计算声子及其相互作用的群论方法
A group theoretical approach to computing phonons and their interactions
摘要：
在这里，我们提出了四个独立的进展，促进声子和它们的相互作用的第一性原理的计算。首先，我们实现了一个组理论的方法来构建纯粹的空间群不可约衍生物（ID），保证对称性的建设，并允许通信和存储声子和它们的相互作用的实际手段的d维晶体的N阶泰勒级数。其次，我们证明了在d维晶体中容纳N个给定波矢量的最小可能的超胞是使用从相应的波矢量形成的矩阵的Smith标准形确定的;除了提供在任意N和d处均匀超胞的最大所需多重性之外，找到所述超胞的计算成本可以忽略不计。第三，我们开发了一系列有限位移方法来计算声子和它们的相互作用，利用前两个发展：孤独和捆绑不可约衍生物（LID和BID）的方法。LID在尽可能小的超胞中一次计算一个ID，或者尽可能少的ID，而BID利用小于N阶的微扰导数（例如赫尔曼-费曼力），以便使用尽可能少的计算在尽可能小的超胞中提取所有ID。最后，我们推导出一个方程的阶N-2体积导数的声子的阶N ID。鉴于前者很容易计算，它们可以被用来作为一个严格的，无限范围的测试ID。我们的一般框架说明石墨烯，产生不可约的声子相互作用的第五阶。此外，我们提供了成本分析的岩盐结构在N=3，展示了大量的加速相比，流行的有限位移法在文献中。
英文摘要：
Here we present four independent advances which facilitate the computation of phonons and their interactions from first-principles. First, we implement a group-theoretical approach to construct the order N Taylor series of a d-dimensional crystal purely in terms of space group irreducible derivatives (ID), which guarantees symmetry by construction and allows for a practical means of communicating and storing phonons and their interactions. Second, we prove that the smallest possible supercell which accommodates N given wavevectors in a d-dimensional crystal is determined using the Smith Normal Form of the matrix formed from the corresponding wavevectors; resulting in negligible computational cost to find said supercell, in addition to providing the maximum required multiplicity for uniform supercells at arbitrary N and d. Third, we develop a series of finite displacement methodologies to compute phonons and their interactions which exploit the first two developments: lone and bundled irreducible derivative (LID and BID) approaches. LID computes a single ID, or as few as possible, at a time in the smallest supercell possible, while BID exploits perturbative derivatives for some order less than N (e.g. Hellman-Feynman forces) in order to extract all ID in the smallest possible supercells using the fewest possible computations. Finally, we derive an equation for the order N-2 volume derivatives of the phonons in terms of the order N ID. Given that the former are easily computed, they can be used as a stringent, infinite ranged test of the ID. Our general framework is illustrated on graphene, yielding irreducible phonon interactions to fifth order. Additionally, we provide a cost analysis for the rock-salt structure at N=3, demonstrating a massive speedup compared to popular finite displacement methods in the literature.
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