Hohenberg-Kohn theorem, wave function calculations

The density functional theory of Hohenberg, Kohn and Sham [173,205] has become the standard formalism for first-principles calculations of the electronic structure of extended systems. Kohn and Sham postulate a model state described by a singledeterminant wave function whose electronic density function is identical to the ground-state density of an interacting /V-clcctron system. DFT theory is based on Hohenberg-Kohn theorems, which show that the external potential function v(r) of an //-electron system is determined by its ground-state electron density. The theory can be extended to nonzero temperatures by considering a statistical electron density defined by Fermi-Dirac occupation numbers [241], The theory is also easily extended to the spin-indexed density characteristic of UHF theory and of the two-fluid model of spin-polarized metals [414],... 

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. 

The development of DFT is based on Kohn and Hohenberg s mathematical theorem, which states that the ground state of the electronic energy can be calculated as a functional of the electron density [18], The task of finding the electron density was solved by Kohn and Sham [19]. They derived a set of equations in which each equation is related to a single electron wave function. From the single electron wave functions one can easily calculate the electron density. In DFT computer codes, the electron density of the core electrons, that is, those electrons that are not important for chemical bonds, is often represented by a pseudopotential that reproduces important physical features, so that the Kohn-Sham equations span only a select number of electrons. For each type of pseudopotential, a cutoff energy or basis set must be specified. 

Density functional theory (DFT), developed within solid state physics, is based on the theorem of Hohenberg and Kohn that the ground state energy of a system depends on the electron density. It can be applied to calculations performed either with localised basis sets or by combination of plane waves. Both approaches have been applied to microporous solids, although the plane wave methods have been used more commonly. The SIESTA code, for example,permits DFT calculations using localised basis sets as does GAUSSIAN. 

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