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Learning Equivariant Non-Local Electron Density Functionals
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Proposes EG-XC, a learnable non-local exchange-correlation functional that compresses the density into an SO(3)-equivariant nuclei-centered point cloud, processes it with an equivariant GNN, and is trained end-to-end by differentiating through an SCF solver using only energy targets.
Problem
Accurately approximating non-local exchange-correlation contributions in density functional theory, which existing ML functionals handle with poor scalability or require costly reference densities, while ML force fields extrapolate poorly.
Approach
Convolve the electron density with equivariant radial-spherical-harmonic filters at nuclear positions to get an atomic-range point cloud, apply equivariant message passing (NequIP-style) for molecular-range interactions, and use the resulting non-local feature density to reweight a semi-local meta-GGA energy density plus a graph readout; train by backpropagating through the SCF iterations using en
Key results
- On 3BPA, EG-XC achieves relative MAE of 0.42 (300K), 0.73 (600K), 1.39 (1200K) mHa, reducing error by 35% to 51% versus the next-best method.
- On MD17 (CCSD(T) reference), EG-XC attains the lowest error on 3 of 5 molecules and reduces errors by a factor of 2 to 4 relative to its semi-local base functional.
- On QM9 size extrapolation, EG-XC trained on QM9(6) beats the best alternative trained on QM9(7) with 5x more samples on the largest molecules.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add EG-XC in place of the strongest listed baseline | KS-DFT energy prediction · MD17 test molecules | MAE: 0.69 vs 1.04; 0.10 vs 0.02; 0.21 vs 0.25; 0.27 vs 0.29; 0.20 vs 0.13 | Mixed |
| use EG-XC instead of the best listed baseline | 3BPA energy prediction · 300K, 600K, 1200K and the three potential-energy slices | relative MAE: 0.42 vs 0.81; 0.73 vs 1.36; 1.39 vs 2.27; 0.35 vs 0.75; 0.23 vs 0.61; 0.20 vs 0.56 | Helped |
| add the mGGA component (ablation) | EG-XC · 3BPA 300K | relative MAE: 7.00 -> 0.42 | Helped |
| add the mGGA component (ablation) | EG-XC · 3BPA 600K | relative MAE: 12.89 -> 0.73 | Helped |
| add the mGGA component (ablation) | EG-XC · 3BPA 1200K | relative MAE: 25.85 -> 1.39 | Helped |
| add the graph readout (ablation) | EG-XC · 3BPA 300K | relative MAE: 0.97 -> 0.42 | Helped |
| add the graph readout (ablation) | EG-XC · 3BPA 600K | relative MAE: 1.38 -> 0.73 | Helped |
| add the graph readout (ablation) | EG-XC · 3BPA 1200K | relative MAE: 2.30 -> 1.39 | Helped |
| add equivariant message passing instead of a node-wise equivariant MLP (ablation) | EG-XC · 3BPA 300K | relative MAE: 0.60 -> 0.42 | Helped |
| add equivariant message passing instead of a node-wise equivariant MLP (ablation) | EG-XC · 3BPA 600K | relative MAE: 0.87 -> 0.73 | Helped |
| add equivariant message passing instead of a node-wise equivariant MLP (ablation) | EG-XC · 3BPA 1200K | relative MAE: 1.59 -> 1.39 | Helped |
Surprising findings
- Force fields trained on energies only performed poorly on MD17 while KS-DFT-based methods reached chemical accuracy.
- The non-local GNN component contributes less than the mGGA and graph readout in ablations; removing the GNN still gave errors well below the full-mGGA baseline.
Limitations
- Relies on nuclear positions to represent the density, so it is not truly universal with respect to the external potential.
- Non-local form allows no known way to enforce most physical constraints; the paper notes missing constraints may allow basis-set error correction.
- Cannot model systems without nuclei (e.g., homogeneous electron gas) and open-shell systems need spin extension; KS-DFT is more expensive than force fields.
Experimental setup
Datasets: MD17 3BPA QM9
Benchmarks: MD17 3BPA QM9
Metrics: MAE relative MAE
Baselines: SchNet PaiNN NequIP Δ-ML (SchNet, PaiNN, NequIP) Dick et al. learnable mGGA functional Nagai et al. learnable mGGA functional
Builds on: NequIP meta-GGA functionals differentiable SCF (Dick et al.) SchNet
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