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

In this manuscript (available at http://arxiv.org/abs/1004.1765), we present a systematically improvable, linear scaling formulation for the solution of the all-electron Coulomb problem in crystalline solids. In an infinite crystal, the electrostatic (Coulomb) potential is a sum of nuclear and electronic contributions, and each of these terms diverges and the sum is only conditionally convergent due to the long-range 1/r nature of the Coulomb interaction. In the all-electron quantum-mechanical problem in solids, there are three distinct divergences that must be addressed simultaneously: (1) the 1/r divergence of the electrostatic potential at the nuclei; (2) the divergence of both potential and energy lattice sums due to the long-range Coulomb interaction; and (3) the infinite self energies of the nuclei.

We achieve linear scaling 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. In so doing, the all-electron problem is decomposed into analytic strictly-local nuclear, and numerical long-range electronic parts; with required numerical solution in the Sobolev space , so that convergence is assured and approximation is optimal in the energy norm. Expressions for the Coulomb energy per unit cell, analytically excluding the divergent nuclear self-energy, are derived. Rapid variations in the required neutral electronic potential in the vicinity of the nuclei are efficiently treated by an enriched finite element solution, using local radial solutions as enrichments (see this paper). 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 quantum-mechanical calculations with an application to crystalline diamond.

For some background material on density-functional theory and all-electron calculations, the discussions in the September 2008 and February 2009 journal club issues are pertinent.