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The DFT formalism exploits certain ground‐state properties of many‐electron systems in an external field, thus in our case the Coulomb field of the nuclei. Following up ideas proposed by Thomas and Fermi in the 1930s, it has been rigorously established by Hohenberg, Kohn, and Sham in the early 1960s [2], who showed that the total energy of the system is a functional of the electronic density. DFT gets rid of the many‐body wavefunction, which depends on 3n electronic spatial coordinates, and replaces it by the simpler electronic density ρ(r) that only depends on three spatial coordinates:

(3)

Kohn and Sham [3] showed that this density can be written as a sum of the density of noninteracting particles, , where ϕ_i are the fictitious one‐particle Kohn–Sham (KS) orbitals, and thus the ground‐state density ρ₀(r) is given by the sum of the ground states of these particles. Hence, the total energy of the system can be expressed as

(4)

where T[ρ] is the kinetic energy of a system of n noninteracting electrons having the density ρ, , and the second term is the Coulomb interactions between electron–electron and electron–nuclei. The third term is the so‐called exchange and correlation energy which accounts for all quantum many‐body effects due to the Pauli exclusion principle which correlates electrons. Since this term cannot be evaluated exactly, approximations have to be made. Even though E_xc[ρ] is usually substantially smaller than the two other terms, the manner it is approximated may become crucial for chemically complex systems [2]. The simplest one, proposed originally by Kohn and Sham [3], relies on the assumption that at each point the exchange‐correlation energy density corresponds to that of a homogeneous gas of electrons. This approximation is called the “local density approximation” (LDA) and is given by:

(5)

Even for such a simple model system, the expression of the correlation energy has to be calculated numerically with Monte‐Carlo methods. A more advanced approximation, the so‐called “generalized gradient approximation” or GGA, is based on a more complex operator making use of the density gradient of mth order:

(6)

However, this higher approximation does not necessarily give more reliable results so that it is a priori not clear which exchange‐correlation functional should be used [2]. Despite this problem one can say that the KS approach and reasonable (simple) approximations for the exchange‐correlation term have opened the door to the calculations of the electronic structure for many‐atom systems relevant to the study of real materials.

Although expressing the full quantum mechanical problem in the language of DFT leads to a significant reduction of the computational effort for calculating the forces on the nuclei, in practice this task is still extremely demanding when dealing with more than a few tens of atoms. Therefore, one usually makes the further approximation that all the core electrons of an atom are lumped together so that their effect is replaced by an effective potential, the so‐called “pseudo‐potential,” for the remaining valence electrons which are described by a pseudo‐wavefunction [2]. The physical motivation for this approximation is that the chemical bonds between two atoms are usually related to the outer valence electrons and thus depend only weakly on the inner core electrons.