Molecular Geometry & Bonding
what the shape decides
Concept

Energy per site — where it appears

The binding energy of an extended structure divided by the number of sites. It is the quantity that has a limit as the structure grows, where the total does not, and it is what a cohesive energy is compared against.

Named by 11 essays across 2 fields — each of them below, with the objects they name alongside it.

How long a chain has to be before its ends stop mattering. The difference in energy per site between a ring and a chain of the same length, against that length. It falls as one over the length, which is what it means for the difference to be an end effect, and the size at which it drops below a thousandth of a β is printed.

Where a molecule stops being one

There is no size at which a molecule becomes a solid, and the useful question is a different one — how large must it be before a given property has stopped changing? The answers differ by a factor of several hundred between one property and the next, and every one of them is a measurement.

solids · Bands in a solid ⟨x²⟩ is the average number of neighbours. For each structure, the mean coordination counted off the edge list beside the mean of x² measured off the eigenvalues. They are equal by an identity about graphs, not by a limit — and the binding each bond supplies, in the last column, obeys no such rule and falls as neighbours are added.

The bond that weakens as neighbours multiply

Wrapped structures with two, four and six neighbours per site give a mean of x² of exactly 2, 4 and 6. Their binding per site goes 1.272, 1.611, 1.979 — and their binding per bond falls from 0.636 to 0.330, which is why a metal atom with twelve neighbours has weak bonds and a great many of them.

solids · Cohesion 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 A mixture is not the average of its ends. The binding per site of a square net whose sites are of two kinds, against how many of each. The straight line is arithmetic rather than a fit: a structure of one kind only has every level shifted by ±δ, so the two ends and the line between them are known before anything is diagonalised. Every mixture lies above it — more bound — by as much as 0.43 per site at the middle, and that departure is the whole of what makes an ordered compound worth forming.

A mixture is not the average of its ends

Half of a structure's sites raised and half lowered, and the line between the two pure ends is arithmetic — no diagonalisation needed. Every mixture lies below it, by 0.42987 per site for the ordered arrangement and 0.23690 for the segregated one at the same composition, so composition fixes neither the binding nor the gap. And the departure is not quadratic in the contrast: it grows as its 1.79 power in a chain and its 1.28 power on a square net.

solids · Cohesion The whole difference lives at the two ends. The energy difference between the two dimerisations of an open chain, held at the same distortion, multiplied by the number of sites. It settles on a constant — 1.09 in units of the hopping — so the difference per site falls as one over the length, with a fitted exponent of -1. An end is a bond that is not there, and it is worth the same amount whatever it is attached to.

The distortion the ends decide

A chain of an even number of sites has an odd number of bonds, so its two dimerisations are different molecules rather than one molecule translated. Held at the same distortion they differ by 1.08715 in units of the hopping, whatever the length — a fixed amount of energy living at the two ends, with the per-site difference falling as one over the length at a fitted exponent of −0.99986. And below a hundred and twenty-eight sites the second dimerisation does not exist at all.

applied · Peierls distortion 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 The chain distorts hardest where it stops. The alternation of each bond along a relaxed chain of 64, at four elastic constants, with the bulk value of each drawn as a dashed line. Every chain alternates more at its end than in its middle — by 1.21 times at the stiffest and 3.15 at the softest — and the excess dies away over a handful of bonds. The uniform alternation usually assumed is the flat part of these curves.

The chain distorts hardest where it stops

Holding the alternation uniform is what made the end energy a clean constant, and it is the one assumption the end-energy calculation had to make. Letting every bond find its own value shows the distortion is largest at the end and decays inwards over a measurable length — one and a half bonds in a strongly dimerised chain, five in a weak one.

solids · Peierls distortion 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.

Thermodynamic limitTight-binding modelsBand widthDensity of statesGraphSecond momentBand gapCohesionFillingBands in a solidExact diagonalisationModel limit

All concepts