objects. Scale a chain, a square net and a cubic structure to one width and what is left varies by a factor of four — a chain puts more than half its levels in the outer thirds of its band and a cubic structure puts more than half in the middle third — and that difference alone decides how strongly each of them binds.

solids · Bands in a solid Where two bands lie, as their centres are pulled apart. The σ band and the π band of a two-orbital chain, drawn as the intervals they occupy, against the difference in site energy between the two orbitals. Below a difference of 3 the two intervals overlap and the filled-band count stops deciding anything.

A full band is not an insulator

Two electrons per atom, two orbitals per atom, and the lower set exactly full: the count says insulator, and magnesium is a metal. The count is not wrong about the count. What it assumes is that the two sets of levels occupy separate ranges of energy, and whether they do is a comparison of four numbers that has nothing to do with how many electrons there are.

wrong · Metal Four fillings, four periods. four fillings of a ring of 120, and for each of them what a distortion of every available period is worth. Every period pays the same elastic cost, so the bars compare what the electrons give back and nothing else. The winner is one over the filling in every case.

The distortion the filling chooses

A half-filled chain of equal bonds is unstable and alternates — long, short, long, short. That is the case everyone is shown, and it is one case. Fill the chain a third of the way instead and the alternation is worthless: what wins is a pattern that repeats every three bonds, and the period is one over the filling at every filling tried.

solids · Peierls distortion A band becomes a bell curve, and the dimension is the sample size. The normalised fourth moment of a wrapped hypercubic structure's density of states against its dimension, with the closed form 3 − 3/2d drawn through it. They agree to eight decimal places at every dimension from one to 6, and the reason is a limit theorem: a hypercubic spectrum is the sum of d independent one-dimensional ones, so its cumulants fall as powers of d exactly as a sum of independent samples does. The Gaussian value of three is approached and never reached.

A band becomes a bell curve

The normalised fourth moment of a hypercubic structure's density of states is 3 − 3/2d exactly, from one dimension to six, to eight decimal places — because the spectrum is a sum of d independent one-dimensional ones and the central limit theorem is what it is obeying. Reach further in one dimension instead and the shape overshoots a Gaussian rather than approaching it: a chain touching its third neighbour has a cubic structure's coordination and a fourth moment of 3.389 against 2.500.

solids · Bands in a solid Every arrangement, and the winner is not the one with the most unlike bonds. All 1820 ways of raising 4 of 16 sites on a wrapped square net, at a contrast of 4, each placed by its count of unlike bonds against the binding it gives. The best arrangement has 12 unlike bonds where 16 is available, and it binds at 1.103953 against 1.080031 for the best of those that do have the most. The count and the spectrum are two different orderings.

The arrangement a count cannot pick

Three arrangements of one composition came out ordered by their count of unlike bonds, which looked like a rule. Enumerating every arrangement instead of three shows it is not one: it holds at every composition on a square net at a small contrast, fails at four of them at a large contrast, and fails on a triangular net at any contrast at all.

solids · Cohesion The carriers a distortion was hiding. A half-filled ring of 40 — one of the 4m rings that carry exactly one pair of carriers at every temperature — allowed to distort. Cold, it alternates by 0.1232, opens a gap of 0.4927 and carries 3.6e-15 carriers rather than one pair. The alternation is undone continuously at kT = 0.1358, and the carrier count comes back as it goes.

The carriers a distortion was hiding

A half-filled ring of 4m carries exactly one pair of thermal carriers at every temperature, which is true only of a ring held rigid. Allowed to move, it does not carry them: it alternates, opens a gap of 0.4927, and carries none at all until a temperature that undoes the distortion.

wrong · Metal One net has a two-colouring and the other cannot. Sixteen sites wrapped into a square net and into a triangular one, with the wrapping bonds left undrawn. The square net is bipartite: its sites split into two classes with every bond running between them. The triangular net is not, and the obstruction is a triangle — three sites in a cycle of odd length cannot be two-coloured. It has thirty-two of them, two per site, and forty-eight bonds against the square net's thirty-two.

A net with no two-colouring

Half filling is the easiest composition to search and the reason given was the two-colouring: on a bipartite net it is the unique arrangement with every bond unlike, so the optimum has nothing competing with it. A triangular net has no two-colouring, its half-filled composition is reached by every one of two hundred starts, and its best arrangement is two bonds short of what counting allows.

solids · Cohesion On the frustrated net the widest gap wins everywhere but just past a change of winner. Every basin found at half filling on the triangular net, at eleven contrasts, placed by the gap it opens at the Fermi level; the winning basin is filled and the rest open, sized by how many of two hundred starts reach them. Up to a contrast of 2.5 there is one basin. From 3 a second appears, and the winner is the one with the wider gap — except at 4, where the winner has a gap of 4.947 and a runner-up has 5.088. Squares mark basins with thirty-two unlike bonds and circles thirty.

The gap follows the winner late

On a triangular net at half filling, the arrangement that binds best was expected to be the one that opens the widest gap at the Fermi level. Across twenty cases it is, eighteen times. The two exceptions are not noise: the frustrated net's winner changes at a contrast of 3.790, from an arrangement with thirty unlike bonds to one with thirty-two, and the gaps of the two do not cross until 4.849. For a whole unit of contrast the better binder has the narrower gap.

solids · Cohesion

Named alongside it

The objects these essays reach for when they reach for this one.

Tight-binding modelsOne-electron modelsBand widthCohesionModel limitSecond momentBand gapClosed formDensity of statesGraphLocal minimumCoordination

All concepts