Electronic Structure: Density Functional Theory

Density functional theory (DFT) is a successful theory to calculate the electronic structure of atoms, molecules, and solids. Its goal is the quantitative understanding of materials properties from the fundamental laws of quantum mechanics. Traditional electronic structure methods attempt to find approximate solutions to the Schrödinger equation of N interacting electrons moving in an external, electrostatic potential (typically the Coulomb potential generated by the atomic nuclei). However, there are serious limitations of this approach: (i) the problem is highly nontrivial, even for very small numbers N and the resulting wave-functions are complicated objects, (ii) the computational effort grows very rapidly with increasing N , so the description of larger systems becomes prohibitive. A different approach is taken in density functional theory where, instead of the many-body wave-function, the one-body density is used as fundamental variable. Since the density n(r) is a function of only three spatial coordinates (rather than the 3N coordinates of the wavefunction), density functional theory is computationally feasible even for large systems. The foundations of density functional theory are the Hohenberg-Kohn and Kohn-Sham theorems which will be reviewed in the following section. In section III, we will discuss various levels of approximation to the central quantity of DFT, the so-called exchange-correlation energy functional. Section IV will present some typical results from DFT calculations for various physical properties that are normally calculated with DFT methods. The original Hohenberg-Kohn and Kohn-Sham theorems can easily be extended from its original formulation to cover a wide variety of physical situations. A number of such extensions is presented in section V, with particular emphasis on time-dependent DFT (section VC).