MLX periodic DFT: 8-atom silicon on M5 Max

This is the canonical performance ledger for the eight-atom conventional silicon cell. The production target is the complete 6 × 6 × 6 k-point SCF, not a small fixed-Hamiltonian or partial-k-point development probe.

Workload identity

Field	Value
Engine	mlx_atomistic MLX/Metal runtime
Host	Apple M5 Max MacBook Pro, low-power mode
Cell	8-atom conventional silicon periodic cell
Electrons / requested bands	32 valence electrons / 16 occupied bands
Functional / pseudopotential	PBE / Si GTH-PBE-q4
Plane-wave cutoff / FFT grid	25 Ha / 56 × 56 × 56
k-point mesh	6 × 6 × 6: 216 explicit, 108 calculated representatives after time reversal
Precision	complex64/float32 full grid; complex128 CPU projected eigensolve
Eigensolver	block Davidson–Rayleigh–Ritz, at most 64 subspace vectors
Runtime boundary	complete SCF timing includes setup and total-energy cycle; raw report publication is outside the measured SCF kernel

Accepted complete-SCF results

State	Wall time	SCF cycles	Representative k-points	Verdict
Earlier retained implementation	152.291 s	13	108	Numerically valid historical baseline
Adaptive-tolerance implementation	73.743 s	14	108	Previous complete baseline
Finite Hψ shape scheduler	59.231 s	14	108	Latest complete result; retained default

The retained scheduler result is 19.68% faster than the immediately preceding complete run and about 2.57× faster than the 152.291-second historical result. It passes the 67.844-second full-run retention gate and reaches the 60-second stretch target without changing the SCF cycle count.

Raw complete-run evidence:

• Earlier diagnostic family: results/mlx-dft-runtime-architecture/diagnostics/
• Previous report: results/mlx-dft-runtime-architecture/diagnostics/20260721-adaptive-tolerance/full-scf/report.json
• Current report: results/mlx-dft-runtime-architecture/finite-buckets/full-candidate-r1/report.json
• Current process-memory trace: results/mlx-dft-runtime-architecture/finite-buckets/full-candidate-r1-memory.json

Current complete-run profile

Phase	Time	Share of 59.233 s observed
Hψ applications	29.32 s	49.5%
Orthogonalization	11.46 s	19.3%
Projected Rayleigh–Ritz, including CPU small solves	6.11 s	10.3%
Eigensolver control	5.66 s	9.6%
Density	2.43 s	4.1%
Setup, persistence, mixing, and unaccounted	4.25 s	7.2%

The finite scheduler maps variable Davidson batches onto only 12 reusable Metal shapes: lane capacities 1, 2, 4, or 8 crossed with vector capacities 4, 8, or 16. This avoids repeatedly compiling and dispatching many nearly unique tail shapes. Against the previous complete run, Hψ time fell from 36.93 to 29.32 seconds and orthogonalization from 13.35 to 11.46 seconds, while logical Hψ work rose only 0.57%. The speedup is therefore primarily better GPU execution shape, not less accurate physics or fewer SCF cycles.

The largest remaining cost is still applying the Hamiltonian. The next optimization should reduce useful solver work, not add more scheduler shapes.

Scientific EOS validation

Runtime convergence is not scientific validation. The repository now carries a separate, source-backed equation-of-state (EOS) gate for the existing zero-temperature, fixed-occupation silicon method. It fits seven energies over 0.94–1.06 V₀ to a third-order Birch–Murnaghan equation and reports:

Quantity	Primary all-electron PBE reference
Conventional lattice constant	5.469916 Å
Bulk modulus	88.511 GPa
Pressure derivative B₀′	4.3118

The primary values are the curated FLEUR/WIEN2k average from the Materials Cloud ACWF verification archive. The pinned bundle also includes the CP2K Quickstep TZV2P/GTH curve from the same archive as a same-pseudopotential-family check, Quantum ESPRESSO SSSP as context, and the NIST experimental lattice parameter as context only. The reference JSON is hash-locked and retains source filenames, DOI, URLs, and CC-BY attribution.

The gate deliberately does not claim exact ACWF protocol parity. ACWF uses 0.0045 Ry Fermi–Dirac smearing and the E-TS free energy; the current product solver uses 16 fixed occupied bands for insulating silicon. This first gate asks whether that existing method produces the correct silicon energy curve.

Admission tier	Δ factor	Lattice	Bulk modulus	B₀′
Verified	≤ 3 meV/atom	≤ 0.5%	≤ 10%	≤ 15%
Excellent	≤ 1 meV/atom	≤ 0.2%	≤ 5%	≤ 10%

The independent numerical-convergence gates are ≤ 1 meV/atom maximum EOS-curve change, ≤ 0.1% lattice drift, ≤ 3% bulk-modulus drift, and ≤ 10% B₀′ drift for the complete 30 Ha / 64³ cutoff curve. A separate three-volume 8³ spot check requires its central energy shape to remain within 1 meV/atom of the 6³ baseline. A combined 30 Ha / 64³ / 8³ profile remains an optional stress diagnostic and is not part of the admitted scientific claim. Tolerances are fixed; a failure is reported rather than widened.

Each geometry runs in a fresh supervised process with a 40 GB process-tree ceiling. Baseline and cutoff points time out at 180 seconds; 8³ points time out at 240 seconds. The runner stops on the first numerical failure or nonphysical central energy triad. Fingerprinted point artifacts make the longer admission ladder resumable without silently accepting mismatched inputs.

The 56³ profiles retain the production solver’s 512 MiB logical compact-batch ceiling. The 64³ convergence profiles use 768 MiB, matching the 1.49× FFT-volume increase; this changes only workspace admission, not the Hamiltonian or physics. The external 40 GB process-tree ceiling remains authoritative.

Inspect the exact three-point screen without running SCF:

uv run python -m mlx_atomistic.benchmarks.dft_silicon validate-eos \
  --manifest results/mlx-dft-science/workload/manifest.json \
  --gth-source results/mlx-dft-science/workload/resources/Si-GTH-PBE-q4.gth \
  --level screen --dry-run \
  --out results/mlx-dft-science/eos-screen --json

Remove --dry-run to execute the bounded three-point screen. Use --level admission only after that screen passes. Admission schedules 17 points: complete seven-point baseline and cutoff curves plus the three central 8³ spot checks. Add --include-combined only to append the optional three-point 64³/8³ interaction stress profile; it is not required for admission. Use --summarize-only to rebuild the partial scientific report from existing fingerprinted point artifacts and bounded-failure traces without launching SCF.

The real baseline screen passed on 2026-07-22:

Volume factor	Lattice	Relative energy	SCF cycles	Wall time	Peak process tree
0.98	5.433491 Å	1.158 meV/atom	14	59.96 s	7.12 GB
1.00	5.470205 Å	0	14	60.64 s	7.85 GB
1.02	5.506433 Å	3.286 meV/atom	14	61.88 s	7.60 GB

All three points passed the residual, overlap, electron-count, timeout, and memory gates. The center is lower than both neighbors. A diagnostic quadratic through only these three points places the minimum near 5.4613 Å, 0.158% below the all-electron PBE reference. This is encouraging screening evidence, not an admitted EOS: seven points are still required for the Birch–Murnaghan fit, bulk modulus, pressure derivative, Δ factor, and basis convergence. Raw evidence is under results/mlx-dft-science/eos-screen/.

The subsequent admission run completed all 17 required points. The first 64³ point exposed a 512 MiB compact-workspace ceiling inherited from the 56³ production profile. The science harness now assigns 768 MiB only to 64³ profiles, proportional to their 1.49× FFT-volume increase. A focused rerun passed in 119.49 seconds with 8.28 GB peak process-tree memory and a stable plateau.

All three central convergence triads then passed:

Profile	Diagnostic minimum	Maximum central-curve change vs baseline
25 Ha / 56³ / 6³	5.461259 Å	—
30 Ha / 64³ / 6³	5.461136 Å	0.0215 meV/atom
25 Ha / 56³ / 8³	5.461261 Å	0.0010 meV/atom

The complete seven-point baseline curve is scientifically verified against the all-electron PBE reference:

Quantity	MLX result	Difference from reference
Conventional lattice constant	5.460859 Å	0.166%
Bulk modulus	88.306 GPa	0.232%
Pressure derivative B₀′	4.3052	0.153%
Lejaeghere Δ factor	1.942 meV/atom	verified tier

The Birch–Murnaghan fit has 0.0051 meV/atom RMSE and 0.0103 meV/atom maximum residual. It passes every verified-tier threshold, but not the stricter 1 meV/atom excellent Δ threshold.

The complete seven-point 30 Ha / 64³ / 6³ curve also passes numerical convergence against the baseline:

Convergence metric	Observed	Limit
Maximum EOS-curve change	0.0583 meV/atom	1 meV/atom
Lattice drift	0.00086%	0.1%
Bulk-modulus drift	0.0793%	3%
B₀′ drift	1.019%	10%

The three-point 8³ spot check passes and tracks the 6³ baseline central curve within 0.0010 meV/atom, far inside its 1 meV/atom limit. Together with the complete cutoff curve and all-electron comparison, this admits the 6³ EOS for the stated practical scope. A full seven-point 8³ curve is deliberately outside that scope and is not a pending task.

One additional outer 8³ diagnostic is retained as supporting evidence. It first reached the 240-second limit in low-power mode, then completed an identical full-power run in 123.19 seconds and 14 SCF cycles with a 9.20 GB peak. No other outer 8³ points are scheduled.

The optional combined 30 Ha / 64³ / 8³ stress point independently reached the same 240-second limit, at an 11.44 GB peak with a stable memory plateau. The timeout was not raised into the excluded 300–445 second range or relabeled as a scientific pass because this optional profile is outside the admission claim. The persisted report therefore says scientifically verified baseline, cutoff-converged, and 8³ spot-check passed. Artifacts are under results/mlx-dft-science/eos-admission/.

The verified claim is an all-electron equation-of-state agreement, not a same-code cross-engine parity. A separate MLX-versus-Quantum-ESPRESSO PWscf plane-wave parity run exists but remains diagnostic, not admitted: its comparison is blocked on case_not_comparable:strain_minus, case_not_comparable:volume_scan, and diagnostic_profile_not_admitted_for_qe_parity. That cross-engine closure is deferred and is not part of this verified result.

Bounded development evidence

These rows are intentionally not comparable to the complete production result.

Probe	Scope	Result	What it proves
Fixed-Hamiltonian A/B	First 8 representatives, one eigensolve, no density loop	1.890 → 1.752 s	Restart changes removed redundant Davidson work
Adaptive SCF diagnostic	First 8 representatives, complete density loop for only those points	7.469 s, 13 cycles	Tail lanes repeatedly carry large active subspaces after most bands converge
One-representative adaptive gate	One representative, partial-zone SCF	5.990 s, 14 cycles	Fast fail-early baseline, not a production timing
Finite-shape one-point gate	One representative, partial-zone SCF	4.904 → 1.135 s, 14 cycles both	Large shape-dispatch benefit without cycle drift
Finite-shape paired gate	First 8 representatives, three interleaved SCF pairs	18.38%, 16.45%, and 9.83% faster; 16.45% median	Candidate was faster in every pair and passed numerical/work gates

The direct fixed-Hamiltonian A/B reduced CholeskyQR2 attempts from 336 to 241, orthogonalized vectors from 3,193 to 1,918, and Hψ vector equivalents from 2,344 to 2,206. All eight lanes passed residual and overlap checks. Later absolute samples varied, so the work counters and matched A/B are stronger evidence than the isolated 1.752-second value.

Reproducer

The supported partial-zone SCF gate requires explicit science inputs and labels its output as non-production:

uv run python -m mlx_atomistic.benchmarks.dft_scf_smell \
  --manifest results/mlx-dft-runtime-architecture/workload-slice1/manifest.json \
  --gth-source results/mlx-dft-runtime-architecture/workload-slice1/resources/Si-GTH-PBE-q4.gth \
  --mode adaptive --representatives 8 \
  --out results/mlx-dft-runtime-architecture/smell/adaptive-8.json --json

Add --shape-profile only for scheduler-design diagnostics. It enables detailed batch events, emits an aggregated Hψ shape inventory and one-tail replay table, and therefore does not produce an admissible timing sample.

Run it through the repository’s bounded process-tree wrapper when collecting performance evidence. The hard process-tree memory limit is 40 GB. A one-point gate precedes three interleaved eight-point baseline/candidate pairs. A candidate must improve every pair and the paired median by at least 10%, without increasing SCF cycles or Hψ vector work, before any complete 108-representative run is allowed.

Rejected experiments

Experiment	Bounded outcome	Decision
Padded multi-lane CholeskyQR2	1.890 → 2.590 s; one residual failure	Removed
GTH overlap chunk 1024 → 2048	2.199 s; more Davidson/Hψ iterations	Removed
Predictive Gram admission	2.055 and 2.013 s	Removed
Ragged padded projected eigensolves	2.010 s; one extra Hψ round	Removed
Compiled GTH contraction	2.169 s; Hψ rose to 1.176 s	Removed
Davidson maximum subspace 64 → 48	2.192 s; more iterations and Hψ	Removed
Hybrid RMM-DIIS prototype	One point: 5.550 → 8.458 s; 14 → 21 cycles	Removed
Safe converged-subspace locking	One point: 5.180 → 6.902 s; 14 → 17 cycles; 4.09 → 3.92 GB peak	Removed before the eight-point gate
One main plus one Hψ tail shape	Stable profile: 17,024 submitted versus 8,980 logical vector equivalents; the best candidate within the call-growth bound removed 23.0%, below the 25% implementation gate	Stopped before scheduler implementation or timing gates

The earlier power-of-two shape prototype was initially removed because its memory tradeoff had not been validated end to end. The reconstructed finite policy is now retained: its complete run peaked at 7.86 GB, passed the 8 GB candidate gate, and showed a stable late-run memory plateau under the hard 40 GB process-tree limit.

The symbols associated with rejected implementations must remain absent: finite_certified, projected_eigh_ragged, prediction_tolerance, predicted_gram, measure_locked_overlap, and compiled_gth_contraction.