As a consequence of Bloch’s theorem, the numerical computation of the fermionic ground state density matrices and energies of periodic Schrödinger operators involves integrals over the Brillouin zone. These integrals are difficult to compute numerically in metals due to discontinuities in the integrand. We perform an error analysis of several widely-used quadrature rules and smearing methods for Brillouin zone integration. We precisely identify the assumptions implicit in these methods and rigorously prove error bounds. Numerical results for two-dimensional periodic systems are also provided. Our results shed light on the properties of these numerical schemes, and provide guidance as to the appropriate choice of numerical parameters.

中文翻译：

---

周期性薛定谔算子在布里渊区中的数值求积

作为布洛赫定理的结果，费米基态密度矩阵和周期性薛定谔算子能量的数值计算涉及布里渊区上的积分。由于被积函数的不连续性，这些积分很难在金属中进行数值计算。我们对布里渊区域积分的几种广泛使用的正交规则和拖尾方法进行了误差分析。我们精确地识别这些方法中隐含的假设并严格证明错误界限。还提供了二维周期系统的数值结果。Our results shed light on the properties of these numerical schemes, and provide guidance as to the appropriate choice of numerical parameters.