Hohenberg-Kohn-Sham formalism

The general relativistic Hohenberg-Kohn-Sham formalism, outlined above, contains the spin degrees of freedom in a complete form. Consequently, the spin and kinetic motion effects are not separable. Indeed, they are contained in the external potential term as one can see if such term is written using the orbital current... 

As in the case of the Schrodinger approach in which spin is introduced by giving a specific form to the wave function, the spin dependence in the Hohenberg-Kohn-Sham formalism in a nonrelativistic framework is introduced by imposing some form of restrictions to the functional. Namely, the total energy can be written as [3,5]... 

The Hohenberg-Kohn-Sham density functional theory provides the common formal framework for various computational methods. Since each of the methods in use involves approximations, the calculated properties are not exact. Nevertheless, these methods proved to be very useful in chemistry and materials science. The huge and ever growing number of applications (see Figure 2-1) speaks for itself. Frequently,... 

After the discovery of the relativistic wave equation for the electron by Dirac in 1928, it seems that all the problems in condensed-matter physics become a matter of mathematics. However, the theoretical calculations for surfaces were not practical until the discovery of the density-functional formalism by Hohenberg and Kohn (1964). Although it is already simpler than the Hartree-Fock formalism, the form of the exchange and correlation interactions in it is still too complicated for practical problems. Kohn and Sham (1965) then proposed the local density approximation, which assumes that the exchange and correlation interaction at a point is a universal function of the total electron density at the same point, and uses a semiempirical analytical formula to represent such universal interactions. The resulting equations, the Kohn-Sham equations, are much easier to handle, especially by using modern computers. This method has been the standard approach for first-principles calculations for solid surfaces. 

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],... 

In Chap. 4, the Kohn-Sham equation, which is the fundamental equation of DFT, and the Kohn-Sham method using this equation are described for the basic formalisms and application methods. This chapter first introduces the Thomas-Fermi method, which is conceptually the first DFT method. Then, the Hohenberg-Kohn theorem, which is the fundamental theorem of the Kohn-Sham method, is clarified in terms of its basics, problems, and solutions, including the constrained-search method. The Kohn-Sham method and its expansion to more general cases are explained on the basis of this theorem. This chapter also reviews the constrained-search-based method of exchange-correlation potentials from electron densities and... 

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