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电子结构计算：从MLWFs到MRWFs

最大局域化Wannier函数（MLWFs）是周期晶体电子结构的精确降阶模型。从布洛赫波函数生成MLWFs通常需要选择初始猜测，这些猜测通常是通过化学直觉和试错来进行的。由于低能电子结构通常可以利用原子轨道的紧束缚模型来描述，因此初始猜测通常选择类氢的s、p、d、f轨道。

Fig. 1 WF shapes and band interpolation of silicon.

然而，当涉及到仅有价带的情况，或者特别是与更高能带混合的导带的情况，可能难以确定好的初始猜测。但许多物理性质（如电极化）依赖于占据多种的Wannier函数（所有价带Wannier函数的Wannier中心之和）。此外，使用专用的MLWFs意味着可以获得更小的紧束缚模型，在计算时更高效。

Fig. 2 WF shapes and band interpolation of MoS 2 .

来自瑞士洛桑联邦理工学院材料理论与模拟中心的Junfeng Qiao等人，自动混合最大局域化Wannier函数（MRWFs）到多目标的方法，通过对具有能系的各个子能带（每个k点）构造MLWFs来实现多能带分离。该方法自然适用于多价带和导带的情况，也能自然扩展到任何其他孤立的能带组。

Fig. 3 Band structure comparison of the single MoS2 top valence band.

作者对硅和二硫化钼的研究结果表明，最终价带/导带MLWFs准确地恢复了成键/反键轨道的化学属性，并准确地再现了多价带/导带混合能带结构的价带/导带。该工作为不同材料的电子结构能带计算提供了新途径。相关论文近期发布于npj Computational Materials 9: 206 (2023)。

Fig. 4 Comparison of DFT and Wannier-interpolated bands for SrVO 3 .

Editorial Summary

Electronic structure calculations: From MLWFs to MRWFs

Maximally localized Wannier functions (MLWFs) are accurate reduced-order models for the electronic structures of periodic crystals. The generation of MLWFs from Bloch wavefunctions typically requires a choice of initial guesses, which are often conjectured from chemical intuition with trial and error. The initial guesses are usually chosen from the hydrogenic s, p, d, f orbitals, since the low-energy electronic structure can often be well described by a tight-binding model of atomic-like orbitals.

Fig. 5 Band distances of 77 insulators.

However, when it comes to the cases of valence bands (VB) alone, or especially conduction bands (CB) which are mixed with higher-energy bands, it might become difficult to identify good initial guesses. Meanwhile, many physical properties (such as the electric polarization) depend only on the Wannier functions (WFs) of the occupied manifold (sum of Wannier centers of all the valence WFs). Using dedicated MLWFs means that one can obtain smaller tight-binding models that are thus more efficient when computing.

Fig. 6 WF spreads of 77 insulators.

Junfeng Qiao et al. from the Theory and Simulations of Materials, École Polytechnique Fédérale de Lausanne, introduced an automated method (manifold-remixed Wannier functions (MRWF)) to separate band manifolds by constructing MLWFs for the respective submanifolds that have finite energy gaps (at each k-point) between them. The method naturally extends to the case of valence and conduction manifolds, but also to any other case of isolated groups of bands. Results on silicon and MoS2 suggested that the final valence (conduction) MLWFs restore faithfully chemical intuition for bonding/anti-bonding orbitals, and accurately reproduced the valence/conduction part of the band structure of the valence plus conduction manifold. The proposed approach provides a new avenue for electronic band structure calculations for a variety of materials.

This article was recently published in npj Computational Materials 9: 206 (2023).

原文Abstract及其翻译

Automated mixing of maximally localized Wannier functions into target manifolds (自动混合最大局域化Wannier函数到多目标)

Junfeng Qiao, Giovanni Pizzi & Nicola Marzari

Abstract

Maximally localized Wannier functions (MLWFs) are widely used in electronic-structure calculations. We have recently developed automated approaches to generate MLWFs that represent natural tight-binding sets of atomic-like orbitals; these describe accurately both the occupied states and the complementary unoccupied ones. For many applications, it is required to use MLWFs that describe instead certain target groups of bands: the valence or the conduction bands, or correlated manifolds. Here, we start from these tight-binding sets of MLWFs, and mix them using a combination of parallel transport and maximal localization to construct manifold-remixed Wannier functions (MRWFs): these are orthogonal sets of MLWFs that fully and only span desired target submanifolds. The algorithm is simple and robust, and is showcased here in reference applications (silicon, MoS2, and SrVO3) and in a mid-throughput study of 77 insulators.

摘要

最大局域化Wannier函数（MLWFs）广泛应用于电子结构计算。我们最近开发了能够自动生成代表自然紧束缚类原子轨道集MLWFs的方法；这些轨道准确描述了占据态和互补的未占据态。在许多应用中，我们需要使用描述特定目标带组的MLWFs：价带或导带，或相关的多种能带。在这里，我们从这些MLWFs的紧束缚集开始，并使用并行传输和最大局域化的组合，将它们混合来构建多种再混合的Wannier函数（MRWFs）：这些是MLWFs的正交集，完全且只跨越所期望的多种子目标。该算法简单而稳健，并在参考应用（硅、MoS2和SrVO3）和77种绝缘体中进行的中通量研究中得到了演示。