Kohn–Sham Method

Abstract

In the field of quantum chemistry, theories on electronic motion states commencing with the Hartree-Fock method have intended to incorporate electron correlation more efficiently. However, due to the long computational times required, quantum chemistry calculations had been mostly restricted to trial applications by theoreticians until the 1980s. In the 1990s, density functional theory (DFT) appeared in quantum chemistry to resolve this situation. After this, DFT has become widespread, so that is now the main theory, which is used in more than 80% of quantum chemistry papers in 2014. The basic concept of DFT is to make it possible to perform high-speed calculations of many-electron systems by representing the potential as the functional not of the orbitals but of the electron density. The name "functional" indicates the function of a function, which is the electron density in this case. This basic concept was suggested in 1927, only one year after the Schrödinger equation was developed and the same year that the Hartree method was proposed. To solve the Schrödinger equation for the electronic motion states of solid crystals, Thomas suggested this basic concept of DFT (Thomas 1927). In his theory, an electronic state of a uniform electron gas is assumed to be a solution of the Schrödinger equation based on electron density. That is, for electronic motions in solid crystals, electrons are distributed uniformly at a proportion of two per unit cell, which is taken to be a cube with each edge length equal to Planck's constant in six-dimensional phase space. It is also assumed that the external potential, which corresponds to the nuclear-electron interaction potential (see Sect. 2.1) in the absence of an electromagnetic field, depends only on the distances from the nuclei and is therefore determined by the nuclear charge and electron density. Based on these assumptions, the kinetic energy functional of the electron density in Eq. (2.76) is formulated as T TF D C F Z d 3 r 5=3 .r/; (4.1) T. Tsuneda, Density Functional Theory in Quantum Chemistry,