ECD: A Machine Learning Benchmark for Predicting Enhanced-Precision Electronic Charge Density in Crystalline Inorganic Materials
Authors: Pin Chen
Zexin Xu
Qing Mo
Hongjin Zhong
Fengyang Xu
Yutong Lu
Venue: ICLR 2025 Oral (2025)
Keywords: Electronic Charge Density
Crystalline Inorganic Materials
Graph Neural Network
Dataset
Primary area: datasets and benchmarks
TL;DR: A Machine Learning Benchmark for Predicting Enhanced-Precision Electronic Charge Density in Crystalline Inorganic Materials

Abstract: Supervised machine learning techniques are increasingly being adopted to speed up electronic structure predictions, serving as alternatives to first-principles methods like Density Functional Theory (DFT). Although current DFT datasets mainly emphasize chemical properties and atomic forces, the precise prediction of electronic charge density is essential for accurately determining a system's total energy and ground state properties. In this study, we introduce a novel electronic charge density dataset named ECD, which encompasses 140,646 stable crystal geometries with medium-precision Perdew–Burke–Ernzerhof (PBE) functional data. Within this dataset, a subset of 7,147 geometries includes high-precision electronic charge density data calculated using the Heyd–Scuseria–Ernzerhof (HSE) functional in DFT. By designing various benchmark tasks for crystalline materials and emphasizing training with large-scale PBE data while fine-tuning with a smaller subset of high-precision HSE data, we demonstrate the efficacy of current machine learning models in predicting electronic charge densities.
The ECD dataset and baseline models are open-sourced to support community efforts in developing new methodologies and accelerating materials design and applications.

Published as a conference paper at ICLR 2025
ECD: A MACHINE LEARNING BENCHMARK FOR PREDICTING ENHANCED--RECISSON ELECTRONIC CHARGE DENSITY IN CRYSTALLINE INORGANIC MA- TERIALS
Pin Chen, Zexin Xu, Qing Mo, Hongjin Zhong, Fengyang Xu, Yutong Lu National Supercomputer Center Guangzhou. School of Computer Science and Engineering, Sun Yat-sen University fchenp85, xuzx25, moqing, Luyutong /mmal. sysu. edu, cn {zhongh j28, xufy @ 12, sysu. edu. cn
ABSTRACT
Supervised machine learning techniques increasingly being adopted speed up electronic structure predictions, serving as alternatives to first-principle meth- ods like Density Functional Theory (DFT). Although current DFT datasets mainly emphasize chemical properties and atomic forces, the precise prediction of elec- tronic charge density essential for accurately determining a system's 's total en- ergy and ground state properties. In this study, we introduce a novel electronic charge density dataset named ECD, which encompasses I 140,64 646 stable crystal ge- ometries with medium-precision Perdew- Beeeww-BBre-eeeeee Ernzeeof (PBE) functional data. Within this dataset, a subset of 7,147 geometries includes high- sigh-precssion elec- tronic charge density data calculated using the Heyd-Scuseria- Cauuserria-Emzzrhooi (HSE) functional in DFT. By designing various benchmark tasks for crystalline materi- als and emphasizing training with large-scale PBE data while fine-tuning with smaller subset of machine learning models in predicting electronic charge densities. The ECD dataset and baseline models are open-sourced to support community efforts in de- veloping new methodologies and accelerating materials design and applications.
HSE data, we demonstrate the efficacy of cur-
INTRODUCTION
The electronic charge density (ECD) is a fundamental yet informative observable in materials physics. Density functional theory (DFT) demonstrates that the properties of materials in their ground state be completely and uniquely determined by the ECD ECDLLewis et al. (2021): 1): Fabrizio et al. (201)). This theory has been widely applied across various physical systems, from individual molecules crystalline solids, significantly advancing our understanding and control of the natural world. For example, bonding characteristics between neighhoring atoms (covalent, ionic, metal- lic bonds) can be fully described through the ECD. Leveraging this property, novel materials with specific structures and target properties can be artificially designed Macchi (2013). Additionally, according modern band structure theory, a wide range of electronic, magnetic, and optical prop- erties, as well as their couplings-such as electrostatic moments, potentials and interaction energies, spin susceptibility, light absorption, and electromagnetic responses-can be directly derived from the ECD Kolb et al. (2017); Tan et al. (202T). Due to these advantages, obtaining the ECD has broad applications.
Obtaining ECD from experimental techniques, such as high-resolution electron diffrac- tion Schmøkel et al. (2014); Chopra (2012), is time-consuming and complex. Theoretically, cal- culating ECD through DFT calculations is more convenient. However, this approach is computa- tionally intensive due to its complexity of O(n²T), where n is the number of electrons and the number of iterations in the structural optimization process Kohn & Sham (1965). Recently, ma- chine learning methods have shown great potential in accelerating DFT computations. For example, various invariant geometric deep learning methods have been developed for representation learn-

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Predicted p(r)
Converged
Pre-training
Fine-tuning
p(r)
SCF loop
Schrödinger Equation
ChurgE3Net
CeurgeE3Net
Large
Properties
Electronic Density p(t)
Hamiltonia
Small
Linergy
Bandgap
PBE data
HSE data
Dos
Datasets
Baselines Changl:3Net
Evaluation Metrics
ECD-PBE
MAE on p(F)
ECD.HSE
Acceleration ratio
Figure 1: The objectives and scope of the proposed ECD dataset and benchmark.
ing of material and molecular structures, providing data-driven approximations to DFT caleulations for faster predictions of physical and chemical properties Choudhaar & DeCost (021 Yan et al.
(2022; Chen et al. (0022). Additionally, extensive DFT datasets have been generated for training machine learning models Jain et al. (2013): [Choudhary et al. (2018) Kirklin et al. (2015). Notably, predicting ECD differs significantly from general physical and chemical property predictions, as the representation of charge density can be uniquely determined by three vectors and scalar matrix in real or reciprocal space. data-driven ECD prediction tasks, maintaining the equivariance of the quantum tensor network architecture is crucial for ensuring that the network can make physically meaningful predictions. Specifically, this equivariance can be represented by the rotational Wigner D-matrix, which may include higher-order rotations beyond three-dimensiones space. Therefore, to conduct systematic and in-depth research on these tasks, is necessary to generate large-scale ECD datasets.
To provide a realistic dataset and enable thorough evaluation, we meticulously constructed a charge density dataset named ECD. This dataset encompasses 140,646 stable crystal geometries calculated using the meedium-precisingg Perdew-Burte Eeikerhof (PBE) functional. Additionally, we carefully selected subset of 7,147 geometries for high-precision electronic charge density calculations using the Heyd-Scuseria- Cicuserri-EEmzzehoff (HSE) functional. The construction of this large-scale dataset was significant challenge due the extensive computational resources required and the careful planning involved. To the best of our knowledge, this dataset represents the largest publicly available col- lection of DFT calculations that includes both PBE and HSE functionals. The creation of the ECD dataset demanded approximately 130 million CPU core hours on cutting-edge supercomputers. No- tably, the HSE functional calculations alone consumed around 25 million core hours, accounting for 19.2% of the total computational resources. This substantial investment of computational time underscores the complexity and resource-intensive nature of generating high-precision charge den- sity data at this scale. To conduct comprehensive studies quantum tensor networks, have designed three types of experiments: Developing and evaluating methodologies for training mod- els capable of accurate charge density prediction, specifically by training large-scale PBE data and fine-tuning with a smaller subset of high-precision HSE data; 2. Investigating the impact of charge density prediction accuracy on related applications: 3. Assessing model performance on out-of- outoof-distributio (OOD) data to evaluate their suitability for real-world applications. To demon- strate the quality of the predicted charge density, we employ two metrics: MAE on charge density to evaluate the accuracy of the model's predictions, and acceleration ratio in DFT calculation to assess whether the predicted charge density can effectively speed up DFT computations. We present the ECD dataset, along with the benchmark objectives and scope, in Fig.
2 BACKGROUND AND RELATED WORKS
2.1 DENSITY FUNCTIONAL. THEORY
DFT functional. DFT functionals are mathematical expressions that describe the exchange- correlation energy, which is crucial component in the total energy calculation. These functionals
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vary in complexity and computational cost, typically following Jacob's Ladder hierarchy Tran al. (016). As one ascends this ladder, functionals incorporate more physical effects and offer higher accuracy but at the expense of increased computational effort as shown in Figure 2 In this study, we focus on the PBE Perdew et al. (1998) and HSE functionals Heydd & Scuseria (004). The PBE functional strikes balance between accuracy and computational efficiency, making it highly cost- effective and widely used for constructing large-scale DFT computation databases. On the other hand, the HSE functional, although computationally intensive, offers higher accuracy in electronic structure information.
Specially, the PBE functional is widely used exchange-correlann functional within the DFT framework. It belongs to the Generalized Gradient Approximation (GGA) PPerdew et al. (1996) class of functionals, which improve upon the Local Density Approximation (L.A))Jackso) Jackson Ped- erson (1990) by including the gradient of the electron density. The PBE functional is designed to balance accuracy and computational efficiency for various systems.
The exchange-correlation energy Eze in the PBE functional is given by:
EPBR
/ PRE (n(r), Vn(r)) + +
(c
PRE (n(r), Vn(r))) n(r) dr
PBE is the exchange energy density, and ce the where n(r) is the electron density at position r, c correlation energy density.
Simplicity
Accuracy
The HSE functional is a hybrid functional that incorporates portion of exact exchange from Hartree-Fock theory Jiménezz Hoyos et al. (2012) with the PBE cooreng- functional. This combination enhances the accuracy for electronic structure calculations, especially for systems with localized d and ellctrons.
Double Hybrid:B2PLYP. PBEO-DH Dependence virtual orbitals
{
Hybrid GGA:HSE, H3LYP, PBEO Dependence on exact exchange
Ex
Meta-GGA: TPSSE, SCAN
T or V pT)
Dependence on kinetic energy densily
The HSE functional modifies the PBE functional by including a screened Coulomb potential, which limits the range of the Hartree-Focd Fock exchange interaction, thus improv- ing computational efficiency. The HSE exchange-correlaton energy given by:
GGA: PBE, PBE sol
Vp(r)
Dependence on the gradient of density
LDA: PZ, VWN
p(r)
Dependence the density
Figure Jacob's ladder of density functional approximations. Higher rungs yield greater accuracy but more complexity.
HISE mc
= &
EHESR (w) +1 a:) -
PBE.S
(w) + EPBELR (w) + EBB
(2)
where HESR (w) represents the short-range Hartree-Fock exchange energy, E EPBE,S (w) and EPPEIL (w) denote the short-range and long-range PBE exchange energies. respectively, and EPER is the PBE correlation energy. The parameter a is the mixing parameter (typically around 0.25), and is the screening parameter.
The combination these terms allows the HSE functional to achieve a better balance between accuracy and computational cost, making it suitable for wide range of materials and electronic properties.
Electronic Charge Density, In DFT, the ECD p(r) and the wave function W(ii, r2,..., FN) related by:
p(r) p(r)= - N sdr... d -
(3)
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The total number of electrons N within unit cell equals the integral of p(r) over the entire volume
Vun:
N = p(
(4)
Vin
After discretization, this relationship expressed as:
NGX.-NGYP-NGZF
V
N
E
p(ri). NGXF F NGZF
(5)
where NG Y, F are the fine Fast Fourier Transform (FFT) grids in reciprocal space along the x, y, and z directions. The discrete values p(ri) at each fine FFT grid point are recorded in the charge density file, which contains all the necessary information about p(r).
In non-spin-polarized calculations for non non-magned magnetic materials, the ECD includes the total electronic charge density p(r) Psgin-up (r) - Pspin-dowa! (r). For spin-polarized calculations of magnetic materi- als, an additional spin electronic charge density Papin(r) = Pspin-up (r) - Pspin-down (r) also provided.
2.2 HIGHER-ORDEE EQUIVARIANT MATRICES
Given a crystal material structure M (A, X, L) where represents atomic species, X denotes atomic positions, and L is the unit cell, we aim to predict the electronic charge density p(7) : R - at any point X. Similar to other tensor properties such as Hamiltonian and forces, the charge density preserves the SE(3) equivariance of the atomic system. This means that rotations and translations of the atomic system Euclidean space result in equivalent transformations of the charge density. Representing graph structures using only invariant scalar features (e.g., interatomic distances) can achieve SF(3) invariance, but lacks angular information, thus limiting the model's accuracy. Morcover, incorporating vector IR3 features to integrate angular information has been shown to enhance the performance of charge density predictions Jørgensen & Bhowmik (2022).
Translation invariance is maintained by utilizing relative atomic coordinates, while rotation equiv- ariance is ensured by confining features to the irreducible representations (irreps) of SO(3), which manipulated by equivariant functions. These features are denoted as V.lep) collection of ten- sors indexed by rotation order l E (0, 1,2, 1,2....) and parity E {-1, 1}. Each tensor has a channel index [0, Nehummeis and index and p dimensions is of size I Nataucc 12f+1), These representations are combined using the equiv- ariant tensor product @ with Clebsch-Gordee coefficients C' as detailed in Thomas et al. (0188 and implemented in e3nn Geiger & Smidt (202))
[-8, E, f]. Consequently, the representation for specific &
(ULE). @ V((2.2))
(Fo.Po) emo
clti,mz)((a.mm (1( cm
my=
where f, and Po are defined by 1&1 f2l lo S 1&1 82 and po P1P2. We retain only those representations with f, S L,, where L is the maximum allowed rotation order.
2.3 DATASETS FOR CRYSTALLINE INORGANIC MATERIALS
Materials databases such as JARVIS Choudhary et al. 2018). OQMD Kirklin al. (2015). NO- MAD Draxl & Scheifler (2019), Materials Project (MP) Jain et al. (013)) and AFLOW Curtarolo al. 2012) primarily provide data on properties such as energy, electronic structure, mechanical, and magnetic properties. However, only NOMAD and MP provide the option to download charge density files. The charge density files in NOMAD are generated during structure optimization cal- culations, not from the self-consistent process, and the parameters used for these calculations are missing, making reliable subsequent processing from these data challenging. MP provides approxi- mately 12,000 electronic structure datasets, accessible via the officially released API. However, the charge density in MP is calculated using the PBE functional, which has limitations for electronic structure caleulations Saßnick & Cocchi (202T), especially when dealing with strongly correlated
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materials. Additionally, there are smaller databases focused on charge density data, such as the ECD-cubic database Wang et al. (2022) with 17,418 entries, but it is limited to cubic inorganic materials, restricting its applicability. MaterialGo Jie et al. (2019) is the only reported database con- taining electronic structure data from approximately 10,000 HSE functional calculations. However, its query and download services are currently unavailable. Additionally, since this database uses the PWmat software package Jia et al. (2013), combining its data with existing PBE data calculated using other software might introduce biases due to differences in the software and computational parameters. In this study, we constructed a dataset of nearly 140,000 entries with PBE precision and subset of approximately 7,000 entries with HSE precision by utilizing supercomputing power. Apart from the specific calculation parameters of these two functionals, all other aspects, such as the software and computational environment, remain consistent. By leveraging this mixed-precision dataset, we aim to develop an electronic charge density prediction model with improved accuracy.
2.4 DATASETS
Dataset Generation. All materials structures are obtained from the Matgen database Chen et al. (2022). We perform DFT computations using the Vienna Ab initio Simulation Package (VASP) ver- All systems are fully relaxed with respect to volume and atomic coordinates using the GGA Perddew et al. (1996) of PBE, with pseudopotentials from the projector augmented wave (PAW) method]Kresse & Joubert ( (199)) at zero temperature and pressure. A unified plane wave cutoff of 520 eV is used for all structures to ensure consistency and compatibility. This cutoff energy also satisfies the condition of being 1.25 times the maximum en- ergy cutoff of the element plane wave basis sets in the pseudopotentials used. Due to the limitations of pure GGA in accurately describing electronic interactions in strongly correlated materials con- taining transition metals or rare earth elements with 3d/4f orbitals, we employ the GGA+U method, with all U values are same Chen et al. (0022). Brillouin-zone integrations are performed using the --centeeed Monkhorst-Paan Pack scheme. We begin with a k-point mesh with a dense sampling mesh of 2T 0.02 -1 For small number of structures that are difficult to converge, we use a medium- dense sampling mesh of 2 x 0.04 -1 The blocked Davidson iteration scheme is used to solve the Kohn-Sham (KS) equations, with convergence criteria set to 0.1 meV for energy and 0.001 eV/ for force.
sion 5.4.4 5.4.4 Kressee & Furthmüller (
For the DFT calculations with the HSE functional (HSE06), hybrid functional calculations are en- abled, with the exact exchange mixing parameter set to 0.25 and the screening parameter set to 0.2. Additionally, the time step for the calculation is specified with 0.4. These settings ensure that the HSE functional provides more accurate electronic structure information with appropriate adjust- ments for exact exchange and screening. Considering that calculations using the HSE functional over ten times slower than those using the PBE functional, we selected a subset of 7,147 struc- tures from an initial pool of 140,646 for computation. The selection criteria is as follows: structures with fewer than 20 atoms, diverse range of element types encompassing all elements, inclusion of unary, binary, and ternary compositions, and representation across various crystal systems. Through this selection strategy, the HSE functional data domain is ensured closely approximate the PBE functional data space. More statistical information is presented in the Appendix A.
We utilized approximately 1,000 computing nodes (64 cores per node) on the supercomputer for large-scale high-throughput computations. In total, approximately 130 million CPU core hours were utilized, with the HSE functional alone consuming around 25 million core hours, consti- tuting 19.2% of the overall computational resources. The DFT calculation workflow is avail- able at: https//anonnymous.eee wwwp:s/s.........eeeeee//eeeeeee 665A. Our benchmark is publicly available at:
Dataset Statistics. The statistical data, including the element distribution and structure size his- togram for ECD, is presented in Figure 2. These material structures include 94 different elements, covering nearly all commonly found elements in the periodic table. Existing datasets like MP and OQMD did not feature the elements Cm, Rn, Ra, Po, and Am. In terms of material size (number of atoms per crystal cell), about half of the structures in ECD have 20 or fewer atoms, and the dataset includes approximately 9,000 large structures with over 80 atoms per unit cell. Specifically, we selected 7, 147 structures for HSE functional DFT calculations. The selection strategy aimed to maximize the resemblance of elemental distribution the 140,646 dataset. To further reduce com-
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putational time, we focused unary, binary, and temary materials with fewer than 20 atoms per unit cell. More detailed information provided in Appendix B.
10000
Cour
Tb
0.1
ston size
(a) The clement distribation of ECD dataset.
(b) The histogram of the strueture size for ECD dataset.
Figure 3: The dataset statics for ECD dataset with 140, 646 entries, including element distribution and structure size histogram.
2.5 TASKS
To thoroughly evaluate the performance of charge density predictions, we define the following tasks utilizing PBE functional and HSE functional precision from the ECD dataset, and the MP PBE precision pretrained model Koker al. (202). The statistics of our defined tasks is presented in Table
Table 1: The statistics of our defined four tasks.
Task
Total entries
Traning/valddation/esstigg entries
ECD-PBE-id
140,646
/38.134/512/200
ECD-MP-id
2000
-/-/2000
ECD-HSE
5,647
4147/500/1000
ECD-PBE HSE-id
1000
-/-/1000
HCD-PBE_LSFE.uninggee
7147
5647/500/1000
EXP-id
41
-/-/41
OOD-id
2000
-1-/2000
ECD-PBE-id. We partition the PBE data in the ECD dataset following the method Koker et al. 2024): 138,134 for training, 512 for validation, and 2000 for testing. This forms the primary evaluation task for predicting charge density matrices.
ECD-MP-id. In this task, we use the model from ECD-PBE-id task to predict the test data of the
ChargE3Net model in the MP dataset for direct comparison with existing models Koker al.) (2024). We perform similarity analysis between the test data in the MP dataset and the training data in the ECD dataset, ensuring no identical data points are present the training set. This task is designed to enable fair comparison with existing charge density models.
ECD-HSF-id. In this task, the dataset comprises 7,147 HSF-calculated data points, which are partitioned into 5,647 for training, 500 for validation, and 1,000 for testing. This task is designed to train a model solely on HSE data, serving as a baseline for comparison with a model pretrained on PBE data.
ECD-PBE HSE-id. We divide the HSE dataset into training, validation, and test sets with 5647, 500 and 1000. We then used the model from the ECD-PBE-id task to predict the HSE test data to determine the gap between PBE functional and HSE functional. This task aims to assess whether model trained PBE data can accurately predict HSE data directly.

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ECD-PBE. Htt.Htniintteeeeeeeeeeeeeeeeeet We fine-tune the model from the ECD-PBE-id task on the HSE dataset.
The trained model is then tested on HSE test set (as in ECD-PBEE HSE-dd. By comparing the results with those of ECD-PBE HSE-id, we determine whether the fine-tuned model achieved improved accuracy.
have compiled dataset consisting of 47 entries that encompass material properties from wet-lab experiments, alongside various machine learning predictions, as well PBE and HSE computational data. This dataset intended assess the potential of electronic charge density prediction models for real-world applications.
EXP-id. Building upon the work of Chen et al. 2002
OOD-id. To further evaluate the model's generalization capability in real-world applications, selected 2,000 material structures from the GNoME dataset and conducted DFT calculations to obtain their electronic charge densities. The structures in the GNoME dataset exhibit novelty and possess a distribution distinct from our existing database, making them suitable for assessing the model's s outtof-distribution (OOD) performance.
2.6 METHODS
Machine learning models have been developed address the challenges of predicting charge den- sity, relying on custom- custom-fitted basis functions. Early approaches utilized symmetry -ndapte Gaussian process regression to predict coefficients for atom-centered basis functions, expressing the structural similarity and geometric relationships between target and training atomic environments through ker- nel functions Fabrizio et al. (2019); Grisaf al. (2018). Recent work has employed invariant and equivariant neural networks to predict these coefficients from atomic features, using characteristics computed from density-functicn
tight-binding calculations as inputs QQio et al. (2022). High- order equivariant neural networks have been used to predict coefficients of small molecules found in self-consistent DFT calculations and directly from atomic features Rackers et al. (2023). Despite achieving high accuracy in specific cases, these methods are limited by the expressivity of the den- sity fitting basis sets. These atom-centered basis sets, typically defined per species, are numerically more challenging converge for solid-state systems than plane-waye basis sets, restricting their applicability to molecular systems vacuum.
As alternative, several methods have been proposed to learn electron density directly from dis- cretized grid of density points. Charge density predictions on grids are agnostic to the basis set used for ground-truth quantum chemistry calculations and serve as natural input format for plane-wave DFT codes. By inserting each grid or "probe" point into the atomic graph. charge density prediction can be modeled as a node regression task [hso et al. (2021); (Sunshine et al. (2023). However, these models mainly focus on small and specialized atomic systems. Invariant graph convolutional net- works have been trained on small datasets of crystalline polymers, demonstrating transferability to networks{Schütt
unseen structures. Invariant graph convolutional networks (Schütt et al. (2018) (Choudharyy & DeCost) (02I), have shown fast and accurate charge density prediction small molecules, lithium-ion bat- tery cathode materials, and electrolytes. Improvements in accuracy have been demonstrated using equivariant graph convolutional networks like PaiNN Schütt et al. (001)), which leverage R3 vector representations and rotationally equivariant operations. While these models perform well on small, specialized datasets, their expressive power on larger, diverse datasets like the MP dataset is not yet fully understood. ChargE3Net|Kokee et al. (2024) constructs rotationally equivariant networks using vector representations and improves model accuracy by incorporating higher-order equivariant fea- tures, showing diverse performance on the MP dataset. Therefore, the equivariant quantum tensor network ChargE3Net is selected as the primary benchmark for the ECD dataset.
2.7 METRICS
To assess the quality of the predicted charge density, we utilize several metrics that evaluate both approximation precision and computational performance.
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MAE p(r). We evaluate models' performance using the mean absolute error (MAE) normal- ized by the total number of electrons within the atomic system's 's volume émae Koker et al. (0244) calculated via numerical integration on the charge density grid points G:
Egec (p(gk) P(gk)
émac
Eg.ea |p(gx))
Unless otherwise stated, the probe points G refer to the complete set of discretized unit cell grid points for which DFT-computed charge density values are available.
Acceleration Ratios. we use achieved ratio and error-level ratio assess the effectiveness of the predicted charge density p(r) in expediting DFT calculations. The achieved ratio measures the num- ber of optimization steps required when initializing with the predicted charge density compared to traditional initial guess methods. A well-predictes charge density enables the Self-Consistent Field (SCF) algorithm to converge more quickly, significantly reducing the number of optimization steps. Conversely, the optimal ratio evaluates the number of single optimization step for each material relative to the total number of steps, serving as a benchmark for ideal performance.
3 EXPERIMENTS
3.1 SETUP
To assess the performance of deep learning approaches the proposed dataset, carry out periments on the four designed tasks as described in Section 2.5 Specifically, we evaluate the performance of ChargE3Net, network specifically designed for efficient and accurate prediction of electronic charge density. ChargE3Net is known for its effectiveness and efficiency in handling the task hand, making a suitable testing method for our benchmark evaluation. For quantitative evaluation, we use the metrics introduced in Section 2.7 GPU and an Intel Xeon Gold 6348 CPU.
We train the models using an Nvidia A800
Following the model setup in ChargE3Net, we employ four node-wise interaction layers to aggre- gate messages from neighboring nodes and update the node irreducible representations in all imple- mented models. We train all models with total training step of 1, 1,000,000 using batch size of 16 with 200 charge density probe points per batch. For each gradient step, a random batch of materials is selected, from which a random subset of the charge density probe points is used. To expedite the convergence of model training, we implement a learning rate scheduler. The scheduler starts with a learning rate of 0.005, which is decayed by 0.96*/# at step s, where 3 is set to x 103. We use error as the loss function for optimization. The Adam optimizer Kingma & Ba (2014) is used for training the models. During the model tuning phase, we fine-tune our model on the ECD-PBE-id task, allowing all parameters to be adjusted during training.
3.2 RESULTS AND DISCUSSION.
Overall performance on ECD dataset. Initially, we assess the model's comprehensive perfor- mance across the designated tasks by examining the accuracy of the predicted charge density matri- ces on the test set. As illustrated in Table the ChargE3Net models employed exhibit satisfactorily low MAE in predicting the charge density matrices for all specified tasks. We observe the following: 1. ChargE3Net demonstrates strong performance on the MP dataset and shows further improvement on our ECD dataset, indicating that an increase in data scale contributes to superior performance; 2. Electronic charge density is an equivariant property, which explains why equivariant models achieve optimal performance across both the MP and ECD datasets; 3. Training on electronic charge density data from the PBE functional, followed by fine-tuning with HSE functional data, helps mitigate the limitations of each dataset in terms of accuracy and data scale.
We evaluate the efficacy of the predicted charge density by examining its impact on accelerating DFT calculations. As outlined in Section 2.7 we compare the number of optimization steps required when starting with the predicted charge density to those needed with conventional initial guess methods Lehtola (2019). Using VASP for DFT calculations, we determine the average optimization step ratio for 19 randomly selected materials in each dataset. Table 3 presents the ratio metries,
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Table 2: The overall performance the testing set on the defined tasks.
Dataset
Model
MAE (eV)
MP-PBE
invDeeDDFT
1.293=10.03 1.212
Jørgensen Bhowmik 2022
MP-PBE
DecpDFT
Jørgensen & Bhowmuk
2022
MP-PBE
ChargeE3Net
0.01 et al.
2024
ECD-PBE.MP-id
ChargeE3Net
0.520
ECD-PBE-id
ChargelB3Net
0.685:10.03
ECD-PBE-id
invChargeE3Net
0.732++0002
ECD-HSE-id
ChargeE33Nt
1.534=10.07
OCD-PBE.HSE-A
ChargeE3Net
2.156:10.08
ChargeE3Net
1.085-10.03
showing the number of optimization steps needed when initialized with the model prediction relative traditional DFT initialization. The results indicate that starting from the predicted charge density matrices provided by ChargE3Net reduces the number of optimization steps required to reach the converged charge density, suggesting that the predicted charge density is close the convergence condition. These findings illustrate the potential of machine learning methods in expediting DFT calculations.
The performance of DFT calculation acceleration. Both models, trained on the ECD-PBL id split and the ECD-PBE HSE tuning-id split respectively, are evaluated on 19 randomly selected materials from the intersection of their test sets.
Table 3:
Training Dataset
Metric
Ratio
ECD-PBE-id
Optimal ratio
0,061=0.016
Achieved ratio +
0.757=0.073
ECD-PBE.HSE.une -inninnig
Optimal ratio 0.071=0.033
Achieved ratio
0.681=0.113
We have conducted additional experiments as suggested to evaluate the sufficiency the HSE dataset and its impact model performance. The results summarized in Table [ As the data indicates, increasing the proportion of high-quality HSE data consistently improves the model's predictive accuracy and accelerates the computation. This trend suggests that further increasing the amount of HSE data could continue to enhance both accuracy and acceleration.
Table 4: Impact of HSE data ratio MAE and acceleration rate.
HSE Data Ratio (%)
MAE (eV)
Acceleration Rate
0
2.156 = 0.08
0.757 0.073
25
2.057 + 0.07
0.743 + 0.081
50
1.524 + 0.10
0.714 0.082
75
1.203 + 0.06
0.692 + 0.098
100
1.085 0.03
0.681 0.113
Evaluation on wet-experimental data. To further demonstrate the potential of predicting elec- tronic charge densities in real- woal-wrrdd material applications, we conducted additional experiments uti- lizing 41 experimentally measured values to emphasize the importance of accurate ECD predictions. Specifically, we focused on the band gap, critical property for applications such as semiconductors, solar cells, and lighting technologies. Additionally, we compared our results with other methods for determining band gaps, including machine learning approaches and DFT calculations using both the PBE and HSE functionals. The results, summarized Table 5 highlight the value of using nigh-precision HSE data for improved band gap predictions, supported by Chen al. (2022). We can observe that the ECD-PBE HSE model demonstrates a significant improvement in accu- racy compared direct GNN predictions, and it also outperforms CrystalNet-TT. model utilizing

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transfer learning with HSE functional data. However, this increased aceuracy comes the cost of approximately 10 times more computational time. Despite this, the charge density prediction ap- proach offers substantial benefits over direct DFT calculations, HSE functional calculations take approximately 10 times longer than PBE functional calculations. Therefore, using electronic charge density predictions strikes an advantageous balance between accuracy and computational efficiency, proving highly valuable for band gap prediction in materials science.
Table 5: Performance comparison of ML and CCD-Based models on wet-experimental data. Model/Dataset
MAE (eV)
Time per structure
Models
(S)
CGCNN
1.45
Chen al.
2019
1.5
MEGNet
1.36
Chen et al.
(011
1.34
CrystalNet
1.19
Chen
(2022)
1.67
CrystalNet-TL
0.70
Chen et al.
(2022)
1.58
ECD-Based Models
(min)
ECD-PBE
1.17
14.4
ECD-PBE.HSE
0.65
14.4
PBE-Based Datasets
(min)
MP
1.38
Choudhary et al. 2018
Matgen
Chen (2022)
24.5
AFLOW
1.20
Choudhary et al. 2018
OQMD
1.09
Choudhary et 2018
ASE-Ba Dataset
(min)
HSE
0.41
Choudhary et al. (0118)
228.1
ECD prediction models are particularly adept capturing material properties such energy, forces, and band gaps in a more comprehensive manner, as demonstrated the ChargE3Net study Koker et al. (2024). These models offer superior accuracy and generalization, especially in complex sys- tems, when compared to direct property prediction methods. Furthermore, ECD-based models are capable of achieving chemical accuracy in non-self-consistent calculations, which enhances their reliability and expands their potential applications within materials science.
OOD evaluation using GNoME dataset. To further validate the model's ability to predict elec- tronic charge densities in truly novel materials, we using GNoME dataset Merchant et al. (202) to evaluation our module for real-world application. Specially, we selected 2,000 structures from the GNoME dataset and conducted DFT calculations using the PBE functional to generate the necessary charge density data. The OOD evaluation on this new dataset yielded an MAE of 0.643 eV, which is comparable to the model's performance eV) same scale, thereby demonstrating the model's robust generalization capability. These find- ings support the robustness and broader applicability of models trained the ECD dataset, further highlighting its value in the field.
the MP dataset (0.523 eV) and the ECD dataset (0.685
CONCLUSION
In this study, we focus on accelerating the computation and prediction of electronic charge densi- ties, which are critical for elucidating the electronic properties of materials and driving the design of novel materials. To this end, we introduce ECD, a novel and publicly accessible dataset comprising 140,646 stable crystal structures computed with the mediumm-peeisst ally, ECD includes a subset of 7,147 crystal structures with nigh-precision electronic charge density data calculated using the more accurate HSE functional. We have undertaken rigorous and com- prehensive experiments to validate the integrity and applicability of the dataset, ensuring that it meets high standards of quality and relevance for real-world applications. The dataset is designed to support large-scale benchmarking and the training of machine learning models, demonstrating their capability in predicting electronic charge densities with remarkable accuracy. Furthermore, this dataset opens up avenues for advancing computational materials science, by providing robust foundation for the development of data-driven models that can accelerate the discovery and optimization of new materials with tailored properties.
PBE functional. Addition-
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ACKNOWLEDGMENTS
This research supported by the National Natural Science Foundation of China (62461 146204).
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A LIMITATION
While this study introduces an extensive, ther improvements in data quality remain essential, particularly through increasing the ratio of high- precision data. Our experimental findings demonstrate that expanding the dataset with additional HSE functional data can further enhance model accuracy, however, this improvement necessitates substantial computational resources. Moreover, due to the high dimensionality of charge density data, which often involves over 1,000 electrons, the complexity of handling node and edge interac- tions within graph neural networks presents significant computational challenges. This complexity underscores the need for more efficient algorithms and optimization strategies. Consequently, en- hancing the efficiency of model training and extending the number of training epochs are critical ar- eas for future research, especially in balancing the trade-off between computational cost and model performance. Additionally, addressing these limitations will enable the scalability of this approach to larger, more complex systems, thereby broadening the applicability the dataset to a wider range of materials science problems.
dataset of electronic charge densities, fur-
B DATA GENERATION
The first part primarily consists of high-quality, literature-reported structures from sources such ICSD and COD Graulis et al. (0012). We have collected experimental crystal structures from the ICSD and COD databases, totaling 621,782 structures, and ultimately acquired 76,463 DFT- calculated data entries, as shown Figure
The second part comprises structures generated through algorithmic methods. representing novel and hypothetical structures not previously observed. Using the DifíCSP Diao al. (2024) crystal structure generation method and ab initio algorithms, we construct a large-scale set of hypothetical crystal structures, which are further refined using the methods described below.
1. Reasonable chemical compositions. We remove number the structure with number of chemical components that more than 10.
2. Electroneutrality approach: We calculate the oxidation states of each element using the SCMAT toolkit Davies et al. (2019) and remove structures with charge imbalance.
3. Exhibiting symmetry: Structures with space group equal will be excluded.
CCD datatase (413.587)
ICBD didebasue (208.425)
Input
siructures (118 (30)
Magnetic Test
(381 4001
Repeat times convergence
Geametry aptimization
Filor organic
stustume
(45.
Static calculation
Magneck calculation
Black calcuratoin
Decuplictton 307)
(37.778)
Band siructure caloutation
Densily of states calculation
Formation snargy calculstion
Deckpticall on
Finel stnudtuve (M))
Band structure
Density of states
Formation Energy
Magnetic properties
Elastic propeties
(76,403)
Motgon Database
Flow chart depicting procces collection.
DFT eslculation workflow.
Figure 4: The flowchart illustrating the data collection process and DFT calculation workflow.
Next, we employ DFT methods for structural optimization and self-consistent calculations on the selected results, as depicted in Figure 4p. All charge density files are obtained from the static cal- culation step within the workflow. To further obtain potentially stable structures, we select about
https://iexd.produccee far-karksuuhe.de
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70,000 hypothetical crystal material calculation results based formation energy. The distribution of formation energy is shown in Figure 5 ranging from -4 to 4 eV. which is consistent with existing databases such MP Jain et al. (2013) and OQMD Kirklin et al. (201).
12000
10000
8000
Count
6000
4000
2000
-4
-2
Formation energy
Figure 5: The distribution of formation energy.
C DATA STATISTICS
Figure iillustrates the statistical distribution of data from HSE functional DFT calculations. This dataset encompasses 94 elements (Figure (6), consistent with the elemental distribution in PBE functional DFT data. Additionally, the dataset includes unary, binary, and ternary compounds, with ternary compounds comprising over half the total (Figure 6p). Regarding the distribution of the number of atoms within the unit cells, all structures contain fewer than 20 atoms, with structures containing more than 10 atoms accounting for approximately one-third of the dataset (Figure (6))
D MODEL DESCRIPTIONS
In Table 2, the performance of ChargE3Net is compared with other models such DeyDeeDFT, DeepDFT, and inv EnvCrhgEE3et. Below, we provide detailed description of these models:
DeepDFT: This model utilizes invariant graph neural networks (GNNs) where the edge features represented by distances between vertices (atoms). is designed capture fundamental pairwise interactions in the system but does not incorporate directional infor- mation.
inv ChrCrgrEE3eet: We implemented this model by following the methodology of DeepDFT ensure a fair comparison. In this invariant version, the edge features represented solely by the distances between vertices, similar to DeepDFT, and the model does not incorporate directional equivariant information.
EquiChargE.3Ne: the equivariant version of ChargE3Net, edge features include both distances and the directions of edges as features. This addition allows the model to capture directional dependencies in the data, which is essential for accurately modeling charge density and other physical quantities.
inyDeepDFT: This is invariant version of the DeepDFT model where, like in- vChargE3Net, the edge features are limited to distances between vertices, ensuring that the model respects invariance properties but lacks directional information.
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5600
Li
Be
Ne
5000
11
Mgg
Ar
4500
K
Sc
Cr
Co
Cu
Zn
Se
Br
Kr
4000
3500
Rbb
Si
Zr
Nb
Mo
Tc
Ru
Rhh
Pd
Ag
Cd
In
Sn
Sb
Te
Xe
3000
76
75
87
Au
Hg
80
81
85
2500
105
Fi
Ra
Lr
R1
D6
5g
8h
Hs
Mt
Ds
Rg
Cn
Nh
F1
Mc
Lv
Ts
Og
2000
1500
La
Ce
Pr
Nd
Pm
5m
Eu
Gd
Tb
Ho
Tm
Yb
Ac
Pa
Np
Am
8k
Cr
20
Fm
100
H01
Md
102
No
500
(a)
The element distribution of HSE data entries.
3,68
2500
binary ternary
1500 38 3335%
1000
08.5)
[0]
(5)
(18 20)
Atom size
(b) The structure component proportion.
(c) The histogram of the structure size.
Figure The dataset statics for ECD dataset with 7147 HSE entries, including element distribution, structure component proportion and structure size histogram.
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By incorporating directional features in EquiChargE3Net, the model achieves higher degree of ex- pressiveness compared to its invariant counterparts, such as DeepDFT and invChargE3Net, thereby enabling improved performance on tasks requiring the representation of directional dependencies.
E HSE CALCULATION DETAILS
We have provided the 41 experimental data points from the HSE-based dataset Choudhary et al. (018)) in the supplementary materials. These data were obtained through first-principes calcula- tions using the HSE functional. Additionally, have included scatter plot comparing the cal- culated results with experimental values. is worth noting KCI and CaO, typically exhibit larger errors. Moreover, HSE functional calculations computationally expensive due to the complexity of solving hybrid functionals, which require a significantly greater number of iterations to converge. We also present the distribution of computation times for each data point. Materials with higher number of atoms and elements located later in the periodic table, such as MoSe2 and ZrO2, exhibit the longest computation times.
KCI
CaO
30
8-
25
HSE
Count
15
2-
10-
0-
0
10
-200
200
400
600
800
1000
1200
1400
1600
EXP.
(min)
Figure 7: The performance characteristics of HSE calculation in wet-experimental evaluation.
F CASE STUDIES
have included a failure for further clarification and analysis (code: ebfc3a7854). We present a comparison plot of the predicted results and DFT results, where the visualization reveals that Charge3Net predicts the electron count of certain atoms, such as Be and Ce, be inconsistent with the target atom.
G DATA FILE SYNTAX EXAMPLE
The general structure of the ECD file is as follows:
System name
Scaling factor
Lattice vector 1
Lattice vector
Lattice vector 3
Element names and atom counts
Coordinate system (Direct/Cartesiaan)
Atomic coordinates
FFT grid dimensions
Charge density data (flattened)
Augmentation occupancies
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2.163
Be
1.449
7.343e-1
2.000e-2
Predicted
DFT calculated
Figure 8: Case of OOD validation experiment, material code in Gnome database: ebfc3a7854.
G.1
SAMPLE DATA
Below is sample ECD file, demonstrating its typical layout:
CeBeAlSi 1.
1.00000000000000
424433
-0.016098
2.765546
-0. 6
7.408746
2.697069
0.066512
-0.029102
11.747799
Ce
Be
A1
Si 1
1
Direct
0.994813
0.994638
0.01C597
0.616360
0.616420
0.766230
0.742351
0.242544
0.515314
0.242735
0.742366
0.515387
0.403740
0.404030
0.192473
120 120
180
0. 0.23427331455E-00
0. 7.28889571492F-03
0. 0.444043710452H03
augmentation occupancies
33
0.1828841E+00 -0.9408141E-01 01 -0.2320062E-02
0.3674660E-02 -0.12127626-02
0.1020137E-02 -0.4797309E-03
0.8535439F-03
0.1785949E-01 0.1289605E-02
G.2 EXPLANATION
Crystal structure information: The first section provides details of the crystal structure, including the system name, scaling factor, lattice vectors, element types, atomic counts, and atomic coordinates.
FFT grid dimensions:
The line 120 120 180 specifies the grid dimensions (NGXF, NGYF, NGZF) for the charge density representation.
Charge density data: The charge density, scaled by the FFT-grid volume, represented multiple real numbers per line and continues until all NGXF XNGYF XNG2F values are listed.
Augmentation occupancies: Additional data required for PAW calculations, if applicable.
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This format ensures compatibility with visualization tools and supports detailed analysis of charge density distributions within the crystal structure.
19


Reviews:
[{"review_id": "TVwwpuJ58e", "reviewer": "Reviewer_dhoo", "summary": "Paper proposes an electronic charge density dataset of 140,646 crystals using DFT PBE functional and 7,147 crystals with HSE functional. Paper trains the ChargE3Net on PBE dataset and with and without fine-tuning on HSE dataset, and compares with original ChargE3Net on MP data. Paper also shows that using the charge density predicted by ChargE3Net (as initialization) can accelerate DFT calculations. Paper compares band gaps predicted by their model with other published results.", "questions": "what is invChargE3Net in Table 2? could authors provide a clear definition or explanation of invChargE3Net in the paper, as this term is not defined.\n\nWhat is ECD-HSE-id in Table 2? could authors clearly define or explain ECD-HSE-id in the paper, as this term is not explained.\n\nReviewer would suggest to authors to include a detailed description of structure of each dataset entry, possibly in appendix, if not already included.\n\nCould authors provide more context about the Choudhary 2018 HSE-based dataset model?, including reasons for its performance characteristics, such as its error and time performance.\n\nFor Table 5, are there error bars? I am assuming the eV is band gap error, what is the acceptable error for band gap prediction? \n\nWhat is reason for discrepancy between optimal and achieved ratio in Table 3, and how was optimal ratio determined?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Paper is clearly written. Paper presents dataset which was generated by significant computation expense. The problem of materials design and prediction is significant.", "weaknesses": "Charge density data is high memory, and may be computationally expensive to train ML models. (authors admit this in limitations section). The optimal vs achieved ratio of Table 3 seem quite far from each other- it seems to indicate that initializing dft calc with ML charge density model prediction does not improve efficiency as much as one would like.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "zwpdkpLmNH", "reviewer": "Reviewer_4VpH", "summary": "The paper introduces a benchmark dataset ECD to enhance machine learning models for predicting electronic charge densities for crystalline materials. ECD includes over 140K crystal structures with medium-precision PBE data and 7147 high-precision HSE data entries. The authors propose tasks for training, fine-tuning, and out-of-distribution testing to improve predictive accuracy and speed up DFT calculations, using ChargE3Net to demonstrate significant improvement. This dataset offers a robust foundation for advancing scalable, high-accuracy electronic structure predictions in materials science.", "questions": "1. ChargeE3Net is compared to other models in Table 2, whereas the detailed description for invDeepDFT, DeepDFT, and invChargeE3Net is missing. Can the authors justify the choice of using DeepDFT for comparison and elaborate on why it is an appropriate baseline? Besides, I guess invChargeE3Net is the invariant version of ChargeE3Net, but it would be better to clarify that in the manuscript. \n\n1. Given the experimental results, it seems that equivariant models do not significantly decrease the errors in ECD prediction. Considering the lower computation complexity of invariant models, equivariant models may not have an edge over invariant models. Can the authors provide the DFT calculation acceleration of invChargeE3Net and make a comprehensive comparison between invariant and equivariant models taking account of both accuracy and speed?\n\n1. Given the results in Table 3, the acceleration from ML-predicted densities is moderate as the achieved ratio is around 0.7. Furthermore, the achieved ratio is around one order of magnitude larger than the optimal ratio. Can the authors provide insight of how to further improve the acceleration or discuss what factors prevent the model from achieving the lower bound on the number of steps?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The ECD dataset is a substantial and valuable contribution, offering over 140K PBE-calculated crystal structures and 7147 HSE-calculated structures, providing a robust foundation for electronic charge density prediction.\n\n1. The paper conducts a comprehensive benchmarking of multiple models, assessing their performance across various tasks and metrics, including both in-distribution and out-of-distribution scenarios, which enhances the reliability and applicability of the findings.\n\n1. The paper demonstrates that fine-tuning models trained on PBE data with high-precision HSE data improves predictive accuracy. This approach can guide future research on leveraging high-cost data sparingly to optimize model performance.\n\n1. The paper highlights the model’s performance on wet-lab experimental data and out-of-distribution data, validating its potential for applications in real-world materials discovery and screening.", "weaknesses": "1. The description of ECD-HSE-id seems missing in Section 2.5. I assume that the data split is consistent with ECD-PBE HSE-id and ECD-PBE HSE tuning-id, but it would be great if the authors could clarify that. \n\n1. ChargeE3Net surpasses its invariant counterparts and other non-equivariant models marginally but significantly increases the computation complexity, thus increasing the time of cost for computation. For example, comparing CrystalNet-TL to ECD-PBE HSE, the MAE is 0.70 vs 0.65 while the time per structure is 1.58s vs 14.4min. \n\n1. In the OOD performance section, while the authors demonstrate the model's generalization, there is minimal analysis of failure cases or instances where the model struggled. Understanding these limitations in detail would be valuable.\n\n1. While the ECD dataset is a major contribution, data augmentation techniques or other methods to improve model generalization are not discussed. Addressing potential improvements for model robustness could make the dataset more valuable for future research.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1I3Ve3saL1", "reviewer": "Reviewer_NK1Z", "summary": "This paper introduces the Electronic Charge Density (ECD) dataset, a benchmark designed for predicting electronic charge densities in crystalline inorganic materials. The dataset comprises 140,646 stable crystal structures computed with PBE functional, alongside a subset of 7,147 structures calculated with HSE functional. Using the ChargE3Net model as a baseline, the authors evaluate its performance across multiple tasks, including prediction accuracy on the ECD dataset and the acceleration of DFT computations. Overall, this paper makes a valuable contribution to machine learning in quantum chemistry and computational materials science, and I recommend acceptance conditional on addressing the concerns outlined below.", "questions": "Although I generally believe this paper is a good contribution to this area, I still have the following concerns and I hope the authors may address:\n1. **Codes and Data Availability**: The links to the workflows and datasets are no longer valid, thus I could not review them. It would be nice if the authors provide new links to them.\n2. **Computational Setup**:\n\n    a) Why did the authors decide to mix charge densities with PBE and PBE+U functionals? Are you sure that the charge densities at these two levels are compatible to be mixed up? More information for you to consider: In public datasets like MP, they mixed PBE and PBE+U functionals mostly because they justified that they can align the formation energies of materials with further compatibility corrections. However, here you are computing charge densities, so I am skeptical if these two levels of theories should be mixed up.\n\n    b) The choice of hybrid functional is not justified. There are multiple hybrid functionals available on the market, including PBE0, B3LYP, the HSE you are using, etc. Why did you pick up HSE as your functional? In addition, you should specify which version of HSE you used, HSE03 or HSE06?\n\n    c) Did you apply any convergence test on the charge densities with respect to the energy cutoff and Brillouin zone sampling? My impression is that the 520eV cutoff you are using might not have converged the charge densities.\n\n3. **Data Storage Schema**: In large-scale calculations used for machine learning, it would be nice if you include the information of the data storage schema in your supplementary information. For example,  what information you stored and what units for each of the information. \n\n4. **Experimental Setups**\n\n    a) **Question on OOD evaluation using GNoME dataset**: The authors *somehow* selected 2k materials from GNoME dataset, which does not make sense to me that these are naturally *out of distribution*. I suggest the authors conduct structure matching test to see how many in the 2k materials are truly out of distribution. Otherwise, this is not good enough to claim that this is an evalutation on OOD materials.\n\n    b) There is a property that the charge densities have to satify:\n    $$\\int \\rho(\\mathbf{r}) \\mathrm{d}\\mathbf{r} = \\text{No. of electrons}$$\n    How did you enforce this in your model?\n\n    c) The evaluation of the prediction of charge densities is based on the MAE with respect to the ground truth, which makes sense in terms of developing ML models. However, I am interested to see how much error in total energy or band gap if the predicted charge densities are put back to DFT iterations. These are the realistic application of the charge densities, otherwise one can not really infer any useful information from predicted charge densities. The authors may consider commenting on this or conducting one or a few examples to show the performance.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "* Originality: Good. The data generated in this work is indeed original.\n* Quality: Good. The functionals PBE and HSE used in this work are indeed the widely accepted ones to compute crystalline inorganic materials. Thus, the data generated in this work are of good qualities.\n* Clarity: The paper is well-structured, with a clear explanation of the dataset’s construction, tasks, and evaluation metrics. \n* Significance: This is a significant contribution to this area. Due to the computational cost to carefully converge the charge densities, it is not easy to generate or to find publicly available data sets. Thus, this work will clearly benefit the researchers in this area.", "weaknesses": "* Limited High-Precision Data: I totally understand that the HSE calculations are much more expensive, but it would be much better if the authors can keep increasing the number of calculations done with HSE functional.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Oczl9JuPN0", "reviewer": "Reviewer_Tq9X", "summary": "The author introduces a new benchmark for predicting electronic charge density for crystalline inorganic materials. The dataset includes 140,646 stable crystal geometries with medium-precision PBE functional, and high accuracy HSE from subsets of the meticulously curated geometries.", "questions": "1. Is there a fundamental difference between predicting the Kohn-Sham Hamiltonian and electron density? The author should discuss the computational challenges and benefits of predicting electron density, as several prior works have focused on predicting the Kohn-Sham Hamiltonian.\n\n2. Can the MAE of electron density be compared to that of a Kohn-Sham Hamiltonian prediction model?\n\n3. In line 528, \"due to the high dimensionality of charge density data, often involving over 1,000 nodes, the complexity of handling node and edge interactions presents significant computational challenges\"—what do \"node\" and \"edge\" refer to here?\n\n4. What does the regression output look like for a given structure? A more detailed example or description of the matrix's size and shape would be helpful.\n\n5. How does the model perform on predicting HOMO and LUMO?\n\n6. Could the author provide a more detailed discussion on error? With an HSE error of 1 eV, what is the impact on practical applications like energy band predictions? For a usable model, is 1 eV sufficient for downstream applications?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The computational efforts used in this study is immense, and the resulting benchmark could be a valuable asset to the ML community.\n2. Interesting task on predicting the gaps across different functionals.", "weaknesses": "1. The work provides minimal justification for predicting electrons instead of the Kohn-Sham Hamiltonian.\n2. There is no comparison with other quantum chemistry benchmarks, like QH9. This will make the paper's calculation comparable in the literature.\n3. The EXP-id lacks detail on its 47 entries, which is essential for practical use. Additionally, the assessment of practical application is sparse.\n4. The writing lacks cohesion, with many GPT-like passages. Figure 2 is unclear: does moving up or down the ladder imply increased simplicity?\n5. The metrics are limited to MAE for charge density and acceleration ratio. Other measures, such as total energy, HOMO-LUMO gap, electronic spatial extent, or dipole moment, should be included, as this is particularly important for practical applications.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Oral)