In solid-state physics, energies of crystals are usually computed with a plane-wave discretization of Kohn-Sham equations. However the presence of Coulomb singularities requires the use of large plane-wave cut-offs to produce accurate numerical results. In this paper, an analysis of the plane-wave convergence of the eigenvalues of periodic linear Hamiltonians with Coulomb potentials using the variational projector-augmented wave (VPAW) method is presented. In the VPAW method, an invertible transformation is applied to the original eigenvalue problem, acting locally in balls centered at the singularities. In this setting, a generalized eigenvalue problem needs to be solved using plane-waves. We show that cusps of the eigenfunctions of the VPAW eigenvalue problem at the positions of the nuclei are significantly reduced. These eigenfunctions have however a higher-order derivative discontinuity at the spheres centered at the nuclei. By balancing both sources of error, we show that the VPAW method can drastically improve the plane-wave convergence of the eigenvalues with a minor additional computational cost. Numerical tests are provided confirming the efficiency of the method to treat Coulomb singularities.

在固态物理学中，晶体的能量通常用 Kohn-Sham 方程的平面波离散化来计算。然而，库仑奇点的存在需要使用大平面波截止来产生准确的数值结果。在本文中，提出了使用变分投影增强波 (VPAW) 方法对具有库仑势的周期性线性哈密顿量的特征值进行平面波收敛的分析。在 VPAW 方法中，可逆变换应用于原始特征值问题，局部作用在以奇点为中心的球中。在这种情况下，需要使用平面波解决广义特征值问题。我们表明，核位置处 VPAW 特征值问题的特征函数的尖峰显着减少。然而，这些本征函数在以核为中心的球体处具有高阶导数不连续性。通过平衡两个误差源，我们表明 VPAW 方法可以显着改善特征值的平面波收敛，而额外的计算成本很小。提供的数值测试证实了该方法处理库仑奇点的效率。