Beyond special quasirandom structures: free energies from energy cumulants

The idea in plain language

A disordered material (say, lithium and metal atoms scattered randomly over a crystal lattice) can be arranged in astronomically many ways, each with a slightly different energy. The standard shortcut, a “special quasirandom structure”, takes the average energy of these arrangements and adds the ideal entropy of mixing. The authors show this is just the first term of an exact series in the cumulants of the energy distribution: the mean, then the variance (how spread out the energies are), then the skew, and so on.

Keeping the variance lets the model capture that, on cooling, atoms prefer low-energy arrangements (short-range order). In the paper’s benchmark on 502 equiatomic alloys of a nine-element refractory system, keeping the variance cut free-energy errors by roughly an order of magnitude. They also screened Li-excess rocksalt oxide cathodes, ranking candidates for the Li₄ clusters that let lithium move, and note the screen is a ranking, not a quantitative prediction.

Interactive demo: how much does each cumulant buy you?

A toy model (not the paper’s data): per-site energies are a constant plus a Gamma-distributed amount, so the exact free energy has a closed form and the paper’s approximations (its Eq. 4, 5, 6) can be compared against it. Everything below is per atom.

Faded dashed lines mark temperatures below the Gaussian truncation’s validity limit β_c = √(2 s₀/κ₂), where the paper notes truncations stop being physical. Regular solution is always an upper bound on F (Jensen’s inequality), which you can see: its error never goes negative.

How to use it

1. Pick a disordered composition and generate a few hundred random atomic arrangements on the lattice (the paper uses random decorations of supercells, not Monte Carlo).
2. Evaluate their energies with a fast model, for example a foundation machine-learned potential (the paper uses GRACE).
3. Compute the energy mean, variance and skew (cumulants). Plug them into the truncated or binomial free-energy formulas above; no further sampling is needed.
4. Compare the resulting disordered-phase free energy with competing ordered phases to estimate where the disordered phase becomes stable, then send only the best candidates to heavier simulation.

Caveats from the paper: truncated series lose validity at low temperature; accuracy is inherited from the underlying energy model; free energies are configurational only (no vibrations); the screening estimate over-predicts Li₄-cluster probability versus Monte Carlo, so near-threshold candidates need verification.