We provide the first rigorous study of the finite-size error in the simplest and representative coupled cluster theory, namely the coupled cluster doubles (CCD) theory, for gapped periodic systems. Assuming that the CCD equations are solved using exact Hartree-Fock orbitals and orbital energies, we prove that the convergence rate of finite-size error scales as $\mathscr{O}(N_\mathbf{k}^{-\frac13})$, where $N_{\mathbf{k}}$ is the number of discretization point in the Brillouin zone and characterizes the system size. Our analysis shows that the dominant error lies in the coupled cluster amplitude calculation, and the convergence of the finite-size error in energy calculations can be boosted to $\mathscr{O}(N_\mathbf{k}^{-1})$ with accurate amplitudes. This also provides the first proof of the scaling of the finite-size error in the third order M{\o}ller-Plesset perturbation theory (MP3) for periodic systems.

翻译：我们首次对最简单且具代表性的耦合簇理论——即耦合簇双激发（CCD）理论——在有能隙周期系统中的有限尺寸误差进行了严格研究。假设CCD方程使用精确的Hartree-Fock轨道和轨道能量求解，我们证明有限尺寸误差的收敛速率标度为$\mathscr{O}(N_\mathbf{k}^{-\frac13})$，其中$N_{\mathbf{k}}$是布里渊区离散化点的数目，用以表征系统尺寸。分析表明，主导误差来源于耦合簇振幅计算，而通过精确振幅可将能量计算中的有限尺寸误差收敛速率提升至$\mathscr{O}(N_\mathbf{k}^{-1})$。该结果也首次证实了周期系统三阶Møller-Plesset微扰理论（MP3）中有限尺寸误差的标度规律。