Can a fast formula over neighbouring atoms stand in for the slow energy model?

EXPERIMENTS.md · lines 612–678

E16 — A cluster expansion works, and it is nearly free

Date 2026-09-12 · Question Can a cluster expansion be built from this data, and would it address the occupancy and finite-size problems? · Provenance …/ce_pilot2.py, icet 4.0

Why CE. A cluster expansion writes the energy of any occupancy on a fixed lattice as a sum over clusters with fitted interactions. Once fitted it evaluates any cell instantly, which addresses four measured defects at once: occupancy noise (E2/E4) becomes a Monte Carlo average rather than a 1-or-10 sample; the finite-size wall (E3/E5) disappears because a fitted CE evaluates 1000-atom cells; the ideal-entropy artefact that makes the 1500 K ranking a component count becomes a real configurational entropy with short-range order; and the composition baseline of E12 is a crude CE, so a proper one is its principled form.

Parameter cost, measured with icet on bcc:

A full eight-element CE therefore needs a few thousand structures - about 300-600 CPU-hours of DFT, which is not available. But MACE-MPA-0 produces one in 0.04 s, so the standard route is to fit the CE to the potential and keep DFT as validation.

Method. Four elements (Mo, Ta, V, W), cutoffs 6.0/4.5, 98 parameters, 775 random occupancies of a 16-site bcc cell at one fixed lattice parameter (3.1593 A, Vegard for equiatomic MoTaVW). Fitted to the mixing energy, not the raw energy: raw energy per atom spans ~7,000 eV across these compositions because V is -2911 and W is -10343 eV/atom, so a raw fit is dominated by which elements are present and reached only 74 meV/atom. Elemental references taken on the same fixed lattice, as a fixed-lattice CE requires.

Result.

meV/atom. The CE reproduces MACE to 0.76 meV/atom mean absolute difference on held-out occupancies. It is an accurate and essentially free surrogate.

Two findings.

1. A reference-convention trap. The CE gives dH of -195 where DFT gives -55 for the same cells. This is not an error: a fixed-lattice CE references elements at the common lattice, while the DFT here references them at their own equilibrium lattice. The ~140 meV/atom difference is the elastic energy of straining each element to the shared lattice. Only quantities on the same convention may be compared, and occupancy-to-occupancy variation is convention-independent because the reference cancels.
2. MACE-MPA-0 may still exaggerate configurational spread. Occupancy standard deviation across the same three cells: DFT 4.03, CE 8.98, MACE 9.83 meV/atom. The CE tracks MACE closely, as it should, so the gap is MACE against DFT. Caveat: the DFT sd comes from n=3, where a standard deviation is very weakly determined. This is a lead worth more occupancies, not a measured factor.

Next for this thread. Extend to all eight elements (344 parameters, a few thousand MACE structures, minutes), then Monte Carlo for configurational entropy and short-range order - which is what would replace the ideal-mixing term that currently makes the 1500 K ranking a count of components.