Unified analysis of finite-size error for periodic Hartree-Fock and second order Møller-Plesset perturbation theory

Abstract

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 rigorous analysis of the finite-size error in the Hartree-Fock theory (HF) and the second order Møller-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 method 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, 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.

Staggered mesh method for correlation energy calculations of solids: Random phase approximation in direct ring coupled cluster doubles and adiabatic connection formalisms

“…In this case, h(q) is said to have removable discontinuity and a trapezoidal rule without q = 0 has super-algebraically decaying error according to the Euler-Maclaurin formula. Otherwise, lim q→0 h(q) does not exist and the quadrature error can be shown to scale as O(N −1 k ) 17 even if q = 0 is not a quadrature node. Similar discussion can give quantitative conditions 17 for the removable discontinuity of the nonsmooth terms Eq.…”

“…Otherwise, lim q→0 h(q) does not exist and the quadrature error can be shown to scale as O(N −1 k ) 17 even if q = 0 is not a quadrature node. Similar discussion can give quantitative conditions 17 for the removable discontinuity of the nonsmooth terms Eq. ( 17) in MP2 calculations.…”

“…In practical calculations, the finite-size error in molecular orbitals and orbital energies at the Hartree-Fock level also contribute to the overall finite-size errors in the standard and the staggered mesh methods. This error can be reduced using other correction methods 4,[15][16][17] or a large MP mesh in mean-field calculations. We will not consider these improvements, and focus on numerical comparisons between the standard and the staggered mesh methods for RPA and RPA-SOSEX energy calculations.…”

“…Despite such progresses, there has been very limited rigorous understanding of the scaling of finite-size errors for general systems. To our knowledge, our recent work 17 accounts for the first rigorous treatment of finite-size errors in periodic Hartree-Fock exchange energy calculations, and periodic second order Møller-Plesset correlation energy (MP2) calculations (which are among the simpest wavefunction methods) for general insulating systems. The key technical difficulty of analyzing the convergence towards the thermodynamic limit boils down to an analysis of quadrature error for a special class of nonsmooth integrands (Corollary 16 in Ref.…”

“…The key technical difficulty of analyzing the convergence towards the thermodynamic limit boils down to an analysis of quadrature error for a special class of nonsmooth integrands (Corollary 16 in Ref. 17). The analysis also suggests a new algorithm for reducing the finite-size error, which employs two staggered Monkhorst-Pack meshes for occupied and virtual orbitals, respectively.…”