Hohenberg-Kohn theorems local density approximation

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. 

Density-functional theory has its conceptual roots in the Thomas-Fermi model of a uniform electron gas [325,326] and the Slater local exchange approximation [327]. A formalistic proof for the correctness of the Thomas-Fermi model was provided by Hohenberg-Kohn theorems, [328]. DFT has been very popular for calculations in sohd-state physics since the 1970s. In many cases DFT with the local-density approximation and plane waves as basis functions gives quite satisfactory results, for sohd-state calculations, in comparison to experimental data at relatively low computational costs when compared to other ways of solving the quantum-mechanical many-body problem. 

One very successful method to solve this Hamiltonian is the density-functional theory (DFT). Within a DFT-based formalism one avoids constructing the many-body wavefunction of the system and instead computes directly ground-state properties from its charge density. However, the exact form of the functional embodied in the Hohenberg-Kohn theorem is unknown and, therefore, approximations are required. One of the most important, local-density approximation (LDA), developed by Kohn and Sham, works successfully for the ground-state properties of weakly... 

It has been argued that the metals do not act as a collection of atoms, but rather as a uniform entity. Thus various properties of the metal, such as elastic constants, are not accurately described by pair potentials [146]. If that is so, a model which assumes a pairwise atom-atom interaction will perhaps have to be replaced with one which incorporates the influence of electron density from a distributed electron gas rather than from localized ones. Such theories can furthermore be based on the Hohenberg Kohn theorem [211], namely that the energy is a functional of the electron density. Thus several approximate theories are based on the electron gas concepts, for example, the embedded atom and diatom methods, or the effective medium approach [210]. 

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