Despite decades of practice, finite-size errors in many widely used electronic structure theories for periodic systems remain poorly understood. For periodic systems using a general Monkhorst-Pack grid, there has been no comprehensive and rigorous analysis of the finite-size error in the Hartree-Fock theory (HF) and the second order M{\o}ller-Plesset perturbation theory (MP2), which are the simplest wavefunction based method, and the simplest post-Hartree-Fock method, respectively. Such calculations can be viewed as a multi-dimensional integral discretized with certain trapezoidal rules. Due to the Coulomb singularity, the integrand has many points of discontinuity in general, and standard error analysis based on the Euler-Maclaurin formula gives overly pessimistic results. The lack of analytic understanding of finite-size errors also impedes the development of effective finite-size correction schemes. We propose a unified analysis to obtain sharp convergence rates of finite-size errors for the periodic HF and MP2 theories. Our main technical advancement is a generalization of the result of [Lyness, 1976] for obtaining sharp convergence rates of the trapezoidal rule for a class of non-smooth integrands. Our result is applicable to three-dimensional bulk systems as well as low dimensional systems (such as nanowires and 2D materials). Our unified analysis also allows us to prove the effectiveness of the Madelung-constant correction to the Fock exchange energy, and the effectiveness of a recently proposed staggered mesh method for periodic MP2 calculations [Xing, Li, Lin, J. Chem. Theory Comput. 2021]. Our analysis connects the effectiveness of the staggered mesh method with integrands with removable singularities, and suggests a new staggered mesh method for reducing finite-size errors of periodic HF calculations.

翻译：尽管历经数十年实践，许多周期系统常用电子结构理论中的有限尺寸误差仍未被充分理解。对于采用一般Monkhorst-Pack网格的周期系统，Hartree-Fock理论（HF）和二阶Møller-Plesset微扰理论（MP2）——分别作为最简单的基于波函数的方法和最简单的后Hartree-Fock方法——其有限尺寸误差至今缺乏全面严谨的分析。此类计算可视为采用特定梯形规则离散化的多维积分。由于库仑奇异性，被积函数通常存在多处间断点，而基于欧拉-麦克劳林公式的标准误差分析会得到过于悲观的估计。对有限尺寸误差解析理解的缺失也阻碍了有效有限尺寸修正方案的发展。我们提出一种统一分析框架，以获取周期HF和MP2理论有限尺寸误差的精确收敛速度。我们的主要技术突破在于推广了[Lyness, 1976]关于一类非光滑被积函数梯形规则精确收敛速度的结果。该结果不仅适用于三维体材料系统，也适用于低维系统（如纳米线和二维材料）。我们的统一分析还证明了马德隆常数修正对Fock交换能的有效性，以及近期提出的用于周期MP2计算的交错网格法[Xing, Li, Lin, J. Chem. Theory Comput. 2021]的有效性。通过将交错网格法的有效性与被积函数的可去奇点性质相关联，我们的分析进一步提出了一种新型交错网格方法，用于减小周期HF计算中的有限尺寸误差。