Introduction

Density functional theory (DFT) has become in the past few decades one of the most widely used methods for the calculation of the properties of complex electronic systems: molecules, solids, polymers. The basic idea, introduced by Hohenberg, Kohn, and Sham in the 1960s (Hohenberg and Kohn (1964), Kohn and Sham (1965)), is to describe the system in terms of the electronic density (and possibly additional densities such as the spin density, the current density, etc.) without explicit reference to the many-body wave function. At first sight, this seems impossible. How can the subtle correlations encoded in an N-electron wave function be adequately represented by a simple collective variable, such as the density? But Hohenberg and Kohn, in their classic paper, were able to show that the ground-state energy of a quantum system can be determined by minimizing the energy as a functional of the density, in much the same way as, in standard quantum mechanics, one can determine the energy by minimizing the expectation value of the hamiltonian with respect to the wave function. Furthermore, the nontrivial part of this functional is universal, that is, it has the same form for all physical systems.

The implementation of the Hohenberg–Kohn minimum principle leads to mean-field-like equations, known as the Kohn–Sham equations, which are simpler than the Hartree–Fock equations, yet in principle exact as far as the calculation of the ground-state density and energy is concerned.