The $GW$ approximation is a widely used method for computing electron addition and removal energies of molecules and solids. The computational effort of conventional $GW$ algorithms increases as $O\left(N^4\right)$ with the system size $N$, hindering the application of $GW$ to large and complex systems. Low-scaling $GW$ algorithms are currently very actively developed. Benchmark studies at the single-shot $G_0{W}_0$ level indicate excellent numerical precision for frontier quasiparticle energies, with mean absolute deviations $<10$ meV between low-scaling and standard implementations for the widely used $\mathit{GW}100$ test set. A notable challenge for low-scaling $GW$ algorithms remains in achieving high precision for five molecules within the $\mathit{GW}100$ test set, namely ${\mathrm{O}}_3, \mathrm{BeO}, \mathrm{MgO}, \mathrm{BN}$, and $\mathrm{CuCN}$, for which the deviations are in the range of several hundred meV at the $G_0{W}_0$ level. This is because of a spurious transfer of spectral weight from the quasiparticle to the satellite spectrum in $G_0{W}_0$ calculations, resulting in multipole features in the self-energy and spectral function, which low-scaling algorithms fail to describe. We show in this paper that including eigenvalue self-consistency in the Green's function ($\mathrm{ev}G{W}_0$) achieves a proper separation between satellite and quasiparticle peak, leading to a single solution of the quasiparticle equation with spectral weight close to one. $\mathrm{ev}G{W}_0$ quasiparticles energies from low-scaling $GW$ closely align with reference calculations; the mean absolute error is only 12 meV for the five molecules. We thus demonstrate that low-scaling $GW$ with self-consistency in $G$ is well suited for computing frontier quasiparticle energies.

中文翻译：

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解决 GW100 基准中的多极挑战可实现精确的低规模 GW 计算


这 $GW$ 近似是计算分子和固体的电子添加和去除能量的广泛使用的方法。传统方法的计算量 $GW$ 算法增加 $O\left(N^4\right)$ 与系统尺寸 $N$ , 阻碍了应用 $GW$ 大型且复杂的系统。低规模 $GW$ 目前算法的开发非常活跃。单次基准研究 $G_0{W}_0$ 水平表明前沿准粒子能量具有出色的数值精度，具有平均绝对偏差 $<10$ 广泛使用的低规模和标准实现之间的 meV $\mathit{GW}100$ 测试集。低规模化的一个显着挑战 $GW$ 算法仍然在实现五个分子的高精度 $\mathit{GW}100$ 测试集，即 ${\mathrm{O}}_3, \mathrm{BeO}, \mathrm{MgO}, \mathrm{BN}$ ， 和 $\mathrm{CuCN}$ ，其偏差在数百 meV 范围内 $G_0{W}_0$ 等级。这是因为光谱权重从准粒子到卫星光谱的虚假转移 $G_0{W}_0$ 计算，导致自能和谱函数中的多极特征，这是低尺度算法无法描述的。我们在本文中表明，在 $G$ 雷恩函数 ( $\mathrm{ev}G{W}_0$ ）实现了卫星峰和准粒子峰之间的适当分离，导致准粒子方程的单一解，其谱权重接近于 1。 $\mathrm{ev}G{W}_0$ 来自低尺度的准粒子能量 $GW$ 与参考计算紧密一致；五个分子的平均绝对误差仅为 12 meV。因此，我们证明低尺度 $GW$ G 中具有自洽性，非常适合计算前沿准粒子能量。