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Helmholtz Fermi Surface Harmonics: an efficient approach for treating anisotropic problems involving Fermi surface integrals

We present a new efficient numerical approach for representing anisotropic physical quantities and/or matrix elements defined on the Fermi surface of metallic materials. The method introduces a set of numerically calculated generalized orthonormal functions which are the solutions of the Helmholtz equation defined on the Fermi surface. Noteworthy, many properties of our proposed basis set are also shared by the Fermi Surface Harmonics (FSH) introduced by Philip B. Allen [Physical Review B 13, 1416 (1976)], proposed to be constructed as polynomials of the cartesian components of the electronic velocity. The main motivation of both approaches is identical, to handle anisotropic problems efficiently. However, in our approach the basis set is defined as the eigenfunctions of a differential operator and several desirable properties are introduced by construction. The method demonstrates very robust in handling problems with any crystal structure or topology of the Fermi surface, and the periodicity of the reciprocal space is treated as a boundary condition for our Helmholtz equation. We illustrate the method by analysing the free-electron-like Lithium (Li), Sodium (Na), Copper (Cu), Lead(Pb), Tungsten (W) and Magnesium diboride (MgB2).