Title: A Recipe for Charge Density Prediction
Venue: NeurIPS 2024
Accepted: yes
Keywords: AI for science

density functional theory

charge density

equivariant

molecule
Abstract: In density functional theory, charge density is the core attribute of atomic systems from which all chemical properties can be derived. Machine learning methods are promising in significantly accelerating charge density prediction, yet existing approaches either lack accuracy or scalability. We propose a recipe that can achieve both. In particular, we identify three key ingredients: (1) representing the charge density with atomic and virtual orbitals (spherical fields centered at atom/virtual coordinates); (2) using expressive and learnable orbital basis sets (basis function for the spherical fields); and (3) using high-capacity equivariant neural network architecture. Our method achieves state-of-the-art accuracy while being more than an order of magnitude faster than existing methods. Furthermore, our method enables flexible efficiency-accuracy trade-offs by adjusting the model/basis sizes.

A Recipe for Charge Density Prediction

Abstract

In density functional theory, the charge density is the core attribute of atomic systems from which all chemical properties can be derived. Machine learning methods are promising as a means of significantly accelerating charge density predictions, yet existing approaches either lack accuracy or scalability. We propose a recipe that can achieve both. In particular, we identify three key ingredients: (1) representing the charge density with atomic and virtual orbitals (spherical fields centered at atom/virtual coordinates); (2) using expressive and learnable orbital basis sets (basis functions for the spherical fields); and (3) using a highcapacity equivariant neural network architecture. Our method achieves state-ofthe-art accuracy while being more than an order of magnitude faster than existing methods. Furthermore, our method enables flexible efficiency-accuracy trade-offs by adjusting the model and/or basis set sizes.

1 Introduction

Density functional theory (DFT) is a computational quantum chemistry method that has enabled countless advancements in the chemical sciences by providing a tractable means to calculate the electronic structure of molecules and materials [1]. The central concept in DFT is the charge density, a fundamental quantity from which all derivable ground-state physicochemical properties of a system, such as energy and forces, can, in principle, be derived. The most widely used Kohn-Sham formalism [2] of DFT offers a reasonable balance between accuracy and computational efficiency among conventional DFT workflows. However, it still scales with a complexity of roughly O ( N 3 e ) where N e is the number of electrons, rendering it computationally expensive and limiting its viability for both large-scale systems and long-timescale ab initio molecular dynamics simulations.

In DFT, the solution to the Kohn-Sham equations are reliant on an iterative calculation to identify the charge density that minimizes the potential energy functional for a given atomic configuration. This process, known as converging the self-consistent field, is the main computational expense within DFT. With a machine learning (ML) model that can effectively bypass the Kohn-Sham equations by accurately and efficiently predicting the charge density, the number of steps required to converge the ground-state electron density can be drastically reduced or potentially eliminated altogether by using the predicted charge density as the initial guess. If accurate enough, a machine-learned charge density could also be used to directly predict electronic structure properties, such as the band gap, band structure, and electronic density of states of a material. Furthermore, the charge density itself can provide an enormous amount of insight into a molecule or material. From the charge density, partial

+

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=

(a)

(b)

High

Low

atomic charges, dipole moments, atomic spin densities, and effective bond orders can all be directly computed through one of several population analysis methods [3, 4]. For some materials discovery tasks, the charge density can also be a crucial descriptor depending on the application area [5, 6]. Therefore, efficient and accurate representations and ML models for charge density prediction are highly desirable as a means of accelerating the discovery of promising molecules and materials.

In machine learning workflows, the charge density is a volumetric, data-rich object, usually represented as voxels with a grid resolution of around 0 . 1 ˚ A [7, 8]. This poses a challenge, as even relatively small molecules and materials can require hundreds of thousands to millions of grid points to represent the charge density at this (relatively coarse) resolution. At the same time, small deviations in the charge density that result from a representation that is too coarse can have a substantial impact on energy and other derivable properties. This need for both efficiency and accuracy creates a significant challenge for ML methods.

The existing literature has mainly focused on two approaches to learning to predict charge density. The first approach (orbital-based), illustrated in Figure 1 (a), is to predict atomic orbital basis set coefficients by regressing over coefficients extracted from DFT data [9, 10, 11, 12, 13]. The atomic orbital basis functions are based on the composition of radial functions and spherical harmonics. Under this scheme, the charge density is represented as a set of spherical fields centered around each atom. The real space charge density voxel can be constructed by overlaying the spherical fields and evaluating at each grid point. For orbital-based ML models, both the prediction of the basis set coefficients and the evaluation of the spherical fields are relatively scalable, making this approach efficient at inference time. However, this approach can suffer from sub-optimal accuracy due to the limited representation power of the chosen basis set. In particular, it is challenging for the atom-centered atomic orbitals to model complex electronic structures between atoms.

The second approach (probe-based) [14, 7, 15, 16], illustrated in Figure 1 (b), is to predict the charge density by inserting 'probe nodes' at all grid coordinates of the charge density voxel and applying graph message passing between the atoms and these probe nodes. Finally, the scalar charge density at each grid coordinate is predicted through node-wise readout over the probe nodes. This approach, while expressive and accurate, is computationally expensive. To see why, recall that the number of grid points in the charge density voxel is usually very large for even a small atomic system. Conducting neural message passing over millions of nodes is both computationally and memory intensive. The large number of nodes usually requires sampling a subset of grid points from the charge density voxel (Figure 1 (b), right) in each training or inference step [7].

This paper aims to address this accuracy-efficiency dilemma with a new recipe for building representations and ML models for charge density prediction. We identify three key ingredients:

We represent the charge density using an atomic orbital basis set (spherical fields centered at each atom) to leverage its efficiency and equivariant properties. Beyond orbitals placed

at the atomic coordinates, we further introduce virtual orbitals to improve expressivity. In other words, we also place spherical fields centered at coordinates other than the atomic centers, while ensuring the placement algorithm is SE(3)-equivariant.

We use domain-informed and expressive basis sets. In particular, we construct an eventempered Gaussian basis from an atomic orbital basis set. This allows us to smoothly control the expressivity of the atomic orbitals and enable flexible accuracy-efficiency trade-offs. We make the basis set exponents learnable to further improve expressivity.

We use a high-capacity equivariant neural network architecture (eSCN [17]), which enables efficient training and inference with features of high tensor order for a large dataset.

We apply our recipe to the widely used QM9 charge density benchmark [18, 19, 7]. Our method outperforms existing state-of-the-art methods while being around 30 × faster. Furthermore, we can flexibly trade off accuracy and efficiency by adjusting the model/basis size; in doing so, we achieve up to 171 × efficiency compared to state-of-the-art methods with only a slight degradation in accuracy. This tunability is valuable, as different applications, material classes, and available computing resources may require drastically different levels of accuracy in the charge density prediction. We conduct an ablation study to justify the significance of each proposed ingredient.

2 Related Works

ML methods for charge density prediction. Orbital-based methods predict coefficients for the orbital basis set functions to recover the target charge density. Past works have explored Gaussian processes [9] and graph neural networks [10, 11, 12, 13] in small molecules, water, and materials systems. [20] used Jacobi-Legendre expansion - a many-body extension of atomic orbitals - for representing and predicting the charge density. These approaches, while efficient, suffer from lower accuracy in benchmarks such as QM9 [18, 19, 7] and the Materials Project [21, 8] charge density datasets. Probe-based methods, on the other hand, predict the charge density by neural message passing between the atoms and probe nodes at all grid points. These methods [14, 7, 15, 16] have shown superior accuracy in both molecules and materials but suffer from poor scalability, as they require neural processing of millions of probe nodes for molecule/material structures of tens of atoms. Recent works also explored a combination of atomic orbitals and probe-based methods [12] or plane-wave basis sets [22]. However, both methods still require neural message passing with a large number of probe nodes, which limits their scalability. In the present work, we combine virtual nodes, even-tempered Gaussian basis, and trainable basis functions to greatly improve the expressivity of orbital basis functions.

Equivariant neural networks . Equivariant neural networks [23, 24, 25, 26, 27, 28, 29, 17] use equivariant representations and processing layers that can preserve rotational and translational symmetries that are critical to atomistic modeling tasks. Equivariant models have shown advantages in ML potentials with respect to the accuracy, sample complexity, and molecular dynamics simulation capabilities [30, 31, 32, 33] in addition to charge density prediction tasks [7, 15]. This is because atomic forces and charge densities are indeed SE(3)-equivariant with regard to the input atomic coordinates. In this work, we leverage recent advances in methods for building more expressive and scalable equivariant architectures [17] to improve the accuracy and scalability of charge density prediction.

3 Methods

Our recipe for building ML charge density prediction capabilities involves two complementary aspects: the charge density representation and the prediction model.

3.1 Charge Density Representation

Gaussian-type orbitals (GTOs) are widely used as basis sets for representing electron configurations in quantum chemistry [34]. They are spherical Gaussian functions centered at atomic coordinates. For an atom i at coordinate r i , a GTO basis function with exponent α , angular momentum quantum number (also called tensor order or degree) l , and magnetic quantum number m is given by the

(b)

600

def2-QZVPPD

β=2.0

β=1.5

400

β=1.3

300

200

B

100

0

N

following expression:

where r = || r -r i || is the distance from a query coordinate r to the atom coordinate r i and Y l,m are real spherical harmonics. z α,l is a normalizing constant, such that ∫ R 3 || Φ || 2 2 d V = 1 . For the purpose of developing a machine learning model based on GTOs, we choose to represent the charge density, ρ , of an atomic system via a linear combination of many basis functions:

where N is the number of atoms (including virtual ones when applicable), N i b is the number of l values ( α values) for atom i . It should be noted that the charge density in Kohn-Sham DFT is not computed in this way; rather, Equation (2) is an artificial representation for the sake of training a machine learning model that is inspired by the orbital-like character of GTOs. The basis functions Φ α i,j ,l i,j ,m, r i are chosen first as the basis set, with a fixed set of l and α values for each element. For example, the values for l and α for hydrogen in the def2-QZVPPD basis set [35] are presented in N i

Appendix A, Table 2. The number of basis functions for an atom i can be derived as ∑ b j =1 (2 · l i,j +1) , because m can be an integer from -l to l . A higher l value corresponds to a more complex angular part of the basis function and allows the corresponding spherical field to be more anisotropic. The number of orbital basis functions for elements H, C, N, O, and F of the def2-QZVPPD basis set (and its even-tempered variant, detailed later in this section) is included in Figure 2. Atoms with more complex electronic structures are often represented with more basis functions. The number of basis functions, l s, and α values are carefully chosen in existing basis sets such as def2-QZVPPD . We refer interested readers to the original papers [35, 36, 37] for more details regarding the construction of atomic orbital basis sets.

In training the machine learning model, after the basis set is determined, the coefficients c i,j,m are then fit such that Equation (2) best represent the charge density. GTOs have been studied in several previous works [9, 11, 12] as a means of representing the charge density with promising results. However, their accuracy still bears significant room for improvement. We next introduce virtual orbitals, even-tempered Gaussian basis, and scaling factors for orbital exponents that greatly improve the expressive power of GTOs for charge density representation.

Virtual orbitals. The atom-centered spherical fields often struggle to capture non-local electronic structures, which induces representation errors. This limitation is effectively addressed with the introduction of virtual orbitals, which define sets of spherical fields located in a position other than the atomic centers. Due to the critical importance of chemical bonds in defining the overall electronic structure, we insert virtual nodes into the midpoint of all chemical bonds for a given molecule (illustrated in Figure 2 (a)). With this method, the coordinates to insert the virtual nodes are SE(3)-equivariant with regard to the input atom coordinates. Therefore, as long as the prediction of the basis set coefficients is SE(3)-equivariant, the overall charge density prediction will still be equivariant after the introduction of the virtual orbitals. We discuss potential extensions to virtual

orbital assignments in Section 5. After the virtual nodes are created, one must decide which basis functions to use for the virtual orbitals. In this work, we use the basis functions of element O for the virtual nodes, which offers a balance in accuracy and efficiency based on preliminary experiments.

Even-tempered Gaussian basis. The number of basis functions in existing basis sets, such as def2-QZVPPD , may be insufficient for representing complex charge densities. At the same time, expanding the number of basis functions requires care in choosing the values of l and α that improve expressivity effectively. As an example, the def2-QZVPPD basis set for hydrogen already contains basis functions with l = 1 and α = 2 . 292 . Extending this basis set with basis functions with l = 1 and α = 2 . 0 will not significantly improve its expressivity because the spherical pattern will be similar to existing basis functions. A general methodology for controlling the basis set size is to use an even-tempered Gaussian basis set [38]. Based on a reference atomic orbital basis set (e.g., def2-QZVPPD ), the even-tempered basis set constructs a series of GTOs with a set of angular momentum quantum numbers l determined by the atomic number and exponents α given by:

For each spherical harmonics degree l , α and N l are chosen such that the exponents in the reference atomic orbital basis set are well-covered 2 . β controls the number of basis functions - a smaller β creates a more expressive basis set with denser exponents. The use of an even-tempered basis set allows us to smoothly control the number of basis functions N i b effectively. Figure 2 (b) shows how the number of orbital basis functions for elements H, C, N, O, and F grows with a smaller β for the even-tempered Gaussian basis derived from the def2-QZVPPD basis set.

Scaling factors for orbital exponents. In existing orbital-based models [9, 10, 11, 12, 13], while the coefficients for the basis functions are predicted by the ML model, the exponents are fixed for each atom type and not trainable. However, atoms in different local atomic environments can exhibit significantly different charge density patterns around them, especially for the virtual orbitals that aim at capturing interatomic interactions. To further improve the expressivity of the basis set, we make the exponents trainable by learning a positive scaling factor s > 0 , such that Equation (1) becomes:

where z α,l,s is a normalizing constant such that ∫ R 3 || Φ || 2 2 d V = 1 . The charge density is now represented with coefficients c i,j,m and scaling factors s i,j as:

the introduction of the learnable scaling factors for the exponents significantly improves the expressive power of our charge density representation but is also prone to instability during training. We resolve the instability issue with a fine-tuning approach detailed in Section 3.2.

3.2 Prediction Model

Using the atomic orbital basis set representation of charge density, the prediction model aims to predict the basis set coefficients c i,j,m and the scaling factors s i,j for each real and virtual node such that the predicted charge density matches the ground truth density obtained from DFT calculations. The model F takes as input the types A = { a i | i = 1 , . . . , N } and coordinates R = { r i | i = 1 , . . . , N } of all real and virtual nodes:

Backbone architecture. Our construction of the ML prediction model is motivated by the following:

Charge density is SE(3)-equivariant with regard to the input atom coordinates. An equivariant model that can preserve this symmetry is desired. Concretely, the basis set coefficients c i,j,m are SE(3)-equivariant with regard to the input atom coordinates. The scaling factors s i,j are SE(3)-invariant with regard to the input atom coordinates.

Charge density is data-rich and very sensitive to the local atomic environment. A highcapacity and expressive model is desired.

Efficiency is key for general applications of charge density prediction. The model should be efficient while being expressive.

Based on these criteria, we consider equivariant model architectures and the balance between capacity and efficiency. For equivariant models, an important aspect of model expressivity is the representation of node and edge features in the form of irreducible representations (irreps) of SO(3) 3 : spherical harmonic coefficients. A higher degree of representation ( L ) is desired for building highcapacity models. Previous works have employed the PaiNN architecture [27, 7] that is based on Cartesian features (equivalent to L = 1 ) or architectures based on irreps of SO(3) and tensor products [10, 15, 12, 28] for charge density prediction. However, these models suffer from limited expressivity or scalability. Cartesian features are limited in representing angular information ( L = 1 ); meanwhile, the O ( L 6 ) complexity of tensor products limits the degree of representation that can be used while remaining computationally feasible.

In this work, we adopt the equivariant spherical channel network (eSCN) architecture [17] as our model backbone. While using SE(3)-equivariant representations and processing layers, the convolution layers in eSCN reduce the SO(3) convolutions [41] or tensor products [23, 31] to convolutions in SO(2) that are mathematically equivalent. It reduces the complexity of the convolution operation from O ( L 6 ) to O ( L 3 ) . Further, the use of point-wise, spherical non-linearity in eSCN also distinguishes itself from e3nn-based equivariant models that only apply non-linearity to the scalar features in the irreps. In our experiments, we also find that eSCN outperforms alternative architectures, such as tensor field networks [15, 23] and MACE [42]. Using eSCN as the backbone architecture, we get the last-layer latent features x i for all real/virtual nodes:

Prediction layers. The features x i are encoded using multi-channel spherical harmonic coefficients (irreps). Note that the prediction target, basis set coefficients c i,j,m , are also encoded as multi-channel spherical harmonic coefficients. For example, for an eSCN with L = 3 and a latent dimension of 128 , the last-layer latent atom features will be 128x0e + 128x1o + 128x2e + 128x3o . For the (uncontracted) def2-QZVPPD basis set of hydrogen described in Table 2, its irreps are 7x0e + 4x1e + 2x2e + 1x3e (even parity as charge density is reflection-invariant). The scaling factors s i,j are SE(3)-invariant and can be seen as scalar features of multi-channel irreps ( 14x0e for the def2-QZVPPD basis set of hydrogen). Therefore, we can make equivariant predictions of the basis set coefficients and invariant predictions of the scaling factors for each atom i through a fully connected tensor product layer over the atom features and additional processing:

The basis set coefficients are directly obtained through the fully connected tensor product. The tensor product also produces scalar features h i ( 128x0e for a 128-channel eSCN), which are used for predicting the scaling factors. The parameterization of Equation (9) allows the prediction to range from ( C 3 , C 1 + C 3 ) , and the scaling factors will be C 1 / (1 + C 2 ) + C 3 when the linear network in Equation (9) is zero-initialized. By setting C 1 = 1 . 5 , C 2 = 2 , and C 3 = 0 . 5 , we can limit the range of the scaling factors to be (0 . 5 , 2) (at most halve or double an exponent) and let initial scaling factors be 1 with a zero initialization of the linear layer in Equation (9).

With the predicted coefficients and scaling factors, the charge density prediction ˆ ρ can be obtained efficiently by evaluating Equation (5) at all grid coordinates of the charge density voxel. We train the model end-to-end with a mean-absolute error loss L over the charge density:

Fine-tuning for scaling factor prediction. The scaling factors at the exponents lead to significant training instability when the network is trained from scratch. Therefore, we use a fine-tuning approach, where we first pre-train the model with fixed basis set exponents (an even-tempered Gaussian basis derived from def2-QZVPPD ) and then fine-tune the prediction model with a small learning rate with the learning for scaling factors enabled. To achieve this, we zero-initialize the linear layer in Equation (9) and freeze its weights until the fine-tuning stage.

4 Experiments

Our experiments on the QM9 charge density benchmark aim to validate the effectiveness of our proposed recipe in both accuracy and efficiency. We refer to our method as SCDP models, which stands for S calable C harge D ensity P rediction models.

Dataset and metrics. The QM9 charge density dataset [18, 19, 7] contains charge density calculations for 133,845 small organic molecules using the Vienna Ab initio Simulation Package (VASP). We adopt the original split, where 123,835, 50, and 10,000 data points are used for training, validation, and testing, respectively. There are on average 18 atoms in each molecule and 666,462 grid points in each charge density voxel. The entire dataset takes 1.1 TB of disk space. Following previous works [7, 15], we benchmark the prediction accuracy with the normalized mean absolute error, defined as:

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Efficiency (Mol. per min. [min

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=3,

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=8,

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=6,

=2.0

K

=8,

L

=6,

=1.5

VO,

K

=8,

L

=6,

=2.0

VO,

K

=8,

L

=6,

=1.5

VO,

K

=8,

L

=6,

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with scaling factors

without scaling factors

where the integration is approximated by summing over the full charge density voxel. We benchmark the efficiency of different methods by the number of molecules predicted per minute (Mol. per min.) on a single NVIDIA A100-80GB-PCIe GPU for the QM9 test split.

In addition to the QM9 charge density dataset, we also benchmark our method on the MD charge density dataset [43, 44, 12] and the Cubic charge density dataset [45]. We use the same data splits as previous works on these benchmarks [12, 22]. Experimental results and comparisons to baselines are included in Appendix A.

Baseline Models. We compare SCDP to several previous works on the QM9 charge density prediction benchmark [18, 19, 7]. i-DeepDFT, e-DeepDFT [7], and ChargE3Net [15] are probe-based methods with different backbone architectures: i-DeepDFT uses SchNet [27], e-DeepDFT uses PaiNN [27], while ChargE3Net uses higher-order equivariant features under the tensor field network framework [23, 28]. InfGCN [12] combines GTOs and a shallow network for probe-based inference. It also has a more efficient but less accurate GTO-only variant. GPWNO [22] combines GTOs and plane-wave basis sets but still requires a large number (64,000) of probe nodes for constructing the plane wave prediction. NMAE results for i-DeepDFT, e-DeepDFT, and ChargE3Net are from [15]. NMAE results for InfGC and GPWNO are also from the original papers, which uses a different test split from the default QM9 test split (last 1,600 molecules from the QM9 test split). We benchmark the efficiency of baseline models on our hardware when the source code and pretrained model are publicly available (ChargE3Net and InfGCN). We do not apply any modification to the original code but use optimized configurations for inference to better utilize our GPU: for ChargE3Net, we process 20,000 probes in each batch instead of the default setting of 2,500 probes per batch, and for InfGCN, we process 40,000 probes in each batch with a batch size of 4.

A significant advance in both accuracy and efficiency. The metrics for all methods are presented in Table 1. We have a series of SCDP models with different model sizes, basis set sizes, as well as options on the inclusion of virtual orbitals and scaling factors. Our best-performing model uses the virtual orbitals described in Section 3.1, an eSCN of 8 layers and feature representation of order L = 6 , an even-tempered Gaussian basis with β = 1 . 3 , and scaling factor fine-tuning. This model achieves an NMAE of 0 . 178 on the QM9 charge density test set, outperforming the state-of-the-art method ChargE3Net [15] - a probe-based method. While being more accurate, our best model also significantly outperforms ChargE3Net by 31 . 7 × in efficiency. Other configurations of our model with smaller model sizes, basis set sizes, and models without virtual orbitals can trade off accuracy for further gains in efficiency. The trade-off curves are visualized in Figure 3. Compared to a more efficient baseline model, InfGCN, all benchmarked configurations of our method are more efficient and significantly outperform in accuracy. These results convincingly demonstrate a significant advance in the accuracy-efficiency trade-off in ML methods for charge density prediction. Figure 5 in Appendix A shows the convergence of validation NMAE during pretraining and fine-tuning of SCDP models. More details on the hyperparameters for model construction and training are included in Appendix A, Table 3.

Ablation Analysis. We discuss the effectiveness of all ingredients through an ablation analysis of the performance of different SCDP models. Starting from the most lightweight model with

VO+Scaling

-0.05

0.05

0.05

7.25

K

= 4,

L

= 3,

β

= 2.0

K

= 8,

L

= 6,

β

= 2.0

K

= 8,

L

= 6,

β

= 2.0

K

= 8,

L

= 6,

β

= 2.0

K

= 8,

L

= 6,

β

= 1.3

VO

VO+Scaling

0.05

7.25

-0.05

0.05

K = 4 , L = 3 , β = 2 . 0 and no virtual orbitals, we first observe that increasing the model size to K = 8 , L = 6 significantly improves the performance, reducing the NMAE from 0.504% to 0.434%. Next, we increase the basis set size by adjusting β from 2 . 0 to 1 . 5 , which further reduces the NMAE to 0.381%. The introduction of the virtual orbitals renders a significant gain in accuracy by reducing the error from 0.434% to 0.237% for β = 2 . 0 and from 0.381% to 0.196% for β = 1 . 5 . In particular, the charge density near chemical bonds is significantly more accurate after introducing the virtual orbitals, as visualized in Figure 4. On the other hand, for models with higher capacity, the improved accuracy comes at the cost of efficiency. As shown in Table 1 and Figure 3, higher capacity consistently improves performance while sacrificing efficiency. At the same time, all SCDP models remain highly efficient compared to baseline models. When the scaling factors are introduced, accuracy further improves at a slight cost on efficiency for all models. As shown in Figure 3, models with scaling factors from the Pareto front of all SCDP models benchmarked.

5 Discussion

Charge density is a fundamental quantity for atomic systems and is central to DFT. ML methods for charge density prediction are promising as a means of greatly accelerating DFT by circumventing the iterative procedure used to find the ground-state charge density given a set of atomic coordinates. In this paper, we propose a recipe that combines three ingredients: (1) virtual nodes; (2) expressive basis sets; and (3) high-capacity equivariant networks that collectively outperform state-of-the-art methods in accuracy while being more than an order of magnitude faster. Nevertheless, there are still many directions for further improving the performance of our proposed model. First, the simple heuristic of assigning virtual node coordinates to bond centers may not be optimal. With recent advances in auto-regressive [46] and diffusion-based [47, 48] equivariant generative models for 3D atomic structures, learning to insert the virtual orbitals may be a promising avenue for optimizing the placement of virtual nodes, thus improving charge density prediction. Due to the 'nearsighted' nature of electronic matter [49], an automated method for placing a higher density of virtual nodes near sites of chemical relevance may also be worthwhile to pursue. Second, we can use basis functions beyond Gaussian-type orbitals (e.g., Slater-type orbitals [50, 51] or non-decay radial basis functions [52]) that may require fewer functions to achieve the same level of accuracy.

There are several limitations of the current paper that we aim to address in future work: (1) Despite substantial improvements in efficiency, the computational cost for training the current model is still significant: our best-performing model was pretrained for six days and fine-tuned for six days over four NVIDIA A100 GPUs for the QM9 charge density prediction task. The scaling factor fine-tuning

stage requires a small learning rate, which prolongs training. The prediction model can benefit from resolving the training instability issues with the scaling factors as well as further improvement on the model architecture [53]. (2) While our approach achieves state-of-the-art performance on the QM9 charge density prediction benchmark, its effectiveness in crystalline materials [21, 8] has major room for improvement. The GTOs and the equivariant network can be applied to materials without modification. The bond-midpoint-based virtual node assignment for molecules can be generalized to crystals through a crystal graph construction algorithm, such as CrystalNN [54]. Alternatively, virtual nodes can be iteratively added to occupy void space inside the unit cell of the material using an algorithm based on the Voronoi diagram [55]. The virtual nodes are expected to play an even more important role in prediction accuracy - this is because the diverse atomic species in materials and their complex interactions induce even more complex charge density patterns. (3) To better validate the practical utility of the predicted charge density, evaluation on the reduction of self-consistent field calculations, or on recovering physical observables, such as energy and forces [7, 56, 15], will be highly valuable.

Acknowledgments and Disclosure of Funding

We thank Teddy Koker, Chaoran Cheng, and Aria Mansouri Tehrani for their helpful discussions and insights. This work was supported by the GIST-MIT Research Collaboration grant funded by GIST and the Machine Learning for Pharmaceutical Discovery and Synthesis (MLPDS) consortium. A.S.R. acknowledges support via a Miller Research Fellowship from the Miller Institute for Basic Research in Science, University of California, Berkeley.

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A Appendix

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Algorithm 1 Pseudo code for the charge density prediction procedure

1: Input : Atomic numbers A : ( N, 1) , atom positions R : ( N, 3) , grid positions R c : ( M, 3)

2: Output : charge density at the grid positions C : ( M, 1)

3: obtain atomic irreps features: X = { x i | i = 1 , . . . , N } = eSCN( A , R )

4: obtain basis set coefficients { c i,j,m , h i | i = 1 , . . . , N, j = 1 , . . . , N i b ; m = -l i,j , . . . , l i,j } and scaling factors { s i,j | i = 1 , . . . , N, j = 1 , . . . , N i b } = C 1 / (1+exp( -Linear( h i )+ln C 2 ))+ C 3 following Equation (8) and (9)

5: for r in R c do

6: obtain ρ ( r ) following Equation (4) and (5)

7: end for

8: return C = { ρ ( r ) | r ∈ R c }

Experimental results on MD and Cubic. We benchmark the proposed SCDP models on the MD [43, 44, 12] and the Cubic charge density dataset [45] in Table 4. We find the SCDP models significantly outperform baseline models. Hyperparameters used for the MD and Cubic experiments are summarized in Table 5 and Table 6. The MD models are trained on 4 GPUs, while the Cubic model is trained on 8 GPUs. For the molecules in the MD dataset, we use bond centers as coordinates for virtual nodes. For the materials in the Cubic dataset, we iteratively insert virtual nodes up to the number of atoms in the unit cell using an algorithm based on Voronoi diagrams [55].

Software. Basis-set-exchange-v0.9.1 [36, 37] and PySCF-v2.5.0 [39] are used to build the orbital basis sets. E3NN-v0.5.1 [28], PyTorch-v1.13.1 [59], and CUDA-v11.6 [60] are used to build the SCDP models. The eSCN [17] implementation is adopted from the Open Catalyst Project [61]. We also acknowledge Numpy [62], ASE [63], Pymatgen [64], wandb [65], Matplotlib [66], and Plotly [67].

▷ in practice, we use batched inference

12

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125

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Mol. per min. [min

1

]

B Broader Impact

This paper proposes an ML method for accelerating charge density prediction, a crucial task in computational chemistry. The adoption of our method is useful for scientific discovery and can yield positive or negative repercussions, contingent on the applications. The proposed method should be used for materials and drug discovery research that benefit our society.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a new method for calculating charge density using orbital based representations along with an equivariant machine learning architecture. The paper starts by introducing density functional theory (DFT) and how charge density plays a central role for DFT. The important role of charge density in DFT motivates the need for a fast, accurate and robust method of calculating it creating an opportunity to apply machine learning to the problem. The introduction also describes how charge density is a volumetric object, which can it difficult to apply ML methods due to data density and how prior methods have aimed to address the overall challenge. After outlining the primary contributions of the paper, Section 2 provides greater detail for related work in ML methods for charge density prediction and equivariant neural networks.\n\nSection 3 describes the primary method, which focuses on charge density representation and the prediction model. The charge density representation includes Gaussian-type orbitals approximated as a linear combination of spherical harmonics basis functions, as well as virtual orbitals to capture non-local electronic structures and interactions. The charge density approximation also includes trainable orbital exponents for each atom type making the representation more expressive. For the prediction model, the paper motivates the need for an equivariant and expressive model architecture and describes the eSCN-based architecture used for the experiments. The last paragraph of the section outlines a pretraining based approach to enhance training stability for the charge density prediction model.\n\nSection 4 describes the experimental results performed on the charge density dataset for the QM9 database of molecules. The results include a comparison to other methods, as well as an ablation analysis related to the different SCPD models' trade-offs for accuracy and computational efficiency. The paper subsequently ends with a discussion along with a description of limitations that can be used for future research work.\n\nSTRENGTHS:\nThe paper proposes a new method for a relevant problem in AI for scientific applications. Its strengths include:\n* A new method for calculating charge density and integrating ML methods into computational chemistry workflows. [Originality, Significance]\n* A detailed description of the method, relevant background and experimental details. [Quality, Clarity]\n* Experimental results show convincing improvement compared to the state-of-the art. [Quality, Significance]\n\nWEAKNESSES:\nThe paper could be improved by providing more detail along the following:\n* Provide more details on how charge density prediction is different from ML potentials and why that distinction matters. Concretely, what advantages and disadvantages does charge density prediction have over ML potentials and vice versa? [Significance, Clarity]\n* Provide greater clarity on how their proposed method for Charge Density Representation differs from prior approaches, such as a summary table. [Clarity, Significance]\n* Address the questions below.\n\n-----------\n\nAssessment updated during discussion period.\n\nQUESTIONS:\n* Were there other options you considered for the def2-QZVPPD basis set? Can you describe your considerations and/or if it's reasonable to perform an ablation related to this? You somewhat discuss this in \"Even-tempered Gaussian basis.\" but its unclear to me if an empirical evaluation is feasible.\n* Can you include traditional DFT throughput numbers in the table for reference? This is to get a sense of the speedup.\n* Can you describe in more detail why an equivariant network is required? Have you tried evaluating non-equivariant networks?\n\nLIMITATIONS:\nThe paper includes a discussion on limitations in Section 5.", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new effective approach to estimating the charge density of molecular systems using machine learning models. Although this topic has been recently actively studied, the present approaches suffer from either a lack of accuracy or scalability. The proposed approach alleviated the problem by identifying three key ingredients: (1) utilizing atomic and virtual orbitals, (2) utilizing expressive and learnable orbital basis sets, and (3) utilizing high-capacity equivariant neural network architecture. The proposed approach achieved state-of-the-art accuracy on QM9 while reducing computational cost.\n\nSTRENGTHS:\n1. By combining domain domain-knowledge (GTOs, Even-tempered Gaussian basis, etc) and new machine learning techniques (virtual orbitals, scaling factors for orbital exponents), the method achieves around 10% error reduction with around 30 times better efficacy in comparison to the previous state-of-the-art model (ChargE3Net).\n\n2. \"Methods\" are well-described with detailed explanations from physical and engineering point of view.\n\nWEAKNESSES:\nThe following (at least major) weaknesses should be approached before acceptance: \n\nMajor weakness (the upper, the higher priority): \n\n1. The paper only performed the experiments on QM9 and I wonder if the method is over-fitted to the QM9 dataset. The previous methods (InfGCN, ChargE3Net, GPWNO) performed their experiments on average 3-dataset, e.g., among NMC, Cubic, MD, MP, and QM9. Since this method's scope is now limited to molecular systems, either MD or a molecule dataset used in [1] can be utilized. Because of the time limitation of the rebuttal period, at least logical validation of the effectiveness of the other dataset would be necessary (Of course the additional experiments are highly recommended, at least before publication).\n\n2. Although not so crucial, the code is not provided in supplemental material. So, at this stage of this review, I do not have confidence that the paper's result can be reproducible. And I wonder why the authors did not submit it even though mentioning they would provide it after acceptance. Are there any proper reasons to hide the code from the reviewers?\n\n3. Even though the author mentions at line 214-215 that eSCN outperforms tensor field networks and MACE, I cannot find the corresponding part in Section 4. An explicit indication of the experiment would be desirable. \n\n[1] Jørgensen, P.B., Bhowmik, A. Equivariant graph neural networks for fast  electron density estimation of molecules, liquids, and solids. npj Comput Mater 8, 183 (2022). https://doi.org/10.1038/s41524-022-00863-y\n\nMinor (for further improvement): \n\n1. The explanation in \"Prediction layers\" can be improved. It would be strongly recommended to provide more information in the Appendix using more space.  \n\n2. From the description of \"Prediction layers\", this method is a little difficult to apply to other models than eSCN. If not, it would be helpful for the readers that the authors provide additional \"recipe\" for the other models. \n\n3. The inference time scaling in terms of the molecular size (atom number) is missing, which would be important to apply for larger molecular systems, for example, protein, on which DFT has difficulty performing the calculation.\n\nQUESTIONS:\n1. The idea of the scaling factors for orbital exponents in Equation (4) seems to have a relation with the D4FT [2] Equations (20) and (21). Is the idea independently developed? It would be better to cite the paper and explain the difference. \n\n2. At line 102, roto-translational --> rotation-translational     (the former seems not strictly accepted term by all the community). \n\n3. In Table 1, the \"Mol. per min.\" shows around 10% increases in SCDP when adding scaling factors. Why? Equation (4) indicates that the scaling factor would not increase the model complexity and size so much. \n\n[2]Li, Tianbo, et al. \"D4FT: A deep learning approach to Kohn-Sham density functional theory.\" arXiv preprint arXiv:2303.00399 (2023).\n\nLIMITATIONS:\nYes. It is provided in the \"Discussion\" section.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-09"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nOverall this is a nice technical contribution towards the goal of machine-learning based prediction of charge densities trained from DFT calculations. The main merits are that the authors combine the recent eSCN equivariant network, which reduces the computational complexity of equivariant message passing compared to SO(3) message passing, and other existing ideas such as virtual atom/sites, and learnable (through the exponent rescaling) GTOs. The paper is well written and easy to follow. I have no doubt this is a valuable contribution. But my concern is lack of original ideas (all ideas were existing or straightforward) and lack of high impact application. To me you need to have at least one for acceptance.\n\nSTRENGTHS:\nMotivations are clear. Technical details are well presented and easy to follow. The experiments are also clear.\n\nWEAKNESSES:\nMain problem is that the ideas are mainly straight application of existing algorithms and perhaps codes. It is not trivial to put all these pieces together but still I do not see much novelty in ML. The main advantage in application is efficiency compared to e.g. chargeE3Net as the accuracy is very similar. First, the issue of efficiency matters most when one has a killer application for this method. There can be but it is not demonstrated or even explained clearly. ML AIMD might be but we already have ML potentials for that. Second, the efficiency test of ChargeE3Net needs to be explained more clearly. Are we comparing both methods with the same batch size, molecular size, hyperparameters, etc?\n\nQUESTIONS:\nPage 5. \"where z_alph... =1\" should be in the preceding sentence.\nPage 6, equation 9. Are the hidden scalar features h_i the direct scalar output of eSCN? Or the output of FCTensorProd.?\nPage 6, lines 230. If I am not mistaken, should be \"c3, c1+c3\" and \"c1/(1+c2)+c3\"\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-11"}

comments: SUMMARY:
This application-oriented paper uses equivariant GNNs to predict charge density, represented orbital basis sets.

STRENGTHS:
- the idea of this paper is sound. Representing the charge density using atomic orbital basis set sounds more efficient than the voxel-based methods.
- the performance of this new method is evidenced by well-designed experiments
- the presentation of this paper is excellent

WEAKNESSES:
- I feel like the architectures are a simple variant of popular backbones. But this shouldn't affect the novelty of this application paper.
- the computation cost is still considerable

QUESTIONS:
You mentioned in the text that if you use a different backbone other than eSCN, the performance is worse. It would be great if you could share some numbers to back up this statement.

LIMITATIONS:
The limitations are sufficiently discussed in the paper.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2024-07-12