密度泛函理論（DFT）能夠?qū)Σ牧闲再|(zhì)進(jìn)行精確的從頭預(yù)測(cè)，解釋廣泛的實(shí)驗(yàn)數(shù)據(jù)。DFT預(yù)測(cè)的可靠性催生了材料基因組工程，通過(guò)DFT計(jì)算篩選大量可能的材料組分和結(jié)構(gòu)，以確定那些具有優(yōu)異性能的材料，進(jìn)而用于實(shí)驗(yàn)合成與評(píng)估。

然而，全精度DFT計(jì)算在計(jì)算成本上可能相當(dāng)昂貴，特別是對(duì)于復(fù)雜結(jié)構(gòu)或者多組分系統(tǒng)，而且計(jì)算方法和程序的選擇通常涉及計(jì)算精度和實(shí)用性之間的權(quán)衡。幸運(yùn)的是，幾種廣泛使用的DFT程序產(chǎn)生的結(jié)果彼此一致，且已有元素周期表中大量元素的高精度計(jì)算作為基準(zhǔn)。然而，對(duì)于更加復(fù)雜的系統(tǒng)或者大型調(diào)查研究，權(quán)衡依然存在，需要在計(jì)算成本、精確度、甚至是計(jì)算方法的選擇之間尋求折中。

來(lái)自美國(guó)北卡羅來(lái)納州立大學(xué)物理系的E. L. Briggs等，描述了一種利用實(shí)空間網(wǎng)格求解DFT方程的方法。通過(guò)使用自適應(yīng)有限差分對(duì)動(dòng)能算符離散化，可以使實(shí)空間的結(jié)果與基于平面波的程序相符。在顯著減少網(wǎng)格密度的同時(shí)，能夠以基于平面波的程序和全電子程序的精度復(fù)現(xiàn)基準(zhǔn)DFT結(jié)果。

這一改進(jìn)的離散化方案能夠顯著降低高精度實(shí)空間計(jì)算的成本，同時(shí)兼具實(shí)空間方法的優(yōu)勢(shì)，易于在多個(gè)節(jié)點(diǎn)上并行計(jì)算，而且避免了需要跨節(jié)點(diǎn)全局通信的快速傅里葉變換算法的使用。研究者對(duì)71種元素進(jìn)行了著名的Δ測(cè)試，用于評(píng)估自適應(yīng)動(dòng)能算符在電子結(jié)構(gòu)計(jì)算中的準(zhǔn)確性，其平均誤差與成熟的平面波程序相同。通過(guò)對(duì)NiO和具有一系列復(fù)雜成鍵排列的十水硼砂進(jìn)行多物種測(cè)試，進(jìn)一步確定了自適應(yīng)算符的準(zhǔn)確性。

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

作者通過(guò)對(duì)含有2016個(gè)原子的NiO超晶胞進(jìn)行高精度計(jì)算，證實(shí)了實(shí)空間網(wǎng)格方法的可擴(kuò)展性。該文近期發(fā)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及其翻譯

Adaptive finite differencing in high accuracy electronic structure calculations?(高精度電子結(jié)構(gòu)計(jì)算中的自適應(yīng)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.

摘要本文開(kāi)發(fā)了一種多階自適應(yīng)有限差分（AFD）方法，用于基于實(shí)空間網(wǎng)格電子結(jié)構(gòu)程序中的動(dòng)能算符。該方法使用相應(yīng)的贗勢(shì)代碼產(chǎn)生的原子贗軌道來(lái)優(yōu)化標(biāo)準(zhǔn)有限差分（SFD）算符，以提高精度。我們給出了各種測(cè)試系統(tǒng)和布拉維晶格類(lèi)型的結(jié)果，包括對(duì)元素周期表中71種元素，莫特絕緣體NiO，以及包含共價(jià)鍵、離子鍵和氫鍵的十水硼砂，進(jìn)行了著名的Δ測(cè)試。測(cè)試結(jié)果表明，8階AFD算符的平均Δ值與基于平面波的程序相同，且比12階SFD算符更加精確、計(jì)算成本更低。我們使用含有2016個(gè)原子的NiO晶胞證明了實(shí)空間電子計(jì)算的可擴(kuò)展性，當(dāng)CPU-GPU節(jié)點(diǎn)從18擴(kuò)展到144時(shí)，計(jì)算時(shí)間幾乎呈線性下降。