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ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals
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Introduces ELECTRA, an equivariant Cartesian tensor network that predicts floating Gaussian orbital positions, weights and covariances to reconstruct molecular charge densities, with a moment-of-inertia symmetry-breaking input and debiasing layers.
Problem
Charge density prediction for molecules is limited by atom-centered basis sets, which need high angular momentum or augmented functions; good floating orbital placement normally requires expert knowledge, and existing density models are slow or less accurate.
Approach
Predict floating Gaussian orbitals (positions, signed weights, covariances) with an O(3)-equivariant HotPP-based network, breaking input symmetry via moment-of-inertia eigenvector initialization of l=1 features plus learned debiasing layers so orbitals can leave symmetry planes while keeping density rotation equivariance.
Key results
- ELECTRA achieves 0.176% NMAE on QM9 test versus 0.432% for SCDP (L=3) and 0.196% for ChargE3Net, with 0.089 s inference per molecule.
- ELECTRA sets new state of the art on all six MD molecules by roughly halving NMAE compared to SCDP.
- Initializing VASP SCF with ELECTRA-predicted densities reduces SCF iterations by 50.72% on average on unseen molecules.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ELECTRA instead of ChargE3Net | QM9 charge-density prediction · QM9 test | NMAE [%]: 0.196 vs 0.176 | Helped |
| Use ELECTRA instead of SCDP (L=3) | QM9 charge-density prediction · QM9 test | NMAE [%]: 0.432 -> 0.176 | Helped |
| Use ELECTRA instead of SCDP+BO (L=6) | QM9 charge-density prediction · QM9 test | NMAE [%]: 0.178 -> 0.176 | Helped |
| Initialize DFT with ELECTRA-predicted density rather than atomic superposition | VASP DFT calculations on unseen molecules · QM9 unseen molecules | SCF iteration reduction: 50.72% | Helped |
| Add negative Gaussian weights (ablation) | ELECTRA · QM9 test | NMAE [%]: 2.027 -> 0.176 | Helped |
| Add predicted scaling factors to displacements and covariance combination (ablation) | ELECTRA · QM9 test | NMAE [%]: 0.356 -> 0.176 | Helped |
| Add debiasing layers (ablation) | ELECTRA · QM9 test | NMAE [%]: 0.584 -> 0.176 | Helped |
| Add debiasing layers and symmetry-breaking (ablation) | ELECTRA · QM9 test | NMAE [%]: 0.705 -> 0.176 | Helped |
| Add floating orbitals (ablation) | ELECTRA · QM9 test | NMAE [%]: 6.069 -> 0.176 | Helped |
Surprising findings
- Removing the ReLU non-negativity clamp produced no measurable difference in training, per the authors.
- Removing floating orbitals raises NMAE from 0.176% to 6.069%, showing the placement matters far more than other components.
Limitations
- Errors are markedly higher on MD dataset than QM9, attributed partly to model error.
- Moment-of-inertia canonicalization may flip signs and degenerate for linear molecules.
- Only evaluated on QM9 and six MD molecules; periodic systems would need specialized replication of floating orbitals.
Experimental setup
Datasets: QM9 charge density dataset MD charge density dataset (six molecules)
Benchmarks: QM9 charge density MD charge density VASP SCF iteration count
Metrics: NMAE [%] inference time per molecule SCF iterations reduction
Baselines: i-DeepDFT e-DeepDFT GPWNO InfGCN ChargE3Net SCDP SCDP+BO
Builds on: HotPP Gaussian Splatting SpookyNet
Closest papers by meaning
- A Recipe for Charge Density Prediction (NeurIPS 2024 · similarity 0.90)
- Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passing (NeurIPS 2024 · similarity 0.87)
- Learning the Electronic Hamiltonian of Large Atomic Structures (ICML 2025 · similarity 0.87)
- Enhancing the Scalability and Applicability of Kohn-Sham Hamiltonians for Molecular Systems (ICLR 2025 · similarity 0.87)
- EDBench: Large-Scale Electron Density Data for Molecular Modeling (NeurIPS 2025 · similarity 0.86)
- High-order Equivariant Flow Matching for Density Functional Theory Hamiltonian Prediction (NeurIPS 2025 · similarity 0.86)
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