Numerical Methods for Electronic Structure Calculations of Materials

Abstract

The goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems. The paper is intended for a diverse scienti£c computing audience. For this reason, we assume the reader does not have an extensive background in the related physics. Our overview focuses on the nature of the numerical problems to be solved, their origin, and on the methods used to solve the resulting linear algebra or nonlinear optimization problems. It is common knowledge that the behavior of matter at the nanoscale is, in principle, entirely determined by the Schrödinger equation. In practice, this equation in its original form is not tractable. Successful, but approximate, versions of this equation, which allow one to study nontrivial systems, took about £ve or six decades to develop. In particular, the last two decades saw a ¤urry of activity in developing effective software. One of the main practical variants of the Schrödinger equation is based on what is referred to as Density Functional Theory (DFT). The combination of DFT with pseudopotentials allows one to obtain in an ef£cient way the ground state con£guration for many materials. This article will emphasize pseudopotentialdensity functional theory, but other techniques will be discussed as well.

Randomized Algorithms for Matrices and Data

“…(Here the transpose actually refers to the Hermitian conjugate since the wavefunction ψ i (r) is a complex quantity.) Thus, the magnitude of this entry as well as other derived quantities like the trace of ρ give empirically-measurable quantities; see, e.g., Section 6.2 of [180]. More practically, improved methods for estimating the diagonal of a projection matrix may have significant implications for leading to improvements in large-scale numerical computations in scientific computing applications such as the density functional theory of many-atom systems [180,181].…”

Ab initio mass tensor molecular dynamics

“…U(q) denotes either the classical potential energy function [29], or the Kohn-Sham total energy [30,31]. In what follows, we will focus on the latter problem.…”

Limited Memory Block Krylov Subspace Optimization for Computing Dominant Singular Value Decompositions

“…This state of affair is evident when examining survey articles and reference books on eigenvalue and/or singular value computations (e.g., [19,7,25,22,10]). On the other hand, large-scale but unstructured matrices do arise from applications, such as image processing and computational materials science (e.g., [27,20]), which involve large amounts of data even though their dimensions are relatively small compared to those of large sparse matrices. This is especially true in recent years when the demand for computing dominant SVDs of large unstructured matrices has become increasingly pervasive in dataintensive applications.…”