Elastic3rd: A tool for calculating third-order elastic constants from first-principles calculations

The third-order elastic constants (TOECs) are fundamental to describe crystal’s nonlinear response to stress, and can be applied to explore anharmonic properties of crystals such as Grüneisen parameters, thermal expansion coefficient, and the effect of pressure on second-order elastic constants (SOECs). Here, we report an open-source python package, Elastic3rd, which is able to calculate the SOECs and TOECs using the strain–energy method for crystals with any symmetry from first-principles calculations. An algorithm to generate necessary strain modes and the corresponding coefficients for a given symmetry is proposed. These strain modes are then applied to the fully relaxed structure to generate the deformed structures. The total energies of the strained structures are calculated by a chosen first-principles code, and the SOECs and TOECs are determined by fitting the resulted strain–energy data. The present code has been validated by several case studies of C, Si and Mg, and the case of MnP4 shows the ability for low-symmetry crystals.三阶弹性常数（third-order elastic constants, TOECs）是描述晶体应力非线性响应的基础物理量，可用于探究晶体的非简谐特性，诸如格林艾森参数（Grüneisen parameters）、热膨胀系数，以及压力对二阶弹性常数（second-order elastic constants, SOECs）的影响。本文报道一款开源Python软件包Elastic3rd，其可基于第一性原理计算，通过应变-能量法对任意对称性的晶体计算其二阶与三阶弹性常数。文中提出一种针对给定晶体对称性的必要应变模式及其对应系数的生成算法。将这些应变模式施加于完全弛豫的晶体结构，即可生成形变后的结构。通过选定的第一性原理计算代码，可获得各形变结构的总能量，再通过拟合所得的应变-能量数据，即可确定二阶与三阶弹性常数。本软件包已通过碳（C）、硅（Si）以及镁（Mg）的多项案例研究得到验证，而MnP4的案例则展示了其对低对称性晶体的适配能力。