Complete Neural Electronic Initialization Accelerates Materials DFT

Felix Ærtebjerg; Jonas Elsborg; Arghya Bhowmik

cond-mat.mtrl-sciSep 18, 2026

Complete Neural Electronic Initialization Accelerates Materials DFT

Authors: Felix Ærtebjerg, Jonas Elsborg, Arghya Bhowmik

Organizations: Department of Energy Conversion and Storage, Technical University of Denmark

Abstract

We present the first complete machine learning method for accelerating plane-wave density functional theory (DFT) in materials under the projector augmented wave (PAW) formalism. We formalize seven criteria that a Complete Neural Electronic Initializer must satisfy for practical end-to-end PAW DFT acceleration. Applying these to prior work reveals two structure-dependent components, augmentation occupancies and spin initialization, whose absence prevents existing acceleration methods from providing complete reference-free initialization. We show that omitting these components can eliminate or reverse the acceleration obtained via models that only predict the smooth valence density. We satisfy the missing requirements by introducing AugNet, a general equivariant model for PAW augmentation occupancies, and the first general spin density model for materials, which predicts the smooth spin-difference density and spin-difference PAW augmentation occupancies using predicted magnetic moments to constrain the global magnetic state. Combined with existing valence density models, our full method satisfies all seven criteria and forms a fully reference-free electronic initializer for materials DFT, requiring no electronic quantities from a converged target calculation. We show that perfect initialization could cut PAW DFT wall time by 40-52%, and our method recovers up to 62% of this saving, reducing end-to-end DFT wall time by up to ~25% on unseen structures while preserving converged energies.

Figures & tables

Figure 1 : Workflow for end-to-end PAW DFT acceleration, illustrated using the models employed in this work. For a given atomic structure and PAW setup, the fixed PAW datasets define the frozen core and basis information, while a Complete Neural Electronic Initializer provides the three structure-dependent components defined in Table 1 : the smooth valence density $\tilde{\rho}^{+}$ , PAW augmentation occupancies $\mathbf{d}_{a}^{+}$ , and spin/magnetic initialization. ELECTRAFI predicts $\tilde{\rho}^{+}$ , AugNet predicts $\mathbf{d}_{a}^{+}$ , and the corresponding spin-dependent components $\tilde{\rho}^{-}$ and $\mathbf{d}_{a}^{-}$ are predicted by spin-ELECTRAFI and spin-AugNet, with magnetic moment predictions constraining the global magnetic state. Together, these components enable complete reference-free initialization of PAW DFT.

	Reference-free initialization components			Demonstrated evaluation
Method	Valence density C1	PAW augmentation C2	Spin / magnetic initialization C3	Density accuracy C4	SCF acceleration C5	Component ablations C6	Reference-free acceleration C7
GPWNO ( Kim and Ahn, 2024 )	$\checkmark$	$\times$	$\times$	$\checkmark$	$\times$	$\times$	$\times$
InfGCN ( Cheng and Peng, 2024 )	$\checkmark$	$\times$	$\times$	$\checkmark$	$\times$	$\times$	$\times$
SCDP ( Fu et al., 2024 )	$\checkmark$	$\times$	$\times$	$\checkmark$	$\times$	$\times$	$\times$
BOA ( Klockow et al., 2026 )	$\checkmark$	$\times$	$\times$	$\checkmark$	$\times$	$\times$	$\times$
EdenGNN ( Li et al., 2025 )	$\checkmark$	$\checkmark$	$\times$	$\checkmark$	$\times$	$\times$	$\times$

Table 1 : Criteria for complete neural electronic initialization for accelerating materials DFT. C1-C3 are structure-dependent quantities. C4-C7 refer to the required evaluation for demonstrating complete initialization.

\checkmark

denotes a capability in the cited work,

\times

denotes that it was not demonstrated.

\dagger

denotes system-specific models: CJM ( Focassio et al., 2024 ) is specific to MoS 2 , while de Blasio et al. (2023) and EAC-Net ( Qin et al., 2026 ) predict spin densities for Na 3 V 2 (PO 4 ) 3 and Fe, respectively, without demonstrating SCF acceleration.

	Spin channel	Aug. only	Valence only	Default	Valence + aug.	mm only	Smooth grids	Val. + aug. + mm	Oracle
Valence density $\tilde{\rho}^{+}$	$\times$	$\times$	$\checkmark$	$\times$	$\checkmark$	$\times$	$\checkmark$	$\checkmark$	$\checkmark$
Spin density $\tilde{\rho}^{-}$	$\checkmark$	$\times$	$\times$	$\times$	$\times$	mm $\checkmark$	$\checkmark$	mm $\checkmark$	$\checkmark$
Valence aug. $\mathbf{d}^{+}$	$\times$	$\checkmark$	$\times$	$\times$	$\checkmark$	$\times$	$\times$	$\checkmark$	$\checkmark$
Spin aug. $\mathbf{d}^{-}$	$\checkmark$	$\checkmark$	$\times$	$\times$	$\times$	mm $\checkmark$	$\times$	mm $\checkmark$	$\checkmark$
Non-mag. [%]	$-9.7$	$-14.3$	$+0.5$	$0$	$+13.5$	$+12.0$	$+10.2$	$+47.6$	$+49.0$
Mag. [%]	$-70.7$	$-65.0$	$-29.4$	$0$	$-11.0$	$-3.1$	$-4.2$	$+24.6$	$+55.4$

Table 2: Component ablation for PAW DFT initialization. The matrix specifies the electronic components in each experiment, with SCF step savings relative to the default SAD initialization reported separately for non-magnetic and magnetic Materials Project structures.

\checkmark

denotes converged (Oracle) initialization,

\times

denotes SAD/default initialization. mm denotes spin initialization using atomic magnetic moments.

Magnetic initialization	Subset	MAE	Oracle valence + augmentation	ELECTRAFI + AugNet	ChargE3Net + AugNet
Oracle $(\tilde{\rho}^{-},\mathbf{d}^{-})$	Non-mag.	–	49.0%	23.3%	29.1%
	Magnetic	–	55.4%	29.3%	31.9%
Oracle moments	Non-mag.	0.028	47.6%	22.6%	29.0%
	Magnetic	4.658	24.7%	6.2%	9.5%
Models
CHGNet moments	Non-mag.	0.347	38.7%	19.4%	25.9%

Table 3 : Spin-difference density accuracy and SCF step reduction for magnetic initializations using Oracle and ML valence and augmentation on the Materials Project test set.

Model:	CJM	AugNet	AugNet	spin-AugNet	spin-AugNet
Data:	$\mathrm{MoS}_{2}$	MP	GNoME	MP	GNoME
MAE $\downarrow$	0.0130	0.0041	0.0062	0.0025	0.0020
RMSE $\downarrow$	0.0459	0.0119	0.0262	0.0098	0.0129

Table 4 : PAW augmentation occupancy prediction errors. CJM is evaluated on its system-specific

\mathrm{MoS}_{2}

structure. AugNet is evaluated on the MP and GNoME test data.

Dataset	Metric	Default	Oracle	CNEI-EFI	CNEI-C3Net
MP	NMAE [%] $\downarrow$	–	–	0.58	0.54
	SCF steps $\downarrow$	22.05	10.41	19.05	18.36
	SCF steps saved [%] $\uparrow$	–	52.78	13.62	16.76
	Total time [s] $\downarrow$	623.84	302.04	530.04	585.26
	Total time saved [%] $\uparrow$	–	51.58	15.04	6.18
	OAC [%] $\uparrow$	0.0	100.0	29.15	11.99

Table 5 : End-to-end performance of our Complete Neural Electronic Initializer (CNEI), using either ELECTRAFI (EFI) or ChargE3Net (C3Net) as the valence backbone. Total time includes ML initialization and DFT execution. SCF step savings are reported relative to Default. Oracle acceleration captured (OAC) is calculated as defined in Equation 10 .

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

	Initialization components				vs. Default [%]		vs. Oracle [%]
	Smooth density		PAW augmentation
Initialization	$\tilde{\rho}^{+}$	$\tilde{\rho}^{-}$	$\mathbf{d}^{+}$	$\mathbf{d}^{-}$	Non-mag.	Mag.	Non-mag.	Mag.
Baselines
Default	$\times$	$\times$	$\times$	$\times$	–	–	–	–
Oracle	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$+49.0$	$+55.4$	–	–
Electronic component ablations

Appendix

Table 6: Paired SCF-step savings (%) on the Materials Project test set, separated into non-magnetic and magnetic structures. The initialization components are the smooth spin-summed valence density

\tilde{\rho}^{+}

, smooth spin-difference density

\tilde{\rho}^{-}

, spin-summed PAW augmentation occupancies

\mathbf{d}^{+}

, and spin-difference PAW augmentation occupancies

\mathbf{d}^{-}

, where

\mathbf{d}^{\pm}

denotes the collection of atom-wise occupancies

\{\mathbf{d}_{a}^{\pm}\}_{a}

. Default denotes standard SAD initialization, while Oracle uses all converged electronic components from the corresponding completed calculation.

\checkmark

denotes a converged component and

\times

denotes SAD/default initialization. OM,

\mathrm{MP}

and

\mathrm{CG}

denote spin initialization from converged atomic magnetic moments, MPRelaxSet, and CHGNet-predicted magnetic moments, respectively. Positive values indicate faster calculations and negative values indicate slower calculations relative to the corresponding baseline.

Variant ( ISPIN=1] )	Default (SAD)	Oracle	ChargE3Net 1,∗
DFT Steps $\downarrow$	15.33 $\pm$ 8.02	9.21 $\pm$ 7.82	12.34 $\pm$ 8.69
DFT Time $\downarrow$	163.30 $\pm$ 326.32 s	114.89 s $\pm$ 276.57	141.49 $\pm$ 298.70 s
DFT Steps Saved $\uparrow$	–	39.90 %	19.51 %
DFT Time Saved $\uparrow$	–	29.64 %	13.35 %

Appendix

Table 7: 1 Koker et al. (2024) . ∗ ChargE3Net is initialized with converged augmentation occupancies. DFT convergence analysis with different initializations. The tests were performed on the non-magnetic part of the MP test set.

Dataset	Training set	RMSE $\downarrow$	MAE $\downarrow$	MaxAE $\downarrow$	MaxAE ${}_{\mathrm{top\,11}}\downarrow$
MP	1k	0.0340 $\pm$ 0.0206	0.0120 $\pm$ 0.0065	4.504	1.487
	10k	0.0183 $\pm$ 0.0113	0.0064 $\pm$ 0.0034	2.829	0.796
	50k	0.0130 $\pm$ 0.0096	0.0046 $\pm$ 0.0028	1.931	0.664
	Full	0.0118 $\pm$ 0.0091	0.0041 $\pm$ 0.0026	1.114	0.646
GNoME	1k	0.0464 $\pm$ 0.3916	0.0120 $\pm$ 0.0287	366.85	1.724
	10k	0.0302 $\pm$ 0.3915	0.0078 $\pm$ 0.0284	366.58	0.868

Appendix

Table 8 : Physical augmentation occupancy prediction errors versus training-set size on the full MP and GNoME test sets. RMSE and MAE are means over per-structure values, while MaxAE is the largest single-coefficient error. MaxAE top 11 reports the largest error after excluding the ten most extreme coefficients.

Model	Steps	Trainable params.	MAE $\downarrow$	RMSE $\downarrow$	MaxAE $\downarrow$
CJM ( Focassio et al., 2024 )	1k	1,758	0.0130	0.0459	0.9137
AugNet, zero-shot	–	–	0.3854	1.3439	14.9708
AugNet, from scratch	1k	3M	0.0010	0.0258	0.4969
AugNet, head fine-tune	1k	73k	0.1801	0.7039	8.2880
AugNet, head fine-tune	10k	73k	0.0122	0.0400	0.9490
AugNet, full fine-tune	1k	3M	0.0157	0.0352	0.8535

Appendix

Table 9: Transfer of MP-pretrained AugNet to the

\mathrm{MoS}_{2}

PAW setup of Focassio et al. (2024) . Their calculations use a different Mo PAW dataset from Materials Project, so the target augmentation representation changes and direct zero-shot transfer is not expected. “Trainable params.” denotes the number of parameters optimized during adaptation.

Dataset	Metric	Default	Oracle	CNEI-EFI	CNEI-C3Net
MP	$\tilde{\rho}^{+}$ NMAE $\downarrow$	–	–	0.58%	0.54%
	ML init time $\downarrow$	–	–	$(0.24+0.37)$ s	$(78.73+0.37)$ s
	SCF steps $\downarrow$	22.05	10.41	19.05	18.36
	DFT time $\downarrow$	623.84 s	302.04 s	529.43 s	506.16 s
	Total time $\downarrow$	623.84 s	302.04 s	530.04 s	585.26 s
	SCF steps saved $\uparrow$	–	52.78%	13.62%	16.76%

Appendix

Table 10 : Detailed comparison of reference-free ML PAW initializations and resulting DFT performance on the test sets of MP and GNoME. Total time includes both ML initialization and DFT execution. For both CNEI-EFI and CNEI-C3Net, we add the time it takes to evaluate spin-ELECTRAFI (0.24s/0.15s) and spin-AugNet (0.05s both) as well as CHGNet (0.03s both) to ELECTRAFI and ChargE3Net.

Dataset	Metric	Default	Oracle	CNEI-EFI	CNEI-C3Net
MP	$\tilde{\rho}^{+}$ NMAE $\downarrow$	–	–	0.67 %	0.78 %
	ML Init Time $\downarrow$	–	–	(0.24 + 0.37) s	(87.25 + 0.37) s
	SCF steps $\downarrow$	27.82	12.43	24.97	24.48
	DFT time $\downarrow$	882.45 s	377.16 s	777.63 s	744.13 s
	Total time $\downarrow$	882.45 s	377.16 s	778.24 s	831.75 s
	SCF steps saved $\uparrow$	–	55.30%	10.22%	11.98%

Appendix

Table 11 : The magnetic subset counterpart of table 10 . The spin-ELECTRAFI model measures (0.24s/0.15s).

Dataset	Metric	Default	Oracle	CNEI-EFI	CNEI-C3Net
MP	$\tilde{\rho}^{+}$ NMAE $\downarrow$	–	–	0.55 %	0.50 %
	ML Init Time $\downarrow$	–	–	(0.17+0.30) s	(72.11+0.30) s
	SCF steps $\downarrow$	16.83	8.58	13.68	12.80
	DFT time $\downarrow$	389.41 s	233.95 s	304.45 s	290.45 s
	Total time $\downarrow$	389.41 s	233.95 s	304.92 s	362.86 s
	SCF steps saved $\uparrow$	–	49.01%	18.71%	23.91%

Appendix

Table 12 : The non-magnetic subset counterpart of table 10 . The spin-ELECTRAFI model measures (0.17s/0.11s) on these subsets.

Site	Partial waves	$\varepsilon$
1	$(2,2,0,0,1,1)$	$4.6\times 10^{-7}$
2–5	$(0,0,1,1)$	$1.1$ – $10.1\times 10^{-7}$

Appendix

Table 13 : Parameter-free verification of the equivariant transformation law over 241 held-out rotations.

Group	Hyperparameter	Value
Backbone	Hidden width	64
	Max. spherical order $\ell_{\max}$	3
	Cutoff radius $r_{\max}$ (Å)	6.0
	Interaction layers	2
	Correlation order	3
	Avg. number of neighbors	64.3

Appendix

Table 14: Hyperparameters used for training AugNet and spin-AugNet. The spin-summed and spin-difference models share all architectural and optimization settings and differ only in the predicted channel and reference baseline.

Group	Parameter	Value
Backbone (EScAIP)	Layers	$2$
	Hidden size	$256$
	Attention heads	$32$
	Atom embedding size	$128$
	Edge distance embedding	$512$ (expansion $600$ )
	Node direction embedding	$256$ (expansion $13$ )

Appendix

Table 15: Hyperparameters for the ELECTRAFI and spin-ELECTRAFI models, mirroring the choices in Elsborg et al. (2026b) .

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Towards Accelerated SCF Workflows with Equivariant Density-Matrix Learning and Analytic Refinement

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Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, Mexico

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We present \textsc{dm-PhiSNet}, a physically constrained \textsc{PhiSNet}-based equivariant model that predicts one-electron reduced density matrices (1-RDMs) directly from molecular geometries in an atomic-orbital (AO) basis for accelerated self-consistent field (SCF) workflows. Training follows a two-stage schedule with progressively introduced physically motivated objectives, and the resulting predictions are refined by a lightweight analytic block. This block enforces electron-number conservation, drives the 1-RDM toward generalized idempotency in the AO metric, and regularizes the occupation spectrum of the Löwdin-orthogonalized density. Across six closed-shell systems -- H

_2

O, CH

_4

, NH

_3

, HF, ethanol, and NO

_3^-

-- the refined 1-RDMs provide SCF initial guesses that substantially reduce iteration steps by 49--81% relative to standard initializations. Beyond SCF acceleration, the learned 1-RDMs yield accurate one-shot total energies and Hellmann--Feynman atomic forces without force supervision, indicating that the model captures chemically meaningful electronic structure. These results demonstrate that combining equivariant learning with analytic constraint enforcement provides a simple, general route to solver-ready density-matrix initializations and accelerated SCF workflows.

Abstract

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