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.

在密度泛函理论中，电荷密度是原子系统的核心属性，可以通过它推导出所有的化学性质。机器学习方法在显著加速电荷密度预测方面具有潜力，然而现有方法要么缺乏准确性，要么缺乏可扩展性。我们提出了一个能够同时实现准确性和可扩展性的方案，其核心是使用原子和虚拟轨道来表示电荷密度，使用具有表达能力和可学习性的轨道基组，并使用高容量的等变神经网络架构。我们的方法在准确性上达到了最先进的水平，同时比现有方法快一个数量级以上。此外，我们的方法通过调整模型 / 基组大小实现了灵活的效率 - 准确性平衡。

电荷密度预测的配方

A Recipe for Charge Density Prediction

The calculation of electron density distribution using density functional
theory (DFT) in materials and molecules is central to the study of their
quantum and macro-scale properties, yet accurate and efficient calculation
remains a long-standing challenge in the field of material science. This work
introduces ChargE3Net, an E(3)-equivariant graph neural network for predicting
electron density in atomic systems. ChargE3Net achieves equivariance through
the use of higher-order tensor representations, and directly predicts the
charge density at any arbitrary point in the system. We show that our method
achieves greater performance than prior work on large and diverse sets of
molecules and materials, and scales to larger systems than what is feasible to
compute with DFT. Using predicted electron densities as an initialization, we
show that fewer self-consistent iterations are required to converge DFT over
the default initialization. In addition, we show that non-self-consistent
calculations using the predicted electron densities can predict electronic and
thermodynamic properties of materials at near-DFT accuracy.

通过使用等变图神经网络 ChargE3Net 来预测原子系统中的电子密度分布，本研究展示了比先前工作更佳的性能，并在规模更大的系统上可行于密度泛函理论 (DFT) 计算，同时能够预测材料的电子和热力学性质以及加速 DFT 的自洽迭代过程。

材料中电荷密度预测的高阶等变神经网络

Higher-Order Equivariant Neural Networks for Charge Density Prediction  in Materials

Equivariant graph neural networks force fields (EGraFFs) have shown great
promise in modelling complex interactions in atomic systems by exploiting the
graphs' inherent symmetries. Recent works have led to a surge in the
development of novel architectures that incorporate equivariance-based
inductive biases alongside architectural innovations like graph transformers
and message passing to model atomic interactions. However, thorough evaluations
of these deploying EGraFFs for the downstream task of real-world atomistic
simulations, is lacking. To this end, here we perform a systematic benchmarking
of 6 EGraFF algorithms (NequIP, Allegro, BOTNet, MACE, Equiformer, TorchMDNet),
with the aim of understanding their capabilities and limitations for realistic
atomistic simulations. In addition to our thorough evaluation and analysis on
eight existing datasets based on the benchmarking literature, we release two
new benchmark datasets, propose four new metrics, and three new challenging
tasks. The new datasets and tasks evaluate the performance of EGraFF to
out-of-distribution data, in terms of different crystal structures,
temperatures, and new molecules. Interestingly, evaluation of the EGraFF models
based on dynamic simulations reveals that having a lower error on energy or
force does not guarantee stable or reliable simulation or faithful replication
of the atomic structures. Moreover, we find that no model clearly outperforms
other models on all datasets and tasks. Importantly, we show that the
performance of all the models on out-of-distribution datasets is unreliable,
pointing to the need for the development of a foundation model for force fields
that can be used in real-world simulations. In summary, this work establishes a
rigorous framework for evaluating machine learning force fields in the context
of atomic simulations and points to open research challenges within this
domain.

本研究旨在通过对 6 个 EGraFF 算法进行评估和分析，以及提出新的基准数据集、指标和挑战任务，系统地评估机器学习力场在原子模拟中的能力和限制，并指出该领域存在的开放性研究挑战。