Molecular Geometry & Bonding
what the shape decides
Series

Metal — the series

12 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · solids
  2. 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.

    part 2 · solids
  3. 60 electrons in 60 levels. The density of states of a ring of 60, drawn with the energy up the page, and the 60 electrons filled in from the bottom. Where the filling stops is what decides whether the system has cheap excitations.

    A half-filled band is not always a metal

    Every band picture rests on an approximation that a whole class of materials refuses — each electron moving in an average field, never seeing another one individually. Where the repulsion is strong enough, a half-filled band describes an insulator, and no amount of care with the band fixes it.

    part 3 · wrong
  4. A gap where band theory says there cannot be one. The exact charge gap of a half-filled four-site ring against the on-site repulsion, with the one-electron gap of the same ring beneath it. The one-electron answer is zero at every U — the ring's half-filled shell is degenerate — while the exact gap reaches 5.99t.

    The insulator band theory cannot see

    A half-filled ring of four sites has a degenerate shell and no gap at all in the one-electron picture, which is the definition of a metal at that size. Its exact charge gap is zero when the electrons do not repel and grows without limit when they do — so a material can have a half-filled band and not conduct, and here is the number.

    part 4 · solids
  5. 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.

    part 5 · wrong
  6. 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.

    part 6 · wrong
  7. 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.

    part 7 · wrong
  8. 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.

    part 8 · wrong
  9. The slope goes to a half, and a window fit stops short of it. The local slope of the alternation against the reduced temperature, between each neighbouring pair of points, on a ring of 40 at K = 1.6. It rises monotonically from 0.4115 to 0.5053 as the transition is approached, crossing a half at about a part in a thousand of the reduced temperature. The fitted 0.44 is the average of the left-hand end of this curve; the exponent is one half, which is what a free energy analytic in one order parameter is obliged to give.

    The exponent was the window's

    A fit over the last decade before a distortion vanishes gives an exponent of 0.44, and running it on larger rings should say whether the number belongs to the transition or to a forty-site ring. It belongs to neither. The local slope runs to 0.5020 as the transition is approached, and 0.44 is what a fit over that particular decade returns — on every ring size and every stiffness, because the whole curve is one curve.

    part 9 · wrong
  10. The alternation a spring buys, three ways. The alternation against the elastic constant, on a logarithmic axis. The middle line solves (2/π)(K − E)/(1 − δ²) = K for the complete elliptic integrals; the lower one is the exponential form every account of a Peierls distortion quotes, which is its own asymptote and is 6.5 per cent low at K = 1.2; the upper one is a ring of 40, which leaves the infinite chain as the spring stiffens because a smaller alternation is a longer coherence length.

    The amplitude the collapse left behind

    Five Peierls curves became one curve when each was divided by the alternation its ring settles at cold, so that amplitude is the whole of what distinguished them — and it was five golden-section searches over diagonalisations with no formula anywhere. It has one, exactly, as a sum of square roots; and writing it down says that one of the five rings was never measuring a long chain.

    part 10 · wrong
  11. The same collapse, with the cases made comparable. The alternation divided by its own cold value against the reduced temperature, for the published five cases and for five chosen so that every ring is the same size in its own alternations. The published set agrees to 3.41 per cent and the matched set to 1.62 — so the residual left was finite size, as suspected.

    Five rings that were five different sizes

    Five warmed rings have scaled alternation curves that lie on one another to 3.41 per cent, and the departure from the bulk amplitude turns out to be a function of the ring measured in its own alternations. The five cases span a factor of seven in that quantity. Choosing sizes that make them comparable halves the residual — and runs into a floor the lattice itself imposes.

    part 11 · solids
  12. Three targets, and the residual keeps falling. The worst spread across the five scaled curves, at three values of the matched product n·δ∞. It falls from 2.51 per cent at 5 to 1.62 at 9.6, monotonically. The unmatched cases sit at 3.41 per cent throughout, because they are the same five rings whatever target is being aimed at — which is what makes the comparison a comparison.

    Three points, and they all go down

    Matching five rings at one value of n·δ∞ tightens the temperature collapse from 3.41 per cent to 1.62, and what is left might be the even-site rounding rather than anything physical. At three targets the residual falls monotonically — and at the smallest one it is a third of what the rounding leaves, which the rounding cannot explain.

    part 12 · solids

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