The lattice sum that depends on the order of adding

Methods that describe a solid neighbour by neighbour are careful about what they can and cannot compute, and one boundary is usually mentioned without being examined. It deserves an essay because it is the clearest case of a computation that is genuinely out of reach for such a method rather than merely inconvenient — and because the reason is interesting.

An ionic solid’s binding energy is a sum over every pair of ions in the crystal. That sum does not converge absolutely, which means that rearranging its terms changes its value, and the order of summation is therefore part of the physics rather than a detail of the arithmetic.

The sum, written out #

Take rock salt. Each sodium ion has six chlorides at distance $r$, twelve sodiums at $r\sqrt2$, eight chlorides at $r\sqrt3$, six sodiums at $2r$, and so on outwards.

The energy of one ion is the sum of $\pm e^2/4\pi\varepsilon_0 d$ over all of them, with the sign set by whether the neighbour has the same charge or the opposite. Factoring out the nearest-neighbour term leaves a pure number:

$$M = 6 - \frac{12}{\sqrt2} + \frac{8}{\sqrt3} - \frac{6}{2} + \cdots$$

That number is the Madelung constant, and for rock salt it is $1.7476$.

Now look at the partial sums, shell by shell:

$$6.000, \quad -2.485, \quad 2.134, \quad -0.866, \quad 9.867, \quad 0.069.$$

The answer is $1.7476$. After six shells the partial sum has been as high as $9.867$ and as low as $-2.485$, and the excursion at the fifth shell is larger than anything before it. The sequence is not converging in any sense a reader would recognise, and stopping it at a shell that happens to be close to the answer — the sixth is $0.069$, which looks encouraging — would be luck rather than accuracy.

Why the terms do not get small #

The usual reason a series converges is that its terms shrink. These do not, and the reason is a competition between two counts.

A shell of neighbours at distance $d$ contains a number of ions proportional to $d^2$ — it is a surface, and surfaces grow as the square of the radius. Each of them contributes an energy proportional to $1/d$.

So each shell contributes about $d^2 \times 1/d = d$, which grows with distance rather than shrinking. What makes the sum finite at all is the alternation of sign: within each shell, and between successive shells, positive and negative contributions nearly cancel, and the sum survives on that near-cancellation.

A series that converges only because of cancellation between terms of growing magnitude is conditionally convergent, and a conditionally convergent series can be rearranged to converge to any value at all. Riemann proved that in general; here it means that “sum over the ions in order of increasing distance” and “sum over the ions cube by cube” are different prescriptions and may give different answers.

They do. Summing rock salt over expanding cubes rather than expanding spheres gives a different sequence of partial sums, and getting to the right answer requires care about how the surface of the summation region is terminated.

What the resolution actually is #

The resolution is physical rather than mathematical, and it is worth stating because it is a good example of a mathematical difficulty being an instruction about the physics.

The right summation region is the one that is electrically neutral at every stage, with no net charge or dipole on its surface. A sphere cut through a rock-salt lattice generally has a net charge on its surface, and that surface charge is a real thing with a real energy — the sum is oscillating because it is describing a sequence of charged objects rather than a sequence of approximations to a neutral crystal.

Choose neutral regions and the convergence is rapid and unambiguous. That is what Evjen’s method does by weighting ions on the boundary fractionally, and what Ewald’s method does more elegantly by splitting the sum into a fast-converging real-space part and a fast-converging reciprocal-space part.

The second of those is worth naming precisely because it marks the boundary of a real-space treatment. Ewald summation is a reciprocal-space method, and reciprocal space is exactly what a neighbour-list treatment of bands does without — see a band with no structure in it for that boundary and its reasons.

Why the tight-binding method cannot be patched #

A reader might reasonably ask why a tight-binding calculation, which handles a chain of two hundred atoms comfortably, cannot simply be given more neighbours.

The answer is that the method’s whole structure depends on the interaction having a range, and the Coulomb interaction has none.

The tight-binding matrix has an entry between two sites when they are bonded and zero otherwise, and that is not an approximation to a longer-ranged interaction — it is a good description of one that really stops. A covalent interaction depends on the overlap of two orbitals, both of which decay exponentially, so the second-neighbour term is a few per cent of the first and the third is a fraction of a per cent. What holds a solid together sets the four decay laws side by side.

The Coulomb interaction falls as $1/r$ and is never negligible at any distance. There is no truncation that introduces a small error; every truncation introduces an error of the same order as the answer, which is exactly what the oscillating partial sums are displaying.

So this is not a case where more computation would help. It is a case where a different method is required, and the different method works in reciprocal space rather than among neighbours.

What a merely slow convergence would look like #

The contrast worth having is with a sum that converges absolutely but slowly, since the two are often described in the same words and behave completely differently.

An absolutely convergent sum has partial sums that approach the answer monotonically or nearly so, with an error bounded by the size of the terms not yet included. Adding more terms always helps, stopping anywhere gives a bound, and the order does not matter.

The Madelung sequence above does none of those. Adding a shell can move the partial sum further away, the error is not bounded by the next term, and the order matters completely. A reader who has been told the sum is “slowly convergent” and treats it as the first kind will summarise it as an approximation problem, and it is not — it is a well-posedness problem, and the fix is not more terms.

This distinction has a practical consequence in any simulation of an ionic or polar system. Truncating the Coulomb interaction at a cutoff radius, which is exactly what a naive implementation does, introduces errors that do not shrink as the cutoff grows and that depend on the shape of the truncation. That is a well-known trap in molecular simulation, and its cause is the sequence printed above.

Where the answer actually comes from #

Since a neighbour sum cannot compute the Madelung constant, it is worth saying what does, because the methods are elegant and because knowing them makes the boundary sharper.

Evjen’s method sums over cubes and gives fractional weight to ions on the boundary — a face-centre counts a half, an edge a quarter, a corner an eighth. The weighting makes each cube neutral, which is exactly the physical condition, and the sequence then converges quickly and monotonically. It is a change in the summation region, not in the terms.

Ewald’s method splits each point charge into a sharply peaked part and a smooth part. The sharp parts are summed in real space, where they converge quickly because the smooth remainder has been removed; the smooth parts are summed in reciprocal space, where a smooth function has few components and converges quickly there instead. The split is exact and the parameter dividing the two is arbitrary, which gives a check: the answer must not depend on it.

The second is a reciprocal-space method by construction, and reciprocal space is what a real-space neighbour method does without. That is not a coincidence — the reason a long-ranged interaction is awkward in real space is the same reason it is convenient in the transformed one, since a slowly varying function has a compact transform.

The kind of convergence a real-space band calculation does have is a histogram of two thousand computed levels agreeing with its closed form to better than 0.005 across the interior, with the disagreement bounded and stated. That is convergence of a quantity that has a limit; a conditionally convergent series has none until an ordering is chosen.

The same shape of problem, elsewhere #

Conditional convergence is not a curiosity confined to ionic crystals, and two other places it appears are worth mentioning because they make the pattern recognisable.

Dipolar sums. A crystal of dipoles has an interaction falling as $1/r^3$, and a shell at distance $d$ contains $d^2$ dipoles, so each shell contributes $d^2/d^3 = 1/d$ — the harmonic series, which diverges. The sum survives only by orientational cancellation, and the result genuinely depends on the shape of the sample. That is not a computational artefact: a needle-shaped and a disc-shaped crystal of the same material have measurably different depolarising fields, and the mathematics is telling the truth.

Gravitational sums have the identical structure and, since gravity does not alternate in sign, no cancellation to rescue them at all — which is one reason cosmology worries about this class of difficulty rather more than crystallography does.

The general pattern: an interaction falling as $1/r^p$ in $d$ dimensions gives shells contributing $r^{d-1-p}$, which converges only for $p > d$. Coulomb in three dimensions has $p = 1$ and $d = 3$ and is far outside that condition; dispersion has $p = 6$ and is comfortably inside it, which is why a dispersion sum can be truncated at a few shells and an ionic one cannot.

The one-dimensional case, which can be done here #

There is a version of the problem small enough to compute completely, and it is worth doing because it shows the cancellation working.

Take an alternating chain of charges — $+, -, +, -$ — at unit spacing. The energy of one ion is

$$-2\left(1 - \tfrac12 + \tfrac13 - \tfrac14 + \cdots\right) = -2\ln 2 = -1.386,$$

so the one-dimensional Madelung constant is $2\ln 2 = 1.386$, and it comes out of the alternating harmonic series.

Everything difficult about the three-dimensional case is visible in that expression. The alternating harmonic series converges, and it converges conditionally: the terms fall as $1/n$, whose sum diverges, so it survives entirely on the alternation. Rearranged, it gives a different answer — the standard textbook demonstration of Riemann’s theorem uses exactly this series, rearranging it to converge to $\tfrac32 \ln 2$.

So a one-dimensional ionic crystal has a Madelung constant that depends on the order in which its ions are counted, and the physically correct order is the one that keeps the counted region neutral. The three-dimensional case is worse only in that the terms grow rather than merely failing to shrink fast enough.

That the answer is $2\ln 2$ is also worth noticing for its own sake. A structural constant of a crystal has come out as a logarithm, which is not a shape anybody would guess from a picture of alternating charges, and it arrives from the same place the logarithm in a chain cannot stay even does — a sum over a system with no length scale in it.

The surface the arithmetic keeps mentioning is a real surface #

The surface charge is a real thing with a real energy is the sentence that resolves the mathematics, and it is worth following where it leads, because it turns a difficulty about summation order into a prediction about crystals that anybody can check by dropping one.

If the value of the sum depends on how the crystal is terminated, then terminating it two different ways gives two different energies — and a crystal has to be terminated somehow. So the conditional convergence is not an artefact of taking an infinite limit. It is the statement that an ionic crystal’s energy depends on which planes its faces are, and by an amount that does not fall off with size.

Rock salt makes the point in two planes. Cut it on a cube face and every layer exposed contains equal numbers of sodium and chloride ions: each layer is neutral, carries no dipole, and the stack of them converges. Cut it perpendicular to a body diagonal instead and the layers alternate — one entirely sodium, the next entirely chloride — so each layer is charged and the stack is a capacitor with as many plates as the crystal is thick. Its electrostatic energy grows with thickness rather than converging at all.

The crystal’s response is the one the arithmetic predicts. Sodium chloride cleaves on its cube faces, cleanly and reproducibly, and that is the face a grain of table salt shows. The polar face does not exist as a clean termination: a crystal forced to expose one reconstructs, faceting into cube planes, taking on adsorbed charge, or reorganising its outermost layers until the dipole is cancelled. Every route is a way of refusing to be the divergent sum.

So the boundary this essay is about has an observable on the other side of it. A conditionally convergent series is usually presented as a technicality to be handled by a summation trick, and it is the reason a class of crystal faces is unstable, the reason ionic crystals cleave where they do, and the reason a polar oxide surface is a research subject rather than a plane.

That also sharpens what is being deferred rather than merely admitting it. What a neighbour-by-neighbour method cannot compute is not one constant of order two. It is anything whose answer depends on where the crystal stops — which includes every surface energy, every cleavage preference and every charged defect — and those are exactly the quantities for which the covalent method’s locality was doing the work.

What is worth taking from a boundary #

A computation that a method cannot do is worth studying for two reasons.

The first is honesty about reach. A tight-binding treatment computes bands, gaps, densities of states, defect levels and surface states, all from finite matrices, and it would be easy to take away the impression that anything about solids is available by the same route. It is not, and the ionic case is the sharpest counterexample: an entire class of materials whose binding cannot be written down at all in that framework.

The second is that the reason for the boundary is more instructive than another computation would have been. A method works because of a property of the thing it describes — here, that the covalent interaction has a range — and finding the case where that property fails is the best way to understand what the method was relying on. Every exact tight-binding result, including the identity between the second moment and the coordination, depends on the interaction reaching one shell and stopping.

That dependence is invisible while every example satisfies it. The lattice sum is the example that does not, which is what makes it worth understanding even by a method that cannot perform it.

A chain’s levels crowd into a fixed interval as it grows, and every quantity summed over them converges because the interval does not move. The lattice sum has no such interval: its terms fall as one over the distance and its partial sums depend on the shape of the region summed over.

A bound state on a defect is the clearest case of a quantity with a range: it lives on a handful of sites and is exact at every system size. Everything a tight-binding treatment computes about extended structures is of that kind, and the Coulomb sum is not.

Two further readings make the boundary sharper, and both are about what a bounded interaction gives that an unbounded one does not.

And the same comparison read as a departure rather than as three values.

What a reader should take away about ionic solids #

Three statements survive the boundary and are worth having.

The pair term is computable and is large, and it is the only one of the three a neighbour sum produces by itself. A sodium and a chloride at $2.8$ ångströms are bound by $5.14$ electronvolts, from Coulomb’s law and nothing else.

The lattice sum multiplies it by a structure-dependent constant of order two, which has to be taken from somewhere else. For rock salt the factor is $1.7476$, for caesium chloride $1.7627$, for zinc blende $1.6381$. Those constants are close together, which is why radius-ratio arguments about which structure an ionic compound takes work as well as they do — the electrostatics barely distinguishes the candidates and the packing decides.

The constant is a property of the arrangement rather than of the elements. Every rock-salt compound has the same $1.7476$, so the whole variation in lattice energy across the alkali halides comes from the separation and the charges. That is a strong and useful regularity, and it is available without performing the sum, provided somebody else has performed it once.

Which is, in the end, the ordinary situation for any argument of this kind. Nothing is lost by quoting a constant that somebody computed properly, so long as the quoting is marked as quoting — the discipline worth keeping is not that everything must be derived, but that a reader should always be able to tell which numbers were.