The width of a band is a count of neighbours

There is one statement about bands that requires no limit, no approximation beyond the model itself, and no assumption that anything repeats. It is exact at two atoms and exact at two million, and it is a fact about matrices that happens to be a fact about solids.

The average of the squared orbital energies is the average number of neighbours.

The formulation is worth reading twice, because it is stronger than it looks. It is not “roughly proportional to”, not “in the limit of large systems”, and not “for a periodic crystal”. It is an equality, for any collection of atoms with any pattern of bonds between them, at any size, including sizes where the word band is an abuse of language.

Why it is true, which takes three lines #

The orbital energies are the eigenvalues of the matrix $A$ whose entries are one between bonded sites and zero elsewhere. Two facts about eigenvalues do the work.

The sum of the squares of the eigenvalues of a symmetric matrix equals the trace of its square. And the trace of $A^2$ is

$$\operatorname{Tr}(A^2) = \sum_i \sum_j A_{ij}A_{ji} = \sum_i (\text{number of neighbours of } i),$$

because $A_{ij}A_{ji}$ is one exactly when $i$ and $j$ are bonded. So the sum over all sites of the neighbour count is the sum over all levels of the squared energy, and dividing both by the number of sites gives

$$\langle x^2 \rangle = \bar{z}.$$

The first step is the one worth pausing on, since it is where all the content is. A symmetric matrix can be written as $A = U X U^{\mathsf{T}}$ with $X$ diagonal, so $A^2 = U X^2 U^{\mathsf{T}}$, and the trace is unchanged by that conjugation. The trace is therefore computable in either basis — as a sum over levels, which requires solving the problem, or as a sum over sites, which requires only reading the bond list. The identity is the statement that those two are the same number, and its usefulness comes entirely from the second being free.

That is the whole derivation. It does not know whether the structure is periodic, whether it is one-dimensional or three, whether it has ends, or whether it is a crystal at all. It would hold for a random tangle of atoms with no order in it whatever.

The same identity appears in a smaller form in Hückel theory and what it gets right: the sum of the squares of a hydrocarbon’s π levels is twice its bond count, which is this statement before dividing by $n$. What changes here is only the reading — the same arithmetic, asked about a band rather than about a molecule.

What it fixes, and what it leaves free #

A band’s shape is a distribution, and the second moment is one number describing it. Knowing $\langle x^2 \rangle$ pins the scale of a band and says nothing about its shape.

Two consequences follow immediately and both are worth stating.

A band cannot get wider by the system getting bigger. The mean coordination of a chain is $2(n-1)/n$, which climbs to two and stops; it is bounded above by the largest number of neighbours any atom has. Adding sites in a way that does not change how many neighbours each has cannot change the second moment, and therefore cannot change the scale of the band.

A band gets wider by atoms getting more neighbours, or by the neighbour interaction getting stronger. Both are local. This is the same statement overlap decides makes about a single bond, promoted to a whole structure: what an orbital energy is worth depends on the integral between two centres, and everything else in the picture is bookkeeping over how many such pairs there are.

Drawing every level of those chains rather than their moments makes the other half of that plain, and a solid is a molecule that did not stop does draw them: the count of levels grows with the count of sites and the interval they lie in does not, because the mean coordination stops changing after the first few sites. The width is fixed before the chain is ten atoms long.

Where the familiar formula comes from, and where it fails #

The formula every textbook gives is that a tight-binding band is $2z|\beta|$ wide, with $z$ the coordination. For a chain, $z = 2$ and the width is $4|\beta|$, which is correct. For a simple cubic connectivity, $z = 6$ and the width is $12|\beta|$, which is also correct.

The formula is correct for those cases because they share a property: the structure is a product of independent one-dimensional chains, so the energies add and the extremes add too. Six neighbours arranged as three perpendicular chains give a top level of $2 + 2 + 2$ in $\beta$ units.

Arrange six neighbours differently and the arithmetic changes. On a tree in which every site has $z$ neighbours and there are no closed loops at all, the band runs from $-2\sqrt{z-1}$ to $+2\sqrt{z-1}$, so its width is $4\sqrt{z-1}\,|\beta|$. For $z = 4$ that is $6.93|\beta|$ against the formula’s $8|\beta|$, a discrepancy of thirteen per cent between two structures with the same coordination number.

Both of them have $\langle x^2 \rangle = z$ exactly.

So the second moment is the robust statement and the width is the fragile one, which is the reverse of the order they are usually presented in. The width depends on how the neighbours are arranged — specifically on the loops, since a loop is what lets amplitude come back to where it started and interfere with itself. The second moment counts only the shortest walk there is.

The bound the width does obey, and the structure that evades it #

The width is fragile, which is not the same as being unconstrained. There is an exact two-sided bound on it, it holds for any finite structure whatever, and it says precisely which family the textbook formula belongs to.

For any finite connected set of sites, the highest level lies between the mean coordination and the largest coordination,

$$\bar{z} \;\le\; x_{\max} \;\le\; z_{\max},$$

in units of β, with equality at both ends exactly when every site has the same number of neighbours. A chain of forty sites has a mean coordination of 1.95 and a maximum of 2, and its top level is 1.9941 — inside the bracket, as it must be, and pushed off the maximum by the two sites that have only one neighbour.

So the familiar $2z|\beta|$ is not an approximation and not a special case of a limit. It is a theorem, and its scope is finite structures in which every site has the same number of neighbours: there the two ends of the bracket close on each other, the top level is exactly $z$, and the width is exactly $2z$. Stating it as a consequence of the coordination number was never quite the error; stating it without the word regular was.

Which leaves the tree looking like a contradiction, because a tree in which every site has $z$ neighbours is regular by inspection and its band edge is $2\sqrt{z-1}$, comfortably below $z$. For $z = 4$ that is 3.46 against a bound of 4.

The escape is worth the detour, because it is the counting argument turned inside out. No finite piece of such a tree is regular. Cut it off anywhere and the outermost shell are sites with one neighbour, exactly as a chain has two ends — and on a chain, or a cube, those boundary sites become a vanishing fraction as the structure grows, which is why the bracket closes in the limit. On a tree they do not. A tree that branches $z-1$ ways has $(z-2)/(z-1)$ of all its sites in the outermost shell at every size: for $z = 4$, two thirds of the atoms are on the boundary, in a piece of any size whatever.

So a Bethe lattice is a structure that is entirely surface, permanently, and the $1/L$ argument that made seven hundred atoms enough for a chain has no purchase on it at all. Its band edge is a boundary property and stays one.

That is the sharper version of what makes the width fragile. It is not that neighbours can be arranged differently in some unspecified way; it is that loops are what let a structure have an interior, and a quantity fixed by the extreme eigenvalue is a quantity fixed by the boundary until the interior outweighs it. The second moment counts walks of length two and never has to ask the question.

Two structures with the same coordination, drawn #

The claim that the width depends on the arrangement and the moment does not is easier to believe when the two arrangements are in front of a reader.

A ring of forty and a chain of forty have almost the same mean coordination — exactly two for the ring, and $2 \times 39/40 = 1.95$ for the chain, the difference being the two bonds the chain does not have because it has ends. Their second moments come out at those numbers to fourteen decimal places. Their level patterns are not the same at all: the ring’s levels are doubly degenerate except at the very top and bottom, and the chain’s are all distinct.

Their widths are also not quite the same, and the direction is instructive. The chain’s top level sits slightly below $2\beta$ — at $2\cos(\pi/41) = 1.994\beta$ — while the ring’s sits exactly at $2\beta$. The ring reaches the edge and the chain does not, because reaching the band edge requires a state with the same amplitude on every site and the same sign, and a chain’s ends prevent that. It is the same finite-size effect that costs a chain its energy per site, arriving as a missing sliver of band width.

Both effects vanish as $1/n^2$ and $1/n$ respectively, and both are invisible to the second moment, which has already converged.

The stronger version of the same claim holds the coordination exactly fixed and varies nothing else at all.

A fifth of the binding energy is not a correction. It is the difference between two materials, and no quantity in this essay’s identity distinguishes the five structures that produced it.

Where the interaction strength enters, and where it does not #

The identity as stated assumes every bond contributes the same $\beta$. When they do not — heteroatoms, alternating bond lengths, anything that makes one interaction stronger than another — the identity generalises rather than failing:

$$\langle x^2 \rangle = \frac{1}{n}\sum_i \sum_{j \sim i} k_{ij}^2,$$

where $k_{ij}\beta$ is the interaction across each bond. It is still a count, now weighted by the square of each interaction, and it is still exact. The weighted form is the one to check against, which is why the same check applies to a chain whose bonds alternate as to one whose bonds are all alike.

That generalisation carries a small surprise. Alternating a chain’s bonds as $1 \pm \delta$ leaves the sum of the squares at $(1+\delta)^2 + (1-\delta)^2 = 2 + 2\delta^2$ per site, which is larger than the uniform case. So dimerising a chain slightly increases its second moment — the band gets marginally broader in the root-mean-square sense while a gap opens in the middle of it. Both things happen at once and neither is visible in the other’s number, which is the clearest possible demonstration that a moment and a gap are different questions.

The next moments, and what they start to see #

Nothing stops the same argument at the fourth moment. $\operatorname{Tr}(A^4)/n$ counts closed walks of length four, and there are two kinds: out to a neighbour and back twice over, and around a four-membered ring. So the fourth moment sees rings and the second does not, which is exactly why the width does.

That is the whole idea behind the moment methods for computing densities of states, and it is worth naming because it makes the structure of the subject legible. Each successive moment is a count of closed walks of one length longer, and each longer walk can visit more of the structure, so the moments are a series in how far an electron has to go before the answer changes. The second moment needs to know the number of neighbours. The fourth needs to know about squares. The sixth needs to know about hexagons.

A chemist meets the same hierarchy from the other side. Whether benzene is aromatic is a fact about a six-membered ring, and aromaticity as a computed shell closure works it out by diagonalising that ring. Nothing about a ring is visible in a count of neighbours, and it should not be: every carbon in benzene has two π neighbours, exactly like every carbon in a long chain.

Aromaticity as a shell closure fills every ring from three to eight and finds entirely different level patterns behind one coordination and one second moment. Everything that distinguishes them is in the fourth moment and beyond, which is a compact statement of why the shell closures that decide aromaticity are invisible to any argument built on counting neighbours.

The chemistry the hierarchy explains #

Reading the moments as a series in distance makes sense of something a chemist knows and a physicist states differently: local structure decides local quantities, and only long walks see long-range order.

A carbon atom in diamond and a carbon atom in an amorphous carbon film have the same four neighbours at nearly the same distances. Their second moments are the same, their fourth moments are close, and the differences appear only at the moment that first notices the ring statistics — which is why the two materials have similar densities, similar bond strengths and similar hardness, and completely different optical properties.

The same reasoning explains why a chemist’s local picture works as well as it does. Bond energies are transferable between molecules because they are second-moment quantities; they depend on which atoms are next to which. Colour and conductivity are not transferable, because they depend on the fine structure near the middle of the spectrum, which is where the high moments live. Group frequencies and where they stop makes the identical argument about vibrations: a carbonyl stretch is transferable for exactly as long as the motion stays local, and stops being transferable at the point where it does not.

The exactness is doing real work #

It is easy to treat an identity like this as bookkeeping. It is not, and the reason is that almost nothing else in the field is exact.

The band edges are exact only for the models where a closed form exists. The density of states is a limit. The claim that a large system behaves like an infinite one is a statement about how fast something converges, measured in a solid is a molecule that did not stop at roughly seven hundred atoms for a chain. Every one of those has an error bar attached, and every one of them can be wrong for a system with some feature the derivation did not consider.

The second moment cannot. If a computed spectrum has a $\langle x^2 \rangle$ that disagrees with the counted coordination, the computation is wrong — not approximate, wrong — and there is nothing to argue about. That makes it the single best check available on any calculation in this family, and it holds on every structure drawn here.

Taken to a thousand levels the same distribution still has ⟨x²⟩ = 2 to within a part in 10¹⁴ — where the states pile up bins them and confirms it — and its width is still a consequence of the chain having no loops, which no counting argument would have given.

What a count of neighbours cannot tell #

The limits are as instructive as the result. A count of neighbours says nothing about the sign structure of the levels, nothing about which of them are occupied, and nothing about whether the band is split into pieces.

The last of those is the important one. A structure whose sites fall into two classes with different energies — every other atom different, or two elements alternating — has a band that separates into two, with a gap between them. Its second moment is unchanged from the uniform case if the coordination is unchanged, so the moment cannot see a band gap at all. It reports the same number for a metal and for an insulator built on the same skeleton.

The same forty levels with eighty electrons rather than forty are every level doubly occupied, with nothing above to be excited into, and the second moment is identical to the half-filled case because filling is not a property of the matrix at all. Which of two systems with the same spectrum conducts is decided entirely by a number the spectrum does not contain.

That is not a defect. It is a boundary, and knowing where it falls is what makes the identity usable: it constrains the scale, it can refute an arithmetic error, and it is silent on the question that decides whether a material conducts. The gap is not the band width takes up the question the moment cannot answer, and what a metal actually is makes the conducting–insulating distinction without appealing to either quantity.

The moment also says nothing about geometry, because there is no geometry in it. Two structures with identical connectivity and completely different bond lengths give identical spectra here, since a bond either exists or does not. Making $\beta$ depend on distance is the obvious repair and it is what a chain cannot stay even does, with a consequence nobody expects: it turns out the evenly spaced chain assumed everywhere in this essay is not the stable arrangement at all.

And it says nothing about how many electrons there are, which is the omission with the largest consequences. Two materials can have the same skeleton, the same coordination, the same second moment and the same band, and one of them be a metal and the other a transparent insulator, because one has an electron per site and the other has two. A quantity that is exact and blind to the thing that decides the answer is a useful instrument and a poor summary, and both halves of that are worth carrying out of this essay.