Molecular Geometry & Bonding
what the shape decides
Concept

Level spacing — where it appears

The energy between adjacent levels, which falls as a system grows. Whether it falls to nothing in the limit is what distinguishes a metal from an insulator, and the rate at which it falls depends on the dimension.

Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

solids · Bands in a solid 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.

solids · Bands in a solid 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 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.

solids · Metal Two impurities at 2 to 14 sites apart. The splitting between the two levels a pair of impurities of strength -2β pulls out of a chain of 61, against how far apart they are, on a logarithmic scale. It falls by a constant factor per site of separation, and that factor is the decay of the isolated bound state computed from its energy alone. The two levels close on the single impurity's level as the pair separates.

Two defects, and the level between them

One impurity pulls a level out of a band at exactly −√(h²+4). Two of them pull out a pair, split by an amount that falls by a factor of 0.41421 for every site of separation — which is √2 − 1, predicted from the isolated level's energy alone and measured to six figures.

solids · Defect water: every level up to J = 4, from a matrix. The rotational levels of water at κ = -0.4322, each J diagonalised in the symmetric-top basis. A symmetric top would show one level per K with everything above K = 0 doubly degenerate; here every degeneracy is split, and the size of each splitting is what the third constant is measured from.

The top that reports all three

A symmetric top hides one of its two rotational constants in every line of its spectrum. Break the symmetry and the hiding stops: for water, twenty-three of the twenty-five levels up to J = 4 move when A is changed, and the two that do not are the ground state and the one at B + C. There is no formula for any of them.

spectra · Rotation One impurity is a level; many are a band. The impurity levels of a ring of 160 with sites of depth -3, drawn as a bar from the lowest to the highest, against the fraction of sites that are impurities. At the lowest concentration every level is at the same energy and the bar has no height at all. By 30 per cent the levels span 2.44 and have closed to within 0.25 of the host band, which is shaded.

One defect is a level, many are a band

A single deepened site in a chain pulls one state out of the band to −√(h² + 4), exactly, and holds it on 1.42 sites. Put in more and the levels spread: at one site in ten they span 1.45 in the same units and have closed to within 0.69 of the host band, and above one site in eight the count of levels stops matching the count of defects, because two defects on neighbouring sites push one of their pair back into the band.

solids · Defect How many sites a state occupies, and whether that depends on the ring. The participation ratio of the states at the middle of the band — the number of sites a state occupies — against the width of the disorder, for rings of 50, 100, 200 sites. With no disorder the three curves are three different numbers, each two thirds of its own ring. At the right they have converged: 7.47 sites on a ring of 50 and 9.82 on a ring 4 times larger.

The third way to be an insulator

A ring of two hundred sites with a half-filled band has its levels crowding together as 1/n, which is the usual electronic-structure criterion for a metal, and it goes on holding at every disorder tested. Meanwhile the states at the middle of the band go from occupying 127 sites to occupying 10 — and at that disorder the number stops depending on how large the ring is at all.

wrong · Metal How many times more a metal carries, and when. The ratio of the carriers in a uniform ring to those in an alternating one of the same size, at four sizes and six temperatures. Down the left-hand column the two are indistinguishable, because a ring of 42 at kT = 0.002 has a level spacing larger than the temperature and is no more a metal than the gapped one is. Along the bottom row they are indistinguishable again, because the temperature is larger than the gap. The word only means anything in the middle.

The metal a thermometer cannot find

A metal is a system with excitations of arbitrarily small energy, which is a claim about a sequence of finite systems rather than about any one of them. Put a temperature on it and the claim needs a second limit, and the two do not commute: a uniform ring of forty-two at kT = 0.002 carries exactly as much as an alternating one, and at kT = 0.1 a ring of three hundred and twenty-two carries only four times as much.

wrong · Metal 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 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 The coupling at the closest pair, and at the typical one. For runs of four to eight low sites at half concentration: the splitting between two runs one site apart, which is the largest coupling any chain can produce, and the splitting at the separation two runs typically have. Both on a logarithmic axis spanning 173 decades. A band width taken from the second is not a small number; it is not a number.

A band that is a hundred and seventy decades of nothing

The chain length at which a defect's levels become a band can be located by asking when the coupling between the closest pair exceeds the level spacing. A band also has a width, and a width is set by the typical coupling rather than the closest one. For runs of eight at half concentration those two numbers differ by a hundred and seventy-three orders of magnitude.

solids · Defect

Named alongside it

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

Bands in a solidThermodynamic limitBand edgeEigenvalueTight-binding modelsDensity of statesModel limitFermi levelLocalisationParticipation ratioBand gapClosed form

All concepts