More bands than there are orbitals

A photoelectron spectrum is a list of energies at which an electron can be removed, with an intensity at each. Every introduction to the technique reads it as a list of orbital energies, one band per occupied orbital, and the reading is so standard that the picture of the molecular orbitals is usually drawn beside the spectrum with lines joining them.

Koopmans’ theorem is exact for nothing took apart the energies: Koopmans’ theorem holds exactly for nothing, and the two errors it makes — relaxation and correlation — go in opposite directions and cancel by an amount that depends on the orbital. This essay takes apart the count.

What a spectrum actually is #

Exactly, the spectrum is the set of poles of

$$A(\omega) = \sum_m \big|\langle \Psi_m^{N-1} | c_{i\sigma} | \Psi_0^N \rangle\big|^2\, \delta\big(\omega - (E_0^N - E_m^{N-1})\big),$$

which is a sum over every state of the ion, not over the orbitals of the molecule. There is no reason for the number of terms with non-zero weight to be the number of orbitals, and the one-electron picture is the statement that it is.

The two wavefunctions are exact within the model here — the full configuration space of a six-site Hubbard ring at half filling, four hundred dimensions for the parent and three hundred for the ion, no truncation and no selection of important configurations. What comes out is the spectrum rather than an approximation to it.

The sum rule, which is the check #

Before any result there has to be a check, and this one is unusually good because it is an identity rather than a comparison.

The anticommutator of an annihilation operator with its adjoint is one, so summing the intensity over every final state gives the occupation of the orbital removed from — and summing over sites as well gives the number of electrons that can be removed. Here that is three, and the calculation returns 3.000000000 at every repulsion tested.

That matters more than it looks, because the quantity most easily got wrong in a calculation of this kind is the fermion sign. Removing an up electron from site i changes the sign of the amplitude once for every up electron on a lower-numbered site, and a version that dropped the counting would produce a smooth, plausible, wrong spectrum with intensities that did not add up. The sum rule is what says the counting is right.

Three lines, then a hundred #

With the repulsion off there are three lines and they are the three occupied orbital energies, with the whole of the intensity in them. That is the one-electron picture, recovered as a limit rather than assumed — and recovering it is what makes everything after it a result rather than an artefact of the counting.

Turn the repulsion on and the count rises immediately: 46 lines at U = 1, 64 at U = 2, 89 at U = 4, 100 at U = 8. The molecule has six orbitals throughout.

Most of those hundred lines are very weak, and it would be reasonable to dismiss them if the intensity stayed where it was. It does not.

Where the intensity goes #

Define the main lines as the strongest three, which is as many as the one-electron picture has, and everything else as satellite intensity. The share held by the main lines runs 1.000, 0.970, 0.895, 0.694, 0.468 across the five repulsions.

By U = 8 the satellites hold 53.22 per cent — more than the main lines. A spectrum recorded there and assigned by the usual rules would be assigning the wrong features.

That is a strong-repulsion statement and the model is a caricature, so the honest version of the claim is the weaker one at moderate repulsion: at U = 2, which is a perfectly ordinary ratio of repulsion to bandwidth for a π system, ten and a half per cent of the intensity is already outside the main lines and there are sixty-four lines rather than six.

What a satellite is #

The mechanism is worth stating in words, because it is the same one that produces every other correlation effect in this collection and it is usually described as though it were mysterious.

Removing an electron leaves an ion, and the ion’s states are not the parent’s orbitals with one emptied. They are states of a system with one fewer electron, and among them are configurations in which the remaining electrons have rearranged — one promoted from an occupied orbital to an empty one, say, alongside the hole. In the one-electron picture such a state has no overlap with the parent minus an electron, so it carries no intensity. Once the parent’s ground state contains some of the same rearrangement, it does.

So a satellite is an ionisation accompanied by an excitation, and its intensity is a measure of how much of that excitation was already in the ground state. That is why the satellites grow with the repulsion: the repulsion is what puts the excitation into the ground state in the first place. It is the same quantity the double occupancy measures from a different direction.

The spectrum also spreads #

There is a second effect and it is more visible in a real spectrum than the satellites are.

The energy span of the lines with any intensity runs 1.0, 6.37, 7.63, 9.65, 13.37 across the five repulsions. The three occupied orbitals of the free ring span 1.0, so by U = 8 the spectrum is thirteen times as wide as the orbital energies it is supposed to be a list of.

That is not the same as the orbitals spreading. The orbitals do not move at all: the matrix they are eigenvectors of has no U in it. What is spreading is the set of ion states, and the width of the spectrum is a property of the ion rather than of the molecule.

The same distinction runs through the width of a band is a bond length, which reads a vibrational width off a single band and is safe because a vibrational profile is a property of the ion too — it is measured as one rather than attributed to an orbital.

The consequence for a practitioner is worth stating plainly. A band width in a photoelectron spectrum is not a band width of the occupied orbitals, and reading a wide spectrum as evidence of a wide valence band overstates it by a factor that grows with the repulsion.

The energies rather than the intensities are where the exact first ionisation departs from what an orbital energy says it should be. That departure and the appearance of satellites are the same failure of a one-electron description, measured in two currencies.

What this means for the received readings #

Three earlier findings about photoelectron spectra sit differently now.

Fewer bands than electrons counted bands by symmetry: degenerate orbitals give one band, so a spectrum has fewer features than a molecule has electrons. That argument is exact and survives — it is about which final states can be reached at all, which is a selection rule and not a matter of intensity. What is added here is that the count it gives is a lower bound rather than the answer.

Water’s lone pairs are not a pair read two bands as two orbitals of different symmetry, and that reading is safe for the same reason: the two features are separated by a symmetry label, and satellites of one symmetry cannot masquerade as a fundamental of another.

A band is a filter on the modes read vibrational structure within one band. That is the reading most at risk here, because a satellite sitting under a fundamental band adds structure that is not vibrational at all, and nothing in the profile distinguishes the two.

Vibrational structure within a single photoelectron band adds lines with an origin and a spacing. A satellite falling in the same energy range adds lines with neither, so the two can be told apart by their pattern where they cannot be told apart by their position.

The one number that still counts electrons #

There is a temptation to conclude from all this that a photoelectron spectrum measures nothing reliable, and that is too strong. One quantity survives exactly, and it is worth separating it out.

The total intensity is the electron count, at every repulsion, to nine decimal places. That is not an approximation that happens to be good; it is an anticommutator, and it holds for any Hamiltonian, any interaction and any wavefunction. So a spectrum integrated over the whole of its range does count electrons, and the counting is exact.

What fails is every attempt to attribute parts of that total. The intensity of one band is not the occupation of one orbital; the number of bands is not the number of orbitals; the width of the spectrum is not the width of the occupied set. Each of those is a decomposition of the exact total into pieces, and the pieces are properties of the ion.

That pattern keeps turning up. A total bond order is conserved and its distribution is not; a charge sums to the molecule’s and its partition among atoms is a convention; a spectrum’s total intensity is a count and its distribution among lines is not. The sum is physics and the partition is bookkeeping, and it is remarkable how often the interesting-looking quantity is the second one.

A real valence-band spectrum assigned in the usual way rests on symmetry rather than on intensity, which is why that assignment survives everything in this essay. What does not survive is the assumption that the number of bands is the number of orbitals.

Methane’s two bands predicted from four coordinates is a count of features by symmetry and is exact. The intensities are not predicted at all, and it is the intensities that satellites take away — so a symmetry prediction of a band count and a measured band count can disagree without either being wrong.

The charge gap is the difference between the first ionisation and the first affinity, and it is the quantity the two halves of a full spectral function are separated by. A satellite on one side of it and a fundamental on the other are never confused; the difficulty is entirely within one half.

What this cannot say #

No shape to the orbitals. Everything a real spectrum uses to assign a band — the symmetry species, the atomic character, the vibrational profile — is absent here. Why a d–d band is weak is the kind of intensity argument this model has no room for at all.

One orbital per site. A Hubbard site has no shape and no shell structure, so nothing here corresponds to a core level, a shake-up in a particular orbital, or a satellite of one band rather than another.

No relaxation of a core. The most famous satellites in real spectroscopy are core-level shake-ups, and they involve a core hole pulling the valence electrons in — a relaxation this model has no room for.

No cross-sections. The intensities here are the matrix elements of an annihilation operator. A real intensity multiplies that by a photoionisation cross-section that depends on the photon energy and on the orbital’s character, which is why a spectrum recorded at two photon energies looks like two spectra.

And six sites is small. The number of final states grows factorially, so the count of a hundred lines is a property of a three-hundred-dimensional ion space and would be a different number for a larger ring. The share of the intensity outside the main lines is the transferable quantity, and it is the one the argument uses.

What was checked #

The sum rule holds at every repulsion, to a nanounit — the identity that says the fermion signs are counted correctly, and the only check available that does not use the same calculation twice.

With no repulsion there are no more bands than occupied orbitals, and the whole of the intensity is in them — the one-electron limit recovered rather than assumed.

The band count exceeds the orbital count at the strong end, which is the sentence the essay is for.

The satellite share never falls as the repulsion rises, at every step of the scan.

And by the strong end most of the intensity is not in the main lines — above twenty per cent required, 53.22 measured.

The satellites are measured, and one of them is an analytical test #

Extra lines with no orbital corresponding to them are a computed result here, and they are not a peculiarity of a model: they are seen, they have a name, and one of them is used as a routine analytical test.

The name is a shake-up satellite. Removing an electron changes the potential the remaining ones feel, and some of the time one of them is promoted to an empty orbital at the same moment. The photoelectron then leaves with less energy than it would have, and the spectrum acquires a line at a higher apparent binding energy — a line that corresponds to no orbital, only to a two-electron process.

The analytical case is carbon. A carbon 1s spectrum of an aliphatic material shows a single peak; the same spectrum of an aromatic material shows the peak plus a satellite a few electronvolts above it, from the promotion of a π electron to a π* orbital at the moment of ionisation. The satellite is present when there are π electrons to shake up and absent when there are not, so its presence identifies aromatic carbon.

That test is used on polymers, on carbon films and on catalyst supports, where distinguishing aromatic from aliphatic carbon by any other means is awkward — and it works on a feature that a one-electron reading of the spectrum says should not exist.

Two consequences follow for reading any such spectrum.

A satellite is information rather than contamination. Its position measures an excitation energy of the ion, and its intensity measures how strongly the ionisation shakes the remaining electrons, which is a correlation effect made visible.

And a spectrum’s total intensity is conserved. The satellites take their strength from the main line, so a molecule with strong satellites has a weakened fundamental — which is the sum rule computed here, and it means the ratio of satellite to main line is the quantity to read rather than either alone.

The second point has a consequence for the practice of assigning spectra that is worth stating plainly. If satellites borrow their intensity from the fundamentals, then a band’s height is not a count of the electrons in an orbital — it is that count reduced by however much has leaked into satellites, and the leakage differs from one orbital to another. So the intensity ratios that identify a degeneracy are reliable where correlation is weak and are not where it is strong, and the fifty-three per cent measured here at large repulsion is the case where they have stopped meaning anything at all.

What to do with a spectrum that has extra lines #

Three practical readings survive everything above, and they are worth separating from the ones that do not.

Symmetry survives. A final state can carry intensity only if it has the right symmetry, and no amount of correlation changes which representations are reachable. So an assignment made by counting features of each symmetry species — which is what fewer bands than electrons and selection rules as one theorem both do — is not at risk from any of this.

A sequence survives. Comparing one molecule against a series of its relatives, and reading the shift of a band rather than its absolute position, is robust for the same reason error cancellation is robust anywhere: the satellite structure of two similar molecules is similar, and a difference removes most of it.

An absolute intensity does not. Reading a band’s height as a count of electrons, or a spectrum’s width as a valence band width, is what fails — and it fails in the direction of overstating both.

There is one more thing the calculation says and it is easy to miss. The number of lines rose sharply between no repulsion and a little, from three to forty-six at U = 1, while the intensity barely moved: 97 per cent still in the main lines. Most of the extra lines are extremely weak, and the ones that matter arrive later. A spectrum with visible satellites is a spectrum from a strongly correlated system, and a spectrum without them is not evidence that the one-electron picture holds — only that the satellites are below the noise.

Still open: the addition spectrum, and telling satellites from bands #

The obvious open question is the other half of the spectral function. Everything here removes an electron; adding one gives the inverse photoemission spectrum, and the two together are the object whose gap is the charge gap band theory cannot see. The satellite structure of the two halves is not symmetric even for a half-filled system, and measuring the asymmetry would say something about the ion that neither half says alone.

The nearer question is about assignment rather than about physics. Given a spectrum with a hundred lines, a practitioner has to decide which are fundamentals, and the tool available is a comparison against a computed one. What has not been asked here is how distinguishable the two kinds of line are: whether the satellites cluster in energy in a way that a fundamental would not, or whether they are spread thinly enough that any one of them could be mistaken for a band. That is a question about the distribution of the intensity rather than its total, and it is answerable with the calculation already in hand.