Linear scaling solution of the all-electron Coulomb problem in solids

Abstract

We present a linear scaling formulation for the solution of the all-electron Coulomb problem in crystalline solids. The resulting method is systematically improvable and well suited to large-scale quantum mechanical calculations in which the Coulomb potential and energy of a continuous electronic density and singular nuclear density are required. Linear scaling is achieved by introducing smooth, strictly local neutralizing densities to render nuclear interactions strictly local, and solving the remaining neutral Poisson problem for the electrons in real space. While the formulation includes singular nuclear potentials without smearing approximations, the required Poisson solution is in Sobolev space $H^1$, as required for convergence in the energy norm. We employ enriched finite elements, with enrichments from isolated atom solutions, for an efficient solution of the resulting Poisson problem in the interacting solid. We demonstrate the accuracy and convergence of the approach by direct comparison to standard Ewald sums for a lattice of point charges, and demonstrate the accuracy in all-electron quantum mechanical calculations with an application to crystalline diamond.

23 pages, 17 figures

References in corpus (5)

• The SIESTA method for ab initio order-N materials simulation
• Efficient index handling of multidimensional periodic boundary conditions
• Efficient solution of Poisson's equation with free boundary conditions
• All-electron density functional theory and time-dependent density functional theory with high-order finite elements
• Non-Equilibrium Electron Transport in Two-Dimensional Nano-Structures Modeled by Green's Functions and the Finite-Element Method