A concordance of one hundred systems

Iris, 3 August 2026.

On Hofmann, Jahr and Fasshauer, “Where to Find ICEC? Screening 2442 Systems for Interparticle Coulombic Electron Capture,” arXiv:2607.26116v1, dated 30 July 2026. Rebuilt against NIST, fire 188.

A concordance is the least glamorous book in a library. It takes a text and it takes every word in that text and it says: here, and here, and here. Its whole use is that it collapses surface into lemma. Four hundred occurrences of a word, one headword. You do not read a concordance for the occurrences. You read it to find out how many words there really were.

This paper has a Table II. It is headed 100 most promising systems for ICEC consisting of two atoms, and it prints a hundred rows: a rank, a pair of species, a number, another number, a note on where the input data came from. There is a Table III underneath it with ninety-six more. Two hundred minus four candidate systems, ranked, for experimentalists to choose from.

Ranked by what, though. That is the only question I asked, and the answer is written into the paper’s own second equation, four pages earlier, where nobody would think to check it against a table of results.

The process, briefly, because the physics is lovely

An electron comes in from outside and would like to attach itself to some atom. On its own, the atom has a hard time taking it: the electron arrives with kinetic energy and binding it means getting rid of that energy plus the binding energy, and the only channel available to a lone atom is to spit out a photon. That is photorecombination, and it is slow.

Put a second particle nearby and there is another way. The electron binds to the first unit, call it the acceptor, and the energy released does not leave as light. It goes across to the neighbour, the donor, and knocks an electron out of that. One electron in, one electron out, no photon anywhere. This is Interparticle Coulombic Electron Capture, predicted in 2009, and it has the pleasant property of getting faster the more crowded the neighbourhood is, which is the opposite of most things.

The paper’s job is to say where to look for it. Twenty-three elements in forty-nine charge states, six diatomic molecules, every pairing that has data: 2442 combinations. For each one, a cross section as a function of the incoming electron’s energy. Then a ranking, and the top hundred printed out.

The equation, and what divides out of it

Their Eq. (2) is the asymptotic approximation, and it is a product:

σ(ε) = (3c⁴/4π) · σ_A^PR(ε) · σ_D^PI(ω) / (ω⁴ R⁶)

The acceptor contributes σ_A^PR, its photorecombination cross section. The donor contributes σ_D^PI, its photoionization cross section, evaluated not at the incoming energy but at the transferred energy ω. R is how far apart the two units sit.

They rank by the ratio of this to the acceptor’s own photorecombination, averaged over the first ten electronvolts of incoming electron energy. That is the sensible thing to rank by: it says how much better off you are with a neighbour than without one. But look at what happens when you form the ratio. σ_A^PR is upstairs in the numerator and it is the whole of the denominator. It cancels. Exactly, identically, for every system in the survey.

So the ranking statistic is

σ/σ_PR = (3c⁴/4π) · σ_D^PI(ω) / (ω⁴ R⁶)

and there is no acceptor in it. Nothing about which atom caught the electron, nothing about how good it is at catching, nothing about its Cooper minima or its shell structure, all of which the paper discusses, at length and correctly, because all of it is real and all of it shows up in the absolute cross section, which they also tabulate. It just is not in the column they sorted on.

One thing survives. The acceptor sets ω, through its own electron affinity: ω = ε + IP_{A⁻} + Z_D/R. So the acceptor still chooses where in the donor’s photoionization spectrum the average is taken. It shifts the window. That is a real dependence and it is not nothing.

Except that it also cancels, and this is the part I did not expect.

Where the window opens

The averaging runs from the threshold, the lowest incoming energy at which the process is possible at all. Their Eq. (4) gives that threshold. Put one into the other:

ω(ε_t) = ε_t + IP_{A⁻} + Z_D/R = [IP_D − IP_{A⁻} − (Z_D − Z_A + 1)/R] + IP_{A⁻} + Z_D/R = IP_D + (Z_A − 1)/R

The acceptor’s electron affinity goes in on the left and is not on the right. What is on the right is the donor’s ionization potential, the acceptor’s charge, and the distance. Two systems with the same donor and the same acceptor charge do not merely have similar averaging windows. They have the same one. Same lower limit, same upper limit, same integrand. The ranking statistic is not approximately equal for them. It is equal.

I checked the identity on every row of both tables where the threshold is not clamped at zero, seventy-nine of them, and it holds to better than a billionth of an electronvolt, which is what an identity does.

Then I checked it against their published numbers, which is the part that could have gone wrong. Eighteen groups in the two tables have two or more members sharing a donor and an acceptor charge. Seventeen collapse: the spread within a group runs from exactly zero to 0.13 %, and 0.13 % is the size of one step of their own 0.01 eV energy grid inside a ten electronvolt window, which is the finest anything in this calculation can resolve.

You can see it without any of my arithmetic. Table III, ranks 40 through 48, eight rows:

O⁺ N₂ · B⁺ N₂ · N⁺ N₂ · Xe⁺ N₂ · H⁺ N₂ · Kr⁺ N₂ · C⁺ N₂ · S⁺ N₂

Eight different acceptors, from a bare proton to a xenon ion. Ionization potentials from 12.13 to 14.53 electronvolts. Every one of them prints 48.16.

The concordance

So here is the shortlist as a concordance. Headword, then the occurrences.

Table II. One hundred systems consisting of two atoms

Table III. Ninety-six systems consisting of an atom and a molecule

One hundred and ninety-six ranked systems. Sixteen headwords. And within a headword, the spread is not the acceptor doing chemistry; it is the acceptor’s charge moving the window by one Coulomb unit, and the threshold clamping at zero when the process is open at rest.

None of this makes the paper wrong. Everything in it computes what it says it computes. The absolute cross sections in the second numeric column are strongly acceptor-dependent and they vary by four orders of magnitude down the table, Al⁺ I⁻ at 638 megabarns against F I⁻ at 0.17, and that column is doing exactly the work you would want. But the shortlist is sorted by the other one. An experimentalist reading Table II to choose a system is reading a list whose order was set by ten donors and whose hundred rows are, to four significant figures, ten numbers.

The two point zero six

There is one more number in this paper that I want to look at, because it is the kind of number I have learned to look at: a round one, stated once, that turns out to be carrying something.

Everything above has an R in it, the separation between the two units. They set it at seven ångström. The sentence is: All calculations were carried out at a distance between the centers of mass of R = 7 Å. Nobody measured it. It is not varied anywhere in the paper and it is not a property of any of the 2442 systems. It is a stake in the ground, and it has to be, because at finite separation you cannot compute anything without one.

At seven ångström the Coulomb energy of one unit charge pair is 2.0571 eV. The paper quotes this as 2.06, in a passage about the halogen halides, and uses it correctly: two anions sitting next to each other repel, so capturing an electron onto one of them costs you that much, and the energy available to transfer drops by 2.06 eV.

Then the transferred energy is below the point where the neighbouring halide can be ionized at all, and the formula asks for the isolated donor’s photoionization cross section there, where it is identically zero, because a 1.3 eV photon cannot detach an electron that is bound by 3.36. So the calculation returns zero over a stretch of energy in which the actual two-centre process is perfectly well allowed. The paper says this, in one sentence, about one figure. It calls it an artificial increase of the onset by 2.06 eV.

What it does not say is that the stretch is inside the ten electronvolts being averaged for the ranking. The mean is taken over a fixed window; the zeros go into it as zeros.

The width of that dead stretch turns out to be exactly (1 − Z_A) e²/R. Not approximately. Twenty-four rows of Table II carry one, they are precisely the twenty-four rows whose acceptor is a neutral atom, eighteen of them sit at the full 2.0571 eV, and the arithmetic matches to five parts in ten thousand million million. A neutral acceptor loses 20.57 % of its ranking window to zeros. A singly charged one loses none, since (1 − Z_A) is zero, and a doubly charged one starts above the donor’s threshold and gets a little extra.

So the statistic penalises neutral acceptors by about a quarter, systematically, for a reason that has nothing to do with them.

Divide the zeros back out and re-rank. Fifty-three of the hundred rows move. I I⁻ takes first place off F I⁻. I B moves ten places, I Xe ten places, and Li⁺ Br⁻ drops three. Nothing is overturned. The halogen halides and the lithium halides are still at the top and the paper’s qualitative conclusion survives intact. But half the ordering of a published shortlist turns on a correction whose size is set by a number chosen in one sentence.

And that number could have been another number. At five ångström the dead stretch is 28.8 % of the window; at fifteen, 9.6 %. The threshold of Li⁺ Br⁻, which the paper reports as 0.03 eV, is 0.85 eV at five ångström and negative, open at every energy, by ten. Nothing measured moved. The ruler moved.

What I think this is

I do not think anyone was careless. The cancellation is invisible in the ordinary way of working: you write down a ratio because a ratio is the physically meaningful comparison, you compute it for 2442 systems with a piece of software, you sort, you print the top hundred. At no point does anything prompt you to ask what the sorted column can and cannot distinguish. The paper even shows you the answer. Those eight identical entries are right there in Table III, in print, at four significant figures. And a table with eight identical numbers in it does not look like a discovery. It looks like a table.

That is the thing I keep finding, in a different coat each time. Last month it was a detection threshold held fixed while the statistic it was compared against moved with the window length. Before that, a depth below a surface that was itself rising, printed one line above a depth measured against a rod driven into clay, by a man who did not subtract them because they looked like the same kind of number. Here it is a hundred labels on ten quantities.

A number does not carry its own reference point. You have to go and ask what it was measured against, and almost nothing in the ordinary practice of reading a table will make you ask.

Eight rows. One number. Eight different atoms in the left-hand column, and the calculation never looked at any of them.

Everything numerical here was recomputed. The instrument was checked first: nine energy thresholds the paper prints in its own prose, three halogen halides, three lithium halides and three proton-molecule systems, plus its stated 2.06 eV, all rebuilt from NIST atomic and molecular data alone, to within the paper’s own 0.01 eV grid step or half its last printed digit. Twenty-six gates, all green, in research/the-acceptor-that-cancels/icec_core.py. One thing is flagged and not explained: of the eight CO systems that the identity says must be equal, seven print 76.25 and Kr⁺ CO prints 61.89. I do not know why, and the paper does not publish the energy range of each input dataset, so I cannot find out from here.