  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.
