Canonical definition

For a substitutional or interstitial solute X at a grain-boundary site in a host H, the segregation energy per defect is

with the stoichiometric balance and per-atom host chemical potential

Sign convention. E_seg < 0 ⇒ the solute X favors the GB site (segregation is energetically favorable relative to the bulk reference). E_seg > 0 ⇒ the solute prefers the bulk.

Term-by-term reference

Every symbol is a total energy (smeared VASP energy, not energy_zero — see the feedback_energy_sigma memory) or an atom count on a specific structure row. Both DFT and MLIP sides pull the same four energies and the same four atom counts, just from different energy columns.

The chemical-potential reference is pure-host bulk (Fe or Ni). It is not the free-atom reference, not an iso-atom offset, and not the per-(composition, ISPIN) corrected energy used elsewhere on the ASSYST single-point track — see Why no E_seg_corrected below.

Reference resolution per variant (KP / KS / OMAT fallback rules, Ni-host variant) is documented on the deep page. Implementation in build_eseg_dfs.py (compute_eseg). The four reference rows (pure GB, X bulk, host bulk, and the seg-GB row itself) are all ingested as ordinary structure rows in the DB and run through the same single-point / relax pipeline as the segregated structures, so the MLIP side reads exactly the same four cells the DFT side does.

Why no E_seg_corrected

E_seg is a difference of total energies taken between systems with the same composition (after accounting for stoichiometry through n_host_diff and μ_host). The per-(composition, ISPIN) offset corrections used on the ASSYST single-point track cancel by construction in this formula. We deliberately do NOT emit E_seg_corrected columns — there is no analogous correction to apply without changing the science.

BFGS-vs-FIRE fallback rule (best-available)

The corpus was first run with BFGS, fmax = 0.01, max_steps = 10000. A subset of structures (~54k across 50 potentials) hit the BFGS step-cap or otherwise errored — almost all on heavy-radioactive solutes (Po/At/Rn/Fr/Ra, Z = 84–88) that the underlying calculators do not embed.

Those errored structures were retried with FIRE, fmax = 0.01, max_steps = 250. FIRE recovered the slow-converging cases that BFGS step-capped on (notably the MatRIS pair on Ni-host structures); the remaining unsupported-element rows continued to error on the same structures with the same KeyError.

For each potential and each structure_id, the headline metrics use the best-available result:

energy_used = energy_BFGS if status_BFGS == 'done'
              else energy_FIRE if status_FIRE == 'done'
              else NaN  (excluded)

Per-row provenance is preserved in the rollup CSVs via algo_used, n_best_from_bfgs, n_best_from_fire.

Aggregation

Per-(potential, host, solute, partition) RMSE/MAE/Spearman is computed as a count-weighted two-step aggregation matching the ASSYST headline pattern (see headline_rmse_provenance memory). Concretely:

1. Per (host, solute, partition) bucket: RMSE / MAE / Spearman ρ over (E_seg^MLIP - E_seg^DFT) for the structures where the MLIP has a best-available done result.
2. Per potential, the top-line E_seg RMSE is the count-weighted mean of (1) across all buckets, weighted by the number of evaluated structures.

Both E_seg RMSE (eV) and E_seg MAE (eV) are emitted in the rollup and selectable as the leaderboard headline.

Partitions

The gbseg-relaxed-* MLIP relaxation is initialised from the unrelaxed (as-constructed / DFT-input) geometry, not the DFT-relaxed frame — the relaxation's initial frame is the unrelaxed structure.

W_sep (gbseg-cleavage) lives on its own page — see GB cleavage.

DFT reference subsets

The reference E_seg numbers in the gbseg corpus come from three independent DFT sub-campaigns on the Fe-host side and one campaign on the Ni-host side. The rollup CSV carries the campaign tag in a dft_method column and the leaderboard's "DFT method" radio filters on it; metrics are NOT pooled across methods.

Why O+F are excluded from KP, KS, and Kfine

The KP/KS DFT references for O- and F-substitutional structures are systematically inconsistent with the way those species sit in the MLIPs' training sets (SMAX / MPtrj) — the training-set sampling for these species is biased toward oxide/halide chemistries rather than dilute solutes in metallic GBs. Headline KP/KS RMSE numbers therefore exclude O+F so the ranking reflects bulk-metal chemistry the MLIPs are actually competent on.

The same systematic flaw applies on the Ni-host side: the SMAX/MPtrj training-set coverage of O/F in metallic Ni grain-boundary environments is just as biased toward oxide/halide chemistries as on the Fe-host side, so Kfine follows the same rule and omits O+F-solute rows from headline aggregation (no Kfine_with_OF companion is reported — the framing is "omit", not "compare").

Why OMAT keeps O+F (and also reports OMAT_no_OF)

OMAT-trained potentials are designed to handle oxide and halide chemistry properly, so for the OMAT subset we report the full headline (with O+F) and a companion OMAT_no_OF recomputation that drops the same two solute columns. The two side-by-side numbers directly quantify how much of an OMAT-trained MLIP's headline performance comes from (or is hurt by) the O+F slice.

Both OMAT and OMAT_no_OF are emitted as separate rows in the rollup CSV with the same (potential, host, solute, partition) keys but distinct dft_method values. Per-solute rows for non-O/F solutes are numerically identical between the two variants; only the __all__ aggregate rows differ.

Outlier removal

A small subset of (potential-independent) DFT E_seg values in the corpus are catastrophic outliers — wrong unit cells, mis-relaxed neighbours, SCF non-convergence — that would dominate any RMSE if left in. We remove them via the Iglewicz & Hoaglin modified z-score test:

A point is flagged when |m_i| > 3.5. Grain: per (dft_method, host, solute, defect_type) bucket. For buckets smaller than 12 rows the test is statistically uninformative; those fall back to a sanity threshold of |E_seg| > 20 eV.

The current denylist removes 1794 DFT structures: KP 422 · KS 603 · OMAT 359 · Kfine 410.

Snapshot date

Every per-potential summary CSV carries a snapshot_date column. The headline date on the leaderboard is the minimum across all rolled-up potentials — if any potential's rollup is stale, the headline flags it.

For deeper provenance (LSN, exporter git sha, build_eseg_dfs.py commit), see Provenance and the gbseg wiki hub.

Code

Segregation energy from four MLIP total energies (compute_eseg in build_eseg_dfs.py). The MLIP evaluates the four cells; the formula assembles E_seg:

from build_eseg_dfs import compute_eseg
# E_* are MLIP total energies (eV); n_host_* the host-atom counts of each cell
eseg, n_host_diff = compute_eseg(
    e_seg=E_seg,        n_host_seg=N_seg,          # segregated GB (host GB + 1 solute)
    e_pure_gb=E_pureGB, n_host_pure_gb=N_pureGB,   # pure-host GB
    e_elem_bulk=E_Xbulk, n_host_elem_bulk=N_Xbulk, # solute-in-host bulk reference
    e_host_bulk=E_host, n_host_host_bulk=N_host)   # pure-host bulk reference
# eseg = segregation energy (eV); negative ⇒ segregation is favourable