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A Recipe for Charge Density Prediction
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Proposes SCDP, a charge density predictor that places Gaussian-type orbitals on atoms plus bond-midpoint virtual nodes, uses even-tempered learnable-exponent bases, and an eSCN backbone, achieving lower QM9 NMAE and ~30x faster inference than ChargE3Net.
Problem
Machine learning charge density prediction for DFT faces an accuracy-efficiency dilemma: orbital-based methods are fast but inaccurate, while probe-based methods are accurate but do not scale to millions of grid points.
Approach
Represent the charge density with atom-centered and SE(3)-equivariantly placed virtual Gaussian-type orbitals (bond midpoints), use even-tempered expanded bases with learnable exponent scaling factors, and predict coefficients with a high-capacity eSCN backbone, avoiding probe-node message passing.
Key results
- Best SCDP model (eSCN + virtual orbitals, K=8, L=6, beta=1.3, scaling fine-tuned) reaches NMAE 0.178% on the QM9 test set, beating ChargE3Net (0.196%).
- The best model is about 31.7x faster than ChargE3Net in molecules per minute (ablation text: up to 171x for the more efficient configurations versus state-of-the-art).
- Adding virtual orbitals reduces NMAE from 0.434% to 0.237% for beta=2.0 in the ablation.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| increase eSCN model size from K=4, L=3 to K=8, L=6 (ablation) | SCDP without virtual orbitals, β=2.0, fixed scaling factors · QM9 test | NMAE [%]: 0.504% to 0.434% | Helped |
| reduce β from 2.0 to 1.5 to enlarge the basis set (ablation) | SCDP eSCN, K=8, L=6, without virtual orbitals · QM9 test | NMAE [%]: 0.434% to 0.381% | Helped |
| add virtual orbitals (ablation) | SCDP eSCN, K=8, L=6, β=2.0, fixed scaling factors · QM9 test | NMAE [%]: 0.434% to 0.237% | Helped |
| add virtual orbitals (ablation) | SCDP eSCN, K=8, L=6, β=1.5, fixed scaling factors · QM9 test | NMAE [%]: 0.381% to 0.196% | Helped |
| reduce β from 1.5 to 1.3 to enlarge the basis set (ablation) | SCDP eSCN + VO, K=8, L=6, fixed scaling factors · QM9 test | NMAE [%]: 0.206 ± 0.001 to 0.196 ± 0.001 | Helped |
| enable learnable orbital exponent scaling with fine-tuning (ablation) | SCDP configurations · QM9 test | NMAE [%]: 0.504 ± 0.001 to 0.432 ± 0.001; 0.434 ± 0.006 to 0.369 ± 0.007; 0.381 ± 0.001 to 0.346 ± 0.001; 0.237 ± 0.001 to 0.207 ± | Helped |
| increase model capacity (ablation) | SCDP · QM9 test | Mol. per min.: 675.47 to 567.19 | Hurt |
| add virtual orbitals (ablation) | SCDP eSCN, K=8, L=6, β=2.0, fixed scaling factors · QM9 test | Mol. per min.: 567.19 to 231.21 | Hurt |
| fine-tune learnable orbital exponent scaling (ablation) | SCDP eSCN + VO, K=8, L=6, β=1.3 · QM9 test | Mol. per min.: 136.92 to 125.29 | Hurt |
| use the proposed SCDP configuration instead of ChargE3Net | QM9 charge density prediction · QM9 test | NMAE [%]: 0.196 ± 0.001 to 0.178 ± 0.001 | Helped |
| use the proposed SCDP configuration instead of ChargE3Net | QM9 charge density prediction · QM9 test | Mol. per min.: 31.7× | Helped |
Limitations
- Training is expensive: about six days pretraining and six days fine-tuning on four A100 GPUs.
- Scaling-factor training is unstable and needs a small learning rate fine-tuning stage.
- Effectiveness on crystalline materials is not yet validated; virtual node placement is a heuristic bond-midpoint rule.
Experimental setup
Datasets: QM9 charge density dataset
Benchmarks: QM9 charge density dataset
Metrics: NMAE molecules per minute
Baselines: i-DeepDFT e-DeepDFT ChargE3Net InfGCN GPWNO
Builds on: eSCN even-tempered Gaussian basis PaiNN ChargE3Net InfGCN
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