As a consequence of Bloch's theorem, the numerical computation of the fermionic ground state density matrices and energies of periodic Schrodinger 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.
翻译:由于布洛奇的理论,对地表温度密度矩阵和定期施罗德操作员能量的数值计算涉及布利柳因区的积分。由于原状的不连续性,这些积分难以在金属中进行数字计算。我们对布利柳因区集成过程中广泛使用的几种二次曲线规则和涂片方法进行了错误分析。我们准确地确定了这些方法中隐含的假设,并严格地证明了误差界限。还提供了二维周期系统的数值结果。我们的结果揭示了这些数字办法的特性,并为适当选择数字参数提供了指导。