EAC-Net: Equivariant Atomic Contribution Network

• EAC-Net is a geometric deep-learning framework that predicts real-space electronic charge density by summing learned, symmetry-preserving atom-centered contributions.
• It employs equivariant atomic descriptors and message passing to efficiently capture local chemical environments across diverse materials.
• The approach enables accurate downstream electronic-structure calculations with normalized density errors typically below 1%, even in disordered systems.

Searching arXiv for the primary paper and closely related equivariant atomistic modeling context. EAC-Net, short for Equivariant Atomic Contribution Network, is a geometric deep-learning framework for predicting the full real-space electronic charge density $\rho(\mathbf r)$ of atomistic systems without solving the Kohn–Sham equations self-consistently for each new structure (Xuejian et al., 6 Aug 2025). Its defining premise is to represent the total density as a sum of learned atom-centered contributions,

$\rho(\mathbf r)=\sum_{i=1}^{N_a}\rho_i(\mathbf r),$

while treating these $\rho_i(\mathbf r)$ not as a unique quantum-mechanical partitioning of electrons onto atoms, but as a learned, structure-aware decomposition. The model is built to preserve the geometric symmetries of three-dimensional atomic systems, to retain directional chemical information through equivariant intermediate representations, and to bridge local atomic environments with a global continuous density field. The reported scope includes amorphous solids, molecular liquids, surfaces, metallic alloys, magnetic systems, and a Materials Project CHGCAR-derived large-scale variant denoted EAC-mp, with normalized density errors typically below $1\%$ and demonstrated utility in downstream non-self-consistent electronic-structure calculations (Xuejian et al., 6 Aug 2025).

1. Conceptual basis and representational viewpoint

The central motivation for EAC-Net is that charge density is the core quantity in density functional theory, but conventional access to it requires an expensive self-consistent-field calculation. EAC-Net addresses this bottleneck by learning a direct structure-to-density map while retaining the symmetry constraints of atomistic geometry. Its atom-centered decomposition is an inductive bias rather than a claim about a formally unique electronic population analysis. The learned contributions are allowed to be anisotropic and orientation-dependent, so that under a rigid rotation or reflection of the structure, the intermediate atomic descriptors and partial densities transform equivariantly, even though the total density at a fixed physical point remains invariant (Xuejian et al., 6 Aug 2025).

This emphasis on learned local contributions placed inside a symmetry-adapted architecture situates EAC-Net within a broader line of equivariant atomistic modeling. In related theoretical treatments, extensive observables are often written as sums of local atomic terms, and message passing can be understood as a multi-centered extension of atom-centered density-correlation constructions (Nigam et al., 2022). Likewise, analyses of the design space connecting ACE-like models and NequIP-like equivariant message passing make explicit that atom-wise readouts and equivariant latent states belong to a common formal framework (Batatia et al., 2022). EAC-Net specializes that general perspective to the prediction of a dense three-dimensional field rather than a scalar extensive observable.

2. Equivariant atom-interaction stage

The first stage of EAC-Net constructs equivariant atomic descriptors from local chemical environments. For each atom $i$, the atomic number $Z_i$ is encoded as an embedding $\tilde Z_i$. The paper supports one-hot encoding, shell-filling encoding, or their concatenation. In the shell-filling representation, electron shell occupancies are normalized by maximum orbital occupancies and linearly projected, with the stated purpose of injecting chemically meaningful electronic-structure information into the type embedding (Xuejian et al., 6 Aug 2025).

These atom-type embeddings initialize equivariant node features $D^l_{icm}$, indexed by atom $i$, angular momentum order $l$, magnetic index $\rho(\mathbf r)=\sum_{i=1}^{N_a}\rho_i(\mathbf r),$0, and channel $\rho(\mathbf r)=\sum_{i=1}^{N_a}\rho_i(\mathbf r),$1. The implementation uses the e3nn framework and NequIP-like message passing on an atom graph with a cutoff radius. Features are carried as direct sums of irreducible representations, so scalar and higher-order tensor content remain distinct but can interact through tensor products. The representation-theoretic core is the equivariant tensor-product rule

$\rho(\mathbf r)=\sum_{i=1}^{N_a}\rho_i(\mathbf r),$2

with Clebsch–Gordan coefficients $\rho(\mathbf r)=\sum_{i=1}^{N_a}\rho_i(\mathbf r),$3 ensuring correct transformation behavior under rotations.

Within each atom interaction block, channels are first mixed inside each irrep through a self-interaction

$\rho(\mathbf r)=\sum_{i=1}^{N_a}\rho_i(\mathbf r),$4

followed by equivariant convolution over neighbors,

$\rho(\mathbf r)=\sum_{i=1}^{N_a}\rho_i(\mathbf r),$5

The kernel is factorized into radial and angular parts,

$\rho(\mathbf r)=\sum_{i=1}^{N_a}\rho_i(\mathbf r),$6

where $\rho(\mathbf r)=\sum_{i=1}^{N_a}\rho_i(\mathbf r),$7 is a radial basis embedding such as Bessel or Gaussian, $\rho(\mathbf r)=\sum_{i=1}^{N_a}\rho_i(\mathbf r),$8 decays to zero at the cutoff, and $\rho(\mathbf r)=\sum_{i=1}^{N_a}\rho_i(\mathbf r),$9 are spherical harmonics. A maximum angular momentum $\rho_i(\mathbf r)$0 truncates the irreps, and residual connections with SiLU activations are applied after each layer.

After $\rho_i(\mathbf r)$1 rounds of message passing, each atom possesses an equivariant descriptor $\rho_i(\mathbf r)$2 summarizing its local environment. A computationally important consequence is that these descriptors depend only on atomic structure, not on the target grid location, so they can be cached and reused across all queried spatial points during inference (Xuejian et al., 6 Aug 2025).

3. Atom-grid coupling and density construction

The second stage, termed atom-grid coupling, is the main methodological innovation of EAC-Net. Standard grid-based models typically rebuild a descriptor at every grid point by aggregating nearby atoms, whereas basis-function methods predict coefficients in a predetermined atom-centered basis. EAC-Net instead keeps atom-wise interactions with grid points explicit through atom-grid edge features $\rho_i(\mathbf r)$3, where $\rho_i(\mathbf r)$4 indexes a real-space grid point $\rho_i(\mathbf r)$5. This retains directional detail, naturally yields atomic contributions, and avoids redundant recomputation (Xuejian et al., 6 Aug 2025).

The edge update mirrors the atom-convolution structure:

$\rho_i(\mathbf r)$6

with $\rho_i(\mathbf r)$7 the vector from atom $\rho_i(\mathbf r)$8 to grid point $\rho_i(\mathbf r)$9. In effect, each atom’s equivariant representation is projected along the atom-to-grid direction using the same kind of tensor-product kernel used in atomistic message passing.

The final scalar density contribution is decoded from the $1\%$0 edge channels. Scalar features are first compressed through an MLP and may be concatenated with residual geometric or chemical features:

$1\%$1

The atomic contribution is then predicted by

$1\%$2

where $1\%$3 is a value net and $1\%$4 is a weight net. The paper characterizes this as acting like a localized attention mechanism over atoms for each point. The total prediction is assembled as

$1\%$5

Several exact forms of the weight net are studied in ablations: Identity, Smooth-only, MLP+S, MLP$1\%$6S, and Softmax$1\%$7S. In the reported main settings, the architecture used for the weight net is MLP+S. The explicit per-atom/per-grid representation is therefore central both to computational efficiency and to the model’s atom-decomposed output structure (Xuejian et al., 6 Aug 2025).

4. Supervision, datasets, and model variants

The principal supervised target is the DFT charge density sampled on a real-space grid. The main reported evaluation metric is the normalized mean absolute error,

$1\%$8

The paper does not provide an explicit separate conservation loss, integral electron-number constraint, or normalization penalty. Electron-number conservation is therefore not imposed as a hard constraint in the method section, but is expected to emerge from accurate density fitting (Xuejian et al., 6 Aug 2025).

The reported datasets span amorphous silicon, aluminum surfaces, Al–Mg alloys, liquid water, crystalline Si, spin-polarized Fe, cubic GaAs under perturbation, and a Materials Project CHGCAR corpus used to train EAC-mp. Labels are DFT charge densities computed with VASP, mainly with PBE and PAW pseudopotentials. For water, the setup used SCAN with a 680 eV plane-wave cutoff and $1\%$9 k-point spacing. For crystalline Si with 64 atoms per cell, the paper used a manually set $i$0 FFT grid and a $i$1 $i$2-centered mesh. For spin-polarized Fe with 54 atoms per cell, it used a 520 eV cutoff, a $i$3 Monkhorst–Pack grid, Gaussian smearing of 0.1 eV, and initialized magnetic moments of $i$4 per atom.

Because charge density is a dense three-dimensional field with strong spatial redundancy, EAC-Net uses grid-point sampling in most large-scale settings. Four schemes are reported:

• Random sampling: within a finite atom-centered radius.
• Density-weighted sampling: proportional to $i$5.
• Core-focused sampling: near nuclei.
• Gradient-weighted sampling: proportional to $i$6.

For the small-system convergence tests on amorphous Si, Al surface, Al–Mg alloy, and water, full charge-density files were used. For the universal predictor, 5,000 grid points were sampled per charge-density file. The general training hyperparameters for performance evaluations were Adam, initial learning rate $i$7, multi-stage exponential decay often down to $i$8, 150,000 training steps, and batch size 50; the atom cutoff radius was $i$9 \AA\ and the atom-grid cutoff radius was $Z_i$0 \AA. For Al–Mg, the paper reports 250,000 steps and end learning rate $Z_i$1; for Fe spin density, 200,000 steps with final learning rate $Z_i$2.

The ablation-study common architecture used $Z_i$3, feature length 36, four atom convolution layers, two edge update layers, and about 1 million parameters. The large-scale variant EAC-mp retained the same basic architecture but increased scope: it used $Z_i$4, feature dimensionality 36, four atom interaction layers, two edge update layers, MLP+S weighting, and about 3.08 million trainable parameters. Training used 48,183 configurations, each with 5,000 sampled points, for 600,000 steps on $Z_i$5 NVIDIA RTX A6000 GPUs with 48 GB each (Xuejian et al., 6 Aug 2025).

5. Empirical behavior, scaling, and downstream physics

Across the main benchmark systems—amorphous Si, Al surface, Al–Mg alloy, and water—EAC-Net is reported to converge with very few training frames, typically 8–12 frames, and with even a single DFT frame for the aluminum surface. Al–Mg is described as the easiest among these systems, with $Z_i$6, whereas amorphous silicon is the hardest, with error approaching $Z_i$7, attributed to structural disorder and strongly anisotropic covalent bonding. The paper states that training difficulty is more sensitive to structural disorder than to compositional disorder (Xuejian et al., 6 Aug 2025).

Accuracy improves as the angular momentum cutoff $Z_i$8 increases, which the paper interprets as evidence that higher-order angular channels are important for non-spherical density features. This is especially pertinent for covalent, metallic, and spin-polarized systems, although larger $Z_i$9 increases computational cost. In efficiency comparisons on crystalline Si, EAC-Net requires roughly $\tilde Z_i$0–$\tilde Z_i$1 less training time than ChargE3Net for comparable parameter counts, and its inference time grows more slowly with model size. The model is also described as less sensitive to increasing angular resolution: moving from $\tilde Z_i$2 to $\tilde Z_i$3 increases training and inference times by less than $\tilde Z_i$4, whereas ChargE3Net incurs about a twofold increase. The stated explanation is the caching of atom descriptors, which amortizes the expensive atom-graph computation over all queried grid points.

The paper also emphasizes downstream utility. Using EAC-predicted densities as input to non-self-consistent DFT calculations on perturbed Si structures, the model achieved mean absolute errors of $\tilde Z_i$5 meV/atom in total energy and $\tilde Z_i$6 meV/\AA\ in forces. Band structures computed non-self-consistently from predicted densities agree closely with DFT for perturbed Si and GaAs. For Si, perturbations up to $\tilde Z_i$7 \AA\ on the central atom still produced excellent band structures. For GaAs, the training data included random displacements only up to $\tilde Z_i$8 \AA, but predictions at $\tilde Z_i$9 and $D^l_{icm}$0 \AA\ remained excellent, with only slight discrepancies at $D^l_{icm}$1 \AA. These demonstrations are presented as evidence of out-of-distribution and zero-shot-like generalization under structural perturbations.

For spin-polarized Fe, EAC-Net predicts total charge density with $D^l_{icm}$2, while the spin density is more difficult, with $D^l_{icm}$3. The paper attributes this difference to sharper features and higher-order angular structure in the spin density; around Fe, the total density is nearly spherical, but the spin density exhibits octahedral symmetry. The model nonetheless qualitatively reproduces this anisotropic pattern.

The large-scale EAC-mp results extend this picture. Evaluated on over 20,000 frames outside the training distribution, EAC-mp achieves $D^l_{icm}$4 for the vast majority of elements, with most elements clustering around $D^l_{icm}$5 average error. Larger errors are reported mainly for radioactive, sparsely represented, or chemically exotic elements such as Np, U, and Pa. The paper further reports zero-shot generalization to high-entropy alloys, including realistic charge-density predictions for relaxed HEAs and successful non-self-consistent total energies for HEAs under $D^l_{icm}$6 \AA\ atomic displacements (Xuejian et al., 6 Aug 2025).

6. Interpretability, limitations, and significance

A distinctive property of EAC-Net is that it outputs explicit atom-resolved partial densities $D^l_{icm}$7 as a byproduct of the architecture. The paper shows examples for H$D^l_{icm}$8O and graphite. These partial densities are physically motivated in being atom-centered, anisotropic, and environment-dependent, but they are not claimed to correspond to a unique observable partitioning, nor are they tied to predefined population analyses such as Mulliken or Bader. They are also not basis-dependent in the same way as basis-expansion methods, because EAC-Net does not introduce a fixed atom-centered orbital basis for the density representation (Xuejian et al., 6 Aug 2025).

The same design choices define the model’s main limitations. The learned decomposition should not be overinterpreted as a rigorous electronic partition. Accuracy degrades in highly disordered covalent systems such as amorphous Si. Long-range effects are only implicitly captured through local cutoffs of 4 \AA\ for atom interactions and 6 \AA\ for atom-grid coupling, so very long-ranged electronic response may require larger cutoffs or additional mechanisms. Grid resolution still matters because the target is a dense real-space field, and the Fe example shows that sharper spin-density channels are harder to reproduce than total density. Although the method performs well on Al surfaces and Al–Mg alloys, the paper notes that strongly delocalized metallic response outside the training regime should still be expected to remain challenging. Universal coverage also continues to require DFT-generated charge-density labels and substantial training.

In broader methodological terms, EAC-Net occupies a specific point in the development of equivariant atomistic ML. It combines atom-centered equivariant descriptors and message passing, familiar from frameworks such as NequIP-like models and the unified ACE/NequIP design-space perspective (Batatia et al., 2022), with an explicit real-space atom-grid interaction mechanism. It also matches the more general view that message passing and atom-centered density correlations can be placed in a single symmetry-adapted representational framework (Nigam et al., 2022). Its significance lies in transferring that machinery from energies and forces to the direct prediction of the full charge density, while preserving an explicit atomic decomposition and supporting downstream electronic-structure workflows. On the evidence reported in the paper, EAC-Net therefore functions as a scalable architecture for learning charge density itself as an intermediate quantity in materials modeling, rather than restricting machine learning to final scalar observables alone (Xuejian et al., 6 Aug 2025).