Abstract
When density functional theory is used to describe the electronic structure of periodic systems, the application of Bloch's theorem to the Kohn-Sham wavefunctions greatly facilitates the calculations. In this paper of the series, the concepts needed to model infinite systems are introduced. These comprise the unit cell in real space, as well as its counterpart in reciprocal space, the Brillouin zone. Grids for sampling the Brillouin zone and finite k-point sets are discussed. For metallic systems, these tools need to be complemented by methods to determine the Fermi energy and the Fermi surface. Various schemes for broadening the distribution function around the Fermi energy are presented and the approximations involved are discussed. In order to obtain an interpretation of electronic structure calculations in terms of physics, the concepts of bandstructures and atom-projected and/or orbital-projected density of states are useful. Aspects of convergence with the number of basis functions and the number of k-points need to be addressed specifically for each physical property. The importance of this issue will be exemplified for force constant calculations and simulations of finite-temperature properties of materials. The methods developed for periodic systems carry over, with some reservations, to less symmetric situations by working with a supercell. The chapter closes with an outlook to the use of supercell calculations for surfaces and interfaces of crystals.
Highlights
Three-dimensional periodic solids were among the first systems for which the theory of electronic structure was worked out
In early computational work, when computer memory was a major limitation, the focus was on an economic choice of kpoints for Brillouin zone sampling that made it possible to carry out calculations for real-space unit cells with many atoms
With the increasing role of “big data” and high-throughput computations in materials physics and chemistry, it is important to guarantee high accuracy for the first-principles data to be stored in materials databases
Summary
Three-dimensional periodic solids were among the first systems for which the theory of electronic structure was worked out. Methods based on machine learning attempt to select k-point grids that are most suitable for the problem at hand (Choudhary and Tavazza, 2019) As another factor driving innovation in the field of k-point sampling, the interest in special properties of bulk materials, in particular in the areas of electronic transport, magnetism and topological states of matter, has lead to improved (e.g., adaptive) schemes. While we restrict ourselves to density functional theory calculations in this review, an ab initio treatment of periodic systems with the wavefunction-based methods of quantum chemistry is an alternative option that is free of any inaccuracies due to approximate density functionals This field has attracted strong interest and significant progress has been achieved (Booth et al, 2012; Gruber et al, 2018). Convergence issues with respect to the plane-wave expansion of the Kohn-Sham wavefunctions (Kresse and Furthmüller, 1996a) or with respect to atom-centered basis sets (Koch and Holthausen, 2001; Blum et al, 2009) should be addressed by practitioners of DFT before turning their attention to periodic systems; again, we refer to the literature
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
