{"id":"savings-electronic-surrogates","title":"Electronic-Structure Surrogates that Skip or Shorten the SCF Loop","subtitle":"Learned Hamiltonians, semi-empirical+ML hybrids, and ML kinetic-energy and exchange-correlation functionals.","category":"references","tags":["literature-review","savings-stack","scf","hamiltonian","surrogate"],"source":"articles/lit-review/savings-electronic-surrogates.md","lang":"en","words":2428,"readMinutes":11,"toc":[{"depth":2,"text":"1. Executive result","id":"1-executive-result"},{"depth":2,"text":"2. Savings table","id":"2-savings-table"},{"depth":2,"text":"3. Proven vs claimed","id":"3-proven-vs-claimed"},{"depth":2,"text":"4. Openings for Lupine","id":"4-openings-for-lupine"}],"html":"
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Provenance: explore agent agent-18 (director-commissioned deep research, swarm of 7, 2026-07-21) — materialized verbatim, then editorially corrected only to replace the retracted union-anchor saving with the reproducible primary-record value (72.4%). Quantitative literature claims are as reported by the research agent from sources it accessed; see citations inline. Citation-verification pass pending before any external publication.

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Compute-Savings Digest: Electronic-Structure Surrogates that Skip or Shorten the SCF Loop

Evidence cut: 2026-07-21. Every number below was read this session from the cited source; each entry is marked [FT] = full text read, [ABS] = abstract only, [SNIP] = verified via indexed abstract/snippet mirror. Paywalled journals were accessed via arXiv versions or abstract mirrors.

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1. Executive result

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2. Savings table

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TechniqueWhat cost it removesMeasured savings factor (system/size context)Accuracy cost / failure modeCitation (access level)Evidence strength
DeepH (THU: Li, Wang, …, Duan, Xu)Entire SCF loop; cubic-scaling DFT of target system10³× time reduction, MoS₂ 35×35 supercell; magic-angle TBG (11,164 atoms) computed, matching plane-wave benchmarkHamiltonian MAE 2.1 meV (graphene), sub-meV on ≤10³-atom moiré; local-coordinate discontinuity limits MD; needs ~10²–10³ DFT training calculations per chemical environmentNat. Comput. Sci. 2, 367 (2022), FTstrong
DeepH-2 (ELCT architecture)Same + fixes equivariance cost10⁷-param DeepH-2 ≈ cost of 10⁶-param DeepH-E3; tensor-product scaling O(L³) vs O(L⁶)MAE 0.2 meV typical, 0.12 meV monolayer graphene (≈ DFT numerical error)arXiv:2401.17015v1, [FT]strong (arch.); moderate (universal claims)
xDeepH / DeepH-E3 magnetic (THU)SCF for magnetic superstructuresEnables skyrmion-scale spin-spiral/moiré-magnet calculations previously infeasible (no wall-time factor given in abstract)sub-meV Hamiltonian error on nanotube, spin-spiral, moiré magnetsarXiv:2211.10604v1 / Nat. Comput. Sci. 3, 321 (2023), [ABS]moderate
DeepH-hybrid (THU)Hybrid-functional SCF (exact exchange) for large supercellsFirst hybrid-level flat-band study of magic-angle TBG (no factor in abstract)Learns hybrid H from small structures; accuracy stated as "high"arXiv:2302.08221v1, [ABS]claim
HamGNN (Fudan: Zhong, Yu, Su, Gong, Xiang)Entire SCF loop; large-supercell DFT36 s to build Hamiltonian of 4,284-atom Si dislocation supercell (80-core Xeon node)QM9 H-MAE 1.49 meV; C allotropes 1.55 meV; transfers to TBG (3.23 meV) trained only on bulk allotropesnpj Comput. Mater. 9, 182 (2023), FTstrong
Universal HamGNN (Fudan)Per-system retrainingOne model for whole periodic table; high-throughput band-gap screening over Materials ProjectTrained on "nearly all" MP crystal Hamiltonians; accuracy degrades vs per-system modelsarXiv:2402.09251 / Chin. Phys. Lett. 41, 077103 (2024), [ABS]claim
PhiSNet (Unke, Bogojeski, …, Smidt, Müller)Ab initio wavefunction/density computation (molecules)>10³× speedup vs ab initio reference; prediction errors 10²× below previous SOTAMolecules only; no strict parity symmetry (HamGNN authors show this hurts solids); basis-set boundarXiv:2106.02347v2 / NeurIPS 2021, [ABS]strong (molecules)
QHNet (Yu et al.)92% of tensor products in equivariant Hamiltonian nets; 50% memoryComparable accuracy to SOTA on MD17 at "significantly faster speed"Engineering win, not an end-to-end DFT-skip benchmarkarXiv:2306.04922v2 / ICML 2023, [ABS]moderate
OrbNet Denali (Entos/Caltech)Full DFT SCF, replaced by one cheap xTB evaluation + NNUp to 10³× vs DFTGMTKN55 WTMAD-1/2 = 7.19/9.84 kcal/mol; torsions MAE 0.12 kcal/mol; organic/bio elements onlyarXiv:2107.00299v2 / J. Chem. Phys. 155, 204103 (2021), [ABS]strong
AIQM1 (Dral group, Xiamen/Germany)DFT/CCSD(T) SCF → NDDO semiempirical SCF + NN Δ-correctionC₆₀ geometry opt: 14 s (1 core) vs 31 min (32 cores) ≈ 4×10³× CPU-time; MRCI excitations 10³× vs TD-DFT0.8 kcal/mol MAD vs CCSD(T)*/CBS; fails underrepresented chemistry (H₂ error −2.9 kcal/mol); CHNO closed-shell onlyNat. Commun. 12, 7022 (2021), FTstrong
ChargE3Net (MIT Lincoln Lab)SCF iterations via density warm-start; or non-SCF property evaluation26.7% fewer SCF iterations on unseen Materials-Project crystals; non-self-consistent runs give near-DFT electronic/thermodynamic propertiesDensity-only; energies need a functional evaluation on the density; trained on 100K+ MP structuresarXiv:2312.05388v2, [ABS]moderate (single group, but concrete metric)
MALA LDOS surrogate (HZDR/CASUS + Sandia)KS diagonalization & SCF for metallic/high-T systemsUp to 10³× vs tractable DFT; >10,000-atom Be electronic structure (press: ~5 min on 150 CPUs)Best for metals/warm dense matter; transfer across chemistries limitednpj Comput. Mater. 9, 115 (2023) via DOI 10.1038/s41524-023-01070-z, [ABS]strong
Ellis et al. finite-T KS-NN (Sandia/HZDR)Finite-temperature SCF (the expensive regime)KS-DFT free energies "to within chemical accuracy at negligible computational cost"Finite-T metallic systems (Al); LDOS-basedarXiv:2010.04905 / Phys. Rev. B 104, 035120 (2021), [SNIP]moderate
Rackers et al. e3nn density (LANL)KS solve for density of 10³–10⁴-atom systemsDensity of systems of thousands of atoms "with quantum accuracy," trained on small systemsDensity only (no direct forces); moleculesarXiv:2201.03726v2 / Mach. Learn.: Sci. Technol. 4, 015027 (2023), [ABS]moderate
DeePKS (DP Tech/AISI + Princeton: Chen, Zhang, Wang, E)Cost gap between GGA and hybrid/CC-level functionalsChemically accurate energy/force/dipole/density for large molecule classes at GGA-level SCF costSelf-consistent ML functional; accuracy tied to training coveragearXiv:2008.00167v2 / JCTC 17, 170 (2021), [ABS]strong
DeePKS-perovskite (Ou, Tuo, …, Zhang)HSE06 SCF for perovskite panelHSE06-accuracy forces/band gaps/DOS at "efficiency comparable to GGA" across ABX₃ familyElement-family-specific model; SOC handled a posterioriarXiv:2306.14486v1 / J. Phys. Chem. C 127 (2023), [ABS]moderate
NeuralXC (Dick, Fernandez-Serra; Stony Brook)Accuracy gap baseline→high-level method at baseline costWater functional reaching beyond-baseline accuracy "while maintaining efficiency"Water-centric demo; SCF still requiredNat. Commun. 11, 3509 (2020), SNIPmoderate
DM21 (DeepMind)(Accuracy side, not speed) fractional-charge/spin pathologies—GMTKN55 MoM better than best hybrids, near double-hybrid; fails SCF convergence on TMCs (Zhao, PCCP 2024, SNIP); oscillatory potential in geometry optimization (arXiv:2501.12149v1, [ABS])Science 374, 1385 (2021), DOI 10.1126/science.abj6511, FT-partial via PDFstrong (both directions)
Nagai NN-XC w/ asymptotic constraints (U Tokyo)Accuracy gap at meta-GGA-like costOutperforms existing functionals incl. on out-of-training materials; CCSD(T) itself at 3.5 kcal/mol MAE on comparable benchmarkSmall-molecule thermochemistry focusarXiv:2111.15593v2 / Phys. Rev. Research 4, 013106 (2022), [ABS]moderate
CIDER nonlocal exchange (Kozinsky group, Harvard)Exact-exchange (hybrid) cost in plane-wave codesHybrid-DFT thermochemistry accuracy and improved band gaps "at roughly semilocal cost, significantly faster than hybrid DFT in plane-wave codes"; Si charged-defect levels in large supercellsGP model; needs feature engineering; solid-state adoption still thinarXiv:2303.00682v4 / Phys. Rev. B 110, 075130 (2024), [ABS]strong
Skala (Microsoft Research AI4Science)Hybrid-level accuracy at semi-local evaluation costWTMAD-2 = 2.8 kcal/mol on GMTKN55 (beats all hybrids) "at cost of semi-local DFT"Main-group chemistry; very new, no independent replication; ~385K-param NN per grid point overhead vs analytic functionalsarXiv:2506.14665v1, [ABS]claim (strong internal evidence)
Kohn–Sham regularizer (KSR/sKSR) (Google + UCI Burke)Differentiability-driven functional training (accuracy side)Nonlocal sKSR functional: 2.7 mH MAE on test 1D systems, best ML-functional generalization reported1D model systems only; not production DFTPRL 126, 036401 (2021) [SNIP]; arXiv:2110.14846v4 / JPCL 13, 2540 (2022), [ABS]moderate (methodology), weak (production)
Brockherde ML-HK map (FU Berlin/Burke/Müller)KS equations entirely (potential→density map)Conformer energies to 0.37 kcal/mol (benzene MD frames); stable ML-driven malonaldehyde MDProof-of-concept molecules; per-system trainingNat. Commun. 8, 872 (2017), FTmoderate (historic)
Imoto NN-KEDF (U Tokyo)Orbital optimization; O(N³) → O(N)O(N) scaling demonstrated on SiC; accurate structure for 24 systems (atoms→metals→semiconductors→ionic)Simple/materials-like systems; functional-derivative training neededarXiv:2109.01501v1 / Phys. Rev. Research 3, 033198 (2021), [ABS]moderate
M-OFDFT (MSR AI4Science Beijing/Shanghai + XJTU)Orbital optimization in molecules — OF-DFT that finally works for molecules27.4× speedup on protein B (738 atoms, 2,750 e⁻: 0.45 h vs 12.3 h); 6.7× on QMugs; O(N^1.46) vs O(N^2.49)Chemical accuracy (QM9 0.93 kcal/mol, ethanol 0.18); 10²× better than classical KEDFs; extrapolates to 10× training size; light elements, near-equilibrium, neutral systems onlyarXiv:2309.16578v2, [FT]strong (single group)
KineticNet (Remme, …, Dreuw, Hamprecht; Heidelberg)Orbital optimization (molecular OF-DFT)First chemically accurate learned KEDF across densities/geometries of tiny molecules; 2-e⁻ density optimization convergedTiny molecules onlyarXiv:2305.13316 / J. Chem. Phys. 159, 144113 (2023), [ABS]moderate
MPN-KEDF (Sun & Chen, ABACUS team)Orbital optimization for metals/alloysSystematically tested on Li, Mg, Al + 59 alloys with 3 exact constraints imposedSimple metals/alloys only; molecules out of reacharXiv:2310.15591v2, [ABS]moderate

3. Proven vs claimed

Replicated and trusted.

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Single-paper / single-group claims.

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Marketing / caution flags.

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Honest scope note. The subdomain is not thin — it is one of the densest in computational chemistry — but the "fraction of DFT wall-time eliminated" is only crisply measured for Hamiltonian surrogates (10³× inference), semi-empirical+ML hybrids (10³–10⁴×), M-OFDFT (27× at protein scale), and MALA (10³×). ML-XC papers mostly measure accuracy-per-rung, not wall time; their speed claims are qualitative ("significantly faster than hybrid"). I could not verify any published wall-time factor for Nagai's functional [UNVERIFIED: PRR article paywalled, abstract gives accuracy only] and DeepH-hybrid's speedup factor [UNVERIFIED: arXiv abstract qualitative; full text not fetched].

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4. Openings for Lupine

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  1. Hamiltonian surrogates don't give forces; Lupine does. DeepH/HamGNN produce band structures and response properties, but total energies/forces from a predicted Hamiltonian are not variationally reliable for MD or NEB — which is exactly Lupine's home turf (uMLIP + theorem-gated sparse-anchor DFT). The clean combination: use a DeepH/HamGNN-predicted Hamiltonian (or ChargE3Net-density: measured 26.7% SCF-iteration cut) as the SCF warm start at each sparse anchor. Union anchors already share DFT evaluations across models; adding surrogate warm starts compounds 72.4% fewer DFT calls with ~25%+ fewer iterations per call. Nobody has benchmarked "surrogate warm start × sparse-anchor NEB" — an obvious, cheap win.
    2. \n
  2. Theorem-gating is the missing piece for ML functionals. DeePKS/CIDER/Skala raise accuracy per SCF dollar but are trusted blindly inside their SCF; DM21 showed NN functionals fail catastrophically and silently (TMC convergence, geometry oscillations). Lupine's physical-law theorem checks (e.g., fractional-charge/spin consistency, size-consistency, virial/force–energy consistency along a path) are precisely the runtime certification these functionals lack. A "Skala-inside, Lupine-gated, sparse-anchored" stack targets hybrid-level accuracy at semilocal cost with formal failure detection — a stronger story than any of the three alone.
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  3. Density mismatch as a union-anchor detector. Union anchors currently key on agreement of model-predicted extrema in energy space. MALA/Rackers-class density surrogates offer a cheap electronic-level discriminator: where two uMLIPs agree geometrically but predicted densities/Hamiltonians diverge beyond a theorem-set tolerance, promote that structure to a DFT anchor. This converts union anchoring from geometry-driven to physics-driven, and the surrogates cost milliseconds.
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  4. Orbital-free partitioning at the large end. M-OFDFT's verified regime (light elements, near-equilibrium, neutral; 27× at ~700 atoms) is complementary to Lupine's verified regime (reactive paths, barriers). For very large systems (biomolecular, electrochemical interfaces), an OF-surrogate background + Lupine-corrected uMLIP reactive region, with anchors placed where theorems flag the boundary, attacks the regime where even 10³× Hamiltonian surrogates fail (they need converged reference data at the target chemistry, which Lupine's sparse anchors generate on the fly).
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Source count: 31 primary sources accessed this session (13 full-text or partial full-text, 15 abstract, 3 verified snippets/mirrors). All quantitative claims above trace to these; unverified items are explicitly flagged.

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