The collection

Every essay — page 24

Page 24 of 42, continuing through the fields in the same order.

Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search

Where the atoms go

VSEPR as a repulsion minimisation rather than a table, the angles that fall out of it, and the case where five sites are not all alike.

With the bonds free to stretch, which three are long depends on where the lone pair goes. The near three bonds' mean length over the far three's, minus one, against the lone-pair weight, with every bond on a spring of stiffness 20 and the six bonds stretched by about 8.7%. If the lone pair keeps its place among the bonds the long three are opposite the face below a weight of 0.725, as with lengths fixed by hand. If it stays where it was, or sits on a bond's spring, the long three are round the face across that range instead.

The lone pair has no bond to stretch

Put every one of xenon hexafluoride's bonds on a spring and let the repulsion set the six lengths, and which three come out long is decided by something the model has never had to say: how far from the xenon the lone pair sits once the bonds have moved. Three equally defensible answers give three different verdicts — and none of them produces a split within a factor of twenty of the measured one without stretching every bond beyond recognition.

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When the molecule does not stop

A chain of two hundred atoms is a molecule and behaves like a solid. Bands, gaps, metals, defects and surfaces, every one of them out of a finite matrix — and a clear account of what that route cannot reach.

Chains of 2, 4, 8, 16, 40: the levels crowd, the edges do not move. Every level of a chain of 2, 4, 8, 16, 40 sites, drawn at its computed energy on one axis. Adding sites adds levels inside a fixed interval rather than widening it, which is what makes the large system a band rather than a wider molecule.

A solid is a molecule that did not stop

Diagonalise a chain of two atoms, then four, then forty. Nothing new happens at any point, and by forty the levels are a band. The passage from molecule to solid is not a change of subject; it is the same matrix at a different size, and every step of it can be watched.

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⟨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 — the full band width beside them obeys no such rule.

The width of a band is a count of neighbours

The mean of the squared level energies equals the average coordination — exactly, for any structure, with no limit taken and no periodicity assumed. It is the one statement in this field that is arithmetic rather than physics, and the usual textbook formula for band width is a special case of something weaker.

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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.

A density of states is not a spectrum

A molecule's spectrum is a list of positions and a solid's is a shape, and the shape is not the density of states. Between the two sits everything the count leaves out — which transitions are allowed, how strongly, and from where to where.

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Chains of 4, 16, 64, 160: the levels crowd, the edges do not move. Every level of a chain of 4, 16, 64, 160 sites, drawn at its computed energy on one axis. Adding sites adds levels inside a fixed interval rather than widening it, which is what makes the large system a band rather than a wider molecule.

A band with no structure in it

Everything in this field is computed from a finite matrix with no periodicity assumed, which is a real method and a real limitation. It produces a band and cannot produce a band structure — and the difference between those two words is worth an essay, because it is the boundary of what a finite matrix can honestly say.

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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.

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A half-filled ring's cheapest excitation goes to zero. The energy of the smallest available excitation of a half-filled ring, against the number of atoms, on log axes. Every doubling at least halves it, so in the limit there is no smallest excitation — which is what a metal is, before any band picture is drawn.

What a metal actually is

Not shiny, not a good conductor, not an element on the left of the table. A metal is a system with excitations of arbitrarily small energy, and that definition can be checked on a sequence of finite rings without drawing a band diagram or mentioning conduction at all.

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A chain of 100 is more stable alternating. The electronic energy gained by alternating the bonds of a half-filled chain, the elastic cost of doing it, and their sum, all per site and all against the alternation. The sum has its best value away from zero, so the evenly spaced chain is not the stable structure.

A chain cannot stay even

Diagonalise a half-filled chain, alternate its bonds slightly, and diagonalise again. The electrons gain more than the springs lose, and they do so for every spring constant whatever — because the gain is steeper than a parabola near the origin and a logarithm beats any constant.

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The gap against size: uniform against δ = 0.15. The HOMO–LUMO gap of a half-filled chain plotted against the number of sites, on log axes, for a uniform chain and for one whose bonds alternate. The uniform sequence falls without limit; the alternating one settles at four times the alternation.

The gap is not the band width

Two numbers describe a band and they answer different questions. The width is set by how many neighbours an atom has; the gap is set by how unequal they are. A structure can have a wide band and no gap, a narrow band and a large one, and changing one leaves the other alone.

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benzene — D6h. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

Counting electrons in an extended structure

The octet rule, Hückel's 4n + 2 and the 8 − N rule that predicts the structures of the main-group elements are one rule counted three ways. Each says the same thing — close the shell — and each stops being reliable at exactly the point where closing it becomes impossible.

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The state on a defect at site 31 of 60. The amplitude of one eigenvector at each site of a 60-site chain whose site 31 has its energy raised by 1.6β. The two colours are the two signs. A state belonging to the whole chain would reach across it; this one does not.

A defect is a level in the gap

Change one site's energy in a chain of a hundred and sixty and a level leaves the band, carrying a state that lives on a handful of atoms. In one dimension it happens for any change however small — the threshold measured on chains of forty, eighty and a hundred and sixty halves with every doubling, so there is no threshold at all.

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The state on the end of a chain of 60. The amplitude of one eigenvector at each site of a 60-site chain whose site 1 has its energy raised by 0.6β. The two colours are the two signs. A state belonging to the whole chain would reach across it; this one does not.

The end is the hardest place to bind

A site in the middle of a chain traps a state for any energy difference however small. The site at the end demands a whole β before it traps anything — measured at 1.025, 1.013 and 1.006 on chains of forty, eighty and a hundred and sixty, converging on exactly one. The intuition runs the other way and is wrong.

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