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Efficient calculation of the Coulomb matrix and its expansion around k = 0 within the FLAPW method

机译：FLAPW方法中库仑矩阵的有效计算及其在k = 0附近的展开

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We derive formulas for the Coulomb matrix within the full-potential linearized augmented-plane-wave (FLAPW) method. The Coulomb matrix is a central ingredient in implementations of many-body perturbation theory, such as the Hartree-Fock and GW approximations for the electronic self-energy or the random-phase approximation for the dielectric function. It is represented in the mixed product basis, which combines numerical muffin-tin functions and interstitial plane waves constructed from products of FLAPW basis functions. The interstitial plane waves are here expanded with the Rayleigh formula. The resulting algorithm is very efficient in terms of both computational cost and accuracy and is superior to an implementation with the Fourier transform of the step function. In order to allow an analytic treatment of the divergence at k = 0 in reciprocal space, we expand the Coulomb matrix analytically around this point without resorting to a projection onto plane waves. Without additional approximations, we then apply a basis transformation that diagonalizes the Coulomb matrix and confines the divergence to a single eigenvalue. At the same time, response matrices like the dielectric function separate into head, wings, and body with the same mathematical properties as in a plane-wave basis. As an illustration we apply the formulas to electron-energy-loss spectra (EELS) for nickel at different k vectors including k = 0. The convergence of the spectra towards the result at k = 0 is clearly seen. Our all-electron treatment also allows to include transitions from 3s and 3p core states in the EELS spectrum that give rise to a shallow peak at high energies and lead to good agreement with experiment.
机译：我们推导了全势线性化增强平面波（FLAPW）方法中的库仑矩阵公式。库仑矩阵是实现多体摄动理论的重要组成部分，例如电子自能的Hartree-Fock和GW近似或介电函数的随机相近似。它以混合产品基础表示，该产品组合了数值松饼锡函数和由FLAPW基础函数的乘积构成的间隙平面波。间隙平面波在这里用瑞利公式扩展。所得的算法在计算成本和准确性方面都非常有效，并且优于具有阶跃函数的傅立叶变换的实现。为了对倒数空间中k = 0处的发散度进行解析处理，我们围绕该点解析地扩展了库仑矩阵，而无需求助于平面波的投影。在没有其他近似的情况下，我们然后应用对角化库仑矩阵并将散度限制为单个特征值的基础变换。同时，诸如介电函数之类的响应矩阵可以分为头，翼和身体，具有与平面波相同的数学特性。作为说明，我们将公式应用于不同k矢量（包括k = 0）下镍的电子能量损失谱（EELS）。可以清楚地看到在k = 0时谱向结果的收敛。我们的全电子处理还允许在EELS光谱中包括3s和3p核心态的跃迁，这些跃迁会在高能时产生一个浅峰，并与实验保持良好的一致性。

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