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Physical structure as an organizing principle for universal Hamiltonian learning

Mingkang Liu, Zhanghao Zhouyin, Guanghui Wang, Daye Zheng, Lei Shen, Qiangqiang Gu

arXiv:2610.36135v1cond-mat.mtrl-scics.AI

TL;DR

DFT’s cost limits large-supercell calculations and materials screening, while universal Hamiltonians must transfer across chemistry, maintain physical spectra, and remain inexpensive for large cells. UniTB corrects a tight-binding prior with a mixture of experts routed by prior-block invariants and a charge-equilibration head, while spectral projection bounds band shifts. Across six application domains, UniTB-Flash screening scores showed no significant difference from UniTB’s, while UniTB was more accurate and had 0.48 meV matrix-element MAE on nonmagnetic test crystals spanning 68 elements.

  • Problem

    DFT’s cost limits large-supercell calculations and materials screening, while universal Hamiltonians must transfer across chemistry, maintain physical spectra, and remain inexpensive for large cells.

    Source: Introduction · pp. 1–2

  • Method

    UniTB corrects a tight-binding prior with a mixture of experts routed by prior-block invariants and a charge-equilibration head, while spectral projection bounds band shifts.

    Source: Introduction · p. 2

  • Results

    Across six application domains, UniTB-Flash screening scores showed no significant difference from UniTB’s, while UniTB was more accurate and had 0.48 meV matrix-element MAE on nonmagnetic test crystals spanning 68 elements.

    Source: Introduction · p. 3

  • Takeaways & Limitations

    A two-stage screen can use UniTB-Flash to rank a large pool and UniTB to evaluate retained candidates, as demonstrated on 16,195 public crystal structures.

    Source: Discussion · p. 13

Abstract

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Density functional theory (DFT) gives the electronic structure of materials, but its cost limits large-supercell calculations and materials screening. Machine-learned Hamiltonians bypass the self-consistent cycle but are usually system-specific; universal ones could cover new compositions and disordered solids. Such models must generalize across chemistry and keep downstream properties physical, while staying inexpensive for large cells. Here we introduce UniTB, a universal E(3) equivariant Hamiltonian representation framework that integrates physical priors rooted in density-functional tight-binding theory with deep learning. UniTB's network, which corrects a prior Hamiltonian, generalizes across chemistry through a mixture of experts routed by that prior and a charge-equilibration head as in self-consistent-charge tight binding. A projection bounds band shifts from the prior, suppressing spurious states and keeping downstream properties physical. We provide the fastest SO(2) tensor-product GPU kernels for the network. The variant UniTB-Flash assembles its prior from two-center tables at $O(N)$ cost for $N$ atoms, against $O(N\log N)$ on a grid. Trained on 29,242 crystals, UniTB reaches a 0.48 meV matrix-element error on nonmagnetic test crystals spanning 68 elements. Without amorphous training data, both models agree with DFT optics and conductivity of amorphous Ge-Sb-Te. UniTB-Flash ranks candidates with no significant difference from UniTB, whose property values are more accurate; together they screen 16,195 crystals for five application domains of industrial relevance.

Introduction

The section presents UniTB and UniTB-Flash as scalable Hamiltonian-learning models that combine physical priors with learned corrections and transfer from crystals to amorphous-material property calculations. It also describes a two-stage screening workflow that uses Flash for ranking and UniTB for more accurate property values to identify promising thermoelectric candidates among large public-crystal pools.

Introduction

  • The models target transfer across chemistry, spectra without spurious states, and low cost for large cells; a two-stage screen evaluates 16,195 crystals across five industrial application domains.UniTB-Flash ranks candidates, UniTB re-evaluates the highest ranked, and DFT checks the final lists.Source: Introduction · pp. 2–3

A universal Hamiltonian model with tight-binding structure

  • UniTB adds a prior-routed mixture of experts and a charge-equilibration head to a learned correction of the neutral-atom Hamiltonian.Bond type and prior-block rotation invariants route bonds to experts; latent charges and local readouts produce site potentials that shift Hamiltonian blocks through the overlap matrix.Source: Introduction · p. 2, A universal Hamiltonian model with… · pp. 3–4

Grid-free construction of the prior Hamiltonian

  • UniTB-Flash assembles its grid-free prior, overlap, and spin-flip blocks in O(N) operations, compared with O(N log N) scaling for the grid-based prior at fixed density.For a 3,456-atom PbTe cell, UniTB-Flash generates H and S in 1.65 s versus 5.11 s for UniTB on one GPU.Source: Grid-free construction of the prior… · p. 5

Matrix-element accuracy across the periodic table

Environment-aware routing of expert networks

  • Prior-descriptor routing improves matrix accuracy over bond-type-only routing, lowering the dense model’s matrix error by 3.49% versus 1.15% for a wider model with a similar parameter count.Removing prior descriptors raises onsite error by 7.17% and hopping error by 2.30%; shuffling expert gates raises those errors by factors of 3.29 and 2.13.Source: Environment-aware routing of expert… · p. 8

Charge equilibration in the learned Hamiltonian

  • The charge-equilibration head lowers matrix error by 2.08% on the dense model and 4.24% on PDQ-MoE with site channels.A local-only head has 1.97% higher error than the charge head; for cells above 22 atoms, the paired gain reaches 6.98% on PDQ-MoE and 3.42% on the dense model.Source: Charge equilibration in the learned… · p. 8

Spectral projection against spurious states

  • UniTB’s spectral projection yields no spurious state in any of 200 band-test structures and bounds each predicted band’s shift from its H(0) counterpart to 10 eV.UniTB-Flash is not projected because its grid-free prior’s deviation from H(0) is not bounded as a local-potential change.Source: Spectral projection against spurious… · pp. 8–9

Transferability from crystals to amorphous materials

  • Trained only on crystals, both models reproduce DFT-Hamiltonian optical constants and conductivity of amorphous GST and lower matrix error on 18 amorphous cells.Relative to dense models with the same prior, errors fall by 12.0% for UniTB (0.766 meV) and 10.9% for UniTB-Flash (1.818 meV).Source: Transferability from crystals to… · pp. 9–10
  • Across twelve GST cells, both models match DFT-Hamiltonian refractive index at 0.8 eV and absorption at 1.5 eV within 2%, and DC conductivity within 0.1 decades.Across seven GST compositions, both models’ Spearman correlations with DFT are 0.96 for refractive index and conductivity and 1.00 for absorption.Source: Transferability from crystals to… · p. 10, Figure 6 · p. 11
  • Valence-band shape errors average 1.00% for UniTB and 1.15% for UniTB-Flash across twelve GST cells, and 1.17% and 2.77% across six amorphous-Si cells.UniTB is closer to DFT in nine of twelve GST cells and all six Si cells.Source: Transferability from crystals to… · p. 10, Figure 6 · p. 11

Materials screening for industrial applications

  • On nonmagnetic test crystals, screening scores differ by at most 0.051 with no significant domain differences, while UniTB lowers all six property-value errors by 42–59%.The comparison covers six application domains; Fig. 7b presents screening scores and relative errors.Source: Materials screening for industrial… · p. 11, Figure 7 · p. 12
  • The two-stage screen covers 16,195 public crystal structures across interconnect, photonics, photovoltaic, semiconductor, and thermoelectric applications; its five pools contain 9,064 distinct structures.UniTB-Flash ranks eligible structures, UniTB reevaluates the top 200 per pool, and DFT checks the 20 ranked highest by UniTB; the thermoelectric pool contains 694 structures.Source: Materials screening for industrial… · pp. 11–12
  • In the n-type thermoelectric screen, 19/20 candidates exceed the reference-median PF/τ and six exceed Mg2Si; LiTe3 and HfTe5 reach 4.30 and 2.66 times its PF/τ.Three candidates have measured thermoelectric properties; LiTe3 and NaTe3, ranked first and third, had not, to the authors’ knowledge, been studied as thermoelectrics.Source: Materials screening for industrial… · pp. 12–13, Figure 7 · p. 12
  • In the four non-thermoelectric domains, median errors are 0.36–1.1% for UniTB and 0.62–3.6% for UniTB-Flash; UniTB also corrects Rb14ZnGe16’s conduction-band mass from 0.083 to 0.401 me.For PF/τ, median errors are 4.0% and 6.0% for n-type doping and 2.5% and 2.7% for p-type doping, respectively.Source: Figure 7 · p. 12, Materials screening for industrial… · p. 13

Discussion

UniTB uses tight-binding structure to address chemistry transfer, spectral reliability, and cost. UniTB-Flash ranks screening candidates close to UniTB, while UniTB provides more accurate property values.

Discussion

  • Prior invariants route bond-wise experts and describe local environments; charge equilibration adds cell-wide latent-charge site potentials, with larger matrix-error reductions in larger cells.The projection bounds UniTB bands relative to H(0) in the overlap metric; UniTB-Flash assembles its prior, overlap, and spin-flip blocks from tabulated integrals at linear cost, though its correction is harder to learn, mostly in hopping blocks.Source: Discussion · p. 13
  • For the six test-set domains, UniTB-Flash screening scores stay close to UniTB’s despite larger matrix errors, while UniTB gives more accurate property values.The authors propose using the faster model to rank a large pool and re-evaluating the shortlist with UniTB.Source: Discussion · p. 13
  • Screening 16,195 public crystal structures across five industrial application domains yielded final candidates whose values DFT confirmed in every domain.The n-type thermoelectric list includes candidates beyond the strongest reference material.Source: Discussion · p. 13
  • Both models, trained only on crystals, reproduce DFT-Hamiltonian properties of amorphous Ge–Sb–Te and Si.The reported results use nonmagnetic PBE calculations with spin–orbit coupling in one numerical atomic-orbital basis.Source: Discussion · p. 13

Methods

The methods combine a tight-binding prior with equivariant networks, charge equilibration, and spectral projection, then evaluate model accuracy, runtime, and screening behavior across crystal and supercell settings.

Equivariant backbone; Prior descriptors and expert gates

  • Separate onsite and hopping networks use three bond-aligned SO(2) interaction layers, with six routed maps and site-symmetry-based channel allocation.Prior descriptors form expert gates from scalar signed channels and higher-order Gram entries; the final model selects two of four experts, alongside one always-active shared map.Source: Equivariant backbone · pp. 13–14, Prior descriptors and expert gates · p. 14

Charge-equilibration head

  • A QEq-inspired head predicts neutral latent charges and adds centered site potentials; the charge solve uses differentiable double-precision Cholesky solves.The charge-equilibration assembly scales as N^2 and its Cholesky solve as N^3, taking 0.04 s for both models at 3,456 atoms.Source: Charge-equilibration head · pp. 14–15

Spectral projection

  • The spectral projection clips residual eigenvalues in the whitened overlap basis to ±b, bounding predicted bands relative to the prior on a fixed nonsingular overlap subspace.Reported projected spectra use b = 10 eV; UniTB-Flash is not projected.Source: Spectral projection · p. 15

Hamiltonian reconstruction; Prior construction

  • Hamiltonian reconstruction adds predicted corrections to the prior, while spin-flip terms and blocks outside the learned graph follow each model’s prior.UniTB uses H(0) and DFT overlap; UniTB-Flash uses HGF and overlap from two-center tables. The trained Flash prior includes exchange–correlation and third-center local-potential terms.Source: Hamiltonian reconstruction · p. 15, Prior construction · pp. 15–16

Cost of the online assembly

  • With bounded neighbor lists, online assembly of S, spin-flip blocks, and HGF costs O(N); H(0) assembly includes grid operations with O(G log G) terms.The counts assume fixed basis, projectors, tables, quadrature orders, and grid spacing, with a minimum interatomic distance.Source: Cost of the online assembly · p. 16

SO2CUDA architecture and performance

  • SO2CUDA has lower median forward-plus-backward time than every tested comparator across 26 channel configurations; its real-batch onsite step takes 0.79 s, versus 1.64 s for PyTorch and 1.66 s for cuEquivariance.The operator benchmark uses one H200 GPU; synthetic outputs and parameter gradients agree with plain PyTorch within 6.8 × 10^-6 and 1.52 × 10^-5, respectively.Source: SO2CUDA architecture and performance · p. 17, Figure 8 · p. 18

Datasets and labels; Training

  • Training data comprise 29,242 crystals from 68 elements, with labels computed using PBE and fully relativistic pseudopotentials; the learning target is Re H↑↑.Final models use strict FP32 with TF32 disabled; MPHam-29K training takes about 37 h on two H200 GPUs.Source: Datasets and labels · p. 17, Training · p. 18

Matrix error; Band metrics; Application domains; Library screen; Inference timing; Statistics

  • Evaluation defines matrix error on released test structures, band error over full paths of 200 structures, and property-domain scores using paired bootstrap intervals.The library screen contains 16,195 structures, and full inference timings include prior assembly and both networks with charge equilibration while excluding loading and spectral post-processing.Source: Matrix error · p. 18, Band metrics · p. 19, Application domains · p. 19

Funding

The work acknowledges support from national and regional research funding bodies and an individual Singapore Ministry of Education grant.

Funding

  • Support came from the Advanced Materials–National Science and Technology Major Project, the National Natural Science Foundation of China, the Natural Science Foundation of Jiangsu Province, and a Singapore Ministry of Education Tier 2 grant supporting L.S.The listed award identifiers are 2025ZD0618401, 12504285, BK20250472, and MOE-T2EP50123-0012.Source: Funding · p. 26

Supplementary Note 1: Extended related work

The related-work discussion situates UniTB among universal Hamiltonian models, conditional expert routing, and physical-prior and charge-response methods.

Universal Hamiltonians and physical descriptors

  • Prior universal Hamiltonian models broadened chemical coverage, added spin–orbit coupling, or used zeroth-step references as features and residual baselines; UniTB builds its dense backbone on LEM/SLEM updates.The discussion also relates UniTB’s prior-operator invariants to edge-wise expert coefficients.Source: Supplementary Note 1: Extended related… · p. 28

Conditional computation

  • Conditional-computation precedents include dynamic kernels, sparse expert routing, adaptive load balancing, and shared-basis low-rank adapters; UniTB routes edges using invariant prior blocks.The related work distinguishes element-level, structure-level, and invariant-feature-based expert approaches.Source: Supplementary Note 1: Extended related… · p. 28

Physical priors and charge response

  • Earlier physical-prior methods include neutral-atom and Slater–Koster constructions, while QEq and SCC-DFTB motivate constrained charge response and overlap-weighted potentials.UniTB infers zero-total-charge latent charges from equivariant features and adds their site potentials in the SCC-DFTB overlap-weighted form.Source: Supplementary Note 1: Extended related… · p. 28

Supplementary Note 2: Architecture details

The architecture uses prior-routed PDQ-MoE maps on bond-frame SO(2) blocks, with shared bases and expert-specific cores. The dense backbone adds separate onsite and hopping networks and element-level tensor-product modulation.

Supplementary Note 2: Architecture details

  • Prior-routed PDQ-MoE maps act on bond-frame SO(2) blocks, with routing reused across three interaction layers and expert outputs mixed before activation.The m = 0 block couples all degrees; each m > 0 block couples x_m and x_-m for degrees ℓ≥|m|.Source: Figure 1 · p. 29
  • PDQ-MoE experts share the always-active maps and bases, differ in their cores, and have expert-dependent updates of rank at most 64.Prior descriptors and bond-type embeddings select two of four experts in each map.Source: Figure 1 · p. 29

Two features of the dense backbone

  • The dense backbone predicts onsite and hopping blocks with separately trained networks, using element-level tensor products to modulate node and edge features.The two networks each have 5,786,384 parameters; the modulation adds 818,410 parameters per model.Source: Two features of the dense backbone · p. 29

Supplementary Note 3: Module ablations

The ablations compare backbone, prior, routing, charge-head, training-recipe, projection, and additional-control choices under the stated evaluation protocols. They report how these choices change matrix errors and band diagnostics, with some comparisons changing multiple settings together.

Supplementary Note 3: Module ablations

  • Unless stated otherwise, retrained comparisons use the fixed MPHam-12K budget and are evaluated on 3,000 test structures, with paired 95% intervals from 10,000 structure resamples.The reported comparison metrics include onsite, hopping, and combined MAE, plus mean fw10 on 200 band structures after projection at b = 10 eV.Source: Supplementary Note 3: Module ablations · p. 30

Backbone

  • Removing both separate onsite/hopping networks and element-level tensor products raises dense combined MAE by 151.9%; the joint model reaches 2.049 meV.The jointly ablated model is 1.236 meV above the dense backbone and 0.111 meV below LEM; the joint network has half the parameters and training compute of the two separate networks.Source: Backbone · p. 30, Table 1 · p. 30

Prior usage

  • Learning the full Hamiltonian instead of the residual raises dense combined MAE from 0.813 to 5.692 meV; removing input initialization raises combined MAE by 1.54% on PDQ-MoE and 0.77% on UniTB.Passing the prior to edge features only, rather than to both nodes and edges, raises combined MAE on all three architectures.Source: Prior usage · p. 31, Table 3 · p. 32

Expert routing

  • PDQ-MoE lowers dense combined MAE by 3.49%, versus 1.15% for a wider dense backbone with a similar parameter count.Other tested expert designs change combined MAE by −1.62% to +29.2%; routing on bond type alone raises it by 2.74%.Source: Expert routing · p. 33

Charge head

  • The charge head lowers combined MAE by 2.08% on dense and 4.24% on PDQ-MoE with site channels.On PDQ-MoE, its recipe lowers MAE a further 2.25% beyond an additive potential-shift head, while a local-only head without charge equilibration is 1.97% above the charge head.Source: Charge head · p. 36, Table 8 · p. 36

Training recipe

  • Along the site-channels-first recipe, site channels, PDQ-MoE, the charge head, and prior perturbations change combined MAE by −1.71%, −3.90%, −4.24%, and −1.92%, respectively.UniTB is 11.29% below dense; the comparison adds four modules and thus changes multiple settings together.Source: Training recipe · pp. 37–38

Spectral projection

  • Projection at b = 10 eV lowers mean fw10 by 15.2–39.1 meV across five MPHam-12K configurations and removes all strict detections.The table compares each prediction before and after inference-time projection on 200 band structures.Source: Spectral projection · p. 39, Table 12 · p. 40

Additional controls

Supplementary Note 4: Datasets and labels

The benchmark extends the released NextHAM training list with additional MP-20 structures and specifies dataset, gauge-alignment, graph, and orbital conventions. The gauge-alignment sensitivity check produces a small change in UniTB matrix MAE on its 200-structure subset.

Supplementary Note 4: Datasets and labels

  • MPHam-29K contains 29,242 structures after adding 17,289 MP-20 structures to the 11,953-structure MPHam-12K set.The added structures use elements within the existing 68-element coverage and are absent from the NextHAM training and test sets.Source: Supplementary Note 4: Datasets and… · p. 42

Supplementary Note 5: Gauge alignment and graph conventions

  • Under the reduced NextHAM L2 gauge convention, UniTB MAE changes from 0.7113 to 0.7018 meV on the 200-structure sensitivity subset.The standard reported main-text matrix errors are unaligned; the sensitivity result is tabulated under the reduced L2 convention.Source: Supplementary Note 5: Gauge alignment… · p. 43

Graph and orbital masks

  • The stored directed graph retains atom pairs within element-dependent radii, twice the orbital cutoffs, so it contains every pair whose orbitals overlap.Species-specific orbitals use a padded 4s2p2d1f vocabulary with 27 spatial orbitals, and the configured common orbital mask sets the effective training subspace.Source: Graph and orbital masks · pp. 43–44

Supplementary Note 6: Training settings, training time and errors by element

Training uses separate onsite and hopping heads with specified budgets and final-checkpoint evaluation, while supplementary diagnostics characterize elementwise errors, charge-head residuals, and spectral protocols. The projection bound is evaluated on a retained overlap subspace, and reported projected spectra use b = 10 eV.

Supplementary Note 6: Training settings, training time and errors by element

  • MPHam-12K training uses 37,500 updates and about 100 effective epochs, while MPHam-29K uses 91,800 updates and about 300 effective epochs.The 200 band-test structures are a random sample of the 3,000 test structures, and reported results use the final checkpoint.Source: Supplementary Note 6: Training… · p. 45

Training time

  • Training timings are reported per epoch and total epochs per head, with two-GPU-per-head runs using data parallelism.Peak reserved memory per GPU with two A100 GPUs per head is 20 GiB for MPHam-12K and 41 GiB for MPHam-29K.Source: Training time · p. 45, Table 21 · p. 46

Matrix error by element

  • The Supplementary Fig. 2 UniTB-Flash elementwise averages are higher than UniTB’s for every element.The averages cover 2,431 nonmagnetic test structures containing all 68 elements; each structure counts once for each element it contains.Source: Matrix error by element · p. 46, Figure 2 · p. 47

Supplementary Note 7: Charge-head diagnostics

  • Without a charge head, site-potential residual RMS is 6.4 meV on training structures and 54.3 meV on test structures.Test residuals are 28.3 meV for formulas present in training and 59.4 meV for new formulas; errors are also higher for test structures containing O or N than those without them.Source: Supplementary Note 7: Charge-head… · p. 48

Supplementary Note 8: Spectral projection: properties and protocols

  • The projection clips the residual spectrum in the overlap metric, bounding predicted eigenvalue shifts relative to the prior within the retained overlap subspace.When the input spectrum lies within [−b, b], the map is the identity; otherwise the clipped eigenvalues lie between −b and b.Source: Supplementary Note 8: Spectral… · p. 49

Band and state-count protocols

  • The overlap solver retains overlap eigenvalues above 10^-5 in its canonical fallback and pads missing high-energy dimensions at 104 eV.The sampled comparison reports 16 fallback wavevectors per model and label among 15,459, with zero removed directions.Source: Band and state-count protocols · p. 49

Label residual spectra

  • For the 200-structure sampled band paths, label-residual eigenvalues lie in [−5.81, +6.15] eV, and projection at b = 10 eV acts on zero of 36,043 full-path wavevectors.The training-label residuals on 3,063 structures and 27 sampled k points have generalized eigenvalues within ±10 eV.Source: Label residual spectra · p. 49

Supplementary Note 9: Band endpoints and projection comparisons

The band comparisons report paired model–dense changes and endpoint statistics, while projection analyses compare band errors with and without projection under sampled and full-path evaluation. Strict spurious-state detections are absent from the reported MPHam tables.

Supplementary Note 9: Band endpoints and projection comparisons

Projection comparisons

  • Projection comparisons cover sampled wavevectors and full band paths, while Supplementary Figure 4 tracks successive model components through UniTB.Supplementary Figure 3 shows band structures and fw10 distributions, including paired changes before and after projection.Source: Figure 3 · p. 50, Projection comparisons · p. 52, Figure 4 · p. 52

Supplementary Note 10: Supercell selection, property protocol and results

Across the selected amorphous supercells, UniTB has lower valence-band shape error in all six Si cells and nine of twelve Ge–Sb–Te cells. The note also reports optical, conductivity and velocity comparisons, alongside the projection's effects on Si band levels.

Selection

  • The six a-Si-24 selections comprise four continuous random networks, one paracrystal and one diamond control, each with 216 atoms; GST-ACE-24 cells span 180–198 atoms and multiple compositions and preparation categories.Selections use category-median density, with two additional density-quartile GST selections and no repeated source frames.Source: Selection · p. 53

Localization descriptors

  • The mobility-gap criterion is three times the median DFT IPR in the 1–2 eV inner-band window, and localized states are sought within 1 eV of either band edge.The same threshold is used for each route on a given structure; optical and localization errors are absolute relative differences, while conductivity error is the absolute base-ten log ratio.Source: Localization descriptors · p. 53

Per-cell errors and valence-band spectra

  • UniTB has lower XPS shape error in all six Si cells and nine of twelve Ge–Sb–Te cells; the models' mean Ge–Sb–Te optical errors differ by less than one percentage point and conductivity errors by 0.03 decades.These are per-cell comparisons against same-cell DFT.Source: Per-cell errors and valence-band spectra · p. 54

Velocity operator

  • For twelve Ge–Sb–Te cells, UniTB errors are 1.77% for n(0.8 eV), 1.73% for α(1.5 eV), and 0.084 decades for σDC; UniTB-Flash errors are 1.98%, 0.79%, and 0.062 decades, respectively.Central finite differences agree with the analytic velocity to a relative Frobenius difference below 5 × 10^-6 in five cells for DFT and UniTB.Source: Velocity operator · p. 54

Property errors and dense comparison

  • Supplementary Table 29 compares both models against dense baselines on the same eighteen cells, while Supplementary Table 30 reports Ge–Sb–Te property errors using projected UniTB eigenpairs and velocities.The table captions identify same-cell DFT as the reference and explain the reported error definitions.Source: Table 29 · p. 55, Table 30 · p. 55, Figure 5 · p. 55

State counts

  • The ±10 eV projection leaves DFT residual spectra unchanged in all 18 cells and removes every UniTB level more than 1 eV below the DFT band bottom in six Si cells.It also removes all but two excess levels below the DFT midgap in those Si cells.Source: State counts · p. 56, Figure 6 · p. 56

Supplementary Note 11: Prior construction details

The prior-construction note details the environmental correction and GPU assembly, and compares dense networks trained with different priors and residual targets. Its grid-free prior uses tabulated two-center integrals and shares a three-center projector-contraction path with the other prior.

Supplementary Note 11: Prior construction details

  • The environmental correction combines pair and environment envelope moments and signed atomic-orbital moment blocks, with safeguards that limit density-derivative growth at low density.The implementation suppresses unresolved signed environment moments, clamps negative envelope moments to zero and applies the correction only on resolved shell overlaps with positive pair and total densities.Source: Supplementary Note 11: Prior… · p. 57

Online assembly on the GPU

  • For a 1,024-atom PbTe supercell on one H200 GPU, online assembly totals 202.5 ms for HGF and 1,038.7 ms for H(0).The reported totals exclude global charge response and spectral solves, which are additional stages.Source: Online assembly on the GPU · p. 57

Supplementary Note 11: Prior construction details

  • UniTB uses H(0) as its prior, whereas UniTB-Flash uses the grid-free HGF prior assembled from tabulated two-center integrals.The online routes share the three-center projector-contraction path; their local and exchange–correlation terms differ.Source: Online assembly on the GPU · p. 57

Prior input and residual target

  • 0.813 meV is the combined MAE for the H(0)-prior network targeting H − H(0), versus 3.08 meV for the HGF-prior network targeting H − HGF and 11.66 meV for the no-prior network targeting H.The three dense networks were trained identically on MPHam-12K; adding each prior back makes errors refer to the same Hamiltonian entries.Source: Prior input and residual target · p. 57

Supplementary Note 12: Operation counts of the online assembly

The operation-count analysis derives linear-cost assembly for bounded-neighbor grid-free terms and contrasts it with the grid-based prior's fixed-density scaling. It also identifies atom-count ranges where a single band eigensolve exceeds the full prior assembly cost.

Supplementary Note 12: Operation counts of the online assembly

  • With interatomic distances bounded below by dmin, a cell list with bins at least as wide as the largest cutoff builds each neighbor list in O(N) operations.The geometric bound limits the number of atoms within any fixed-radius neighborhood.Source: Supplementary Note 12: Operation… · p. 58

Grid-free prior

Supplementary Note 12: Operation counts of the online assembly

  • At fixed density, the grid-based prior costs O(N log N), while the grid-free prior's bounded-neighbor construction is linear in atom count.The grid-based estimate is O(N + G log G), with G proportional to cell volume.Source: Supplementary Note 12: Operation… · p. 58
  • One band-calculation wavevector exceeds the whole prior assembly above 13–17 atoms for HGF and 51–64 atoms for H(0).A dense generalized eigenproblem costs O(D^3) per wavevector; the reported thresholds include assembly, overlap and spin-flip blocks.Source: Supplementary Note 12: Operation… · p. 59, Table 34 · p. 59

Supplementary Note 13: Implementation and timing

Supplementary Note 13 describes SO2CUDA’s layouts, execution routes, benchmark protocols, and costs, alongside numerical checks and projected band-solve timings. Its benchmarks distinguish operator-level, real-crystal, and synthetic whole-model runs.

Supplementary Note 13: Implementation and timing

  • SO2CUDA–PyTorch maximum differences across six model/edge-count pairs are 6.8 × 10−6 for training outputs, 6.2 × 10−6 for inference outputs, and 1.5 × 10−5 for parameter gradients.The comparisons pass the stated maximum- and RMS-error tolerances.Source: Supplementary Note 13: Implementation… · p. 60

Synthetic whole-model protocol

  • Synthetic whole-model timings compare median forward-plus-loss-backward calls on shared randomly initialized models, excluding setup, input cloning, buffer restoration, JIT compilation, and optimizer updates.The replicated workloads span 20,000, 50,000, and 130,000 directed edges; outputs and gradients are checked against plain PyTorch.Source: Synthetic whole-model protocol · p. 60, Table 35 · p. 60

SO(2) layouts and execution routes

  • SO2CUDA’s four implementation comparisons use equivalent linear multiply-adds but differ in rotation, layout, and copy operations.The figure contrasts SO2CUDA, EquiformerV3, cuEquivariance, and plain PyTorch; execution routes depend on channel layout and weight structure.Source: Figure 8 · p. 61, Figure 9 · p. 62

Operator benchmarks and implementation costs

  • The operator benchmarks report forward-plus-backward time and peak memory relative to SO2CUDA, with stored Wigner data included in memory.The figures compare uniform and nonuniform channel allocations; the accompanying cost analysis reports Wigner storage, kernel counts, time, and peak memory.Source: Figure 10 · p. 62, Figure 11 · p. 63

Model benchmarks

  • Real-crystal training-step benchmarks measure onsite-network timing and peak allocated memory, while synthetic whole-model benchmarks split timing into forward and backward passes.The real-crystal step includes HybridMuon in fast optimizer mode and uses batches averaging 31.7 structures and 55,893 directed edges.Source: Figure 12 · p. 63, Figure 13 · p. 64

Spectral post-processing

  • 46.10 ms without projection and 56.64 ms with it are the mean solve times across 1,000 projected band-solve calls.The calls cover five MPHam-12K predictions on 200 structures; the reported projection overhead is 22.85%.Source: Spectral post-processing · p. 64

Supplementary Note 14: Application domains, library screen and inference timing

The note describes a five-domain library screen in which UniTB-Flash ranks candidates, UniTB reevaluates a shortlist, and DFT checks the final candidates. Across the reported comparisons, UniTB is generally closer to DFT on final property values, while UniTB-Flash supplies the faster first-stage inference.

Library screen

  • For 67% of the 73 absorbers with DFT gaps of 0.5–2.5 eV, UniTB’s SLME is closer to DFT.On the 198 crystals with values from all three Hamiltonians, SLME R2 is 0.986 for UniTB and 0.982 for UniTB-Flash.Source: Supplementary Note 14: Application… · p. 65
  • Across three n-type test crystals, the mean Seebeck-magnitude deviation from DFT over 25 carrier densities is 3.8% for UniTB and 4.2% for UniTB-Flash.The curves use the same three Hamiltonians for each crystal’s bands and Seebeck coefficients.Source: Supplementary Note 14: Application… · p. 66
  • The screen draws on a library of 16,195 crystals and uses a staged workflow: UniTB-Flash ranks pools, UniTB reevaluates the 200 highest-ranked candidates, and DFT recomputes the 20 best by UniTB.The five final lists contain 120 entries and 114 distinct structures.Source: Library screen · pp. 68–69
  • 3.99% (n-type) and 2.50% (p-type) are the median DFT-relative differences for UniTB thermoelectric values, versus 6.04% and 2.68% for UniTB-Flash.By DFT, 19 n-type and 13 p-type candidates exceed their respective median reference-material lines.Source: Library screen · p. 69
  • Final interconnect, photonic, mobility, and photovoltaic values differ from DFT by medians of 0.36%, 0.62%, 1.12%, and 0.61% for UniTB, versus 0.95%, 0.77%, 3.62%, and 0.62% for UniTB-Flash.UniTB is closer for 14, 12, 13, and 11 of the 20 candidates in those domains, respectively.Source: Library screen · p. 69
  • At 3,456 atoms, full inference takes 5.11 s for UniTB and 1.65 s for UniTB-Flash.The prior assembly takes 4.08 s and 0.66 s, respectively; network forward-pass times differ by 0.6–3.8% across cell sizes.Source: Library screen · p. 74

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