Lennard-Jones clusters: where gradients should win

The Lennard-Jones study extends Foundations from small teaching functions to one continuous problem family whose dimension grows from 33 to 294 while its physics stays unchanged. It is intentionally also a boundary example: Lennard-Jones energy has an inexpensive analytic gradient, so a gradient solver is the correct first choice. The fcmaes-core arms measure how its bounded, gradient-free methods scale; they are not presented as the preferred tools for molecular structure prediction.

The source-cited targets are putative minima from the Cambridge Energy Landscape Database table. Neither this guide nor the database table proves global optimality.

The model

For atoms at positions x_i and reduced units epsilon = sigma = 1, the minimized energy is

E(x) = 4 Σ_{i<j} (r_ij^-12 - r_ij^-6),  r_ij = |x_i - x_j|.

The implementation evaluates every pair once and accumulates energy and the analytic Cartesian gradient in the same traversal. It uses a finite quadratic continuation in squared distance below r = 0.75. The continuation matches value and first derivative at the boundary and becomes monotonically worse toward coincidence. overlap_pairs makes use of the guard visible in every artifact. The compact initializer itself keeps atoms at least 0.75 apart.

Two encodings expose rather than hide geometric redundancy:

• free optimizes all 3N coordinates. Translation and rotation leave the objective unchanged, so six flat directions remain for a non-linear cluster.
• fixed-frame places atom 0 at the origin, atom 1 on positive x, and atom 2 in the positive-y half of the xy plane. This leaves 3N-6 decisions. Encoding selects non-degenerate anchors and canonicalizes any supplied structure by a rigid transform.

The frozen cube half-width is 1.5 N^(1/3). Every arm uses the same compact initializer family: rejection-sampled candidates from a sphere of radius 0.8 N^(1/3), under deterministic arm-specific seed streams, not uniform samples from a mostly dissociated cube. Population methods project a candidate back to the declared box if numerical optimizer steps cross it, and the artifact records projected_candidates.

What is compared

The selected sizes alternate relatively accommodating and difficult landscapes:

The Cambridge overview describes N=38 and the Marks-decahedral sizes as stiff tests for putative global optimization. The LJ38 landscape discussion also explains why one mean energy would hide the important funnel behavior.

Each row is one independent root seed. The arms are:

The adapter dependency is a non-default feature of Foundations, not a dependency of fcmaes-core. This is the interoperation pattern described in The optimizer boundary: an external solver uses the same model and evidence contract without expanding the core crate.

One energy-only call and one combined value/gradient evaluation each count as one full pair_traversal. The requested cap is 20,000 within each cluster size, but converged gradient runs may stop early. Across the 100 cells per arm, the measured traversal use is:

L-BFGS therefore wins every best-achieved row below while consuming only 37.7% of the derivative-free traversal cap on average. pair_terms_evaluated multiplies traversals by N(N-1)/2, preventing a reader from mistaking equal calls across sizes for equal compute. Objective calls, gradient calls, measured pair time, wall time, overlap pairs, projections, and estimated optimizer overhead are separate columns. The overhead value is max(wall - measured_pair_time, 0) and is only an estimate: initialization, locking, timing, and serialization are not perfectly separable.

Exact success means energy <= source-cited target + 1e-3. Best achieved objective and target-relative gap remain visible when that strict indicator is sparse; 1%, 5%, and 10% attainment are secondary summaries. Ten independent seeds are required before the results can be described as a comparison. The smoke preset has two seeds and is only a deterministic adapter/conformance check.

Run and audit

Inspect a model without enabling the gradient dependency:

cd foundations
cargo run --release --locked -- \
  --suite lennard-jones --atoms 38 --parameterization fixed-frame

Run the small conformance protocol or the ten-seed publication protocol:

cargo run --release --locked --features gradient-reference -- \
  --lj-campaign --preset smoke --workers 2 --seed 42 \
  --output results/smoke

cargo run --release --locked --features gradient-reference -- \
  --lj-campaign --preset publication --workers 0 --seed 42 \
  --output results/publication

--workers 0 uses available parallelism for independent case rows. Each population optimizer and each retry is single-threaded inside that outer layer, preventing nested oversubscription. Seeds come from (root_seed, run_id), and rows are sorted before serialization.

No Cambridge coordinate file is redistributed. If you obtained an XYZ or plain coordinate file under its own terms, audit it explicitly:

cargo run --release --locked -- \
  --suite lennard-jones --atoms 38 --parameterization fixed-frame \
  --reference-file /path/to/LJ38.xyz

The command prints measured energy, target, absolute error, and whether it meets 1e-6, together with the file’s SHA-256 hash. Missing or malformed files are errors; there is no embedded or identity fallback.

To reproduce the publication audit without retaining the coordinates:

audit_dir=$(mktemp -d)
for n in 13 38 55 75 98; do
  curl -fsSL "https://www-wales.ch.cam.ac.uk/~jon/structures/LJ/points/$n" \
    -o "$audit_dir/$n"
done
cargo run --release --locked --features gradient-reference -- \
  --lj-campaign --preset publication --workers 0 --seed 42 \
  --output results/publication --reference-directory "$audit_dir"

The checked-in run independently reproduced all five targets within 4.9e-7. Its audit manifest retains source URLs, hashes, and residuals while the coordinate files remain outside the repository.

Population and CR-FM-NES configuration

The comparison intentionally uses each core optimizer’s declared policy, not one matched population size. Each retry receives 5,000 of the 20,000 pair-loop traversals:

This is consequently evidence about these explicit policies, not abstract algorithms under matched populations. In particular, DE’s 15d default leaves only one full generation per large-cluster retry.

The first run also exposed a poor CR-FM-NES scale choice. A disjoint, three-seed sensitivity check crossed normalized sigma {0.05, 0.15, 0.50} with population {16, 32, 64} at LJ13, LJ38, and LJ98. It selected sigma=0.05, population 16 by pooled median target-relative gap, replacing the initial 0.15/32 configuration before the primary ten-seed rerun. The full 81-row sensitivity table is retained as a post-hoc configuration diagnostic, not ranking evidence.

Measured scaling evidence

The checked-in publication run contains ten seeds, 20,000 full pair-loop traversals per arm, both encodings, and all five sizes. Because exact success is rare, the most informative compact comparison is the best achieved objective across the 20 runs for each size and arm. Lower is better; bold indicates the best arm, not a newly established minimum.

The strict and tiered attainment counts aggregate 100 cells per arm: five sizes × two encodings × ten seeds.

Thus the exact criterion remains honest, while the objective and target-relative gap expose the quality ordering. This does not show that L-BFGS is a complete cluster-search strategy: it reaches the exact threshold in only 2% of its cells. It does show that ignoring the inexpensive analytic gradient would be especially hard to justify at this budget.

Full precision is in results/publication/lennard-jones/scaling.csv and the complete decisions and protocol metadata are in run.json. Wall times describe the recorded machine and run, not a machine-independent library property. The host was not reserved exclusively for this run, so the overhead panel is diagnostic instrumentation rather than a cross-library timing claim; rerun on an idle host before using its crossover points for a deployment decision.

Descriptor pilot: rejected

The optional quality-diversity proposal pre-registered normalized radius of gyration and mean coordination at cutoff 1.35. Three deterministic arms of 64 compact candidates reached only 3.01% mean coverage of the 12×12 grid, below the frozen 8% gate. All other gates passed, including 92.71% same-niche retention after an independent N(0, 0.01) coordinate perturbation. The verdict is therefore rejected, and the QD campaign is explicitly skipped.

The complete measurements and thresholds are in pilot.md, with every candidate in pilot.csv. Bounds were not narrowed after seeing the sparse archive. The rejection is useful evidence: morphology descriptors are scientifically interpretable, but this compact candidate generator does not support the intended repertoire claim.

What this study can and cannot establish

It can test the model, analytic gradient, parameterizations, adapter boundary, schedule-independent retry, high-dimensional optimizer overhead, and success frequency under one frozen protocol. It cannot prove a global minimum, establish an all-purpose optimizer ranking, replace specialist cluster-search methods, or support conclusions about Bayesian optimization: the pair potential is cheap and parallel.