The bond that weakens as neighbours multiply

A carbon atom in diamond has four neighbours and a C–C bond worth about 350 kJ/mol. A copper atom in its metal has twelve neighbours and a Cu–Cu bond worth about 30. The two structures have cohesive energies within a factor of two of each other, and their individual bonds differ by more than ten.

That pattern — more neighbours, weaker bonds, comparable totals — is general enough to have a name in metallurgy and a rule of thumb attached to it, and the rule of thumb is that the cohesive energy goes as the square root of the coordination number.

The square root is a second-moment argument, and the second moment can be computed exactly.

Comparing coordinations fairly #

The comparison needs three structures that differ in coordination and in nothing else, which is harder to arrange than it looks.

A finite lump of any shape has a surface, and surface sites have fewer neighbours than interior ones. The fraction of them falls as the lump grows, so a comparison across dimensions on finite lumps measures the surface as much as the coordination — and how slowly that fraction falls is computed in the end is the hardest place to bind.

So the structures used here are wrapped in every direction: a ring of 64 sites, a 16 × 16 torus, an 8 × 8 × 8 three-torus. Every site has exactly two neighbours along each direction, so the coordination is exactly $2d$ and there is no surface, no edge, and no site that differs from any other.

That is a graph and not a lattice. There are no lattice vectors, no reciprocal space and no Bloch theorem anywhere in it: the adjacency matrix is diagonalised whole, and every quantity below is a sum over its eigenvalues. That is the same line this field has held since a band with no structure in it worked out what the omission costs.

The one quantity that is exactly the coordination #

The mean of $x^2$ over all the levels is

$$\langle x^2\rangle = \frac{1}{n}\mathrm{Tr}(A^2)$$

and $\mathrm{Tr}(A^2)$ counts closed walks of length two — one for each ordered pair of bonded sites. So the second moment is the average number of neighbours, exactly, for any structure whatever, with no periodicity, no limit and no approximation. It is a statement about a graph.

Measured on the three structures it comes out at $2.000000000$, $4.000000000$ and $6.000000000$, agreeing with the counted coordination to within $3\times10^{-12}$. The width of a band is a count of neighbours established that identity for chains and rings, where the coordination is two in every case; this extends it to structures where the number is something else.

The full width of these bands is also proportional to the coordination — $4$, $8$ and $12$, running from $-Z$ to $+Z$ — but that is a special feature of a product structure rather than a general result. A tree in which every site has two neighbours has a band running from $-2\sqrt{1}$ to $+2\sqrt{1}$ by the same count, and the width is the fragile statement where the moment is the robust one.

The square root, and where it comes from #

If a band of $n$ levels has a shape that does not change and only a scale that does, then every measure of its spread is proportional to every other, and the scale is fixed by the moment: $\sqrt{\langle x^2\rangle} = \sqrt{Z}$.

The energy at half filling is the sum of the lower half of the levels, which is the scale times a shape-dependent number. So

$$\text{energy per site} \propto \sqrt{Z}, \qquad \text{energy per bond} \propto \frac{\sqrt{Z}}{Z} = \frac{1}{\sqrt{Z}}$$

and there is the rule. Bond strength falls as one over the square root of the number of bonds. It is the tight-binding version of bond-order conservation, and it is the reason a twelve-coordinate metal has bonds a third the strength of a four-coordinate covalent solid’s.

The assumption is the phrase “a shape that does not change”, and it is checkable.

The exponent, measured #

Binding per site, from the sum over occupied levels at one electron per site:

The two directions are unambiguous and are the essay’s title. More neighbours bind a site more strongly, and each bond is worth less.

The exponent is not a half. Between two and four neighbours it is $0.341$; between four and six it is $0.507$. So the square-root rule is good where the coordination is high and poor at the bottom of the range, and it is the chain that is the outlier.

The reason is the shape assumption failing exactly where the figure above shows it failing. A chain’s density of states diverges at both band edges, so its occupied half is concentrated at the most bonding energies and it binds more at half filling than a band of the same moment with weight spread evenly would. The square-root rule underestimates the chain, which is the same as saying the measured exponent from two to four is below a half.

The check is written so that this is required rather than excused: at least one measured exponent must depart from a half by more than five hundredths. Requiring the square root would be requiring the approximation.

Why the exponent should approach a half rather than equal it #

The measured exponents rise towards a half as the coordination grows, and the reason is a limit theorem rather than an accident.

As the coordination rises the density of states of a tight-binding band becomes more and more nearly Gaussian. That is a central-limit statement: a site’s level is a sum of contributions from its neighbours, and a sum of many similar independent contributions tends to a Gaussian whatever the individual distribution looks like. A Gaussian is fixed entirely by its second moment, so the shape-does-not-change assumption becomes exact in the limit and the exponent becomes exactly a half.

At two neighbours the sum has two terms and there is nothing central-limit about it: the density of states is the arcsine distribution, with divergences at both edges, and its shape is as far from Gaussian as a bounded distribution gets. At six the shape is already much closer, which is why the exponent has risen from $0.341$ to $0.507$.

Overshooting a half slightly at four-to-six is within the accuracy of a two-point fit on a finite system, and is not a claim this essay makes.

The chemical form of the statement is worth having. The square-root rule is a rule for close-packed structures and a poor one for chains. Metals, which are the systems it is quoted for, are twelve-coordinate or eight-coordinate and are in the range where it works. A polymer backbone, which is two-coordinate along its chain, is in the range where it does not — and the corresponding chemical fact, that conjugated chains bind more per bond than the rule predicts, is the same statement.

What the numbers do and do not explain #

The pattern the essay opened with is now half explained, and being clear about which half is the point.

Explained: the direction and roughly the size. A twelve-coordinate structure has bonds worth roughly $\sqrt{4/12} = 0.58$ of a four-coordinate one’s per bond, and three times as many of them, giving a cohesive energy $\sqrt{3} = 1.73$ times larger. Diamond’s cohesive energy is 7.4 eV per atom and copper’s is 3.5, which is the wrong way round — and that is the next paragraph.

Not explained: everything about which atoms these are. The comparison above holds the interaction strength fixed and varies only the coordination. Carbon and copper differ in the strength of the interaction between neighbours by a large factor, and nothing here computes it. The $\beta$ in the tight-binding model is a parameter with no value, exactly as $\beta$ is in every Hückel calculation.

So the honest reading is that the coordination scaling explains the ratio between structures of the same element, and says nothing about a comparison across the periodic table. Carbon in graphite against carbon in diamond is a comparison the scaling addresses; carbon against copper is not.

Not explained: repulsion. A tight-binding band energy is entirely attractive and would have every structure collapse. What stops it is a repulsive term this model does not contain, and what holds a solid together is explicit that the equilibrium of a real solid is a balance between the two. Everything here is a statement about the attractive half at a fixed geometry.

Half filling, and why it is the case computed #

Every energy above is at one electron per site, which is half filling, and the choice deserves a sentence.

Half filling is where the binding is greatest, which half filled is as bonded as it gets computes: no filling binds more, and a full band binds exactly nothing because the sum over every level of an adjacency matrix is its trace, which is zero. So the comparison is being made at each structure’s best, which is the fair place to make it.

It is also where the second-moment argument is cleanest. At other fillings the answer depends on the shape of the band in the occupied region specifically rather than on its overall spread, so the shape assumption is doing more work and the square-root rule is worse.

For real metals the filling is whatever the electron count gives, and the cohesive energy across a transition series follows the band filling in a curve that peaks in the middle — tungsten and molybdenum at the top, the alkalis and the coinage metals at the bottom. That parabolic trend is the filling dependence rather than the coordination dependence, and it is a different figure.

The largest structure worth diagonalising densely, and why that is the limit #

The three structures here have 64, 256 and 512 sites, and the largest is close to the ceiling.

The solver is a cyclic Jacobi diagonalisation, which costs about $n^3$ and holds an $n \times n$ matrix. At 512 sites that is a matrix of a quarter of a million entries and a diagonalisation of a few seconds, which is quick. At 2,000 sites — the size the chain calculations reach, because a chain’s matrix is tridiagonal and much cheaper — a dense diagonalisation would take minutes.

So the three-dimensional case is the constraint, and it is worth checking whether 512 sites is enough. The second moment is exact at every size, so that half is unaffected. The band energy per site is not: at $6^3$ it comes out at $2.0000$ and at $8^3$ at $1.9787$, a change of one per cent. That is small against the thing the essay measures — the exponent departs from a half by a third at the low end — and it is not nothing, so the exponents quoted are for the structures computed rather than for the infinite limit.

The limit that would bite is a higher coordination. A twelve-coordinate close-packed structure is the case metals are actually in, and building one as a wrapped graph is easy; diagonalising a large enough one is not, because the coordination itself does not raise the cost but the size needed for convergence does. The essay therefore reports two, four and six and extrapolates in words rather than reporting twelve and pretending the arithmetic reached it.

What the identity is worth beyond this #

The second-moment identity is the most portable thing in this essay and it is worth separating from the cohesion argument.

It says that one number about a band is a count and not a measurement. Given any structure at all — crystalline, amorphous, a cluster, a molecule, a random graph — the mean of the squared level energies is the average coordination, exactly. That is a constraint every model of that structure’s electronic structure has to satisfy, and it can be checked without solving anything if the moment is available.

It also says which quantities are safe to reason with. The moment is exact; the width depends on how the neighbours are arranged; the shape depends on the topology in detail. Arguments built on the moment carry over between structures and arguments built on the width do not, which reverses the usual emphasis — the width is what gets quoted and the moment is what is true.

The same argument in a molecule #

The scaling is a statement about graphs and it does not need a solid, which is worth showing on something small enough to check by hand.

Take a carbon in a chain against a carbon in a branched skeleton. In a linear polyene each carbon has two π neighbours; in a fused ring system such as naphthalene the interior carbons have three. The second moment of the π band is two in the first case and, averaged over the molecule, higher in the second — and naphthalene’s spectrum has already been computed for a different reason.

What follows is that the π bond orders in a fused system are lower than in a chain, which is a well-known fact usually stated as delocalisation being spread more thinly. Bond order from the eigenvectors computes benzene’s uniform 0.667 and naphthalene’s non-uniform set running from 0.518 to 0.725, and the average of the second is below the first for the same reason a six-coordinate site’s bonds are weaker than a two-coordinate one’s.

So the coordination scaling and the bond-order arithmetic are two readings of one identity, one applied to a solid and one to a molecule. That is the continuity claim the beyond field has been making since where two-centre bonding stops, arriving from the solid-state side.

How much of the opening comparison this actually explains #

The essay opens with carbon and copper: four neighbours against twelve, bonds of 350 kilojoules a mole against 30, a factor of nearly twelve. It is worth returning to that with the exponent in hand, because the scaling accounts for much less of it than the framing suggests.

Binding per site goes as the square root of the coordination, so binding per bond goes as its inverse square root. Between four neighbours and twelve that is a factor of $\sqrt{3} = 1.73$ — and the measurement here, across two neighbours to six, gives 1.93, in the same place.

So the coordination explains a factor of about two of a factor of twelve. The remaining six lie entirely in $\beta$, which is a property of the atoms rather than of how many of them are adjacent: a carbon 2p orbital overlapping another carbon 2p at 1.54 ångström is simply a far stronger interaction than a copper 3d overlapping a copper 3d at 2.56.

That is worth stating because the rule of thumb invites the opposite reading — that a metal’s bonds are weak because there are many of them. Some of it is, and most of it is not. The scaling is a fair comparison only between structures of the same element, where $\beta$ is held fixed and the coordination is the only thing moving.

What the neighbour count adds to cohesion #

Two arguments mark out the boundaries of cohesion. What holds a solid together is about the kinds of binding and the balance between attraction and repulsion; the lattice sum whose answer depends on the order of adding is about a periodic sum that cannot simply be added up, and says why.

This essay is the one thing between them that a graph can settle: how the attractive half scales with how many neighbours a site has, computed rather than assumed, with the exponent measured and the approximation behind the usual rule named.

The open question needs the repulsive half. An equilibrium structure is where attraction and repulsion balance, and a comparison between structures at their own equilibria is a different and more useful comparison than one at a fixed geometry. That requires a repulsive term with a form and a parameter, both of which would have to be quoted — and a quoted parameter in a comparison is exactly the thing that makes the comparison uninformative. Naming it as the gap is better than filling it with a fitted number.