This is the second part of a project on the foundations of first-principle calculations of the electron transport in crystals at finite temperatures, aiming at a predictive first-principles platform that combines ab-initio molecular dynamics (AIMD) and a finite-temperature Kubo-formula with dissipation for thermally disordered crystalline phases. The latter are encoded in an ergodic dynamical system $(\Omega,\mathbb G,{\rm d}\mathbb P)$, where $\Omega$ is the configuration space of the atomic degrees of freedom, $\mathbb G$ is the space group acting on $\Omega$ and ${\rm d}\mathbb P$ is the ergodic Gibbs measure relative to the $\mathbb G$-action. We first demonstrate how to pass from the continuum Kohn-Sham theory to a discrete atomic-orbitals based formalism without breaking the covariance of the physical observables w.r.t. $(\Omega,\mathbb G,{\rm d}\mathbb P)$. Then we show how to implement the Kubo-formula, investigate its self-averaging property and derive an optimal finite-volume approximation for it. We also describe a numerical innovation that made possible AIMD simulations with longer orbits and elaborate on the details of our simulations. Lastly, we present numerical results on the transport coefficients of crystal silicon at different temperatures.

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来自第一性原理 II 的无序晶体：传输系数

这是关于有限温度下晶体中电子传输的第一性原理计算基础项目的第二部分，旨在建立一个结合了从头算分子动力学 (AIMD) 和有限温度 Kubo 的预测性第一性原理平台-具有热无序结晶相耗散的公式。后者被编码在遍历动力学系统 $(\Omega,\mathbb G,{\rm d}\mathbb P)$ 中，其中 $\Omega$ 是原子自由度的配置空间， $\mathbb G$是作用于 $\Omega$ 的空间群，${\rm d}\mathbb P$ 是相对于 $\mathbb G$-action 的遍历吉布斯测度。我们首先演示了如何从连续统 Kohn-Sham 理论过渡到基于离散原子轨道的形式主义而不破坏物理可观测值的协方差 $(\Omega,\mathbb G, {\rm d}\mathbb P)$。然后我们展示了如何实现 Kubo 公式，研究它的自平均特性并为其推导出最优的有限体积近似值。我们还描述了一项数值创新，它使更长轨道的 AIMD 模拟成为可能，并详细说明了我们的模拟细节。最后，我们给出了不同温度下晶体硅传输系数的数值结果。