How far from converged is ggen's random structure search?

ggen's explore gives every stoichiometry 15 random PyXtal cells spread over 5 space groups, relaxes all of them with ORB, and keeps the lowest energy. Hull distances are relative, so if that budget undersamples a competing phase, a mediocre neighbor looks stable. Before trying to make the search smarter, we wanted to know how far from converged it is.

Setup

Seven formulas. Five are known compounds whose ORB-relaxed Materials Project ground state we have, so there is a ground truth. Two are Fe-Co(-Mn) alloy cells with no known phase. For each, 200 candidates were proposed exactly as explore proposes them (4 seeds × 50 trials over 5 space groups each) and all were relaxed with the pooled torch-sim relaxer (FIRE, 400 steps, fmax 0.01 eV/Å, ORB v3 conservative, one A10G). Every trial keeps its final energy, the energy after 20, 50 and 100 steps from the same starting cell, its space group, and a basin id from pymatgen's StructureMatcher. The whole study cost under a dollar of GPU time.

The headline

Resampling the 200 outcomes gives the chance that the best of n trials lands within 5 meV/atom of the best of 200:

Two things at once. The proposer is fine: with 200 trials every known ground state was recovered to within a meV. The budget is not: at the default 15 trials, a 12 to 16 atom ground state is missed by more than 5 meV/atom about half the time, and the 90th-percentile miss is over 100 meV/atom. Emulating explore exactly (3 trials in each of 5 space groups), the four seeds for Fe8B4 missed by 0, 0, 127 and 133 meV/atom. It is a lottery. About 75 trials gets to 95%.

Where the ground state lives

For the known compounds the ground-state basin is hit by 3 to 5% of trials and almost nothing else is close. Fe4Sn8 has exactly one basin within 50 meV/atom of the best, hit 8 times out of 200. Fe8B4 has three, Fe12B4 three.

The basin was reached from several different proposed space groups: Fe8B4 from SG 79, 131, 165 and 84; Fe4Sn8 from 26, 38, 112, 162 and 74; Fe12B4 from 19 and 200. The symmetry you start in does not decide the outcome; the basin of attraction under relaxation does. One consequence is free: resampling 15 trials as 15 space groups × 1 instead of 5 × 3 raises the hit chance from 29% to 42% for Fe12B4, with no prior about which groups to prefer.

Alloy cells are a different problem

Co3Fe5Mn12 found 197 distinct basins in 200 trials, 37 of them within 50 meV/atom of the best, every single one hit exactly once. Co2Fe15 had 13 low basins, 10 of them singletons. These are site-ordering variants of a bcc or close-packed solid solution. There is no ground-state basin to converge to, the orderings differ by a few meV/atom, and random restart will keep finding new ones indefinitely. The Co3Fe5Mn12 winner beat the database entry by 27 meV/atom and would keep going. Claims about compositions like these belong to a cluster expansion or special quasirandom structures, not to "the lowest random cell we found".

Fe10Si6

The best Fe10Si6 cell is Cmcm at -7.7266 eV/atom, 40 meV/atom below the ORB energy of the Materials Project Fe5Si3 phase. On the ORB Fe-Si hull it sits 6 meV/atom above the Fe3Si (Fm-3m, -8.0131) to FeSi (P2_13, -7.4522) tie line, so it is a near-hull ordered bcc-derived arrangement rather than a new stable phase. MP has Fe5Si3 at 34 meV above hull in DFT, which is consistent. Worth a look, not a discovery.

Early pruning by energy cannot work, and is not needed

Spearman correlation between the energy after k relaxation steps and the final energy:

For cells of 12 atoms or more, the ranking after 20 or 50 steps carries no information, and the eventual winner is often near the bottom early on: the Fe8B4 winner ranked 192nd of 196 at step 20. The trajectories explain it. Trials that have converged by step 100 are the junk (mean gap 300 to 1000 meV/atom to the best), while the trials still descending are the ones that end low. torch-sim already drops converged structures from the batch, so the compute is already going to exactly the trials that matter. There is nothing to prune.

A prior-free "are we done" signal

The raw rate of landing in a never-seen basin never falls below 56%, because most basins are distinct junk, so it says nothing about convergence. Restricted to trials within 50 meV/atom of the current best, the Good-Turing estimate of unseen basin mass (the singleton fraction) is 0 to 0.22 for the compounds and 0.62 to 1.0 for the alloys. That is the per-composition stopping rule we want: it converges for compounds, never for alloys, and it needs no prior about what the answer looks like.

Other observations

• Inflated cells (volume more than 1.3× the winner's): 56% of trials at 4 atoms, 10 to 16% at 12 to 16 atoms, 4% at 17 to 20. A starting-volume problem for small cells only.

• Unconverged after 400 steps: 0 to 2%.

What this means for explore

1. 15 trials is roughly a coin flip at 12 to 16 atoms. Hulls built at the default budget are not reliable at the cell sizes where novelty lives.

2. The budget should scale with cell size or adapt per composition using the restricted new-basin signal, and alloy-like compositions need a different treatment.

3. Spread trials over as many space groups as possible, one each.

4. Early pruning comes off the roadmap.

5. This study only measures random restart. The fact that the ground-state basin is reached from many different starting symmetries suggests it has a large basin of attraction, which is what minima hopping relies on. Kick-and-redescend from near-hull winners is the next experiment.

Data

All 1,390 relaxed trials are in the trials dataset