Hohenberg-Kohn-Sham equations electronic energy

The relativistic correction for the kinetic energy in the Dirac equation is naturally applicable to the Kohn-Sham equation. This relativistic Kohn-Sham equation is called the Dirac-KohnSham equation (Rajagopal 1978 MacDonald and Vosko 1979). The Dirac-Kohn-Sham equation is founded on the Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem on the basis of QED (Rajagopal and Callaway 1973). In this theorem, two theorems are contained The first theorem proves that the four-component external potential, which is the vector-potential-extended external potential, is determined by the four-component current density, which is the current-density-extended electron density. On the other hand, the second theorem establishes the variational principle for every four-component current density. See Sect. 6.5 for vector potential and current density. Consequently, the solution of the Dirac-Kohn-Sham equation is represented by the four-component orbital. This four-component orbital is often called a molecular spinor. However, this name includes no indication of orbital, which is the solution of one-electron SCF equations moreover, the targets of the calculations are not restricted to molecules. Therefore, in this book, this four-component orbital is called an orbital spinor. The Dirac-Kohn-Sham wavefunction is represented by the Slater determinant of orbital spinors (see Sect. 2.3). Following the Roothaan method (see Sect. 2.5), orbital spinors are represented by a linear combination of the four-component basis spinor functions, Xp, ... 

The Hohenberg-Kohn theorems find a very important application in the derivation of the Kohn-Sham equations, in which the problem of approximating the noninteracting kinetic energy (Ts) is eliminated by introducing single-particle orbitals 9,. The exact electron density is written as the electron density of a Slater determinant,... 

The resulting single-particle eigenvalue equations are the Kohn-Sham equations. The Hohenberg-Kohn theorems ensure that the exchange-correlation energy in Eq. (9) is a functional of the electron density. 

The original density functional theory (DFT), based on Hohenberg-Kohn theorems [1], Kohn-Sham equations [2] and the Levy constrained search formulation [3], is a rigorous approach for determining the ground-state density and ground-state energy for any A/ -electron system. Here the electron number... 

The Hohenberg-Kohn principles provide the theoretical basis of Density Functional Theory, specifically that the total energy of a quantum mechanical system is determined by the electron density through the Kohn-Sham functional. In order to make use of this very important theoretical finding, Kohn-Sham equations are derived, and these can be used to determine the electronic ground state of atomic systems. 

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. 

Kohn-Sham equations. According to the Hohenberg-Kohn theorem the density (r) in Eq. (16) can be also the density of an interacting system of electrons moving an external (to be found), i.e. PG " (r) = p (r). The total-energy functional of the interacting-system of electrons can be rewritten as... 

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