A density of states is not a spectrum

A molecule’s spectrum is a list. Water has three vibrational frequencies, methane has four distinct ones, and the whole of how many frequencies, not how many modes is about getting the length of that list right before measuring anything.

A solid’s spectrum is not a list. It is a curve, and the temptation on first meeting it is to say that the curve is the density of states — that the levels have become so numerous that counting them is the spectrum.

That is wrong in a way worth being precise about, because the density of states is a genuinely useful object and the mistake is not that it is useless. The mistake is a category one: a density of states counts states and a spectrum counts transitions, and a transition needs two states and something connecting them.

The confusion is encouraged by the fact that the two objects are often drawn on the same axes and frequently do look alike. They look alike because a transition from a dense region of occupied states into a dense region of empty ones is usually strong, so the peaks often coincide. Usually is not always, and the cases where they part company are the interesting ones.

What the count actually is #

The density of states $g(E)$ is defined so that $g(E)\,\mathrm{d}E$ is the number of levels between $E$ and $E + \mathrm{d}E$. For a system of finitely many levels it is a sum of spikes; for a large system it is smooth enough to draw as a curve, and the curve is what the histogram above is estimating.

It is a complete description of the level structure and it is very far from a complete description of the system. Two facts it does not contain:

Which states. The count says how many levels are at an energy and nothing about what those levels look like — where their amplitude sits, what their symmetry is, whether they are localised on a defect or spread over the whole structure. Two systems can have the same $g(E)$ with entirely different states behind it.

How many electrons. The count is a property of the matrix, and the filling is a separate input. A solid is a molecule that did not stop makes the point at length: the same forty levels with forty electrons and with eighty electrons behave completely differently, and the density of states is identical in both cases.

What an experiment measures instead #

An absorption experiment moves an electron from an occupied level to an empty one and asks how much light was absorbed at each frequency. Three things have to be true for a contribution at photon energy $\hbar\omega$:

• an occupied level at some energy $E$;
• an empty level at $E + \hbar\omega$;
• a non-zero transition moment between those two particular states.

The count enters twice, once for each level, and the third condition is not a count at all. What the experiment sees is a joint density of states weighted by the square of a matrix element, and the two weightings do very different work.

The joint count already differs in shape from the ordinary one. Where the ordinary density diverges at a band edge, the joint density diverges wherever an occupied and an empty band run parallel, which need not be at any band edge at all. A shoulder in an absorption curve is therefore not, in general, at the energy of any feature in $g(E)$.

The matrix element, which is where the symmetry lives #

The third condition is the one a whole field of these essays is about. Selection rules are one theorem shows that every selection rule in spectroscopy is the same statement: a transition moment is an integral, and if the integrand is antisymmetric under an operation that leaves the system unchanged, the integral is zero. Not small — zero, and such integrals, computed, come out at $10^{-17}$.

Nothing about that argument cares how large the system is. An integrand odd about a mirror plane integrates to zero whether the mirror belongs to a four-atom molecule or a four-thousand-atom one. So the selection rules do not weaken as a molecule grows into a solid.

What changes is the visibility of their consequences. In a molecule, a forbidden transition is a line that is not there, and its absence is the sharpest evidence available — what an absence proves is built on exactly that. In a solid, a forbidden transition removes one contribution from a continuum of allowed ones, and what is left is a curve slightly different in shape from the curve that would otherwise have been there. The information is still present and it is enormously harder to read.

This is the honest reason a solid-state spectrum is harder to interpret than a molecular one, and it is not that the physics is harder. It is that a continuum hides the evidence that a discrete list displays.

A worked case where the two disagree completely #

The abstract statement is easier to trust with a concrete pair in front of it, and one is available in the same one-orbital chain.

Take a ring of six sites and a chain of six. Their level patterns are different — the ring has two degenerate pairs and the chain has six distinct levels — but their densities of states, blurred at all, look much alike: six levels spread across the same interval, thicker in the middle.

Their spectra are not alike at all. The ring has a centre of inversion and the chain does not, so in the ring every level is either symmetric or antisymmetric under that inversion, and an electric-dipole transition must connect one kind to the other. Half the transitions a naive joint count would allow are exactly zero. In the chain there is no inversion, no parity label, and no transition forbidden on those grounds.

The same six atoms, the same six π electrons, the same rough distribution of levels, and one of the two systems has half of its transitions removed by an argument that never mentions an energy. That is the gap between a count and a spectrum, at a size small enough to check every entry by hand.

Degeneracy, which the count flattens #

That last point deserves its own paragraph because it is the sharpest thing a density of states throws away.

Two levels at the same energy contribute two to the count. Whether they are at the same energy because a group requires it or because two unrelated quantities happened to coincide is invisible in the count and completely decisive for everything else. A symmetry-required degeneracy survives any perturbation that keeps the symmetry and splits under any that does not; an accidental one splits under essentially everything. The first is what makes a Jahn–Teller distortion inevitable and the second guarantees nothing at all.

Reading a density of states as though the two were the same kind of feature is how a broad peak in a computed curve gets described as “a degenerate band” when it is a dozen distinct levels within a hundredth of a $\beta$ of each other. The distinction is recoverable — it is in the states, not the energies — and it is gone from the histogram.

Photoelectron spectroscopy, which comes closest #

There is one experiment whose result really is close to a density of states, and it is worth naming because it is the exception that makes the rule legible.

In photoemission an electron is removed from an occupied level and its kinetic energy is measured. The final state is a free electron rather than another bound level, so the joint-count problem largely disappears: every occupied level can contribute, and what is recorded is a count of occupied states against binding energy. That is the occupied density of states, smeared by the instrument’s resolution and weighted by cross-sections that depend on which orbital an electron came from.

Even here the weighting matters. The cross-section for removing an electron from a $d$ orbital differs from that for an $s$ orbital by a large factor and depends on the photon energy, which is why the same material measured at two photon energies gives two differently shaped curves from one unchanged density of states. What a photoelectron spectrum measures sets out the molecular version of the same caution: the spectrum is evidence about orbital energies, and it is evidence rather than a photograph.

The edges, which is where the count is least like a spectrum #

The density of states of a one-dimensional band diverges at both edges, and that can be computed rather than assumed: the levels are $2\cos(k\pi/(n+1))$, the cosine turns over at the ends of its range, and many consecutive levels therefore land at nearly the same energy.

A reader who took the count for a spectrum would predict that a one-dimensional material absorbs most strongly at its band edges. What actually happens depends on the transition moment, and near a band edge the states at the top of the occupied band and the bottom of the empty one are frequently of the same symmetry — which is precisely the case in which the moment between them vanishes.

So the strongest feature in the count can be a feature that is absent from the spectrum. That is not a rare pathology; it is the ordinary situation in a centrosymmetric material, where states alternate in parity up the band and a transition between adjacent ones is forbidden.

The same distinction, in vibrations #

Everything above has been about electrons, and the identical argument runs for vibrations with nothing changed but the words.

A molecule of $N$ atoms has $3N - 6$ vibrational modes, and how many distinct frequencies appear in its spectrum is decided by symmetry before any force constant is chosen. A crystal has three modes per atom too, and there are so many of them that their frequencies form bands with a density of states of their own.

The infrared spectrum of that crystal is not its vibrational density of states. Light carries essentially no momentum on the scale of an atomic spacing, so it can only excite modes in which every repeat unit moves in phase — which is a vanishingly small fraction of the modes present, selected by a rule that has nothing to do with how many modes exist at each frequency. The neutron-scattering experiment that does measure the full density of vibrational states is a different and much more expensive measurement for exactly that reason.

The molecular statement of the same fact is normal modes are not bond stretches together with what an absence proves: the count of modes and the count of visible bands are different numbers, and the second is smaller for reasons that are exact.

Methane’s nine vibrations sort into four symmetry species, with the infrared and Raman activity of each read off the character table: four levels, nine modes, and not all four appearing in both spectra. A count of states would give the nine and could not give the activities, which is the same omission one level down.

The curve that gets compared to a measurement is not this one #

There is a second gap between the count and the experiment, and it is not about transitions at all. It is about which count.

The object drawn above is the total density of states: every level, weighted once. What a photoemission experiment produces is resolved — by element, because the photon energy can be tuned to a particular core threshold; by orbital character, because the cross-section for ionising a d electron and an s electron differ by more than an order of magnitude at the same photon energy. So the curve a measurement is compared against is almost never the total. It is a projected density of states: the same levels, each weighted by how much of it sits on a chosen atom, or in a chosen kind of orbital.

That projection is the useful object and it carries a defect the total does not have. The total density of states is the distribution of the eigenvalues of a matrix, and eigenvalues do not care what basis the matrix was written in. Change the basis and every level stays where it is. A projected density of states is not basis-independent at all: it asks how much of each eigenvector lies along a chosen direction, and how much that is depends on what the chosen directions are.

This is the same object, and the same objection, as a population analysis. Deciding that a level is “sixty per cent metal d and forty per cent ligand p” requires the metal d and the ligand p to be defined, and they overlap, so the split depends on how the overlap is apportioned — which is a convention. Two calculations of one solid, in two basis sets, agree exactly about the total density of states and can differ substantially about the projected one.

So the ordering of trustworthiness runs opposite to the ordering of usefulness. The total count is exact and answers no spectroscopic question. The projected count answers the question and is convention-dependent. The measured spectrum is the thing that actually exists and contains a matrix element neither of them has.

None of which makes a projected curve worthless — it is very often the only way to say what a band is made of, and the qualitative answer is usually stable across reasonable conventions. It makes a projected curve a quantity that has to be quoted with its convention attached, in exactly the way a delocalisation energy has to be quoted with its reference. A percentage of d character with no basis named is half a number.

Why the count is worth having anyway #

None of this makes the density of states a bad object. It makes it a thermodynamic object rather than a spectroscopic one.

Quantities that are sums over all states with no pairing and no matrix element are exactly the ones $g(E)$ answers directly: the electronic heat capacity, the magnetic susceptibility of a free-electron metal, the total energy, the number of electrons below a given level. Every one of those is an integral of $g(E)$ against something, and every one of them is insensitive to which state is which.

Quantities that involve a transition are not. Absorption, emission, conductivity at a frequency, and every kind of resonance need the pair and the moment, and for those the count is a starting point that has to be weighted before it means anything.

The clean way to hold the distinction: the density of states answers questions about how many electrons are where, and answers no question about what an electron can be made to do.

What this field can and cannot compute #

This site builds its bands out of finite matrices with no periodicity assumed — see a solid is a molecule that did not stop for the method and a band with no structure in it for its limits. The density of states is comfortably inside what that route can produce: it needs eigenvalues and nothing else, and the histogram above is a count of two thousand of them.

Transition moments are not, quite. Computing one needs the states as well as the energies, which a diagonalisation supplies, and it needs to know where the atoms are, which a chain of neighbours does not — it is a list of neighbours with no positions attached. So the calculation can say which transitions symmetry forbids, and does; it cannot say how strong the allowed ones are.

There is one more thing worth being explicit about, since it is the commonest way a computed curve gets over-read. A density of states drawn from a finite system is a histogram, and a histogram has a bin width. Every feature narrower than that bin is invisible and every feature is broadened to at least it — so a small gap, a sharp spike, or a defect level sitting just outside a band can all be smoothed into the continuum by a choice made for the drawing rather than by the physics. The figures here state their bin count for that reason, and the density check excludes the band edges precisely because the divergence there is a feature no bin width can represent.

That is a real boundary and it is the ordinary one for a tight-binding argument. What it leaves is still the more important half for a reader coming from molecules, because the forbidden transitions are where the certainty is: a vanishing integral is exact, and an intensity is a calculation with an error bar.