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

N. Sukumar's picture

In this manuscript (available at, 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.


Mike Ciavarella's picture

This divergence due to 1/r singularity, reminds me of Olber's paradox. Is there a similarity?  Mike


Olbers' paradox
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Olbers' paradox in action

In astrophysics and physical cosmology, Olbers' paradox is the
argument that the darkness of the night sky conflicts with the
assumption of an infinite and eternal static universe. It is one of the pieces of evidence for a
non-static universe such as the current Big Bang model. The argument is also referred to as the "dark
night sky paradox
" The paradox states that at any angle from the
Earth the sight line will end at the surface of a star. To understand
this we compare it to standing in a forest of white trees. If at any
point the vision of the observer ended at the surface of a tree,
wouldn't the observer only see white? This contradicts the darkness of
the night sky and leads many to wonder why we do not see only light from
stars in the night sky (see physical paradox).




Michele Ciavarella,
Editor, Italian Science Debate,
YouTube Channel

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