A hundred lines and no way to sort them

Count the lines first. Three at no repulsion, a hundred at a repulsion of eight, with 53 per cent of the intensity outside the six lines the one-electron picture allows — and a sum rule holding at 3.000000000 throughout, so the extra lines are real and not an artefact.

It closed on the question a spectroscopist actually faces, which is not how much satellite intensity there is but which lines are the fundamentals. Given a spectrum with a hundred lines in it, something has to decide.

Two tests are available, both about the distribution rather than the total: sort by height, or sort by position. Measured across eight repulsions, each works over a range and then stops — and the finding is that they stop at the same place.

The two tests #

The system is the same: six sites in a ring, half filled, one electron removed. The parent and the ion are both diagonalised exactly, and every pole of the spectrum — its energy and its weight — comes from an overlap between the parent’s ground state with an electron taken out and each of the ion’s eigenstates.

The fundamentals are the six strongest poles, because six is how many lines the one-electron picture has. That is a definition rather than a result, and it is the only honest one available: an exact calculation has no orbitals in it to label lines with. Fewer bands than electrons is the case where the count itself is wrong for a symmetry reason; here the count is right and the identification is not.

The first test is a contrast. How many times stronger is the weakest of those six than the strongest of everything else? A large ratio means a threshold can be drawn and the spectrum sorted by height. A ratio near one means it cannot.

The second test is an overlap in energy. The six fundamentals span a range. A satellite outside that range can be recognised as an extra, because it is somewhere no band should be; a satellite between two fundamentals cannot, because a spectrum with an unexpected line among the expected ones is a spectrum whose assignment is a guess.

What they say #

The contrast falls by a factor of twenty and the second test switches from clean to muddled, and both happen between a repulsion of 4 and one of 6.

At a repulsion of 6 the tallest satellite is 87 per cent as tall as the shortest band. That is not a threshold that has become difficult to place; it is a spectrum in which the six tallest lines are the six tallest lines and nothing about them says they are the fundamentals.

Why they fail together #

The two tests look independent and they are not, and the reason is worth having because it says the failure is structural rather than a coincidence of this ring.

Both are consequences of the same thing: the ion’s states stop being one configuration with a bit of admixture and become mixtures with no dominant term. While a final state is mostly one determinant, the pole that reaches it is strong and the ones that reach its neighbours are weak, and the strong poles sit where a one-electron picture puts them. Once the final states are genuine mixtures, the removal amplitude is spread over many of them — which lowers the tallest and raises the rest, and moves the weight to wherever the mixtures are rather than to where an orbital was.

That is the same mechanism Koopmans’ theorem fails through, taken one step further. There the failure was that an ionisation energy is not an orbital energy; here it is that an ionisation line is not an orbital at all, because the state it reaches is not made of one.

So the contrast falling and the satellites moving inwards are two views of one loss. There is no combination of the two tests that survives it, and there is no third test of the same kind: the spectrum’s own information about which line is which has gone.

The contrast does not fall, it dips #

One detail in the table refuses the simplest reading and is worth keeping rather than smoothing over.

The contrast is 2.01 at a repulsion of 4, 1.15 at 6, 1.51 at 8 and 1.24 at 12. It is not monotone: it has a minimum near 6 and comes partly back.

That is where the repulsion is comparable to the band width — four resonance integrals for a ring at this hopping — and it is the region where a state made of one configuration and a state made of another are nearest in energy and mix most completely. Far above it the two sets of states pull apart again, the removal weight re-concentrates on a smaller set, and a contrast of a sort returns.

It does not return usefully. The satellites are still among the bands at every repulsion above the boundary, so the second test stays failed while the first partly recovers — which is a good reason to have two tests rather than one, and a bad reason to trust either alone. The insulator band theory cannot see opens in the same region, so the repulsion where a spectrum stops being assignable is the repulsion where the system stops being a band material at all.

The total is the wrong thing to watch #

This is the practical conclusion, and it contradicts the natural reading of the count of lines.

The satellite share grows smoothly — 3, 10, 20, 31, 46, 53, 66 per cent — with no feature in it anywhere. A spectroscopist watching that number would conclude that assignment gets steadily harder and that a spectrum with 30 per cent satellite intensity is about half as difficult as one with 60.

That is not what happens. At 31 per cent satellite intensity both tests still work: the weakest band is twice the tallest satellite and no satellite is among the bands. At 46 per cent neither works. The property that matters is not how much intensity is in the satellites but whether the removal amplitude is still concentrated, and those two part company.

The same shape of mistake is what a band’s width can hide: a total is a summary, and the assignment problem is about a distribution.

What a spectroscopist should take from it #

A quoted satellite percentage says nothing about assignability. Two spectra with the same satellite share can be on opposite sides of the boundary, because the share is an integral and the boundary is about where the weight sits.

An unassignable spectrum is not a noisy one. It has exactly the same number of lines as an assignable one at the same repulsion, computed to the same precision. What it lacks is a property of the pattern, and no improvement in resolution recovers it.

And a calculated spectrum is the only way to tell which regime a measurement is in. The tests above both need to know which lines the fundamentals are, which is the thing being asked. That circularity is not an accident of the method — a spectrum counts environments rather than atoms, and what it does not carry is a label saying which environment each line came from.

What a measurement would see #

Three things follow for anybody reading a real spectrum, and all three are consequences of the boundary rather than of the model’s details.

A spectrum can be complete, exact and unassignable. Nothing is missing from the calculation at a repulsion of 6: every line is there, every weight is right, and the sum rule holds. What has gone is a property of the pattern. A measurement of the same system would be equally complete and equally unassignable, and no better instrument would change that.

The first band is safe and the rest are not. The lowest-energy pole keeps a large weight throughout, because the ion’s ground state stays mostly one configuration for longer than its excited states do — so the first ionisation energy survives the boundary and the assignments above it do not. That is the same asymmetry water’s lone pairs turned out to have, where the outermost band is the one everybody agrees about.

And a computed spectrum has to be compared as a pattern, not line by line. Matching a computed line to a measured one by energy is exactly the operation that stops being possible here. What survives is the shape of the whole distribution — its centre, its spread, how much weight is where — and comparing those does not need any line to be identified.

What is quoted, and what is computed #

Nothing is quoted. There is no molecule and no measurement in this essay. The ring, its filling and the repulsions are the model’s parameters; every level, every pole energy, every pole weight and every count is computed from a full diagonalisation of the many-electron Hamiltonian for the parent and for the ion.

The sum rule is the check that runs under all of it: the total removal weight must equal the number of electrons available to remove, exactly, and it does to eight decimal places at every repulsion. A distribution of the wrong numbers would still have a shape and would still produce a plausible essay, which is why the check is on the numbers rather than on the conclusion.

Two handles the spectrum does not have #

Sorting by height and sorting by position both fail, and both are properties of the same one-dimensional object — a list of positions and intensities. A real photoelectron measurement has two further variables, and each of them separates the two kinds of line by a mechanism the energies cannot see.

The angle. Photoelectrons do not leave a molecule isotropically. The distribution has a shape characterised by one parameter, and that parameter depends on the character of the orbital the electron came from — whether it was s-like or p-like, and how the outgoing wave interferes with itself. A fundamental line carries the parameter belonging to its orbital. A satellite is a two-electron process, its outgoing electron has a different energy and reaches the detector through a different channel, and its parameter is generally different.

So measuring the intensity at two angles rather than one gives a second number per line, and two lines at similar energies with different angular parameters are two different kinds of process.

The photon energy. The probability of ionising a given orbital varies with the energy of the light, and it varies differently for different orbitals — a d-derived level and a p-derived one have cross-sections that cross over as the photon energy is raised, which is how the two are told apart in solids. Satellites vary differently again, and some of them are strongly enhanced when the photon energy is tuned to a core resonance, at which point a line that was a percentage becomes a major feature.

So sweeping the photon energy gives a third number per line, and a line whose intensity behaves unlike its neighbours’ is a line of a different kind.

Both are standard and both are outside this model, which contains energies and intensities at one angle and one photon energy. That is worth stating precisely: the sorting problem measured here is a real limitation of a spectrum and not of spectroscopy. A list of positions and heights genuinely stops distinguishing satellites from fundamentals at the repulsion located here; the experiment does not, because it has variables the list has thrown away.

That distinction is worth carrying because it applies to every result that reports what a spectrum can and cannot determine. A spectrum is a projection of a much richer measurement onto one axis, and a statement of the form this cannot be told apart is a statement about the projection unless the other axes have been checked. Here they have not, and could not be — the model produces energies and intensities and has no angular variable in it at all — so the honest reading is that a one-dimensional spectrum stops sorting at the repulsion measured, and that the experiment carries two more dimensions the model does not.

What this cannot say #

Six sites, one orbital each. A real molecule has orbitals of different energies and different symmetries, and a satellite that belongs to a different symmetry species from every fundamental can be assigned, by a selection rule rather than by intensity. That is a route this model has no room for, since every site here is alike.

No vibrations. A real photoelectron band has vibrational structure, and a band’s shape is a filter on the modes; a satellite with an unexpected vibrational envelope is recognisable as one. Nothing here has a nucleus in it.

No angular information. Measuring the photoelectron’s angular distribution gives a second number per line, and two numbers separate cases that one cannot. That is genuinely how the problem is attacked in practice and it is outside this model entirely.

And no comparison with what a spectrum actually measures. Everything here is a pole strength, which is the quantity a photoelectron spectrum reports only after a cross-section that depends on the photon energy has been divided out. Two lines of equal pole strength are not two lines of equal height in a measurement.

And the six-strongest rule is a definition. Calling the six tallest poles the fundamentals is what makes the tests computable, and it builds in the answer at small repulsion. At large repulsion it is arbitrary — which is the finding, stated as an assumption.

What was checked #

With no repulsion the spectrum is the one-electron one — as many lines as there are electrons to remove, and every one carrying the whole of one electron to eight decimal places. That is the tripwire: a calculation that had lost the amplitudes would still produce a hundred lines at a large repulsion and would fail here.

At a small repulsion a fundamental is many times stronger than any satellite, above five, or the first test would never have worked.

And at a large one it is barely stronger, below 1.6 — so no cut in intensity separates them.

There are repulsions on both sides of the second test as well, and it fails above a repulsion rather than at scattered ones, which is what makes it a boundary rather than noise.

Where satellites do sit among the fundamentals they carry a real share of the intensity, above 30 per cent of the satellite weight — a check that would fail on a calculation where one negligible pole had strayed inside the range.

The two tests fail at the same repulsion, within two units, which is the finding.

And the sum rule still holds at every repulsion tested, to eight decimal places, so every distribution above is a distribution of the right numbers.

Still open: whether a second observable can sort them #

The obvious open question is the second number a measurement can supply. An angular distribution gives each line an asymmetry parameter that depends on the symmetry of the orbital the electron came from, and two numbers per line can separate what one cannot. Whether the satellites here would have a different asymmetry from the fundamentals is a question about the removal amplitudes’ angular structure, which this model does not have — but the question of how much separating power a second observable buys is answerable in the abstract, by asking how correlated the two would have to be to help.

The nearer question is about the boundary itself. It has been located here between two repulsions on one ring at one filling, which is enough to say it exists and not enough to say what it depends on. Running the same two tests at other fillings and other ring sizes would say whether the boundary is a property of the repulsion alone or of the repulsion measured against the band width — and if it is the second, a spectroscopist could estimate which side of it a real material is on from quantities that are already measured.