Where a molecule stops being one

Ethene is a molecule. A diamond is a solid. Somewhere between two carbon atoms and $10^{23}$ of them the language changes, and the natural question is where.

The natural question has no answer, and the reason is not that the boundary is fuzzy. It is that there is no single boundary to be fuzzy about. Different properties of a growing chain approach their limits at different rates, so the size at which the system “is a solid” depends entirely on which question is being asked — and the sizes differ by a factor of several hundred.

Three rates, measured #

Take a chain and grow it, watching three quantities.

The energy per site converges as $1/n$. The reason is a counting argument: a chain of $n$ sites has $n-1$ bonds, so its energy per site differs from a ring’s by roughly one bond’s worth divided by $n$. Measured across chains of twenty to three hundred and twenty, the difference times the length settles at $0.71\beta$ and stops moving, which is the signature of exactly that law.

The band edges converge as $1/n^2$. The top level of a chain is at $2\cos(\pi/(n+1))$, and expanding the cosine gives $2 - \pi^2/(n+1)^2$. At $n = 40$ that is $1.994$, at $n = 200$ it is $1.99976$. Two hundred sites is a part in ten thousand.

The density of states near the edges converges slowest of all, and in a sense never does: the limiting density diverges at the band edge, and a finite system has finitely many levels there, so no finite chain reproduces the divergence. What converges is the density away from the edges, which is why this site’s check on it compares the interior and excludes the outer three-tenths of a $\beta$ explicitly rather than quietly.

So “large enough” is $n \approx 700$ for the energy, $n \approx 200$ for the edges, and no $n$ at all for the divergence.

Every level behind that figure is a closed form rather than a diagonalisation, which is what makes six hundred and forty sites affordable and the slopes worth fitting. The check the claim actually needs is the fourth — that the three powers differ — because three quantities converging at one rate would make “large enough” a single number, and the argument here is that it is three.

Watched directly rather than as a rate, the growth looks deceptively finished. A solid is a molecule that did not stop draws chains of eight to a hundred and sixty sites: the interval is essentially fixed by twenty sites and the number of levels inside it keeps growing, so a picture of the band converges long before any quantity computed by summing over the band does. That mismatch is why a figure is a bad instrument for this question and a slope is a good one.

Why the rates differ, which is the useful part #

The pattern behind the three rates is worth extracting, because it generalises past this model.

A quantity converges as $1/n$ when it is an average over sites and the sites near the ends are different. There are two such sites out of $n$, so their contribution to any per-site average is proportional to $1/n$. Energy per site, average bond order, average charge: all $1/n$.

A quantity converges as $1/n^2$ when it is an extremum of a smooth function of a discrete index. The top of the band is the largest of $2\cos(k\pi/(n+1))$; near a maximum a smooth function is flat, so the error from sampling it at discrete points goes as the square of the spacing. Band edges, ionisation thresholds, any highest or lowest anything: $1/n^2$.

A quantity does not converge at all when the limiting object is singular. Divergences in a density of states, sharp band edges, anything with an infinite derivative.

That taxonomy is more useful than any single number, because it says which of a new quantity’s convergence to expect before computing it. It also explains the pattern of behaviour that puzzles people running cluster calculations: total energies converge smoothly and monotonically, gaps oscillate and converge slowly, and densities of states look wrong near the edges no matter how large the cluster is. Those are three different laws, not one calculation behaving inconsistently.

Rings behave the same way and are drawn the same way in where the states pile up: levels crowding into an interval that has already stopped moving, with the density of states beside them as the limit the columns are approaching. And at the smallest sizes nothing has converged to anything at all — aromaticity as a shell closure fills rings of three to eight and finds completely different level patterns, different shell closures and different chemistry, with no statement about a limit available for any of them. Convergence only becomes an askable question somewhere above those.

The part that never converges #

There is a fourth category, and it is the one with the most chemistry in it.

The two ends of a chain cost about $0.72\beta$ in total, and that number settles rather than shrinking. At twenty sites it is $0.492\beta$, at eighty $0.668$, at six hundred and forty $0.719$: it is approached from below, quickly, and then stops. It is not a finite-size artefact and it does not vanish in any limit. What shrinks is its share of the whole.

So the statement “surface effects vanish in the thermodynamic limit” is true of the fraction and false of the quantity, and which of the two matters depends entirely on the measurement. A calorimeter measuring the heat of formation of a gram of material is measuring the whole, and the surface is a rounding error. A reaction happening on the surface is measuring only the part that never converged.

This is why a nanoparticle is not a small piece of bulk material in any useful sense. Its surface fraction is large, and the sites in that fraction are the ones with an entire bond’s worth of binding missing. The end is the hardest place to bind works out what an end site does to the spectrum, and the answer is not what the missing bond suggests.

What produces that cost is visible in a single state and needs no impurity to see it. The lowest state of a plain uniform chain has amplitude everywhere, largest in the middle, tapering to nothing at both ends — the end is the hardest place to bind draws it — because no state can have full amplitude at a site with one neighbour. The 0.72β is that taper, integrated.

The gap, which behaves worst of all #

One quantity deserves separate treatment because it converges badly enough to have caused real confusion, and because it is the quantity a chemist cares most about.

The gap between the highest occupied and lowest empty level of a half-filled chain is not a fixed quantity approaching a limit. It is the spacing between two adjacent levels near the middle of the band, and the middle of the band is where the levels are densest — so the gap falls as $1/n$, all the way to nothing.

Measured on chains of twenty, forty, eighty and a hundred and sixty sites, it goes $0.299$, $0.153$, $0.078$, $0.039$: halving with every doubling, exactly as a $1/n$ law requires, with no sign of settling.

That is the correct behaviour and it is a trap for anyone who computes the gap of one cluster and quotes it. A conjugated polymer’s absorption does redshift as the chain grows, and it does not go to zero — real polyacetylene has a gap of about 1.5 electronvolts however long the chain is. The difference is that a real chain does not have equal bonds, which is a chain cannot stay even’s subject, and once the bonds alternate the gap converges on a finite value instead.

So the same computation, run on two chains differing only in whether their bonds are equal, gives one sequence heading to zero and one settling at four times the alternation. Which of the two a material follows is decided by an instability the model has to be asked about separately.

What a “molecule” and a “solid” actually distinguish #

If no size separates them, what does the distinction pick out?

Three things, and each is a comparison rather than a size.

The level spacing against something else. A system is molecular when its levels can be resolved individually, and that depends on temperature, on the lifetime of a state and on the resolving power of the instrument as much as on the system. A chain of forty at room temperature has a level spacing comparable with the thermal energy; the same chain at a millikelvin has forty distinguishable levels.

Whether the ends are a correction or the subject. A benzene molecule is entirely edge. A crystal grain is entirely interior with a thin skin. The interesting regime is the one in between, and it is where most of nanoscience lives.

Whether the number of particles is fixed. A molecule has a definite formula; a solid is described by a composition and can absorb defects, vacancies and dopants without becoming a different substance. This is the distinction that does the most work in practice and it is chemical rather than quantum-mechanical.

None of the three is a threshold. All three are comparisons, and stating which one is meant is usually more informative than the word itself.

Three dimensions change the arithmetic and not the argument #

Everything measured here is one-dimensional, and a reader is entitled to ask how much of it survives in a real grain of material.

The convergence laws survive with their exponents changed in the obvious way. A three-dimensional cube of side $L$ has $L^3$ sites and $6L^2$ surface sites, so the surface fraction goes as $1/L$ — which in terms of the number of atoms $N$ is $N^{-1/3}$ rather than $N^{-1}$. Surface effects therefore die away far more slowly with atom count in three dimensions than in one.

The numbers that follow are sobering and are worth stating. A cube of a thousand atoms is ten on a side and has $488$ of them on its surface — nearly half. A cube of a million atoms is a hundred on a side and still has six per cent of its atoms on the surface. Reaching a surface fraction of one part in a thousand takes a cube six thousand atoms on a side, which is around $2\times10^{11}$ atoms and a grain about two micrometres across.

So the answer to “how big before the surface stops mattering” is bigger than most things people study. Every catalyst particle, every quantum dot, every pigment grain and most of what is called a thin film is well inside the regime where the surface is a substantial fraction of the material rather than a correction to it.

What does not change in three dimensions is the shape of the argument: a per-site average converges as the surface fraction, an extremum converges as its square, and a singular feature does not converge. Those follow from what kind of quantity each is, not from how many dimensions it lives in.

The lesson for the model, not just for the material #

There is a methodological point buried here, and it is the reason this essay sits in the middle of the field rather than at its end.

Every claim in this field is computed on a finite system. Some of those claims are statements about the finite system and some are statements about the limit, and telling them apart is not optional. “$\langle x^2\rangle$ equals the mean coordination” is exact at every size and needs no limit — the width of a band is a count of neighbours makes that its whole subject. “A half-filled chain has excitations of arbitrarily small energy” is a statement about a sequence, and the essay that makes it, what a metal actually is, states it as one.

Between those two lies the class of claim that gets made carelessly: a number computed on one system of one size, presented as though it were the limit. The defence against it is the one this essay is built on — compute the same quantity at several sizes, plot it against the size, and state the law it follows. That is more work than quoting a single number and it is the difference between a result and an anecdote.

Two other figures are the two ends of that spectrum and are worth naming as such. What a metal actually is plots the level spacing at the Fermi level for five chain lengths, falling as 1/n with no limit — a statement about a sequence, and the definition it is built on. Hückel theory draws benzene’s six levels, which are six numbers about one molecule and converge to nothing because there is nothing for them to converge to. Every claim about chains and rings is one or the other.

The rates are measurable, and the slowest one became a technology #

Every convergence above is computed on a model chain. The claim that “large enough” is a different size for every question is testable on real material, because particles can now be made at a chosen size and measured, and the two ends of the rate hierarchy behave exactly as the arithmetic says they should.

The fast end is invisible. Cohesive energy per atom, lattice constant, density — the quantities that converge as one over the linear size — have essentially reached their bulk values in a particle of a few thousand atoms, which is four or five nanometres. Nobody buys a nanoparticle for its cohesive energy, and the reason is that there is nothing left to buy: it converged before the particle was small enough to be interesting.

The slow end is an industry. The gap converges worst of all, and a semiconductor nanocrystal is sold on precisely that. Bulk cadmium selenide has a gap of about 1.74 electron volts and emits in the deep red; a crystal of it two nanometres across emits green, near 2.5 electron volts. Same material, same bonding, same coordination for almost every atom in it — and three quarters of an electron volt of difference, because the one quantity that had not converged is the one being looked at.

The direction of the hierarchy is what makes that possible. If the gap had converged as fast as the binding energy there would be no size-tuned emitter, because by the time a particle were small enough to shift its colour it would no longer be the same material. The property that converges slowest is the property a size series can be used to engineer, and it is slow for the reason the chain calculation gives: a gap is a difference between two particular levels near the middle of the spectrum, and the spacing between neighbouring levels is the last thing to stop changing.

The metals give the crossover a number rather than a trend. A metal is a metal because its levels near the top of the filled set are closer together than the thermal energy; a cluster whose spacing exceeds $kT$ has discrete levels and behaves like a large molecule. The spacing of a metal with $N$ atoms is roughly its Fermi energy divided by $N$, so the condition is

$$\frac{E_F}{N} \approx kT.$$

For gold, with $E_F$ near 5.5 electron volts and $kT$ at room temperature of 0.026, that gives $N \approx 210$ atoms — a particle about 1.7 nanometres across. Measured, gold clusters cease behaving as metals and start showing discrete optical transitions and a real gap at very close to two nanometres, and clusters of a few tens of atoms have gaps above an electron volt and fluoresce.

An estimate from one division landing within a factor of a nanometre of a measured crossover is not a coincidence, and it is the same argument this essay makes throughout: the size at which a system becomes a solid is the size at which the quantity being asked about stops moving, and for a metal the quantity is a level spacing, so the answer comes out at hundreds of atoms rather than at the seven hundred a chain’s energy needed or the many thousands its gap would.

An old dispute the arithmetic settles #

The question this essay refuses to answer with a number has a history, and the history is instructive because both sides were arguing about the wrong thing.

Through the middle of the twentieth century there was a genuine methodological split between chemists computing molecules and physicists computing solids, each convinced the other’s methods did not apply to their systems. Chemists pointed out that a solid-state calculation could not describe a defect, a surface or a molecule adsorbed on one — all true, and all consequences of a method organised around a label that those things destroy. Physicists pointed out that a molecular calculation could not be extended to $10^{23}$ atoms — also true, and a consequence of the cost of diagonalising a matrix.

Neither objection was about the physics, and the arithmetic in this field shows why. The same matrix, the same eigenvalue problem and the same filling rule describe both, and the two disciplines had specialised their methods to the systems they cared about rather than discovering different laws. What separated them was that one had a symmetry to exploit and the other had a boundary to describe, and each built its tools around what it had.

The convergence measurements above are the quantitative version of that reconciliation. A chain of forty is close enough to the limit for its band edges and nowhere near it for its gap, so a chemist computing forty atoms and a physicist computing an infinite chain will agree about one quantity and disagree about another — not because either is wrong, but because the two quantities converge at different rates and only one of them has arrived.

The exception is the quantity neither side has to wait for. The width of a band is a count of neighbours draws a chain of forty against one of eighty against one of a hundred and twenty, and the second moment equals the mean coordination in every row to fourteen decimal places — no rate, no sequence, no limit taken. A field with one exact identity in it and three different convergence laws around that identity is exactly the shape of subject where a single computed number is worth least.

Where this leaves the field #

The rest of the field can now be read with the right kind of scepticism. When an essay here says a chain of a hundred behaves in some way, the question to ask is which of the three convergence laws the quantity obeys and whether a hundred is enough for that law.

Mostly it is, and where it is not, the figures show the sequence rather than the endpoint. The Peierls calculation in a chain cannot stay even is done at a hundred sites and the quantity it turns on — whether the energy gain is steeper than quadratic — is a statement about the shape of a curve near zero, which converges quickly. The gap sequences in the gap is not the band width are shown at four sizes precisely because a single size would have been uninterpretable.

And the honest boundary from a band with no structure in it applies throughout: none of these convergences says anything about a wavevector, because there is none here to converge.