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Physics-Aligned Electronic Ground-State Learning Improves Generalization

Eike S. Eberhard, Xaver Kainz, Viktor Kotsev, Abdulrahman Aldossary, Stephan Günnemann

arXiv:2610.10298v1cs.LGphysics.chem-phquant-ph

TL;DR

Ground-state descriptor models (GSMs) have been explored to accelerate costly KS-DFT calculations, but existing models are not well aligned with learning physical ground states. The paper aligns GSM objectives and architectures with KS-DFT, and reports substantially lower size-extrapolation energy errors for density and Hamiltonian models.

  • Problem

    Existing GSMs that predict Hamiltonians or density matrices are not well aligned with learning physical ground states, despite their potential to accelerate costly KS-DFT calculations.

    Source: §1 · p. 2

  • Method

    The authors align GSM training with KS-DFT using orthonormal-basis loss, gauge-aware Hamiltonian training, and density predictions constrained to the Grassmannian.

    Source: §1 · p. 2

  • Results

    Across size extrapolation, ON-Loss consistently reduces errors for density and Hamiltonian GSMs; on QMugs, GROOTproj and ROCKET achieve 99.6% and 99.9% lower energy errors than NON baselines, respectively.

    Source: §1 · p. 2, §5 · p. 7

  • Takeaways & Limitations

    The authors identify GSM advantages over MLIPs that may justify their increased evaluation costs for some applications.

    Source: §1 · p. 2

    Evaluations are restricted to closed-shell Kohn-Sham systems and the B3LYP/def2-SVP level of theory.

    Source: §6 · p. 9

Abstract

from arXiv · show

Machine-learned interatomic potentials (MLIPs) excel at in-distribution tasks, accelerating drug and material development, yet they struggle to generalize out-of-distribution. We propose to push the cost-accuracy Pareto frontier by designing observable-agnostic electronic ground-state descriptor models (GSMs) with computational costs situated between MLIPs and Kohn-Sham density functional theory (KS-DFT). We align the learning objectives and architectures of GSMs with the governing equations of KS-DFT by enforcing physical constraints and removing optimization pressure on unphysical or irrelevant degrees of freedom. In our size-extrapolation experiments from QM9 to QM40, our combined contributions OrthoNormal-Loss (ON-Loss) and Grassmann Restricted Occupied-Orbital Training (GROOT) reach a 79.1% energy and 83.4% force mean absolute error (MAE) reduction over previous state-of-the-art density GSMs. For Hamiltonian GSMs, ON-Loss and Residual Optimal-gauge Conditioning-aware KS-Eq. Training (ROCKET) together reduce the energy and force MAEs of the strongest baseline by 99.8% and 95.9%, respectively. Using a self-consistency rejection criterion, we filter out extrapolation errors on QMugs, rejecting fewer than 0.4% of predictions while reaching an energy MAE of 0.07 mHa. Finally, we demonstrate the efficiency of label-free self-consistency fine-tuning, and transfer GSMs to reactive chemistry in Transition1x, reaching energy errors below chemical accuracy.

1 INTRODUCTION

The introduction frames electronic ground-state prediction as a trade-off between accuracy and computational cost, and proposes GSMs that better align learning with KS-DFT physics. It presents loss, training, and uncertainty methods to improve physical validity and generalization while retaining costs between MLIPs and multi-Fock-build approaches.

1 INTRODUCTION

  • ON-Loss measures matrix errors in an orthonormal basis, correcting the spectral misalignment of non-orthonormal-basis supervision with physical observables.The introduction reports that ON-Loss alone reduces QMugs energy, force, dipole, and density errors by 87–99% for density GSMs and 93–99.5% for Hamiltonian GSMs.Source: §1 · p. 2
  • ROCKET addresses the non-uniqueness of Hamiltonian matrices that yield the same ground state by accounting for their physical invariance in training.It is introduced as Residual Optimal-gauge Conditioning-aware KS-Eq. Training.Source: §1 · p. 2
  • GROOT restricts density-matrix predictions to the Grassmannian of physically valid densities, removing the leading-order energy error term relative to unconstrained baselines.It is a matrix readout for SO(3)-equivariant graph neural networks.Source: §1 · p. 2
  • A practical framework operates under a one-Fock-build budget, using a first-order energy correction, GSM-native uncertainty rejection, and label-free self-consistency fine-tuning for new chemical domains.The framework is proposed for applications where GSMs’ evaluation costs, between those of MLIPs and methods requiring multiple Fock builds, may be justified.Source: §1 · p. 2
  • On QMugs size extrapolation, GROOTproj and ROCKET reach mean energy errors of 0.21 and 0.56 mHa, below chemical accuracy, while rejecting fewer than 0.4% of predictions.Models train on molecules with at most 150 electrons and evaluate on molecules with up to 660 electrons. After label-free self-consistency fine-tuning on Transition1x, QMugs-trained GROOTproj and ROCKET reach energy errors of 0.66 and 0.74 mHa, rejecting fewer than 1% of predictions.Source: §1 · p. 2
  • For QM9-trained models evaluated on QM40, ON-Loss lowers density GSM ERIC from 0.82 to 0.70, and GROOT lowers it further to 0.67.The introduction reports that the GROOT result requires a third fewer iterations than the traditional MINAO initial guess.Source: §1 · p. 2

2 BACKGROUND

KS-DFT ground states are self-consistent solutions of a generalized eigenvalue problem, with the density matrix encoding the physical state. Although Hamiltonian and orbital representations contain redundant degrees of freedom, valid density matrices correspond to occupied subspaces on a Grassmann manifold.

2 BACKGROUND

  • The KS-DFT ground state solves the nonlinear generalized eigenvalue problem HC = SC diag(ϵi); its lowest O = Nelec/2 orbitals define the density, and C*, P*, and H* are complete descriptors.Because the Hamiltonian depends on the density, the problem is solved iteratively until the ground state is a fixed point.Source: §2 · p. 3
  • Transforming to an orthonormal basis converts the generalized eigenvalue problem into a regular eigenvalue problem.The basis transformation uses the overlap matrix S to form orthonormal representations of the coefficients, density, and Hamiltonian.Source: §2 · p. 3
  • Many Hamiltonian and orbital representations yield the same density: constant or virtual-level shifts and orbital rotations leave the physical state unchanged.The virtual orbitals do not enter the density, and occupied orbitals can be rotated without changing their occupied subspace.Source: §2 · p. 3
  • Valid orthonormal-basis density matrices are symmetric and idempotent with trace O, and form an OV-dimensional Grassmann manifold.Idempotency and the trace condition characterize densities that factor into an orthonormal occupied block.Source: §2 · p. 3

3 RELATED WORK

The related work contrasts observable prediction with KS-DFT descriptor learning, then identifies gaps in basis supervision, density physicality, and molecular-Hamiltonian shift handling that this work addresses.

3 RELATED WORK

  • MLIPs predict geometry-dependent energies in linear system-size time, while GSMs predict KS-DFT Hamiltonians or densities from which most observables require at most one Fock build.GSMs can also accelerate SCF calculations; the paper compares GSM observables under a one-Fock-build budget. (Yu et al., 2023b; Liu et al., 2026)Source: §3 · p. 4
  • QH9 has saturated and its very-small-molecule scope limits size-extrapolation assessment; this work compares GSM observables under a one-Fock-build budget.Kim et al. (2026) report energy errors more than two orders below chemical accuracy on QH9; Kim et al. match MLIP force accuracy on in-distribution MD trajectories at higher cost, requiring two Fock builds.Source: §3 · p. 4
  • Although NON-basis prediction is standard and motivated by spatial localization, its supervision has weak observable justification, motivating direct ON-Loss versus NON-baseline comparisons.Kaniselvan et al. (2026) report weak correlation between NON matrix-element and eigenvalue errors, also observed for downstream observables; prior ON-basis work leaves its physical-observable implications poorly understood, while Li et al. (2025b) perturb an ON Hamiltonian loss and combine it with NON supervision.Source: §3 · p. 4
  • Unrestricted density prediction can violate physicality, whereas GROOT transports or projects onto the valid density manifold; ROCKET handles unconstrained molecular-Hamiltonian shifts with a shift-invariant loss.Prior density corrections use projection (Zhang et al., 2026) or charge rescaling and McWeeny purification, applied at evaluation or through loss terms; molecular Hamiltonian losses had left the shift degree of freedom untreated.Source: §3 · p. 4

4 METHODS

The methods align descriptor learning with KS-DFT by supervising orthonormal matrix errors and restricting density and Hamiltonian training to physically relevant degrees of freedom.

ON-Loss.

  • ON-Loss minimizes orthonormal-basis matrix-error norms, avoiding the overlap-spectrum weighting of observable-error directions induced by non-orthonormal-basis supervision.In the orthonormal basis, the overlap matrix is the identity and error propagation to observables becomes approximately isotropic.Source: ON-Loss · p. 5
  • GROOT constrains predicted densities to the Grassmannian, making them physically valid and rendering total-energy error quadratic while dipole and real-space density errors remain linear.The real-space density metrics identified are L1 and L2.Source: ON-Loss · p. 5
  • GROOTrot predicts densities through a Grassmann-preserving orbital-rotation flow, while GROOTproj differentiably projects a symmetrized readout onto the Grassmannian.GROOTrot uses a closed-form unit-time map, avoiding numerical ODE integration; GROOTproj passes parameter gradients through the projection.Source: ON-Loss · p. 5
  • ROCKET removes Hamiltonian supervision pressure along gauge and irrelevant directions, allowing molecule-specific energy shifts while softly restraining cross-molecule gauge variation.It uses a closed-form optimal shift and an exponential moving average reference gauge.Source: ON-Loss · pp. 5–6
  • ROCKET masks errors within the orthogonal complement of the lowest O+k reference eigenvectors; retaining k extra eigenvectors separates occupied states from the first unsupervised state.This provides a robustness tradeoff between a smooth, higher-dimensional Hamiltonian target and a discontinuous, lower-dimensional density target; k = 1 is the default.Source: ON-Loss · p. 6

5 EXPERIMENTS

The experiments test observable-agnostic ground-state models on size and conformational extrapolation, then assess self-consistency filtering and label-free transfer to reactive chemistry. Across these settings, the proposed loss and training methods improve extrapolation, while Grassmannian constraints support reliable adaptation.

Size-Extrapolation

  • On QM9-to-QM40 size extrapolation, ON-Loss consistently lowers errors versus NON, ROCKET beats tested Hamiltonian GSMs, and GROOTrot improves on ROCKET for every observable except the Kohn–Sham energy.GROOTproj has higher energy errors than ROCKET but surpasses it on the other observables; post-hoc manifold projection rejects 66% of molecules as unreliable versus 0.6% for GROOTproj.Source: Figure 2 · p. 7, §5 · p. 7

Loss Observable Alignment

  • ON-Loss makes relative loss sensitivities approximately direction-independent across observables, and its QM40 models strictly outperform NON-trained models on all tested observables.The NON loss emphasizes directions according to its sensitivity rather than observable impact, while ON-Loss concentrates matrix error in high-impact directions; ON-Loss residuals also have higher mutual information with downstream observable errors across evaluated models.Source: §5 · p. 7

Size-Extrapolation

  • On QMugs extrapolation from at most 150 to as many as 660 electrons, GROOTproj achieves an order of magnitude better energies than energy-only linear-scaling MLIPs while rejecting less than 1% of its predictions.The figure reports observable errors for accepted molecules and brackets MAEs including rejected predictions; GSMs remain orders of magnitude below NON baselines as system size grows.Source: §5 · p. 6, Figure 3 · p. 8
  • On the 3BPA torsional surface, ROCKET reduces energy errors by 8–12× and force errors by 5–7× versus PaiNN, NequIP, and MACE; Pbase (ON) reduces them by 148–234× and 20–29×, respectively.The study isolates conformational generalization by training on QMugs and evaluating the 3BPA torsional surface.Source: §5 · p. 8

Towards foundational GSMs

  • After label-free self-consistency fine-tuning on Transition1x, GROOT and ROCKET reach energy errors below chemical accuracy on accepted predictions, while NON-pretrained models fail under the occupied–virtual objective.Pbase (ON) has a first-order corrected energy error of 0.52 mHa but rejects 25% of test frames; GROOTproj and ROCKET reject 0.2% of predictions, and GROOT avoids unconstrained densities leaving the Grassmannian.Source: §5 · pp. 8–9, Figure 4 · p. 9

6 DISCUSSION · APPENDIX CONTENTS · A KOHN–SHAM THEORY IN A FINITE BASIS

The discussion attributes GSM gains to aligning training with KS-DFT constraints, while noting evaluation limits and possible self-consistency-based workflows. The appendix documents theoretical foundations and reproducibility details.

6 DISCUSSION

  • On QM9-to-QM40 extrapolation, ON-Loss with GROOT reduces the previous state-of-the-art density GSM’s energy error by 79.1%, while ON-Loss with ROCKET reduces the strongest Hamiltonian baseline’s by 99.8%.Under GROOT, 98% of molecules have energy errors below 0.5× chemical accuracy.Source: §6 · p. 9
  • For QMugs systems up to 660 electrons, GROOTproj and ROCKET reduce energy errors by 99.6% and 99.9% versus corresponding NON baselines, and self-consistency fine-tuning transfers QMugs-pretrained models to Transition1x below chemical accuracy.Fock builds raise inference cost relative to MLIPs, but self-consistency residuals can reject extrapolation errors; accepted Transition1x predictions remain below chemical accuracy across the reaction coordinate.Source: §6 · p. 9
  • Evaluations cover only closed-shell KS-DFT at B3LYP/def2-SVP; unrestricted systems are unevaluated, and transfer to other functionals and basis sets remains untested.The authors suggest a GROOT-based active-learning pipeline that accepts one-iteration observables below a residual threshold, otherwise falls back to iterative SCF and can update the GSM.Source: §6 · pp. 9–10

REPRODUCIBILITY STATEMENT

  • The appendix provides proofs of GROOT’s reachability, geometric validity, and expressivity, plus observable-error bounds and quadratic energy scaling for valid density projectors.It documents dataset sources, reference protocols, splits, architectures, and hyperparameters; code is planned for release with final publication.Source: REPRODUCIBILITY STATEMENT · p. 10

A KOHN–SHAM THEORY IN A FINITE BASIS

  • The finite-basis theory overview omits explicit spin degrees of freedom and applies to closed-shell systems, where spin-up and spin-down densities are equal and electron count is even.For open-shell generalization, it refers readers to Szabo & Ostlund (1996) and Lehtola et al. (2020a).Source: A KOHN–SHAM THEORY IN A FINITE BASIS · p. 18
  • KS-DFT constructs ground-state electron density from molecular orbitals that solve the Kohn–Sham equations, whose terms include kinetic, nuclear, electron-electron, and exchange-correlation potentials.The orbitals are indexed by eigenvalue and are orthonormal, with occupied and virtual orbital counts defined by electron count and basis size.Source: A KOHN–SHAM THEORY IN A FINITE BASIS · p. 18
  • Representing orbitals in a finite basis defines a density matrix, and density integration uses the overlap matrix, which is constant for a given basis set and positive definite.In finite basis sets, the stated discretized formulation is equivalent to solving the Kohn–Sham equations.Source: A KOHN–SHAM THEORY IN A FINITE BASIS · p. 18

B ROTATIONAL GROOT: TRANSPORT ON THE GRASSMANNIAN

GROOTrot transports between physical densities of equal rank through occupied–virtual rotations, while its readout produces valid generators without restricting which generators are reachable. The construction also separates density-changing rotations from redundant orbital rotations that leave the density unchanged.

Rotational GROOT: Transport on the Grassmannian

C OBSERVABLE BOUNDS FOR THE ORTHONORMAL LOSS … G AO VERSUS LÖWDIN SPATIAL DECAY

This section analyzes observable-error bounds for density and Hamiltonian predictions, including how overlap-spectrum directions, energy gaps, and different matrix-error components affect loss sensitivity. It also distinguishes orthonormal loss and ROCKET from WALoss and examines how AO and Löwdin entries vary with atom-pair distance.

C OBSERVABLE BOUNDS FOR THE ORTHONORMAL LOSS

  • Density errors bound one-body observables linearly, while valid-projector energy errors are quadratic; Hamiltonian-derived bounds additionally depend on the HOMO–LUMO gap.AO and orthonormal norms weight density and Hamiltonian error directions oppositely, so orthonormal bounds are not always tighter; full-rank orthonormalizations preserve Frobenius loss, and symmetric Löwdin minimizes squared basis-function displacements.Source: C OBSERVABLE BOUNDS FOR THE… · pp. 22–24

D DIRECTIONAL SENSITIVITY IN THE OVERLAP SPECTRUM

  • In the overlap eigenbasis, the orthonormal loss weights each perturbation direction by (λ_iλ_j)^(s/2), whereas the non-orthonormal loss weights directions equally.Here s = +1 for density matrices and s = −1 for Hamiltonians; this basis diagonalizes the overlap matrix and both loss transformations.Source: D DIRECTIONAL SENSITIVITY IN THE… · pp. 26–27

E A UNIFYING PERSPECTIVE ON HAMILTONIAN AND DENSITY LEARNING

  • ROCKET with level shifts unifies Hamiltonian and density learning: as the shift grows without bound, Hamiltonian targets reduce to density targets, exposing a smooth, higher-dimensional versus discontinuous, lower-dimensional tradeoff.The analysis does not establish an optimal level shift.Source: E A UNIFYING PERSPECTIVE ON… · p. 28

F RELATION OF THE ORTHONORMAL LOSS TO THE WAVEFUNCTION ALIGNMENT LOSS

  • Li et al. (2025b)’s initial WALoss formulation shares ON-Loss’s orthonormal matrix geometry, but its final paper formulation uses state-wise weights and retains only diagonal Hamiltonian-error terms.Those diagonal terms capture first-order orbital-energy shifts but omit off-diagonal occupied–virtual couplings that drive first-order orbital rotations; the preceding formulation’s full Frobenius norm includes them.Source: F RELATION OF THE ORTHONORMAL LOSS TO… · p. 28
  • Eq. (70) is not a uniformly weighted Frobenius norm: separate means weight each diagonal entry B −1 times as much as each off-diagonal entry, and released code omits Eq. (69) weighting.The omitted weighting is the occupied–LUMO ρ/ξ weighting.Source: F RELATION OF THE ORTHONORMAL LOSS TO… · p. 29
  • WALoss is not a standalone orthonormal loss; its combined Hamiltonian losses retain overlap-dependent anisotropy, basis-dependent entrywise MAE, and penalties on physically irrelevant gauge shifts.The AO Frobenius term retains overlap-dependent anisotropy, and diagonal terms penalize shifts that leave generalized eigenvectors and density unchanged.Source: F RELATION OF THE ORTHONORMAL LOSS TO… · p. 29
  • ROCKET first identifies Hamiltonians representing the same physical ground state, unlike WALoss, which reweights a fixed reference Hamiltonian.ROCKET quotients out the gauge freedom before introducing further conditioning or occupied-subspace structure.Source: F RELATION OF THE ORTHONORMAL LOSS TO… · p. 29

G AO VERSUS LÖWDIN SPATIAL DECAY

  • AO entries decay with atom-pair distance, whereas Löwdin entries remain large after AO entries have dropped.Figure 6 compares H, P, P − PMINAO, and H − HSAP − cS in both bases for QM40 molecule 68311 at B3LYP/def2-SVP, scaled by dataset RMS σ0.Source: Figure 6 · p. 30

H DATASETS · I MODEL HYPERPARAMETERS

The sections specify dataset roles, evaluation conventions, reference calculations, and compute stages, then document output-scale normalization and model-baseline architectures.

H DATASETS

  • Five datasets cover QM9 training, QM40 size extrapolation, QMugs training and size extrapolation, and transfer to conformational chemistry (3BPA) and reactive chemistry (Transition1x).QMugs tests size extrapolation up to 660 electrons; 3BPA and Transition1x test transfer from QMugs-trained models.Source: H DATASETS · p. 31
  • Evaluations use only atomic numbers and coordinates, recompute electronic-structure quantities, and restrict elements to each checkpoint’s training vocabulary.Evaluation subsets are frozen identifier lists; size-stratified subsets sample equally from electron bins.Source: H DATASETS · p. 31, Table 5 · p. 34
  • All reference labels use restricted Kohn–Sham DFT at B3LYP/def2-SVP, computed with GPU4PySCF; Transition1x self-consistency training uses no labels.The reference pipeline also produces evaluation references for all five datasets and training labels for QMugs-trained checkpoints.Source: H DATASETS · p. 31, Table 2 · p. 31
  • Predictions are evaluated in up to three increasingly costly stages, with the cheap one-Fock-build stage used to compare all models.On larger test sets, SCF and force stages run on nested subsets sampled equally from electron bins to preserve the full-set size distribution.Source: H DATASETS · pp. 31–32, Table 3 · p. 32
  • Dataset splits enforce the intended transfer conditions: QM40 excludes molecules containing S or Cl, QMugs splits are molecule-disjoint, and Transition1x separates formulas and reactions.3BPA is scored as complete fixed-dihedral scans without subsampling; Transition1x retains reactant, transition-state, product, and selected path images around the transition state.Source: Table 5 · p. 34, Table 6 · p. 35, Table 7 · p. 36
  • The output-scale normalization fits σ0 on all 103,790 training molecules, and σ0 anticorrelates with system size (r = −0.86 against basis dimension).The table reports ¯c = 0.404129 Ha and ¯cAO = 0.382862 Ha; the accompanying paragraph gives b1 = 2.03 × 10−3 and b2 = 4.70 × 10−3.Source: Table 9 · p. 38, H DATASETS · p. 38

I MODEL HYPERPARAMETERS

  • The QHFlow2 backbone uses SO(2)-equivariant edgewise–nodewise updates, replacing its generative flow with the GROOT readout (Kim et al., 2026; Fu et al., 2025).The cited architecture is described as being in the style of eSEN.Source: Table 10 · p. 39
  • The PaiNN, NequIP, and MACE baselines sum per-atom energies; their architectures and model-specific scales, shifts, and readouts are specified.MACE uses 256×(0e+1o+2e) hidden features, edge spherical harmonics through l = 3, and readouts after each interaction layer.Source: Table 11 · p. 39

J MODEL IMPLEMENTATION DETAILS

The implementation combines an equivariant matrix-learning architecture with a JAX port that corrects two reference-code bugs, improving rotation equivariance and preserving decoder feature ordering.

J MODEL IMPLEMENTATION DETAILS

  • Following Kim et al. (2026), the eSCNMD backbone and QHNet readout construct blockwise Hamiltonian, density, or GROOTrot-input matrices from equivariant node features.The backbone builds on eSEN and eSCN, combines equivariant matrix encoders for initial-guess and overlap matrices, and symmetrizes decoded Hamiltonian and density matrices.Source: J MODEL IMPLEMENTATION DETAILS · p. 40
  • The JAX port preserves atom-indexed features and contracts each block into matching degree ℓ=0, 1, 2 channels, reducing relative rotation-equivariance error from 10^-3 to 10^-6.The reference implementation reshaped atom-indexed features as graph-indexed, causing atoms to receive incorrect features dependent on batch and atom ordering.Source: J MODEL IMPLEMENTATION DETAILS · p. 40
  • The JAX port explicitly transposes backbone features into the decoder’s required memory layout, and its decoder matches the PyTorch decoder to double-precision round-off.The reference raw reshape misordered components for every degree ℓ≥1 by omitting blockwise transposition between magnetic-index-first and channel-first layouts.Source: J MODEL IMPLEMENTATION DETAILS · p. 40

K TRAINING COSTS

Completed 24-epoch QMugs learning-rate sweep runs took 35–39 hours on one NVIDIA H200 GPU, or about 1.5 hours per epoch.

K TRAINING COSTS

  • 35–39 h: completed 24-epoch QMugs learning-rate sweep runs took this long on one NVIDIA H200 GPU, or about 1.5 h per epoch.Figure 10 shows validation-loss trajectories for Hamiltonian and density models during learning-rate selection.Source: Figure 10 · p. 41

L TRAINING AND OPTIMIZATION

This appendix documents training configurations and validation/probe-loss evaluations across density and Hamiltonian models, and describes label-free checkpoint selection for Transition1x self-consistency training.

Training hyperparameters

  • All reported runs share the Muon optimizer configuration, while model-initialization and training-PRNG seeds are drawn from OS entropy and logged per run.Family-specific optimizer deviations are given in the notes.Source: Table 12 · p. 42

ROCKET restraint hyperparameters

  • ROCKET’s restraint weight is αROCKET = α/9, with α fitted on the full training split so a three-standard-deviation gauge offset costs as much as the mean shift-minimized loss.The restraint is specified as a three-sigma gauge-offset penalty.Source: Table 13 · p. 42

Density-model training and ablations

  • Density-model tables evaluate NON, Löwdin-frame ON-Loss, and GROOT variants on QM9 validation and fixed QM40 probes, with prediction-frame and auxiliary-supervision ablations also reported.The density targets are Δ-MINAO; the tables describe validation/probe objectives and do not provide cell values in the supplied passages.Source: Table 14 · p. 43, Table 15 · p. 43, Table 16 · p. 44

Hamiltonian-model training and ablations

  • Hamiltonian-model tables cover NON, WALoss, ON-Loss, and ROCKET training on Δ-SAP, evaluating validation and fixed QM40 probes; the AO-frame ablation lacks a QM40 probe.The NON table notes that its covariant loss is gauge-dominated and nearly flat, so it is included for completeness rather than ranking.Source: Table 17 · p. 44, Table 18 · p. 45, Table 19 · p. 45

Self-consistency learning-rate selection

  • Transition1x self-consistency checkpoints are selected without reference labels by choosing the learning rate with the lowest validation residual at each scored epoch, then evaluated on the test split.Table 22 reports mean training-objective residuals on the validation split; its ROCKET third-stage values omit the gauge-restraint term, and residual blocks are not comparable across models.Source: L TRAINING AND OPTIMIZATION · p. 48, Table 22 · p. 48

M SELF-CONSISTENCY TRAINING

Self-consistency training fine-tunes pretrained GSMs on molecular geometries without converged electronic reference labels, using one Fock build per step. The section compares matrix-residual and occupied–virtual-gradient objectives on Transition1x.

M SELF-CONSISTENCY TRAINING

  • SCT fine-tunes pretrained GSMs on molecular geometries by penalizing self-consistency deviations rather than errors against converged electronic reference labels.Each training step performs a single Fock build; the two objectives are compared on Transition1x.Source: M SELF-CONSISTENCY TRAINING · p. 49
  • The Hamiltonian and density-matrix residual objective compares predictions with values reconstructed from their eigenvectors, extending Zhang et al. (2024) to density GSMs.Residuals are evaluated in the non-orthogonal basis and backpropagated through the Fock build; equal matrix-entry weighting inherits overlap-dependent spectral distortion, asymmetric target scaling, and possible eigensolver instability near orbital degeneracies.Source: M SELF-CONSISTENCY TRAINING · p. 49
  • The occupied–virtual-gradient objective supervises GSMs using the electronic-energy gradient from SAIL, with its multi-step trajectory reduced to one step under a one-Fock-build budget.This avoids directly evaluating matrix differences.Source: M SELF-CONSISTENCY TRAINING · p. 49

O eH(ˆP) eCV . (74) · N LOSS VS OBSERVABLES · WALoss

The OV-gradient loss gives density and Hamiltonian models a shared, self-consistency-aligned objective, while Löwdin-basis losses better track observable errors on QM40. LOV also reduces Transition1x rejection for idempotent predictions.

O eH(ˆP) eCV . (74)

  • For density models, occupied and virtual orbitals come from the predicted Löwdin density; for Hamiltonian models, they come from the predicted Löwdin Hamiltonian.The Hamiltonian prediction defines its density as P(Ĥ).Source: O eH(ˆP) eCV . (74) · p. 49
  • The occupied–virtual gradient measures energy change under occupied–virtual rotations, vanishes at self-consistency, and is invariant to rotations within either orbital subspace.Its energy units provide a unified physical objective for density and Hamiltonian GSMs without an ad hoc matrix norm.Source: O eH(ˆP) eCV . (74) · p. 49
  • For idempotent densities, the commutator’s diagonal blocks vanish, so its Frobenius norm is determined by the occupied–virtual gradient and gives the rejection rule’s DIIS residual.Thus the OV gradient directly minimizes the commutator rejection statistic used at inference.Source: O eH(ˆP) eCV . (74) · p. 49
  • For GROOTrot, LOV lowers Transition1x rejection from 13.5% under the matrix residual to 2.6%; for GROOTproj, it falls from 1.2% to 0.2%.For unconstrained Pbase, fractional eigenvalues leave commutator contributions unpenalized and rejection remains 25.0%.Source: O eH(ˆP) eCV . (74) · p. 49

N LOSS VS OBSERVABLES

  • The QM40 production-run probe compares each model family’s own training loss only within its block and evaluates downstream observable errors against matrix-level losses.The probe reports one-Fock-build electronic-energy MAE and catastrophe counts alongside the loss.Source: Table 23 · p. 50, WALoss · p. 51

WALoss

  • On QM40, Löwdin loss predicts observable errors better than atomic-orbital loss in 22 of 24 correlation pairs and all 24 mutual-information pairs.The force rows use 200 molecules per seed, versus 1,000 molecules per seed for the evaluation overall.Source: Table 24 · p. 51, WALoss · p. 51

O MODEL EVALUATION DATA · P RUNTIME

The evaluation tracks observable errors across in-distribution and size-extrapolation tests, resolving performance by electron count on QM40 and QMugs. On QMugs, GROOTrot remains accurate across the full range, while ML-started SCF converges faster than the MINAO baseline.

O MODEL EVALUATION DATA

  • The evaluation uses QM9 test errors as an in-distribution baseline, then resolves density and Hamiltonian errors by electron count on QM40 and larger QMugs molecules.QMugs reaches about 650 electrons versus about 270 on QM40; the QM9-trained model figures use the same evaluation protocol, with QM40 shown before QMugs.Source: O MODEL EVALUATION DATA · p. 54, Figure 16 · p. 55
  • On QMugs up to about 650 electrons, GROOTrot alone keeps error panels flat, leads every panel by up to three orders of magnitude on energies, and stays below the reference guess in ERIC.The Löwdin ∆-readout also stays below the reference guess in ERIC; GROOTproj rises above it from about 320 electrons and approaches the cap on the largest molecules.Source: Figure 16 · p. 55, O MODEL EVALUATION DATA · p. 55
  • On QM40, Löwdin-basis ROCKET Hamiltonian models separate from AO-basis models by two to three orders of magnitude on every energy panel and stay below the reference guess in ERIC throughout.The comparison varies representation, loss, gauge treatment, and an occupied-plus-LUMO masked final variant.Source: Figure 17 · p. 56
  • On QMugs, the gauge-EMA occupied-plus-LUMO masked ROCKET variant leads every Hamiltonian error panel, but all Hamiltonian models exceed the reference guess in ERIC beyond about 500 electrons.The AO-basis model has errors one to two orders of magnitude above the Löwdin-basis ROCKET models and remains at the ERIC cap.Source: Figure 18 · p. 56
  • QMugs-trained checkpoints are also evaluated on an OOD QMugs test set by electron-count bins, with density-readout variants in Figure 19 and Hamiltonian-readout variants in Figure 20.The protocol spans 160–660 electrons and reports dataset-level means in Table 28.Source: Figure 19 · p. 59, Figure 20 · p. 59, Table 28 · p. 60
  • The evaluation includes separate conformational-transfer scans for 3BPA and a Transition1x test of QMugs-trained parent checkpoints before and after self-consistency training.The 3BPA scans isolate conformational generalization across three fixed-dihedral grids; Transition1x geometries were not encountered during parent-checkpoint pre-training.Source: Table 29 · p. 61, Table 30 · p. 62

P RUNTIME

  • On out-of-distribution QMugs, median time to SCF convergence is 0.70 of the MINAO baseline for both GROOT models and 0.72 for ROCKET; all runs converge to within 3 × 10^-11 Eh of baseline energy.The timed workflow includes the initial guess, model input, forward pass, and SCF, excluding only per-molecule system setup.Source: Figure 21 · p. 63

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