Two structures with the same neighbours

There is an exact identity: the mean of the squared levels of any structure is the average number of neighbours per site. Not approximately, not for a lattice — for any graph at all, because the trace of the square of an adjacency matrix counts closed walks of length two and there is exactly one of those per bond in each direction.

Its consequence is easy to read off: the band’s width goes as the square root of the coordination, so a bond in a twelve-coordinate structure is worth less than a bond in a four-coordinate one, and cohesive energies do not go as the number of neighbours.

Both of those are about a single number extracted from a distribution of levels. This essay asks what the number leaves out, and the answer is everything except the first term of a series.

Five structures, four neighbours each #

Take five structures with sixty sites and four neighbours per site: a wrapped six-by-ten lattice and four rings in which every site is joined to the ones a fixed pair of steps away.

Each is four-connected, so each has exactly the same second moment. The identity is not approximate and holds to nine decimal places in every case, which is a check on the construction rather than a result.

The spread is the finding. If coordination decided the binding, that column would be constant.

What the moments are counting #

A moment of a level distribution is a count of walks, and the correspondence is exact rather than an analogy.

$\langle x^k \rangle$ is $\mathrm{Tr}(A^k)/n$, and $\mathrm{Tr}(A^k)$ counts closed walks of $k$ steps on the structure. So:

$\langle x^2 \rangle$ counts out-and-back, one walk per bond per direction, which is the coordination.

$\langle x^3 \rangle$ counts triangles. It is zero for any structure with no odd cycles at all, which is what bipartite means, and it is 6 for the ring joined at one and two, where every site sits in two triangles.

$\langle x^4 \rangle$ counts squares — and also the out-and-back-and-out-and-back walks, which every structure has, so it is never zero. For a four-connected structure with no four-cycles it is 36; the ring joined at one and twenty-nine has 48, because that joining creates four-cycles in abundance.

The series continues, and each term counts a longer loop. The binding is not any one of them: it is a sum over the whole distribution of levels, and the distribution is determined by all of the moments together.

Why more four-step walks means less binding #

The direction of the effect is not arbitrary and can be argued before it is computed.

Binding at half filling is the sum of the magnitudes of the occupied levels, which for a symmetric distribution is the mean of $|x|$. The second moment fixes $\langle x^2 \rangle$; among distributions with a given second moment, the one with the largest $\langle |x| \rangle$ is the one concentrated at $\pm\sqrt{\langle x^2 \rangle}$, and any spreading away from those two points lowers $\langle |x| \rangle$ while raising $\langle x^4 \rangle$.

So at fixed coordination, more weight far from the two shoulders means a larger fourth moment and a smaller binding. The structure with $\langle x^4 \rangle = 48$ binds at 1.2756 against 1.63 for the ones at 36, and the ordering is what the argument predicts.

That is one direction of an inequality rather than a rule, and the essay is careful not to claim more, because the four structures at or near 36 do not order by their fourth moments at all.

Two structures the series cannot separate #

Two of the five agree on the second moment, on the third and on the fourth: the wrapped six-by-ten lattice and the ring joined at one and seven both have 4.000, 0.000 and 36.000.

Their bindings are 1.63148 and 1.60929, differing by 1.4 per cent.

Their sixth moments are 402 and 400.

So three moments do not settle the binding, and the fourth term that separates these two is a count of six-step closed walks — hexagons, and every longer combination of shorter loops. There is no finite number of moments that settles it in general, because the binding is a functional of the whole distribution and matching $k$ moments constrains a distribution without fixing it.

That is the honest form of the result. The moments are an expansion, the first term is exactly the coordination, the second and third are named quantities a chemist recognises — triangles and squares — and the expansion does not terminate.

The field usually meets this statement in a coarser form. Where the states pile up draws the level distributions of one-, two- and three-dimensional structures, which differ in shape while the second-moment identity holds for all of them — so dimensionality is one of the things the higher moments encode. What the five structures here add is that dimension is not the only thing: all five are one dimension or two, and they differ anyway.

What a bond is worth, and why it is not transferable #

The most common way of using a cohesive energy is to divide it by the number of bonds and call the result a bond energy, then add up bond energies to estimate something else.

Divide each of the five bindings by four and the spread survives intact: a bond in the least bound structure is worth 0.319 and one in the most bound 0.409, in the same units, in structures whose atoms have identical numbers of identical neighbours.

So a bond energy in this model is not a property of a bond. It is a property of a bond and its surroundings out to some distance, and the distance is however far the loops go that the moments are counting. That is an unfamiliar way of saying something familiar: the reason bond additivity works as well as it does in organic chemistry is that the structures it is applied to are locally alike out to several bonds, and the reason it fails for strained and bridged systems is that they are not.

The same failure shows up in this collection where a surface is counted as broken bonds: an atom in the surface keeps 91.2 per cent of its binding while keeping 83.3 per cent of its bonds, because the bonds that survive get stronger when their competitors are removed. That is the second moment’s square root doing the work. This essay’s twenty-two per cent is the next term of the same expansion.

The square-root rule’s own illustration is the layer-by-layer binding of an open block, where the coordination changes with depth and the binding does not follow it linearly. Every atom in its interior has the same neighbours as every other and the same topology, which is exactly why that argument could stop at the second moment and this one cannot: the surface changes a count and the five structures here change a shape.

What survives of the square-root rule #

None of this overturns the square-root rule, and saying exactly what it does to it is the useful part.

The rule compares structures of different coordination, where the second moment differs and dominates: a twelve-coordinate structure’s band is $\sqrt{3}$ times as wide as a four-coordinate one’s, and that factor swamps everything the higher moments do. The correction this essay measures is twenty-two per cent within one coordination; the effect the rule is about is a factor of nearly two between coordinations.

So the two results compose into one statement. Coordination sets the scale of the binding and the topology sets a correction of order twenty per cent — which is exactly the size of difference that separates one crystal structure of an element from another, and is therefore not a correction anybody can ignore when the question is which structure a material takes.

Every structure in this essay has a distribution of one general kind — bounded, symmetric, with its second moment pinned — and differs only in where the weight sits between the edges. The limiting case computed on two thousand sites is in where the states pile up, and it is the shape all of this is a perturbation of.

The same argument, in a molecule #

The moments are not a solid-state device, and the smallest case where they say something is a molecule with four atoms in it.

A four-site chain and a four-site ring have the same number of atoms and different numbers of bonds, so their second moments differ and the comparison is the easy one. A more interesting pair is two structures with the same number of bonds: cyclobutadiene’s ring and two isolated double bonds have four π electrons and four carbons each, and the comparison between them is what a delocalisation energy measures. The ring has more neighbours per site and a wider band; the isolated pairs have all their weight at the shoulders and bind better per bond.

Aromaticity as a shell closure fills rings from three to eight, every one of them two-connected and therefore every one with the same second moment — and their π energies per site differ, for exactly the reason this essay is about.

The molecular version of the finding is therefore already familiar under another name: the shape of a level distribution matters and the count of bonds does not fix it, which is what makes a shell closure a property of the ring size rather than of the bond count.

Which moment first tells two structures apart, and what that names #

That two of these five agree through the fourth moment and first differ at the sixth is not an accident of which five were chosen. It is decidable in advance, from the graphs alone, and saying how turns the moment series into a statement about something a chemist already has a word for.

The $n$-th moment counts closed walks of $n$ steps. A closed walk either retraces itself — out and back, out and back — or it goes round a cycle. The retracing walks are fixed entirely by the degrees, so two structures with the same coordination have the same number of them at every order. Everything by which two equally-coordinated structures differ is a difference in their cycles, and a cycle of length $\ell$ first contributes to the moment of order $\ell$.

So the first moment that separates two structures is the length of the shortest ring they hold in different numbers. Two four-connected structures with no three-, four- or five-membered rings, differing only in how many six-membered rings pass through each site, agree exactly through the fifth moment and part company at the sixth — which is the pattern this essay measured and could not otherwise have expected.

That renames the whole residue. Beyond the coordination, what a structure’s binding depends on is its ring statistics: how many rings of each size run through an atom. It is not an abstract property of a matrix; it is the quantity crystallographers count in a zeolite framework and the quantity used to characterise an amorphous network, where the distribution of five-, six- and seven-membered rings is the standard description of what distinguishes one continuous random network from another.

It is tempting to read the order as a size as well, and this is where the essay has to be careful, because the arithmetic refuses it. A term further down a converging series ought to matter less, so two structures agreeing until the sixth moment ought to be closer in energy than two parting at the fourth. Across the ten pairs these five structures make, that is false.

The reason is the size of the coefficient rather than the order of the term. The sixth-moment pair differs by thirty in four hundred, which is seven per cent of that moment; the fourth-moment pair differs by two in thirty-six, which is five per cent — and the sixth moment weights the far tails of the distribution, where the binding integral is most sensitive. So the order at which two structures separate names which ring length distinguishes them and says nothing about what that ring is worth.

What the order does say is which questions are structurally easy. Two structures differing in their three-membered rings differ already at the third moment, so there is a difference to find at the coarsest level of description — a triangulated arrangement is visibly different from an open one. Two agreeing to the fifth are alike in every ring shorter than six, which is a strong statement about their local geometry however far apart their energies turn out to be. For the close-packed metals, whose ring statistics agree down to the third order, that is why the structures are so hard to tell apart by any local description — and their energy differences are millielectronvolts for reasons this expansion locates rather than predicts.

Where the model stops #

Every structure here is at half filling. The comparison at other fillings is different and can reverse: the binding is an integral over the occupied levels, so which distribution is best depends on how much of it is filled. A structure whose weight sits near the band edges is the best at half filling and the worst at a quarter.

Binding against filling for two structures is a curve rather than a number — half filled is as bonded as it gets draws two of them — and comparing two structures at one filling is a slice through a picture like that. A comparison at a different slice can come out the other way.

Every bond is the same strength. These structures differ only in which sites are joined, never in how strongly, so nothing here is about bond lengths, and a real comparison between two crystal structures has different neighbour distances as well as different topologies.

Only one filling was compared and it is the symmetric one. Half filling is where a bipartite structure’s electron–hole symmetry makes the comparison cleanest, and it is also the case that is as bonded as it gets.

And it is one orbital per site, with no repulsion. The caution this whole field carries applies: a half-filled band is a metal in this model and a strongly correlated material with a half-filled band is not.

The rings joined at distant steps are not structures anybody could build. A ring of sixty joined at one and twenty-nine is a graph with the right coordination and no realisable geometry — the neighbours are not equidistant in any embedding. It is included because the argument is about connectivity, and because the extreme case is what makes the fourth moment’s effect visible.

What a chemist would call this #

The quantity this essay is measuring has a name in two literatures and the names do not know about each other.

In the tight-binding literature it is the moments expansion, and the practice built on it is to compute a handful of moments from the local topology and reconstruct enough of the level distribution to compare two structures. The second moment gives the width, the third the skewness — a structure with triangles has more weight on one side — and the fourth the shape of the middle: whether the distribution is peaked or has two humps. Reading it that way, the twenty-two per cent spread above is a shape effect, and the ring joined at one and twenty-nine has a distribution with the wrong shape for half filling.

In chemistry the same statement is about rings: a four-membered ring is destabilising, a six-membered one is not, and the reason has to do with what the ring does to the level pattern rather than with the bonds it contains. Those two rules are the third and fourth moments of the same expansion, arrived at by looking at molecules instead of at traces.

Neither literature usually says that its rule is one term of a series with no last term, which is what the two tied structures above demonstrate.

What is quoted, and what is computed #

Nothing is quoted except the measured cohesive energies in the comparison figure, which are there for scale.

Every structure is constructed from a rule, every level found by diagonalisation, every moment computed as a sum of powers of the levels, and every binding as a sum over the occupied half. The coordination identity is checked rather than assumed on all five.

Three quantities the second moment does fix #

It is worth listing what the identity settles outright, because the list is short and every item on it is used constantly.

The band width, to within the factor that relates a distribution’s spread to its extremes — which is where the square-root rule comes from.

The energy scale of everything else. A gap, a defect level, a distortion energy: all of them are quoted in units of the hopping and are compared with a band width that is a count of neighbours. A band gap is not a bond energy, but both of them are measured against the same width.

And the fact that the distribution is bounded at all. A structure of finite coordination has levels in a finite range, which is what makes an edge exist and what makes a chain’s density of states have the shape it has.

What it does not fix is the one thing a chemist most often wants from it, which is which of two structures is lower.

What the comparison requires #

Every structure in the set has the same coordination and therefore exactly the same second moment, to a part in a billion.

Their bindings differ by more than two per cent, so the coordination is not the whole of the story.

The structure with the largest fourth moment is the least bound of the set — one direction of the inequality argued above, checked rather than assumed.

And two structures agreeing on the second, third and fourth moments still differ in binding, and first differ at the sixth. Without this the essay would be claiming that a finite series settles the question, which is false and would be the more comfortable result.

Still open: the difference between two structures #

The next thing to compute is the one this essay keeps gesturing at: not the binding of a structure but the difference between two structures of one element, which is what decides a crystal structure.

That difference is small — tens of millielectronvolts per atom separates the close-packed structures of most metals — and it is a difference between two numbers each of which is several electronvolts. Everything above says that a calculation of it has to be doing two things right at once: the coordination term, which is easy and dominant, and the topology term, which is small and is what the answer consists of. That is the reason structural energy differences are the standard hard case for an electronic structure method, and it is visible in miniature in the third decimal place of two of the five numbers above.