The dielectric response function and its inverse are crucial physical quantities in materials science. We propose an accurate and efficient strategy to invert the dielectric function matrix. The GW approximation, a powerful approach to accurately describe many-body excited states, is taken as an application to demonstrate accuracy and efficiency. We incorporate the interpolative separable density fitting (ISDF) algorithm with Sherman--Morrison--Woodbury (SMW) formula to accelerate the inversion process by exploiting low-rank properties of dielectric function in plane-wave GW calculations. Our ISDF--SMW strategy produces accurate quasiparticle energies with $O(N_{\mathrm{r}}N_{\mathrm{e}}^2)$ computational cost $(N_{\mathrm{e}}$ is the number of electrons and $N_{\mathrm{r}}=100$--$1000N_{\mathrm{e}}$ is the number of grid points) with negligible small error of $0.03$~eV for both complex molecules and solids. This new strategy for inverting the dielectric matrix can be \(50\times\) faster than the current state-of-the-art implementation in BerkeleyGW, resulting in two orders of magnitude speedup for total GW calculations.

翻译：介电响应函数及其逆是材料科学中的关键物理量。我们提出了一种精确且高效的策略来对介电函数矩阵求逆。以GW近似（一种精确描述多体激发态的强有力方法）作为应用来展示其精度与效率。我们将插值可分离密度拟合算法与谢尔曼-莫里森-伍德伯里公式相结合，通过利用平面波GW计算中介电函数的低秩特性来加速求逆过程。我们的ISDF-SMW策略能够以$O(N_{\mathrm{r}}N_{\mathrm{e}}^2)$的计算代价（其中$N_{\mathrm{e}}$为电子数，$N_{\mathrm{r}}=100$--$1000N_{\mathrm{e}}$为网格点数）获得精确的准粒子能量，对复杂分子和固体的误差仅为$0.03$电子伏特。这种介电矩阵求逆新策略比当前伯克利GW软件包中最先进的实现快50倍，从而使GW总计算速度提升两个数量级。