We present a method to construct an efficient approximation to the bare exchange and screened direct interaction kernels of the Bethe–Salpeter Hamiltonian for periodic solid state systems via the interpolative separable density fitting technique. We show that the cost of constructing the approximate Bethe–Salpeter Hamiltonian can be reduced to nearly optimal as $\mathcal{𝒪}\left(N_k\right)$ with respect to the number of samples in the Brillouin zone $N_k$ for the first time. In addition, we show that the cost for applying the Bethe–Salpeter Hamiltonian to a vector scales as $\mathcal{𝒪}\left(N_{k}log{N}_k\right)$. Therefore, the optical absorption spectrum, as well as selected excitation energies, can be efficiently computed via iterative methods such as the Lanczos method. This is a significant reduction from the $\mathcal{𝒪}\left(N_k^2\right)$ and $\mathcal{𝒪}\left(N_k^3\right)$ scaling associated with a brute force approach for constructing the Hamiltonian and diagonalizing the Hamiltonian, respectively. We demonstrate the efficiency and accuracy of this approach with both one-dimensional model problems and three-dimensional real materials (graphene and diamond). For the diamond system with $N_k=2197$, it takes $6$ hours to assemble the Bethe–Salpeter Hamiltonian and $4$ hours to fully diagonalize the Hamiltonian using $169$ cores when the brute force approach is used. The new method takes less than $3$ minutes to set up the Hamiltonian and $24$ minutes to compute the absorption spectrum on a single core.

我们提出了一种通过插值可分离的密度拟合技术，构造周期性固态系统的Bethe-Salpeter Hamiltonian的裸交换和筛选的直接相互作用核的有效近似方法。我们证明，构建近似的Bethe–Salpeter哈密顿量的成本可以降低到几乎最优，因为$\mathcal{𝒪}（{ñ}_{ķ}）$ 关于布里渊区的样本数量 ${ñ}_{ķ}$首次。此外，我们证明了将Bethe–Salpeter哈密顿量应用于向量的成本为$\mathcal{𝒪}（{ñ}_{ķ}日志{ñ}_{ķ}）$。因此，可以通过诸如Lanczos方法之类的迭代方法有效地计算光吸收光谱以及选定的激发能。与$\mathcal{𝒪}（{ñ}_{ķ}^2）$ 和 $\mathcal{𝒪}（{ñ}_{ķ}^3）$缩放与蛮力方法相关，分别用于构造哈密顿量和对角化哈密顿量。我们通过一维模型问题和三维真实材料（石墨烯和钻石）展示了这种方法的效率和准确性。对于钻石系统${ñ}_{ķ}=2\mathrm{1个}97$， 它需要 $6$ 小时来组装Bethe–Salpeter哈密顿量和 $4$ 小时以完全对角化哈密顿量 $\mathrm{1个}69$使用蛮力方法时会产生核心。新方法花费少于$3$ 分钟设置汉密尔顿和 $24$ 分钟以计算单个纤芯的吸收光谱。