What holds a solid together

Everything in this field so far has been about one kind of binding: an electron shared between neighbouring atoms, described by an interaction $\beta$ and a matrix of neighbours. That is the covalent case and it is where tight-binding arithmetic lives.

It is also the kind a chemist thinks of first, and it accounts for a minority of the matter anybody handles. It is not the only way matter holds together, and the others are not merely weaker versions of it. They are different in kind — different distance dependence, different origin, different sensitivity to what the atoms are — and putting them on one scale is worth doing because the scale spans a factor of several hundred.

The four, and where each number comes from #

A shared pair. The covalent case, quoted as $\beta \approx 2.5$ electronvolts — the value fitted to benzene’s electronic spectrum, and the unit every energy in this field is expressed in. It is a fitted parameter and it is labelled as one.

Two opposite charges. Coulomb’s law at contact: $e^2/4\pi\varepsilon_0 r$, which at $2.8$ ångströms — the sodium–chloride distance — gives $5.14$ electronvolts. Nothing is fitted; the only input is the separation.

Two aligned dipoles. The interaction of two point dipoles head to tail, $2\mu^2/4\pi\varepsilon_0 r^3$. Taking water’s dipole moment of $1.85$ debye and a separation of $3.0$ ångströms — roughly a hydrogen bond — gives $0.158$ electronvolts. The dipole moment is quoted from measurement; the arithmetic is not.

Two induced dipoles. The London expression for dispersion, $\tfrac34 \alpha^2 I / r^6$, with argon’s polarisability of $1.6$ cubic ångströms, its ionisation energy of $15.6$ electronvolts and its nearest-neighbour separation of $3.8$ ångströms. That gives $0.0099$ electronvolts, or about ten millielectronvolts.

The division between what is quoted and what is computed matters here. Dipole moments, polarisabilities and ionisation energies are measurements and should be quoted; the energies that follow from them are arithmetic and are done here.

The ordering, which was wrong when it was assumed #

The expected ordering is covalent above ionic above dipolar above dispersion, the sequence in which bond types are usually taught.

The numbers refuse it. Ionic comes out at $5.14$ and covalent at $2.50$, so a single ion pair at contact is worth twice a shared pair.

That is not a defect in the numbers. It is the comparison being wrong, and it is worth keeping rather than tidying away, because the wrongness is where the physics is.

An ionic solid’s binding is a lattice sum. Every ion feels every other ion in the crystal, the contributions alternate in sign as the shells of neighbours alternate in charge, and the series converges to the Madelung constant times the nearest-neighbour term. For rock salt that constant is $1.7476$, so the actual binding per ion pair is substantially larger than the pair term above.

A covalent solid’s binding is not a sum of that kind. $\beta$ is a two-centre integral between neighbouring orbitals, and a third atom two shells away contributes essentially nothing to it — the interaction falls off with overlap, which decays exponentially.

So the two entries are answers to different questions. One is a term in a series that has to be summed and the other is essentially the whole thing. The lattice sum that depends on the order of adding takes up what the summation involves and why it is not attempted here.

The distance dependence is the real classification #

A more useful way to sort the four is by how fast each falls off, because that decides almost everything about the resulting material.

Coulomb, $1/r$. The slowest possible decay, and slow enough that the sum over a lattice does not converge absolutely — the order of summation matters, which is a genuine mathematical difficulty and not a technicality.

Dipole–dipole, $1/r^3$. Fast enough to converge but slow enough that orientation matters enormously: two dipoles head to tail attract and side by side repel, so a dipolar solid’s structure is decided by getting the orientations right.

Dispersion, $1/r^6$. Effectively a nearest-neighbour interaction. Doubling the distance divides it by sixty-four, so only the closest neighbours matter and the number of them is what counts.

Covalent, exponential. Falls off faster than any power, because it depends on the overlap of two exponentially decaying functions. This is why the tight-binding model of this field can get away with a list of neighbours and nothing else: the second neighbours contribute a few per cent and the third essentially nothing.

That last observation is what licenses everything else in the field. The matrix of ones and zeroes used throughout is not a simplification of a longer-ranged interaction; it is a good description of an interaction that really does stop.

What each kind of solid is like #

The four interactions produce four recognisably different classes of material, and the differences follow from the numbers above.

Covalent solids — diamond, silicon, silicon carbide — are hard, high-melting, and often brittle. Their bonds are strong and directional, because they depend on orbitals overlapping in particular directions, so deforming them means breaking bonds rather than sliding.

Ionic solids — the halides, many oxides — are high-melting and brittle in a different way. Their binding is strong and non-directional, but sliding one plane past another brings like charges into contact and the crystal cleaves.

Molecular solids — argon, ice, most organic compounds — are soft and low-melting. Argon melts at 84 kelvin, and the arithmetic says why: ten millielectronvolts per pair, twelve neighbours, and a thermal energy of about seven millielectronvolts at that temperature.

Metals are the odd one out and are not on this scale at all — not because their binding is too weak or too strong to fit, but because it is not a pair interaction and cannot be quoted per pair, because their binding is not a pair interaction. It is the delocalisation energy of a partly filled band, and what a metal actually is is about the state rather than about the bond. Metals are dense, malleable and conducting for reasons that all trace back to the binding being shared over many neighbours rather than assigned to pairs.

Why dispersion is not negligible #

The smallest number on the scale deserves defending, because “negligible” is the usual word and it is wrong.

Ten millielectronvolts per pair is small. Multiply it by twelve neighbours in a close-packed arrangement and it is $120$ millielectronvolts per atom, which is about $12$ kilojoules per mole — comparable to argon’s actual heat of sublimation of $7.7$ kilojoules per mole, so the estimate is right to within a factor that a two-parameter model has no business achieving.

More to the point, it is the only interaction available between two non-polar molecules. Everything about a molecular crystal — which structure it takes, what it melts at, how it dissolves — is decided by an interaction of this size, because there is nothing else. So is protein folding, so is the operation of a chromatography column, and so is why a gecko can walk up glass.

The interaction that scales worst with distance and is smallest per pair is also the one that is always present, between everything, and that combination is why it accounts for more chemistry than its magnitude suggests.

Melting points, which the numbers predict badly and usefully #

A crude test of whether the scale means anything is to compare it with melting points, since melting is where thermal energy overcomes cohesion.

Argon melts at 84 K, where the thermal energy is about 7 millielectronvolts. Its computed pair interaction is 10 millielectronvolts, and it has twelve neighbours, so the binding per atom is about 120 millielectronvolts — seventeen times the thermal energy at melting.

Sodium chloride melts at 1074 K, thermal energy about 93 millielectronvolts, against a pair term of 5.14 electronvolts. That is a ratio of about fifty-five.

Diamond melts above 3800 K, thermal energy about 330 millielectronvolts, against 2.5 electronvolts per bond and two bonds per atom. Ratio about fifteen.

Those three ratios — 17, 55, 15 — are not equal, and it would be surprising if they were, since melting involves entropy as well as energy and the number of neighbours differs between the three structures. What they do establish is that the ratio is of order tens in every case, which means the scale is measuring something real. A model that produced ratios of 2 and 500 for two materials would be measuring an artefact.

The one that stands out is sodium chloride, and it stands out in the direction the previous section predicts: the pair term understates the ionic binding by the Madelung factor, so the true ratio is smaller and closer to the other two. The discrepancy is the missing lattice sum making itself visible in a comparison it was not invited into.

Hydrogen bonding, which is not on the scale and should be #

There is an omission worth naming, because it is the interaction most chemists care about most.

A hydrogen bond is stronger than the dipole–dipole estimate above suggests — typically 0.2 to 0.3 electronvolts rather than the 0.16 computed for two point dipoles — and the excess is not dipolar in origin. It comes from a genuine, if weak, sharing of electrons between the hydrogen and the acceptor’s lone pair, which is to say it is partly covalent.

That is why it does not fit the classification. The four rows on the figure are meant to be four mechanisms, and a hydrogen bond is two of them at once in proportions that depend on the pair involved. Putting it on the scale as a fifth row would suggest it was a fifth mechanism.

A one-electron treatment has a partial view of it. Waters lone pairs are not a pair computes what the acceptor’s orbitals actually look like, and the answer — that they are not two equivalent rabbit ears — matters for the directionality that makes hydrogen bonding structurally decisive. What it cannot do is compute the interaction energy, for the same reason it cannot compute any of the others: no positions.

The covalent case is the one that cannot be written as a pair interaction at all. A chain’s levels come from an interaction that reaches one neighbour and stops, and its binding is the sum of what every occupied level gained by being part of an extended system — a quantity with no pairwise decomposition, which is why a filled band belongs on the scale above and does not belong in the same column as the other three.

What “at contact” is hiding #

The figure’s caption says per pair, at contact, and the ordering it produces was called a fact about the comparison rather than about the numbers. It is worth saying precisely which feature of the comparison did it, because the answer is in three words of the caption.

Contact is not one distance. A covalent bond sits near one and a half ångström, an alkali-halide ion pair near two and a half to three, and a pair of molecules in van der Waals contact near four. So the four rows are evaluated at four different separations, and the axis carries a second variable that does not appear on it.

That matters because the four fall off at wildly different rates — which is the essay’s own classification. Reading the same four interactions at one common distance instead gives a different picture, and the direction is predictable from the exponents. A covalent interaction decays exponentially, with a decay length of a few tenths of an ångström, so at three ångström it is a small fraction of its value at a bond length. A Coulomb attraction between unit charges has fallen only in proportion to the distance and is still worth several electronvolts there. Comparing at three ångström would therefore put the ionic case further ahead than the figure does, not less.

So the choice of comparison is not neutral, and neither available choice is obviously right.

At a common distance the comparison is one function evaluated four ways, which is what a graph normally means — and it flatters whichever interaction has the longest reach, by evaluating the short-ranged ones somewhere they were never meant to act.

At each one’s own contact the comparison is fair to each mechanism on its own terms, and the price is that the axis is four separate statements sharing a scale rather than one measurement repeated. Nothing on the figure is being varied; four numbers from four models are being placed side by side.

The second is the right choice and the figure makes it, which is why the caption names it. What the caption cannot show is that the horizontal variable moved between rows, and a reader who takes the scale as a curve sampled at four points has read a comparison that was never offered.

The general form of the caution arrives from many directions. A number is a number about a process, and putting two on one axis claims that the two processes are comparable. Here they are — barely, and only because each was evaluated where its own mechanism actually operates.

Why the strongest is not the one that decides #

A final observation that the scale makes available and that is easy to miss.

The largest number on the figure belongs to the interaction with the least say in what a material’s structure is. Coulomb attraction between opposite charges is enormous and completely non-directional — it depends on distance and not at all on angle — so it tells a crystal to pack its ions as closely as possible and nothing else. The structures of the alkali halides are decided almost entirely by the relative sizes of the ions, which is why they are predictable from radius ratios and why so few structure types account for so many compounds.

The smallest number belongs to an interaction that is also non-directional, and molecular solids are correspondingly packed according to shape.

The interaction in the middle of the scale — the covalent one — is the one that is directional, because it depends on orbitals pointing at each other. That is what makes carbon capable of diamond and graphite and buckminsterfullerene from the same atom, and it is what the whole of where the atoms go is about at molecular scale.

So the ordering by strength and the ordering by structural importance are different orderings, and the second is arguably the more useful one. What decides a structure is not how much energy is available but how much the energy varies with the arrangement, and those are not the same quantity.

What tight binding can and cannot say about the four #

Being explicit about the reach matters here more than in most places.

The covalent case is what tight binding computes, and it computes it as a matrix of neighbours with an exponential interaction, which is the right shape.

The ionic case can be stated per pair, as above, and cannot be summed here. The lattice sum needs positions and this model has none.

The dipolar case can be stated per pair from a measured dipole moment, and the orientational sum over a structure needs the same positions. The dipole is not a sum of bonds sets out where a molecular dipole comes from, and its warning applies here: a dipole is a property of the whole charge distribution, and treating it as a point at the centre of a molecule is an approximation that gets worse as the molecules get closer.

The dispersion case is quoted from a model with two measured inputs, and nothing about it is derived here. The London expression itself comes from second-order perturbation theory on two fluctuating dipoles, which is well outside a tight-binding model.

There is a fifth kind that does not appear on the figure at all, and its absence is deliberate. Metallic binding is not a pair interaction — it is the energy an electron gains by being delocalised over a whole structure, which is exactly what every band calculation in this field computes. It cannot be drawn on a scale of pair energies because it is not one, and putting it there at some notional per-neighbour value would be inventing a quantity to make a figure tidy.

So one of the four is native, one can be stated but not summed, and two are quoted. That distribution is honest about what a tight-binding model is for, and the figure labels each row with the model it came from so a reader can see which is which without taking it on trust. A scale on which one entry is derived and three are borrowed is worth having, provided it says so on its face.