Elements of density-functional theory

An alternative to Hartree-Fock theory is density-functional (DF) theory in which certain elements in the Hamiltonian are evaluated at fixed points directly from the electron densities at those points. This can circumvent the need for calculating the enormous numbers of electron-electron repulsion integrals encountered in Hartree-Fock calculations. We here briefly describe density-functional theory based upon reviews by von Barth (1986) and Srivastava and Weaire (1987). 

During the 1970s and 1980s, density-functional theory became an important tool for calculating static electronic and structural properties of solids. The theory represents in principle an exact formulation of the many-electron problem in terms of a single particle moving in the mean field of the other electrons. All the difficulties associated with the solution of the many-electron problem are enclosed in this mean field, for which some approximation must be adopted. In practice, most calculations have been carried out using the local-density approximation (LDA), which has 

The Hohenberg-Kohn theorem was of great importance for the development of density-functional theory, but a practical implementation of DF theory was first presented by Kohn and Sham (1965). This paper contains two important advances of the theory (1) the exact conversion of the many-electron problem to an effective one-electron problem, and (2) the local-density approximation. 

The conversion to a one-electron problem was achieved by a simple reshuffling of terms in the functional for the total energy normally written 

T[p] is the total kinetic energy, t/[p] is the interelectronic Coulomb interaction, and the last term represents the interaction with the external potential V(r). Introducing the kinetic energy Tofp] in the ground state of a fictitious system of noninteracting electrons also having the density p(r), we can write  

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