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Where the Error Lives: Spectral Representations of Learned Kohn-Sham Hamiltonians

Fang Sun, Hekai Cheng, Ruoyan Li, Yuanzhou Chen, Wei Wang, Jason Cong, Yizhou Sun

arXiv:2610.37843v1physics.chem-phcs.AI

TL;DR

Entry-wise Hamiltonian losses do not reliably yield accurate eigen-properties, while existing physics-informed losses see only parts of prediction error. FermiScope fine-tunes Hamiltonian networks with a cumulative-LDOS loss that compares predicted and reference electron counts below energy thresholds. Across three architectures and four QH9 splits, this loss improves every eigen-derived metric in all 12 settings, reducing standard-split gap error by 49–79% while element-wise error rises at most 0.3%.

  • Problem

    Entry-wise Hamiltonian losses do not reliably yield accurate eigen-properties, while existing physics-informed losses see only parts of prediction error.

    Source: §1 · pp. 2–3

  • Method

    FermiScope fine-tunes Hamiltonian networks with a cumulative-LDOS loss that compares predicted and reference electron counts below energy thresholds.

    Source: §1 · p. 3

  • Results

    Across three architectures and four QH9 splits, this loss improves every eigen-derived metric in all 12 settings, reducing standard-split gap error by 49–79% while element-wise error rises at most 0.3%.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The results support training representation, not only architecture, as a determinant of whether predicted Hamiltonians are useful.

    Source: §1 · p. 3

    Density matching cannot distinguish Hamiltonians differing only in orbital-energy shifts or within-space rotations when no level crosses the gap, limiting its use for direct one-shot predictions.

    Source: §3.1 · p. 6

Abstract

from arXiv · show

Kohn-Sham density functional theory (DFT) is the standard method for computing the electronic structure of molecules and materials, but its self-consistent-field (SCF) iterations make it expensive for large systems and high-throughput screening. Machine-learned Hamiltonian models offer a shortcut: an equivariant network predicts the Kohn-Sham Hamiltonian matrix in one forward pass, and diagonalizing the prediction yields orbital energies, the HOMO-LUMO gap, molecular orbitals and the electron density. These models are trained to match the reference matrix entry by entry, yet a small entry-wise error does not guarantee accurate eigenvalues or eigenvectors. Two physics-informed alternatives go partway: a recently proposed wavefunction-alignment loss adds a spectral penalty without diagonalizing the prediction, and matching the electron density of the diagonalized prediction supervises it directly. Neither is complete: by representing the prediction error in the eigenbasis of the reference Hamiltonian, we show that each sees only part of the error, and that shifting the predicted orbital energies leaves the electron density exactly unchanged. The same analysis identifies a complete training signal: the cumulative local density of states (cumulative LDOS), the number of electrons at each point in space with energy below a given threshold. The cumulative LDOS is a single physical observable that contains the electron density as a special case, responds to every component of the error, determines the Hamiltonian uniquely, and yields gradients that do not vanish for large orbitalenergy errors. Fine-tuning three Hamiltonian networks on QH9 in this way, which we call FermiScope, reduces the HOMO-LUMO gap error without increasing the element-wise error. The gains carry over to out-of-distribution and molecular-dynamics splits, where the wavefunction-alignment loss helps less.

1 INTRODUCTION

Kohn–Sham DFT is widely used but its repeated self-consistent-field iterations limit large-system and high-throughput calculations. This section shows why entry-wise Hamiltonian accuracy can fail to deliver accurate eigen-properties, then introduces cumulative LDOS training and its QH9 results.

1 INTRODUCTION

  • Entry-wise error does not ensure accurate eigen-properties: continued element-wise training raises QHNet’s standard-split gap error by 53% despite lowering matrix error.The models’ element-wise losses treat entries alike even though eigenvalues and eigenvectors respond unevenly to errors.Source: §1 · p. 2
  • In reference-orbital coordinates, prediction errors divide into orbital-energy shifts, occupied–virtual mixing, and rotations within occupied or virtual spaces.Properties and training signals respond selectively to these sectors, explaining why element-wise error alone is a poor guide to eigen-properties.Source: §1 · p. 3
  • Wavefunction-alignment and density losses each miss error components: density is unchanged by orbital-energy shifts and within-space rotations, which span about 69% of QH9 error degrees of freedom.The alignment loss either weights all entries uniformly or, in its diagonal-only form, penalizes orbital energies only to first order.Source: §1 · p. 3
  • The cumulative LDOS loss compares predicted and reference electron counts below energy thresholds, providing a signal that responds to every error sector and uniquely determines the Hamiltonian.Its density slice is at the Fermi level, and unlike a peak-shaped LDOS it continues pulling orbital energies toward their targets however far inside the energy window they start.Source: §1 · p. 3
  • Across three architectures and four QH9 splits, cumulative-LDOS training improves every eigen-derived metric in all 12 settings and reduces standard-split HOMO–LUMO gap error by 49–79%.Element-wise error rises by at most 0.3% on the standard split; under distribution shift, gap error falls by 36–77%.Source: §1 · p. 3

2 BACKGROUND

The background defines the Kohn–Sham Hamiltonian and its role in self-consistent DFT, then describes learned Hamiltonian prediction and physics-informed losses.

Kohn–Sham Hamiltonians

  • In a nonorthogonal atom-centred basis, the overlap matrix accompanies the Kohn–Sham Hamiltonian, and molecular orbitals are obtained from its generalized eigenvalue problem.The molecular orbitals are orthonormal even though the basis functions are not.Source: §2 · p. 4
  • For a closed-shell molecule, the occupied subspace consists of the lowest Ne/2 orbitals, and the density matrix is twice the projector onto that subspace.The remaining orbitals are virtual.Source: §2 · p. 4
  • SCF alternates between obtaining a density matrix by diagonalizing and filling a Hamiltonian and building a Hamiltonian from that density matrix.The reference is the fixed point of these two maps; the iterations dominate calculation cost for hybrid functionals.Source: §2 · p. 4

Learning Hamiltonians

  • Hamiltonian networks predict the matrix from atomic numbers and positions, and downstream quantities follow from the predicted Hamiltonian’s eigenpairs.Evaluation combines element-wise Hamiltonian error with orbital-energy errors and orbital-coefficient similarity.Source: §2 · p. 4

Physics-informed losses

  • WALoss evaluates a prediction in the reference orbital basis, whereas density matching diagonalizes it and penalizes differences from reference density at sampled points.Density matching down-weights regions near nuclei and is a special case of the cumulative-LDOS loss.Source: §2 · p. 5

3 SPECTRAL REPRESENTATIONS OF HAMILTONIAN ERRORS

The paper analyzes Hamiltonian prediction errors in the reference-orbital basis, showing that existing losses see only selected error components. It identifies cumulative LDOS matching as a signal that retains density information while also resolving orbital energies and responding to every error sector.

3.1 WHERE THE PREDICTION ERROR LIVES

  • In the reference-orbital basis, errors split into diagonal energy shifts, occupied–virtual mixing, and within-occupied or within-virtual rotations, which affect eigen-properties differently.To first order, orbital energies depend on diagonal entries, while density depends only on occupied–virtual couplings and is most sensitive to pairs near the HOMO–LUMO gap.Source: §3.1 · p. 5
  • Density matching is exactly blind to energy shifts and within-subspace rotations when no level crosses the gap; these directions comprise a median 69.0% of error degrees of freedom on 4,000 QH9-stable molecules.The blind sectors include every orbital-energy error, including the HOMO–LUMO gap error; the 69.0% figure counts directions, not error magnitude.Source: §3.1 · p. 6

3.2 FROM THE DENSITY TO THE CUMULATIVE LDOS

  • Cumulative LDOS represents each orbital by an energy-positioned step whose height is its spatial density, retaining both orbital energies and shapes for comparison across sampled points and energies.FermiScope combines this loss with element-wise loss and computes the cumulative LDOS from the prediction’s exact eigenpairs, propagating gradients through the eigensolver.Source: §3.2 · p. 7

3.3 THEORETICAL PROPERTIES

  • Cumulative LDOS contains density matching as an energy slice, responds to every error sector, uniquely determines the Hamiltonian, and retains a non-vanishing gradient for large orbital-energy errors.At an energy inside the HOMO–LUMO gap, the cumulative LDOS recovers the density up to exponentially small tails; its gradient approaches a nonzero limit as the energy displacement grows.Source: §3.3 · pp. 7–8
  • The loss concentrates supervision on valence and low-virtual orbitals, while errors between orbitals far outside the energy window contribute mainly through Gaussian tails and remain covered by the element-wise term.Unlike a peak-shaped loss, cumulative matching continues pulling displaced orbital energies toward their targets beyond a few smoothing widths; the element-wise error need rise only slightly as error moves out of sensitive sectors.Source: §3.3 · p. 8

4 EXPERIMENTS

The experiments fine-tune three pretrained Hamiltonian networks on QH9 with the cumulative LDOS loss and compare it with WALoss under a matched protocol. The cumulative LDOS loss improves eigen-derived metrics across the tested splits while preserving element-wise accuracy, with ablations examining energy resolution and loss-weight sensitivity.

4.1 SETUP

  • Fine-tuning uses the same pretrained checkpoints, optimizer, schedule, batch size and data for 30k steps, varying only the auxiliary loss and its weight.The cumulative loss adds about 5% to QHNet pretraining cost and no inference cost.Source: §4.1 · p. 9

4.2 MAIN RESULTS

  • On stable-id, the cumulative LDOS loss lowers HOMO–LUMO gap error by 49–79% across QHNet, SPHNet and WANet, while Hamiltonian MAE stays within 0.3% of pretrained values or falls by 3.5%.It gives the best orbital energies and coefficients; its gap error is 1.6–2.9 times lower than WALoss, which raises Hamiltonian MAE by 0.6–2.3%.Source: §4.2 · p. 10

4.3 GENERALIZATION ACROSS SPLITS

  • On the three shifted splits, the cumulative LDOS loss lowers gap error for every backbone and split, by 36–77%, without raising element-wise error by more than 0.3%.WALoss increases gap error in all three shifted QHNet settings; on dynamic-mol, QHNet’s orbital similarity rises from 79.7% to 90.6%.Source: §4.3 · p. 10, Figure 3 · p. 10

4.4 ABLATION STUDIES

  • Adding energy resolution to density matching lowers gap error from 1101 to 791 µHa on WANet and from 970 to 789 µHa on SPHNet.On QHNet, it removes the element-wise cost of density matching (H from 76.82 to 74.81 µHa) while further lowering LUMO and occupied-orbital errors.Source: §4.4 · p. 10
  • On QHNet, cumulative-loss gap error stays between 1811 and 2231 µHa as its weight varies tenfold, while diagonal-only WALoss degrades as its weight grows.Source: §4.4 · p. 10

5 RELATED WORK

Related work covers learned Hamiltonian prediction, physics-informed spectral objectives, and learned electron density or LDOS. This paper positions cumulative LDOS as a training signal for predicted Hamiltonians.

Hamiltonian prediction

  • Hamiltonian networks including QHNet, WANet and SPHNet use element-wise losses, while the proposed loss applies to each without modification.Earlier models include SchNOrb, PhiSNet and DeepH; QH9 established the benchmark used here.Source: §5 · p. 11

Physics-informed and spectral objectives

  • WALoss avoids an eigensolver by evaluating the predicted Hamiltonian in the reference orbital basis, whereas the paper’s error-sector analysis maps which entries matter for each property.Self-consistency training enforces an SCF fixed point rather than the spectrum and is described as complementary.Source: §5 · p. 11

Electron density and LDOS in machine learning

  • Prior work predicts electron density or LDOS, while this work uses LDOS to train a predicted Hamiltonian and argues for optimizing its cumulative form.LAGNet introduced the sampling grid and core suppression used here.Source: §5 · p. 11

6 CONCLUSION

The conclusion presents cumulative LDOS as a complete signal for Hamiltonian error and reports reduced gap error without sacrificing element-wise accuracy. The remaining statements disclose AI use, reproducibility materials and the paper’s expected societal impact.

6 CONCLUSION

  • Cumulative LDOS sees every part of the error, weights it by spectral effect and uniquely determines the Hamiltonian; FermiScope reduces gap error by 49–79% while leaving element-wise accuracy intact.Source: §6 · p. 11

AI USE STATEMENT

  • Generative AI assisted with software implementation, literature identification and editing, while authors designed and ran experiments and checked reported numbers against DFT reference data.Source: AI USE STATEMENT · pp. 11–12

REPRODUCIBILITY STATEMENT

  • The supplementary material provides loss code, cached quadrature grids, evaluation scripts and proofs; the three backbones and QH9 are publicly available.Source: REPRODUCIBILITY STATEMENT · p. 12

ETHICS STATEMENT

  • The work uses public QH9 data and reports no known direct negative societal impact; it may help reduce molecular-simulation costs by improving predicted-Hamiltonian accuracy without extra data or architecture changes.Source: ETHICS STATEMENT · p. 12

A KOHN–SHAM DFT IN BRIEF

This background section introduces the quantum-chemistry terms used in the paper and outlines the iterative SCF procedure and its cost. It also states mathematical properties of basis functions used in the additional theory.

A KOHN–SHAM DFT IN BRIEF

  • The section collects a glossary of quantum-chemistry terms and describes the self-consistent-field procedure and its cost.Table 2 defines the terms used in the paper.Source: A KOHN–SHAM DFT IN BRIEF · p. 15, Table 2 · p. 15
  • SCF iteratively builds and diagonalizes the Kohn–Sham matrix until input and output density matrices agree, while a Hamiltonian network predicts the matrix in one forward pass.For 50 QH9-stable-ood molecules, median SCF convergence takes 9 iterations from minao versus 17 from the core-Hamiltonian guess; a QHNet pass and eigensolve take 0.1 s together.Source: A KOHN–SHAM DFT IN BRIEF · p. 16, Figure 4 · p. 16

B PROOFS AND ADDITIONAL THEORY

  • The proof assumptions state that nonzero basis-representable functions cannot vanish on an open set, and that a product of two such functions vanishing everywhere implies one factor is zero.Source: B PROOFS AND ADDITIONAL THEORY · p. 17

B.1 THE ERROR IN THE ORBITAL BASIS

The error’s location in the reference orbital basis determines which properties it affects: density matching and existing spectral losses each miss some error directions. The cumulative LDOS responds to every sector, uniquely identifies the Hamiltonian, and can retain a strong gradient for large energy errors within its training window.

B.1 THE ERROR IN THE ORBITAL BASIS

  • Whitening and rotating the prediction error into the reference eigenbasis yields ∆, whose Frobenius norm is bounded by κ(S) times the original matrix-error norm.The bound is loose on QH9: median amplification is 3.0 versus a median bound of 1878, making error location more informative than its size.Source: §B.1 · p. 17

B.2 PERTURBATION THEORY FOR EACH PROPERTY

  • Density depends only on occupied–virtual errors, while orbital energies and gaps also respond to diagonal and off-diagonal errors, with nearby orbital pairs more influential.A HOMO–LUMO coupling changes the LUMO about 75 times more than an equally large coupling to a deep core orbital on QH9.Source: §B.2 · p. 18
  • At the QH9 median, 69% of residual directions are density-blind, although physically realized blind-sector error is smaller in norm and grows with displacement.Blind sectors carry 15.6% of ∥∆∥^2_F at 0.02 Å and 30.3% at 0.30 Å.Source: §B.2 · p. 18

B.3 PROPERTIES OF THE CUMULATIVE LDOS LOSS

  • Within the HOMO–LUMO gap, cumulative LDOS equals half the electron density up to exponentially small tails, so it includes density matching as a special case.The training grid includes density-matching slices near the Fermi level, while other energies add information about how density is distributed across orbitals.Source: §B.3 · p. 19
  • The cumulative LDOS responds to diagonal and off-diagonal error components, and matching it over an energy interval uniquely determines the Hamiltonian.By contrast, density matching and diagonal-only WALoss have blind directions.Source: §B.3 · p. 19
  • The all-space injectivity result does not guarantee that 256 sampled points resolve every direction; denser sampling or the element-wise term addresses unresolved directions.Unresolved sampled directions are mostly occupied–virtual couplings to virtual orbitals above the energy window.Source: §B.3 · p. 20
  • Unlike a Gaussian-peak mismatch, the cumulative-step gradient remains near its limit for displaced levels inside the energy window, but vanishes after levels leave it.On the K = 64 grid, it stays within 0.01 of erf(δ/2σ) for |δ| ≤ 0.1 Ha when levels are at least 0.18 Ha inside the window.Source: §B.3 · p. 20

B.4 ERROR REALLOCATION

  • Weighting sensitive error sectors can lower their weighted error while increasing total residual norm, unless weighted and unweighted objectives share a minimizer.The cumulative-LDOS gradient concentrates on close and frontier predicted orbital pairs inside the energy window, unlike the uniform element-wise gradient.Source: §B.4 · p. 21

C THE QH9 DATASETS AND SPLITS

The section describes the QH9 reference data, its generalization splits, and the real-space and spectral quantities used to interpret the losses. The examples show a large density-blind residual subspace, localized Hamiltonian structure, and complementary density and spectral views.

C.1 PROVENANCE AND QUANTUM-CHEMICAL SETTINGS

  • QH9 supplies converged B3LYP/def2-SVP Hamiltonians for QM9 molecules, with stable equilibrium geometries and short molecular-dynamics trajectories.The dynamic release contains 99,900 records from 999 molecules sampled at 100 geometries each.Source: §C.1 · p. 22

C.2 SPLITS

  • The four splits test random interpolation, size extrapolation, unseen geometries, and unseen molecules; the size-test molecules are larger and have wider median gaps than training molecules.Median atom count and basis size shift from 17 atoms and n = 166 in training to 23 atoms and n = 196 in testing.Source: §C.2 · p. 22, Table 4 · p. 22

C.3 COMPOSITION, SPECTRA, AND THE SIZE OF THE BLIND SUBSPACE

  • 69.0% of residual directions are density-blind at the QH9-stable median, with a 66–71% 5–95% range; the figure’s partition assigns the remaining 31% to the density-changing occupied–virtual slice.This is a count of residual parameters, not a measurement of how much realized error lies in those directions.Source: §C.3 · p. 23, Figure 5 · p. 23, Table 5 · p. 23

C.4 A GALLERY

  • The gallery samples chemically diverse QH9-stable molecules, spanning planar and non-planar structures, varied ring systems, and reference gaps of 0.14–0.30 Ha.The panels share length and colour scales; open nuclear markers indicate atoms displaced from the best-fit plane.Source: §C.4 · p. 24, Figure 6 · p. 24

C.5 THE CACHED AUXILIARY QUANTITIES

  • Cached reference densities, quadrature weights, sampled points, and orbital values support auxiliary losses without basis-function evaluation during training.The overlap reconstruction differs from the released matrix by 2.2 × 10^-7; integrating the cached density gives 59.978 electrons for a 60-electron molecule.Source: §C.5 · p. 24

D.1 THE HAMILTONIAN AND OVERLAP MATRICES

  • Hamiltonian elements are strongly local and span a wide dynamic range, with same-atom blocks helping explain why diagonal-block MAE is about twice off-diagonal-block MAE.The figure plots matrix magnitudes, overlap, and Hamiltonian-element magnitude against inter-atomic distance.Source: §D.1 · p. 25, Figure 7 · p. 25

D.2 DENSITY, FRONTIER ORBITALS, AND CORE SUPPRESSION

  • The occupied–virtual sector mixes frontier orbitals, while core suppression compresses density peaks and leaves the valence region unchanged.For the illustrated planar molecule, the frontier π orbitals vanish in the nuclear plane and are sampled 1 a0 above it.Source: §D.2 · p. 26, Figure 8 · p. 26

D.3 THE SPECTRAL PICTURE AND THE FERMI-LEVEL IDENTITY

  • Cumulative-LDOS curves encode orbital weights as staircase risers, and their Fermi-level slice matches half the reference density to a maximum relative deviation of 6 × 10^-8.Unlike isolated Gaussian-peak profiles, the staircase has no energy intervals with zero gradient for displaced levels inside the window.Source: §D.3 · p. 27, Figure 9 · p. 27

D.4 WHERE A REAL RESIDUAL LIVES

Across 640 frame pairs from 214 trajectories, occupied–virtual changes dominate the residual, while density-blind sectors grow with displacement. The overlap-matrix bound substantially overstates the amplification observed in these frame pairs.

D.4 WHERE A REAL RESIDUAL LIVES

  • The occupied–virtual sector carries a median 70–84% of ∥∆∥2 F while occupying only 31% of the parameters.Density matching is therefore well aimed and Lρ alone achieves a large share of the gap improvement in Table 1.Source: §D.4 · p. 28
  • At 0.02 Å, density-blind sectors hold 15.6% of ∥∆∥2 F; at 0.30 Å, they hold 30.3%, with essentially all growth in the virtual–virtual block.The virtual–virtual block contains the LUMO energy error, which the electron density cannot resolve.Source: §D.4 · p. 28, Figure 10 · p. 28
  • Observed amplification has median 3.0, versus a median whitening bound of 1878, which overstates amplification by a median factor of 580.The bound is loose on real data and cannot certify that small element-wise error implies small spectral error.Source: §D.4 · p. 28

E METRIC DEFINITIONS

The section defines how matrix, orbital, shape, and SCF outcomes are evaluated, then describes the implemented auxiliary-loss and SCF protocols. It identifies the shifted-split results table and its reported metrics, without giving the table’s cell values.

E METRIC DEFINITIONS

  • Evaluation includes Hamiltonian entry-wise error, frontier and occupied-orbital energy errors, occupied-orbital cosine similarity, and SCF cost ratios.SCF ratios compare iteration count and wall time against a minao initialization; total time includes model inference.Source: E METRIC DEFINITIONS · p. 29

F THE LOSSES AS IMPLEMENTED

  • For each selected molecule, the implementation whitens the predicted Hamiltonian, diagonalizes it in float64, and compares prediction and reference spectra using the same code path.Gradients flow through the Cholesky factorization and eigensolver.Source: F THE LOSSES AS IMPLEMENTED · p. 30
  • The density and energy-resolved losses use cached sample points; the cumulative and peak-shaped terms average mismatches over an energy grid.The grid is anchored to the reference Fermi level, and eight molecules per optimizer step contribute to the auxiliary term.Source: F THE LOSSES AS IMPLEMENTED · p. 30
  • The SCF protocol uses the predicted Hamiltonian to form an initial density for an otherwise unmodified B3LYP/def2-SVP calculation, compared with a minao-started calculation.Reported ratios use the same 50 molecules and include model inference in total time.Source: F THE LOSSES AS IMPLEMENTED · p. 30

G.1 ALL METRICS ON THE SHIFTED SPLITS

  • Table 6 reports shifted-split test results in µHa for energies and percent for ψ, with bold marking the best downstream metric per block.The supplied passages identify the table’s scope but do not give its cell values.Source: Table 6 · p. 31, §G.1 · p. 31

G.2 ELEMENT-WISE-ONLY CONTROL

The element-wise-only control improves matrix error without reliably improving eigen-derived quantities, while cumulative-LDOS fine-tuning performs well across tested weights. Peak-shaped loss can drift on deep orbitals, and the predicted Hamiltonian still offers SCF acceleration with modest training overhead.

G.2 ELEMENT-WISE-ONLY CONTROL

  • Continued element-wise training lowered QHNet Hamiltonian MAE on all four splits but raised gap error by 53%, 246%, 236%, and 19%, respectively.The cumulative-LDOS loss improved every eigen-derived metric while keeping Hamiltonian MAE at most 0.3% above pretrained; the control comparison spans three backbones on stable-id and QHNet across four splits.Source: §G.2 · p. 32, Table 8 · p. 33, Table 7 · p. 33

G.3 WEIGHT SENSITIVITY ON QHNET

  • Across tested weights, density-only and cumulative losses give 2–3 times lower gap error than WALoss; at w = 30, cumulative loss has 7.3 times lower gap error than the control.Its gap error is 2231, 2027, and 1811 µHa at w = 10, 30, and 100, respectively; its element-wise error stays essentially constant.Source: §G.3 · p. 34, Table 9 · p. 34, Figure 11 · p. 35
  • As diagonal-only WALoss weight grows, QHNet gap error rises from 12,373 to 26,813 µHa and occupied-orbital error rises from 797 to 1573 µHa.The tested form sees sector A only, and only to first order.Source: §G.3 · p. 35, Table 10 · p. 35

G.4 PEAK-SHAPED LDOS VARIANTS

  • On QHNet, peak-shaped LDOS loss is unreliable: identical settings end at gap errors of 6917 and 1994 µHa, while deep occupied orbitals can drift.For HOMO−23, error grows from 1,049 to 20,979 µHa; the cumulative loss shows no drift and improves every orbital.Source: §G.4 · p. 36, Table 11 · p. 36

G.5 SCF ACCELERATION

  • On 50 stable-id test molecules, every model saves about a third of SCF iterations and 13–15% of total time versus minao initialization.Total time includes model inference, and the models’ iteration ratios are 0.671–0.674.Source: §G.5 · p. 37, Table 12 · p. 37

G.6 TRAINING COST

  • On QHNet, the auxiliary term adds 0.72 s per training step and no GPU memory; over 30k fine-tuning steps, this is about six hours.That is roughly 5% of the cost of pretraining QHNet for 300k steps, and inference cost is unchanged.Source: §G.6 · p. 37, Table 13 · p. 37

H NUMERICAL VERIFICATION

Numerical checks validate the energy-shift and orbital-rotation perturbations and test recovery from random Hamiltonian errors. Cumulative-LDOS mismatch recovers orbital energies much more closely than density mismatch in the reported recovery test.

H NUMERICAL VERIFICATION

  • The energy-only perturbation shifts the intended orbital by 2.000 mHa while eigenvector overlap changes only at double-precision round-off; the shape rotation changes eigenvalues by 2.4 × 10−13 Ha.These perturbations separate energy errors from shape errors.Source: H NUMERICAL VERIFICATION · p. 38, Figure 12 · p. 38, Table 14 · p. 39
  • Minimizing cumulative-LDOS mismatch recovers every orbital energy to 0.03 mHa, while minimizing density mismatch leaves orbital energies wrong by 10 mHa.For the cumulative-LDOS recovery, the residual is 20–30% of the initial ∥∆∥F and lies mostly in core–virtual couplings.Source: H NUMERICAL VERIFICATION · p. 38, Table 14 · p. 39

I LIMITATIONS AND FUTURE WORK

The sector analysis is exact only under stated perturbation assumptions, and does not predict how a network moves among sectors during training. The paper also identifies evaluation scope and cost boundaries, alongside two directions for future work.

I LIMITATIONS AND FUTURE WORK

  • The sector analysis is exact as a change of basis under first-order perturbation theory, with non-degenerate reference energies for orbital energies and shapes and an open HOMO–LUMO gap for density.It diagnoses residual error but does not predict how a network moves among sectors during training.Source: I LIMITATIONS AND FUTURE WORK · p. 40
  • Evaluation covers QH9 (B3LYP/def2-SVP), three architectures, and all four official splits; the energy window targets ground-state properties, while excited-state targets would require widening it.The smoothing width σ was fixed rather than tuned, and testing on larger molecules, other theory levels, and periodic systems is planned.Source: I LIMITATIONS AND FUTURE WORK · p. 40
  • Training adds one eigendecomposition per sampled molecule per step, costing 0.72 s per QHNet step on QH9, while inference is untouched.At thousands of basis functions, dense solvers would give way to iterative and fragment-based solvers already used by DFT codes.Source: I LIMITATIONS AND FUTURE WORK · p. 40
  • The cumulative LDOS could be used from the start of training, while validation-set sector analysis could identify residual errors and help select a training signal.The authors propose these as future directions.Source: I LIMITATIONS AND FUTURE WORK · p. 40

J IMPLEMENTATION DETAILS AND PROTOCOL

The implementation uses matched fine-tuning settings within each backbone, with auxiliary loss and its weight as the varying choices. It specifies sampling, eigensolving, energy grids, evaluation, and verification procedures.

J IMPLEMENTATION DETAILS AND PROTOCOL

  • Each molecule uses a cached fixed random subset of 256 atom-centred radial–Lebedev quadrature points at PySCF level 0, with core suppression fitted from geometry.The suppression factor uses the element-wise fit of Ushenin et al. (2025).Source: J IMPLEMENTATION DETAILS AND PROTOCOL · p. 41
  • For each of eight molecules per step, the generalized Hamiltonian is formed using the overlap factorization, diagonalized by a CPU float64 symmetric eigensolver, and back-transformed with gradients through the eigensolver.The passage specifies forming L^-1 H L^-T with S = LL^T.Source: J IMPLEMENTATION DETAILS AND PROTOCOL · p. 41
  • The energy grid uses 64 points from μ − 1.5 to μ + 0.3 Ha with σ = 0.02 Ha; orbitals below the window contribute a constant step, and N(r, E) = ρ/2 inside the gap.The loss weights are listed in Table 15, and the density identity holds up to the bound of Proposition B.4.Source: J IMPLEMENTATION DETAILS AND PROTOCOL · p. 41
  • Evaluation uses basis-permutation-invariant element-wise MAE and eigen-derived metrics from the predicted generalized eigenproblem with DFT overlap, using matched backbone-specific evaluators.SPHNet and WANet use GPU evaluation with TF32 disabled and float64 solves, matching QHNet's CPU float64 evaluation.Source: J IMPLEMENTATION DETAILS AND PROTOCOL · p. 41
  • Within each backbone, fine-tuning methods share settings except for auxiliary loss and weight, which is selected on the validation split; unlisted settings follow the released training recipe.The optimizer, schedule, batch size, clipping, step budget, and data split are held fixed.Source: Table 15 · p. 41, J IMPLEMENTATION DETAILS AND PROTOCOL · p. 41
  • Verification perturbs the reference Hamiltonian by eigenvalue-only shifts and HOMO/LUMO-plane rotations, then checks the stated identity, closed forms against quadrature, and sector sensitivities.The passage reports that Appendix H gives the outcomes of these checks.Source: J IMPLEMENTATION DETAILS AND PROTOCOL · p. 41
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