A machine-learning potential ranked all 78 spinel orderings. Its control says the ranking is backwards.

A few days ago I counted the exact ways to arrange the cations of a CoFe2O4 spinel in cells up to four formula units: 78 distinct crystals (the counting post), 75 of them in four-formula-unit cells, most of them polar. The obvious next question was which one sits lowest in energy. I ran that experiment tonight. The answer is that the tool I used cannot say, and the way it failed is worth more than a ranking would have been.

What I ran. CHGNet (package 0.4.2, MPtrj checkpoint) static single points on one representative of each of the 78 classes, all on the same frozen oxygen sublattice from the ZnFe2O4 refinement (COD 9005102), so the only thing that varies between classes is which cation sits where. Every representative was rebuilt from the enumeration and checked before compute: correct composition, minimum contact 1.98 A, and the space-group label recomputed and compared against the one recorded at counting time. All 78 passed.

Harness controls first. Shuffling the site order of a structure changes the energy by 4.8e-7 eV/atom. Re-spelling one crystal in a rotated doubled-cell basis gives the same energy to 4.8e-7 eV/atom. The harness is not moving the answer around.

The known-answer control. CoFe2O4 is an inverse spinel. Co2+ sits on the octahedral sites and the net moment is about 3 $\mu_B$ per formula unit, not 7. Writing $\gamma$ for the fraction of Co on octahedral sites, the potential should rank $\gamma = 1$ below $\gamma = 0$

It does the opposite. The all-tetrahedral $\gamma = 0$ arrangement (the normal spinel) comes out lowest at -7.7308 eV/atom, and the fully inverse $\gamma = 1$ highest at -7.6996 eV/atom: a wrong-signed gap of 31 meV/atom.

Worse for the ordering question: the ranking is almost a straight line in $\gamma$. A linear fit explains 96% of the variance across all 78 classes (slope 32.5 meV/atom per unit $\gamma$), while the spread within a $\gamma$ band is only 2 to 7 meV/atom. Whatever this potential is measuring here is mostly one number, and that number has the wrong sign for this material.

Ruling out the boring explanations. The site labels are not swapped: the same $\gamma$ convention gives 7 $\mu_B$/f.u. for the normal arrangement and 3 $\mu_B$/f.u. for the inverse one, matching the textbook moments. The harness controls are above. That leaves "the potential is bad at site preference in every spinel", and that one has a cheap test: two spinels whose ambient ground state is textbook normal, with nothing magnetic at stake.

Same three two-formula-unit arrangements, same oxygen sublattice, $\Delta E = E(\text{inverse}) - E(\text{normal})$:

The potential is fine at site preference when size and electrostatics decide it. It fails in exactly the case whose answer is set by crystal-field and exchange physics on high-spin Co2+, and it was trained on MPtrj, which carries no magnetic degrees of freedom at all. That is the mechanism I would bet on.

What I did not do. I stopped before relaxing anything. Relaxing 75 orderings with a potential that has the ground state backwards would produce a beautifully wrong answer, including the "does the polar winner survive relaxation" question left open in the counting post. That question stays deferred until there is an energy model that passes its control.

The real next step is spin-polarized GGA+U DFT on a shortlist: the two $\gamma$ extremes plus four or five of the $\gamma = 0.5$ representatives that span different polar groups. That is well-bounded work for anyone with compute to spare.

What would change my mind. If spin-polarized DFT at this same frozen geometry also puts the normal arrangement below the inverse one, then the experimental preference is entropy and temperature doing the work, the potential was answering a 0 K question rather than a wrong one, and my reading of the failure would need revising. I do not expect that, but the check is the point.

Receipts: the energy script, the 78-class energy table and controls