随机密度泛函理论：实空间和能量空间的碎片化降噪
Stochastic Density Functional Theory: Real- and Energy-Space Fragmentation for Noise Reduction
摘要：
随机密度泛函理论（sDFT）正成为研究扩展材料基态性质的重要工具。描述Kohn-Sham轨道的计算复杂性被引入一组随机（随机）轨道所取代，导致在引入统计误差的情况下，某些基态可观察到的线性和通常次线性缩放。减少噪声的方案是必不可少的，例如，使用从sDFT获得的力来确定结构。最近，我们引入了两个嵌入方案，以减轻统计波动的电子密度和合力的核。这两种技术都是基于在真实空间中分割系统或将占用的空间切割成能量窗口，从而显著减少统计波动。为了化学准确性，需要进一步减少噪音，这可以通过增加随机轨道的数量来实现。然而，收敛相对较慢，因为根据中心极限定理，统计误差缩放为$1/\sqrt{N_χ}$，其中$N_χ$是随机轨道的数目。在本文中，我们结合了上述嵌入方案，并介绍了一种新的方法，建立在重叠片段和能量窗口。这种新方法显着降低了噪音的基态属性，如电子密度，总能量，和力的核，所证明的G-中心在体硅。
英文摘要：
Stochastic density functional theory (sDFT) is becoming a valuable tool for studying ground state properties of extended materials. The computational complexity of describing the Kohn-Sham orbitals is replaced by introducing a set of random (stochastic) orbitals leading to linear and often sub-linear scaling of certain ground-state observable at the account of introducing a statistical error. Schemes to reduce the noise are essential, for example, for determining the structure using the forces obtained from sDFT. Recently we have introduced two embedding schemes to mitigate the statistical fluctuations in the electron density and resultant forces on the nuclei. Both techniques were based on fragmenting the system either in real-space or slicing the occupied space into energy windows, allowing for a significant reduction of the statistical fluctuations. For chemical accuracy further reduction of the noise is required, which could be achieved by increasing the number of stochastic orbitals. However, the convergence is relatively slow as the statistical error scales as $1/\sqrt{N_χ}$ according to the central limit theorem, where $N_χ$ is the number of random orbitals. In this paper we combined the aforementioned embedding schemes and introduced a new approach that builds on overlapped fragments and energy windows. The new approach significantly lowers the noise for ground state properties such as the electron density, total energy, and forces on the nuclei, as demonstrated for a G-center in bulk silicon.
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