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Despite decades of practice, finite-size errors in many widely usedelectronic structure theories for periodic systems remain poorly understood.For periodic systems using a general Monkhorst-Pack grid, there has been nocomprehensive and rigorous analysis of the finite-size error in theHartree-Fock theory (HF) and the second order M{\o}ller-Plesset perturbationtheory (MP2), which are the simplest wavefunction based method, and thesimplest post-Hartree-Fock method, respectively. Such calculations can beviewed as a multi-dimensional integral discretized with certain trapezoidalrules. Due to the Coulomb singularity, the integrand has many points ofdiscontinuity in general, and standard error analysis based on theEuler-Maclaurin formula gives overly pessimistic results. The lack of analyticunderstanding of finite-size errors also impedes the development of effectivefinite-size correction schemes. We propose a unified analysis to obtain sharpconvergence 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 classof non-smooth integrands. Our result is applicable to three-dimensional bulksystems as well as low dimensional systems (such as nanowires and 2Dmaterials). Our unified analysis also allows us to prove the effectiveness ofthe Madelung-constant correction to the Fock exchange energy, and theeffectiveness of a recently proposed staggered mesh method for periodic MP2calculations [Xing, Li, Lin, J. Chem. Theory Comput. 2021]. Our analysisconnects the effectiveness of the staggered mesh method with integrands withremovable singularities, and suggests a new staggered mesh method for reducingfinite-size errors of periodic HF calculations.

Unified analysis of finite-size error for periodic Hartree-Fock and second order M{\o}ller-Plesset perturbation theory