Molecular Geometry & Bonding
what the shape decides
Concept

Second moment — where it appears

The mean of the squared level energies of a structure, equal to its average coordination. Both count closed walks of length two, so the width of a band is a count of neighbours rather than an energy.

Named by 10 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 The same neighbours, and a fifth of the binding between them. five structures in which every site has 4 neighbours. Their second moments are identical — 4 for every one, which is the coordination and is what the band width is read from. Their bindings per site are not: they run from 1.28 to 1.64, and the least bound is the one with the most four-step walks.

Two structures with the same neighbours

Five structures in which every atom has exactly four neighbours. Their second moments are 4.000 to nine decimal places, because that identity is the coordination and nothing else. Their bindings per site run from 1.2756 to 1.6363 — a spread of twenty-two per cent — and two of them that agree on the second, third and fourth moments together still differ in the third decimal place.

solids · Cohesion 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 How anisotropic a structure has to be. The fourth moment of a layered band, divided by the square of its second so that the width drops out, against the coupling between layers. The curve is the closed form that follows from adding cumulants — 3 − 3(2 + λ⁴)/(2(2 + λ²)²) — with nothing fitted; the points are eigenvalues. At λ = 1 it is the three-dimensional value 2.5 and at λ = 0 the two-dimensional 2.25, and it is halfway between them at λ = 0.47 — so the question of when a band is effectively two-dimensional has a number rather than an opinion.

Two bands, and the shape of each

A band's shape has a closed form when the structure it belongs to factorises. Put a second band beside it and the form survives exactly while the two do not talk, and fails as the square of the coupling once they do. Make one direction weaker instead and the form survives everywhere — 3 − 3(2 + λ⁴)/(2(2 + λ²)²), agreeing with eigenvalues to nine decimal places — so how anisotropic a structure has to be before its band is two-dimensional is a number: λ = 0.46518.

solids · Bands in a solid Where the two halves of an alloy come apart. The gap at the centre of a binary alloy's band against the contrast between its two components, for a chain of 2048 sites arranged three ways. The ordered arrangement's gap is exactly twice the contrast and opens at once; the segregated one's is exactly the contrast less the band width and opens at 2; and the random one, which has no closed form, opens a little before the segregated one and stays a little wider. Below the openings the curves sit at one level spacing rather than at zero, which is what a finite chain has instead of a gap.

Two bands, if the chain is short enough

Take a chain, raise half its sites and lower the other half, and ask when the band comes apart into two. The ordered arrangement splits at once, the segregated one at a contrast equal to the band width, and the random one splits earlier than either — and then closes again as the chain is made longer, because a long chain contains a long run of like atoms and a long run is a narrow sub-band.

solids · Defect 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 Two curves that cross, and two that do not. Every pair of parameters that reproduces one measured number, for three numbers, with the true system marked at Δ = 12 and t⊥ = 0.06. The lower band's shape is a function of the ratio of the two, so its curve is a straight line through the origin; the excess gap is a function of the coupling squared over the separation, so its curve bends. The two cross at one point. The upper band's shape draws a line almost on top of the first, because it is a function of the same ratio — a second measurement lying along the first fixes nothing the first had not already fixed.

Two ways of being second order

A band's shape and a band's gap are both second order in the coupling that mixes two bands, which sounds like a reason to measure only one of them. They are second order in different ways — one goes as the square of the ratio and the other as the square over the separation — and that single difference of one power is what turns a curve of possible answers into a point.

solids · Bands in a solid Two bands, square below, triangular above. The 128 levels of a structure carrying two orbitals on every one of its 64 sites, where the lower orbitals hop on a square net and the upper ones on a triangular net, separated by 12 and coupled at 0.06. The two bands have different widths and different shapes before anything mixes, which is the situation a single net cannot construct. Every level here is also reproduced by a two-by-two problem, one per mode, to 3e-13.

The constant that belonged to one net

Divide one second-order departure by another and the coupling cancels, leaving one constant times the gap — which reads as arithmetic. It was arithmetic about a square net. On a triangular one — the same graph for both bands, nothing else changed — the quotient drifts by a hundred and twenty-eight per cent.

solids · Bands in a solid How many places a local search can stop. The number of distinct arrangements a steepest-ascent search settles at, for each net and contrast, with the fraction of starts reaching the best of them written beside it. 16 sites, contrast 1: 2 from 200 starts, best reached 67 per cent of the time; 16 sites, contrast 4: 2 from 200 starts, best reached 59 per cent of the time; 36 sites, contrast 1: 4 from 20 starts, best reached 80 per cent of the time; 36 sites, contrast 4: 12 from 20 starts, best reached 10 per cent of the time. The larger net at the larger contrast is a different kind of landscape.

Twelve basins where there were two

Sixteen sites can be searched exhaustively and thirty-six cannot, so the only instrument available past about twenty sites is a local search — and how much weaker it is has never been measured against the answer on a net where both can be run. On sixteen it is barely weaker at all. On thirty-six it fails nine starts in ten, and the arrangement it is beaten by has fewer unlike bonds than the one it starts from.

solids · Cohesion

Named alongside it

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

Tight-binding modelsBand widthDensity of statesClosed formEnergy per siteCohesionCoordinationGraphBand fillingExact diagonalisationModel limitThermodynamic limit

All concepts