Numerical Methods for Electronic Structure Calculations of Materials

This is the published version. Copyright 2010 Society for Industrial and Applied MathematicsThe 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 scientific 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 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 five or six decades to develop. In particular, the last two decades saw a flurry 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 efficient way the ground state configuration for many materials. This article will emphasize pseudopotential-density functional theory, but other techniques will be discussed as well

Dynamic tight binding for large-scale electronic-structure calculations of semiconductors at finite temperatures

Calculating the electronic structure of materials at finite temperatures is important for rationalizing their physical properties and assessing their technological capabilities. However, finite-temperature calculations typically require large system sizes or long simulation times. This is challenging for non-empirical theoretical methods because the involved bottleneck of performing many first-principles calculations can pose a steep computational barrier for larger systems. While machine-learning molecular dynamics enables large-scale/long-time simulations of the structural properties, the difficulty of computing in particular the electronic structure of large and disordered materials still remains. In this work, we suggest an adaptation of the tight-binding formalism which allows for computationally efficient calculations of temperature-dependent properties of semiconductors. Our dynamic tight-binding approach utilizes hybrid-orbital basis functions and a modeling of the distance dependence of matrix elements via numerical integration of atomic orbitals. We show that these design choices lead to a dynamic tight-binding model with a minimal amount of parameters which are straightforwardly optimized using density functional theory. Combining dynamic tight-binding with machine learning molecular dynamics and hybrid density functional theory, we find that it accurately describes finite-temperature electronic properties in comparison to experiment for the prototypical semiconductor gallium-arsenide

Main-group test set for materials science and engineering with user-friendly graphical tools for error analysis: Systematic benchmark of the numerical and intrinsic errors in state-of-the-art electronic-structure approximations

Understanding the applicability and limitations of electronic-structure methods needs careful and efficient comparison with accurate reference data. Knowledge of the quality and errors of electronic-structure calculations is crucial to advanced method development, high-throughput computations and data analyses. In this paper, we present a main-group test set for computational materials science and engineering (MSE), that provides accurate and easily accessible crystal properties for a hierarchy of exchange-correlation approximations, ranging from the well-established mean-field approximations to the state-of-the-art methods of many-body perturbation theory. We consider cohesive energy, lattice constant and bulk modulus of a set of materials that representatives for the first- and second-row elements and their binaries with cubic crystal structures and various bonding characters. A strong effort is made to achieve high numerical accuracy for cohesive properties as calculated using the local-density approximation (LDA), several generalized gradient approximations (GGAs), meta-GGAs and hybrids in all-electron resolution, and the second-order Møller-Plesset perturbation theory (MP2) and the random-phase approximation (RPA) both with frozen-core approximation based on all-electron Hartree-Fock, PBE and/or PBE0 references. This results in over 10,000 calculations, which record a comprehensive convergence test with respect to numerical parameters for a wide range of electronic structure methods within the numerical atom-centered orbital framework. As an indispensable part of the MSE test set, a web site is established http://mse.fhi-berlin.mpg.de. This not only allows for easy access to all reference data but also provides user-friendly graphical tools for post-processing error analysis

Simulation of electron energy loss spectra with the turboEELS and thermo-pw codes

For some materials like noble metals, electron energy loss spectra have a complex structure that makes them difficult to analyze without the help of ab initio calculations. Various theoretical approaches can be used for this purpose, among which the time-dependent density functional perturbation theory (TDDFPT) which has been widely used to study plasmons in a number of bulk and surface systems. In the present paper we present a comparison of the results and performance of two different numerical implementations of TDDFPT: the Sternheimer and Liouville-Lanczos methods. The former approach is implemented in the thermo-pw module and the latter one in the turboEELS code of the QUANTUM ESPRESSO package for electronic structure calculations. In the present paper a comparison is made for bulk bismuth, a semimetal, taking into account spin-orbit coupling, as well as for bulk gold, a noble metal. We show that for these two examples, both codes gives identical results and the turboEELS code has a better performance than the thermo-pw code, and point out in which cases the usage of thermo-pw alone or of both codes can be advantageous

Periodic Pulay method for robust and efficient convergence acceleration of self-consistent field iterations

Pulay's Direct Inversion in the Iterative Subspace (DIIS) method is one of the most widely used mixing schemes for accelerating the self-consistent solution of electronic structure problems. In this work, we propose a simple generalization of DIIS in which Pulay extrapolation is performed at periodic intervals rather than on every self-consistent field iteration, and linear mixing is performed on all other iterations. We demonstrate through numerical tests on a wide variety of materials systems in the framework of density functional theory that the proposed generalization of Pulay's method significantly improves its robustness and efficiency.Comment: Version 2 (with minor edits from version 1