自适应有限差分：高精度电子结构计算

密度泛函理论（DFT）能够对材料性质进行精确的从头预测，解释广泛的实验数据。DFT预测的可靠性催生了材料基因组工程，通过DFT计算筛选大量可能的材料组分和结构，以确定那些具有优异性能的材料，进而用于实验合成与评估。

然而，全精度DFT计算在计算成本上可能相当昂贵，特别是对于复杂结构或者多组分系统，而且计算方法和程序的选择通常涉及计算精度和实用性之间的权衡。幸运的是，几种广泛使用的DFT程序产生的结果彼此一致，且已有元素周期表中大量元素的高精度计算作为基准。然而，对于更加复杂的系统或者大型调查研究，权衡依然存在，需要在计算成本、精确度、甚至是计算方法的选择之间寻求折中。

来自美国北卡罗来纳州立大学物理系的E. L. Briggs等，描述了一种利用实空间网格求解DFT方程的方法。通过使用自适应有限差分对动能算符离散化，可以使实空间的结果与基于平面波的程序相符。在显著减少网格密度的同时，能够以基于平面波的程序和全电子程序的精度复现基准DFT结果。

这一改进的离散化方案能够显著降低高精度实空间计算的成本，同时兼具实空间方法的优势，易于在多个节点上并行计算，而且避免了需要跨节点全局通信的快速傅里叶变换算法的使用。研究者对71种元素进行了著名的Δ测试，用于评估自适应动能算符在电子结构计算中的准确性，其平均误差与成熟的平面波程序相同。通过对NiO和具有一系列复杂成键排列的十水硼砂进行多物种测试，进一步确定了自适应算符的准确性。

Fig. 4 Differences in total energies between the reference energy obtained using Quantum Espresso and RMG, and timings.

作者通过对含有2016个原子的NiO超晶胞进行高精度计算，证实了实空间网格方法的可扩展性。该文近期发布于npj Computational Materials 10: 17 (2024).

Fig. 5 Scaling of the 2,016-atom NiO high-accuracy calculations with adaptive (AFD) and standard (SFD) finite difference operators with the number of Frontier nodes.

Editorial Summary

Density functional theory (DFT) has enabled accurate, ab initio predictions of material properties and explanations of a wide range of experimental data. The reliability of DFT predictions has led to materials-genome-type projects, in which a large set of possible material compositions and structures are screened by DFT calculations in order to identify those with promising properties. Those with the most potential are then suggested or selected for experimental synthesis and evaluation. However, full-precision DFT calculations can be computationally expensive, especially if complex structures or multi-component systems are involved, and the choice of approach and code often involves a tradeoff between computational accuracy and practicality. Fortunately, the methodology has advanced to the point where the results of several widely-used DFT codes agree well with each other and a set of benchmark high-precision calculations for a large set of elements across the periodic table. However, the tradeoffs remain for more complex systems or large survey studies, requiring practical compromises between the computational expense, accuracy, and even the choice of the computational method. 

E. L. Briggs et al. from the Department of Physics, North Carolina State University, described an approach that uses real-space grids to solve DFT equations. By using adaptive finite differencing to discretize the kinetic energy operator, the real-space results agree with those of plane-wave-based codes using much lower grid densities than those previously required and reproducing the benchmark DFT results at the same accuracy level as those of plane-wave-based and all-electron codes. The improved discretization enables high-precision real-space calculations at a substantially reduced cost while leveraging the well-known advantages of real-space methods of easy parallelization across many nodes, and avoiding the use of Fast Fourier Transform algorithms, which require global communication across nodes. The authors tested the accuracy of this adaptive kinetic energy operator in electronic structure calculations using the well-known Δ test for 71 elements, and the average error of the calculations is the same as those of the well-established plane-wave codes. The accuracy of the adaptive operator was further established with multi-species tests on NiO and borax decahydrate, which exhibit a range of complex bonding arrangements. The scalability of real-space grid methodology was then confirmed in highly accurate calculations for a 2,016-atom NiO supercell.

This article was recently published in npj Computational Materials 10: 17 (2024).

原文Abstract及其翻译

Adaptive finite differencing in high accuracy electronic structure calculations (高精度电子结构计算中的自适应有限差分)

E. L. Briggs,Wenchang Lu & J. Bernholc 

Abstract A multi-order Adaptive Finite Differencing (AFD) method is developed for the kinetic energy operator in real-space, grid-based electronic structure codes. It uses atomic pseudo orbitals produced by the corresponding pseudopotential codes to optimize the standard finite difference (SFD) operators for improved precision. Results are presented for a variety of test systems and Bravais lattice types, including the well-known Δ test for 71 elements in the periodic table, the Mott insulator NiO, and borax decahydrate, which contains covalent, ionic, and hydrogen bonds. The tests show that an 8th-order AFD operator leads to the same average Δ value as that achieved by plane-wave codes and is typically far more accurate and has a much lower computational cost than a 12th-order SFD operator. The scalability of real-space electronic calculations is demonstrated for a 2016-atom NiO cell, for which the computational time decreases nearly linearly when scaled from 18 to 144 CPU-GPU nodes.

摘要本文开发了一种多阶自适应有限差分（AFD）方法，用于基于实空间网格电子结构程序中的动能算符。该方法使用相应的赝势代码产生的原子赝轨道来优化标准有限差分（SFD）算符，以提高精度。我们给出了各种测试系统和布拉维晶格类型的结果，包括对元素周期表中71种元素，莫特绝缘体NiO，以及包含共价键、离子键和氢键的十水硼砂，进行了著名的Δ测试。测试结果表明，8阶AFD算符的平均Δ值与基于平面波的程序相同，且比12阶SFD算符更加精确、计算成本更低。我们使用含有2016个原子的NiO晶胞证明了实空间电子计算的可扩展性，当CPU-GPU节点从18扩展到144时，计算时间几乎呈线性下降。