A solid is a molecule that did not stop

Two carbon atoms with a p orbital each give two molecular orbitals, one bonding and one antibonding. That calculation is the first one anybody does, and it is a two-by-two matrix.

Two hundred carbon atoms in a row give two hundred molecular orbitals. That calculation is a two-hundred-by-two-hundred matrix, and it is otherwise identical: the same entries, the same solver, the same filling rule. Between those two there is no point at which the method changes, no size at which a new principle is required, and no step where the word molecule has to be exchanged for the word solid.

That claim is easy to make and worth watching happen.

The matrix, which is a list of neighbours #

In the simplest useful model an atom contributes one orbital, an orbital on its own has energy $\alpha$, and two neighbouring orbitals interact with strength $\beta$. Write those down as a matrix — $\alpha$ on the diagonal, $\beta$ wherever two sites are bonded, zero elsewhere — and the molecular orbital energies are its eigenvalues.

For two sites the matrix is

$$\begin{pmatrix} \alpha & \beta \\ \beta & \alpha \end{pmatrix},$$

whose eigenvalues are $\alpha + \beta$ and $\alpha - \beta$. Since $\beta$ is negative, the first is the lower one and holds the bonding pair. That is ethene’s π system and it is the base case of everything below.

For a chain of $n$ sites the matrix has ones on the two off-diagonals and nothing else. Hückel theory and what it gets right sets out where this comes from and what it leaves out, which is a great deal: there is no electron repulsion in it and no distinction between one element and another. The point here is not that the model is accurate. The point is that it does not change.

The band edges do not move #

Take the chain’s eigenvalues seriously. They come out at

$$x_k = 2\cos\!\left(\frac{k\pi}{n+1}\right), \qquad k = 1, \ldots, n,$$

in units of $\beta$ above $\alpha$, and this site derives that rather than quoting it — the closed form is checked against the diagonalised matrix at every size drawn.

Read what the cosine does. Whatever $n$ is, the argument runs over part of a half-turn and the cosine stays between $-1$ and $1$, so every level lies between $-2\beta$ and $+2\beta$. The lowest level of a two-atom chain and the lowest level of a two-hundred-atom chain are within a hair of each other. Adding atoms does not widen the range of energies available; it adds levels inside a range that was fixed by the first pair.

That is the single most counterintuitive fact in the subject, and it is worth being blunt about why. A chemist’s instinct is that a bigger molecule has more spread-out energies, because a bigger molecule has more of everything. The instinct is wrong because the width of the band is set by how strongly two neighbours interact, and a chain of two hundred atoms still has each atom next to two others. Nothing about being long changes what a neighbour is.

The band limit makes the same argument for rings, where the closed form is cleaner and the degeneracies are exact. The chain is worth doing separately because a chain has ends, and what ends do is the subject of half this field.

Where a set of levels starts being a band #

For four sites the levels are at $1.618$, $0.618$, $-0.618$ and $-1.618$: obviously discrete, obviously countable, and a spectroscopist would expect to resolve them. For forty the spacing near the middle of the band is under a twentieth of a $\beta$. For two thousand it is a thousandth.

There is no size at which the set of levels becomes a continuum, and that is the honest answer rather than an evasion. What there is instead is a size at which the spacing falls below whatever else is smearing the levels — thermal energy, the lifetime of a state, the resolution of the instrument. A band is a set of levels closer together than anything that could tell them apart.

Which means “band” is a statement about the comparison and not only about the system. At a hundredth of a $\beta$, room-temperature thermal energy already exceeds the spacing for a chain of a few hundred atoms. A chain of forty at a millikelvin is a molecule with forty resolvable levels.

What β is, and why the band width is not a measurement #

Everything above is quoted in units of $\beta$, which deserves a paragraph of its own, because the habit of expressing answers in units of an unmeasured quantity is exactly the habit this site spends most of its time refusing.

$\beta$ is the matrix element between orbitals on two neighbouring atoms. It is not an observable and it has no experimental value; what it has is a range of values fitted, in different contexts, to different measurements — around 2.5 electronvolts if the fit is to benzene’s electronic spectrum, appreciably different if the fit is to a heat of hydrogenation or to a photoelectron spectrum. Hückel theory and what it gets right is where the fitting question is worked through.

The reason that is tolerable here is that almost nothing in this essay is a claim about a number of electronvolts. “Every level lies between $-2\beta$ and $+2\beta$” is a claim about a ratio, and so is “the width does not change with $n$”, and so is “$\langle x^2\rangle$ equals the coordination”. A ratio survives not knowing the unit. The moment a claim is made about an actual energy — a band gap in electronvolts, a melting point, a colour — the fitted parameter enters and the claim inherits its uncertainty. Those claims are marked where they occur.

Filling, which is the step everything downstream turns on #

The levels are half the calculation. The other half is how many electrons there are, and this is where a chain of forty carbons and a chain of forty of something else stop being the same problem.

One electron per site fills the lower half of the band and leaves the upper half empty. Two per site fills all of it. The first case has an occupied level immediately below an empty one and the second has no empty level at all, and that difference — not the band width, not the density of states, not the size of the system — is what decides whether the material conducts.

The density of states, which is where the levels pile up #

Once the levels are too close to count individually, the useful question changes from where is level fifty-seven to how many levels lie between these two energies. That count, per unit energy, is the density of states.

For the chain it has a closed form, and the derivation needs nothing about crystals. The levels are $2\cos(k\pi/(n+1))$ with $k$ evenly spaced, so the number of them landing in a small energy interval is however many values of $k$ map into it — which is the derivative of $k$ with respect to $x$, and that derivative is $1/(\pi\sqrt{4 - x^2})$ once it is normalised.

The interesting feature is at the ends. As $x$ approaches $\pm 2$ the cosine is turning over, so many consecutive values of $k$ give nearly the same energy and the levels pile up. The density diverges at both band edges.

Two thousand computed levels binned across the band agree with the closed-form density to better than 0.005 in absolute density across the interior. That is the limit the sequences above are heading for, and reaching it is what makes the last column of this essay a solid rather than a large molecule.

The pile-up is not a curiosity. It is why the optical and thermal properties of a solid are dominated by what happens at the band edges, and why the middle of a band is comparatively featureless. It also arrives here with no reference to a lattice at all — a chain of two thousand atoms of unspecified spacing, diagonalised.

The one exact statement: width is a count of neighbours #

Almost everything above is a limit, and limits invite the objection that a large finite thing is not an infinite thing. There is one statement in this field that is exact at every size, needs no limit, and holds for any structure whatever.

The mean of the squared level energies equals the average number of neighbours.

The reason is a fact about matrices rather than about physics. The sum of the squares of the eigenvalues is the trace of the square of the matrix, and the trace of $A^2$ counts the ways of stepping from a site to a neighbour and back again — one for each ordered pair of bonded sites. Divide by the number of sites and it is the mean coordination.

$$\langle x^2 \rangle = \bar{z}.$$

A chain has $\bar{z} \to 2$, a ring has exactly 2, a three-connected sheet has 3, a four-connected network has 4. No approximation, no periodicity, no limit.

The mean coordination counted off the edge list and ⟨x²⟩ measured off the eigenvalues agree to machine precision, because they are the same quantity computed two ways. That identity holds at every size in this essay, which is the reason a quantity like a band width is available on a molecule of six atoms as well as on a solid.

That distinction is worth holding on to. The root mean square width of a band is $\sqrt{\bar{z}}\,|\beta|$ and is a count. The full width is $4|\beta|$ for a chain, and getting from the count to the full width requires knowing how the neighbours are arranged, which is a different and much harder question. Textbook statements that a band is “$2z\beta$ wide” are quoting a result for one particular family of structures as though it were general.

What the ends cost, and how quickly they stop costing #

A ring and a chain of the same length are not the same system. The ring’s levels are doubly degenerate and the chain’s are not; the ring has no ends and the chain has two. The energy per site differs between them, and that difference is precisely an end effect.

So it should fall as one over the length, since the number of ends stays at two while the number of sites grows. Multiplying the difference by the length should therefore settle on a constant, and it does.

The energy per site of a ring minus that of a chain of the same length, times the length, settles at about 0.71β — a fixed amount of energy living at the two ends whatever the chain’s length. That is the last quantity in the sequence to converge, and it converges to a constant rather than to zero.

The constant it settles on is worth reading as well as the fact that it settles. About $0.71\beta$ is what two ends cost, in total, once the chain is long enough for them not to interact — a little over a third of a $\beta$ each. That is a substantial fraction of one bond’s worth of binding, missing from the sites at either end, which is a compact statement of why a surface atom is a reactive atom.

Seven hundred atoms is the answer to a question usually waved at rather than asked. “Surface effects are negligible for a large enough sample” is true, and the number attached to it here is that a one-dimensional chain has to be several hundred atoms long before its ends contribute less than a thousandth of a $\beta$ per site. For a real three-dimensional grain the arithmetic is kinder, since surface sites scale as the two-thirds power of volume — but the shape of the statement is the same, and it is a measurement rather than a reassurance.

Seven hundred atoms, or a third of a micrometre #

The kindness of three dimensions was stated above and left there, and it is worth taking the extra step, because the answer changes what kind of quantity “large enough” is.

In one dimension the deficit per site went as $0.71\beta/n$: two ends, $n$ sites, so a reciprocal in the count. Write the same accounting for a cube of side $L$ atoms. The number of sites is $L^3$; the number on the surface is $L^3 - (L-2)^3$, which for large $L$ is $6L^2$. Each face atom on a simple-cubic lattice is missing one of its six neighbours, so the missing bonds per site work out at about

$$\frac{6L^2}{L^3} \times \frac{1}{6} = \frac{1}{L},$$

and the deficit is again of order $\beta/L$. The functional form did not change at all. What changed is what $L$ counts.

That is the whole of it, and it inverts the usual way the question is asked. The criterion for bulk behaviour is a linear size, not a number of atoms. Reaching a deficit of a thousandth of a β per site needs $L$ near a thousand in one dimension and $L$ near a thousand in three; the chain therefore needs about seven hundred atoms and the cube needs about a thousand along each edge, which is $10^9$ of them. A billion atoms and seven hundred atoms are the same condition stated in different variables.

A thousand atoms across is roughly three hundred nanometres at ordinary spacings — a third of a micrometre, which is a useful number to carry because it sits exactly where the experimental literature says it should. A grain a micrometre across is bulk by every measurement anybody makes on it. A particle ten nanometres across is thirty atoms on a side, its surface holds a fifth of its atoms, and it melts lower than the bulk, absorbs at a different wavelength and catalyses reactions the bulk does not. Nothing about that transition is a new physical principle; it is $1/L$ with $L$ small enough to notice.

Two qualifications keep the estimate honest.

The coefficient is a lattice’s and not a universal. A simple-cubic count was used because it makes the missing-neighbour arithmetic a single fraction. A close-packed lattice has twelve neighbours and loses three at a flat face, and a real surface reconstructs rather than simply losing bonds, so the coefficient moves by a factor of order one. The reciprocal in $L$ does not.

And a thousandth of a β is an arbitrary bar. It was chosen in one dimension to make a number available, and everything above inherits it. What is not arbitrary is the ratio between the two cases, because both were measured against the same bar and the bar cancels: the linear sizes agree, whatever value it takes.

So the reassurance that surface effects vanish in a large sample turns out to be a statement with a length in it rather than a mass. A gram of anything contains enough atoms; a film four atoms thick does not, however many grams of it there are.

What has and has not been shown #

What has been shown is that the molecular orbital picture does not break down as a system grows. It produces a band, a density of states, band edges in the right place, and an exact relation between the second moment and the coordination — all from finite matrices with nothing periodic assumed.

What has not been shown is anything about a direction in a crystal. Every system here is a chain or a ring, which is to say a one-dimensional connectivity with no geometry attached; the atoms have neighbours but no positions. That is a real limitation and it is the subject of a band with no structure in it, which sets out what this route cannot reach and what the other route buys.

Nor has anything been shown about why the atoms sit where they do. The chain here is a list of neighbours; the spacing between them never appears, and the model would give the same answer if the atoms were a nanometre apart or a metre. Everything in this essay is therefore a statement about connectivity, and the connectivity has been assumed rather than derived. Where the atoms go is the site’s field on that question, and its methods do not scale to two hundred.

It is also worth saying what has been assumed. Every electron here moves in a fixed potential and does not see the others. Orbitals are not where the electron is makes the general case for treating that as a description rather than a fact; in this field it bites harder than anywhere else on the site, because the one-electron picture predicts that a half-filled band is a metal, and there are materials with a half-filled band that are insulators. A half-filled band is not always a metal is where that failure is made explicit rather than left in a caption.

The rest of the field takes the band as built and asks what can be done with it: what a metal actually is makes the metal–insulator distinction without drawing a band diagram at all, a chain cannot stay even shows that the evenly spaced chain assumed throughout this essay is not in fact stable, and a defect is a level in the gap puts one site out of step and watches a state appear.