The Basics of Electronic Structure Theory for Periodic Systems

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.