Kohn-Sham 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],... 

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. 

The simpler PW methods are the most popular in the Kohn-Sham periodic-systems calculations. Plane waves are an orthonormal complete set any function belonging to the class of continuous normalizable functions can be expanded with arbitrary precision in such a basis set. Using the Bloch theorem the single-electron wavefunction can be written as a product of a wave-like part and a cell-periodic part ifih = exp(ifer)Mj(r) (see Chap. 3). Due to its periodicity in a direct lattice Uj(r) can be expanded as a set of plane waves M<(r) = C[Pg.281]

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