We present the detailed formalism for stress calculations within the framework of numerical atomic orbital (NAO) bases, as implemented in the first-principles electronic code package ABACUS. The validity and numerical precision of our implementation are benchmarked against reference results obtained using the finite difference method as well as those calculated using plane wave (PW) bases. The equation of state for α-quartz calculated using our NAO-based implementation is in excellent agreement with the experimental data. We further show that the stresses calculated in terms of NAO bases converge much faster with respect to the energy cutoff than those calculated using PW bases. This enables accurate stress calculations at relatively low cost. An analysis is provided to elucidate the origin of this behavior.

我们在数值原子轨道 (NAO) 基础框架内展示了应力计算的详细形式，如在第一原理电子代码包 ABACUS 中实现的。我们实施的有效性和数值精度以使用有限差分方法获得的参考结果以及使用平面波(PW) 基础计算的结果为基准。α的状态方程使用我们基于 NAO 的实现计算的 -quartz 与实验数据非常吻合。我们进一步表明，与使用 PW 基础计算的应力相比，根据 NAO 基础计算的应力在能量截止方面收敛得更快。这使得能够以相对较低的成本进行准确的应力计算。提供了一个分析来阐明这种行为的起源。