Generalized Kohn-Sham method

Therefore, as shown in Eq. (4.10), the total electronic energy in the Kohn-Sham method is separately calculated using the exchange-correlation energy functionals. Note, however, that no such difference should be included in a universal functional, and there is a possibility in the future to develop an exchange-correlation functional giving no difference. In this case, process 4 is not necessary. 

The Kohn-Sham equation is also transformed into a matrix equation on the basis of the Roothaan method in Sect. 2.5. Similar to the Hartree-Fock equation, the Kohn-Sham-Roothaan equation is written as 

Terms other than the exchange-correlation potential are the same as those in Sect. 2.5. In this method, the total electronic energy is separately calculated as different to that in the Roothaan method. 

Although the Kohn-Sham method has been the basic procedure in DFT calculations, many exchange-correlation functionals used in conventional DFT calculations have no strict theoretical basis upon which to be used in the Kohn-Sham method. Since the Kohn-Sham method is based on the constrained search formulation, it is proven to be applicable to pure exchange-correlation energy density functionals. 

The first Hohenberg-Kohn theorem for this method is established if there is a universal functional,, that uniquely determines the electron density, giving the least action  

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Stein T, Eisenberg H, Kroiuk L, Baer R (2010) Fundamental gaps in finite systems from eigenvalues of a generalized Kohn-Sham method. Phys Rev Lett 105 266802... 

Density functional theory (DFT) provides an efficient method to include correlation energy in electronic structure calculations, namely the Kohn-Sham method 1 in addition, it constitutes a solid support to reactivity models.2 DFT framework has been used to formalize empirical reactivity descriptors, such as electronegativity,3 hardness4 and electrophilicity index.5 The frontier orbital theory was generalized by the introduction of Fukui function,6 and new reactivity parameters have also been proposed.7,8 Moreover, relationships between those parameters have been found, and general methods to relate new quantities exist.9... 

The idea of the Kohn-Sham method is best understood as follows. Consider a generalized Hamiltonian of Eq. (2) in which the term 4e is scaled by an electron-electron coupling constant A. We are interested in values of A between 0 and 1. Each value of A corresponds to a distinct universal functional of the density. In Levy s constraint search formulation [36] of the Hohenberg-Kohn principle, this is explicitly stated as... 

Differently to the Hartree-Fock method, the total electronic energy in the Kohn-Sham method is generally calculated using not the exchange-correlation potential functional but the exchange-correlation energy functional, as... 

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... 

Section 11.2 presents the non-variational theory. In Sect. 11.3 Kohn-Sham-like equations are obtained through adiabatic connection. Density scaling is applied to obtain a generalized Kohn-Sham scheme in Sect. 11.4. The optimized potential and the KLI methods are generalized in Sect. 11.5. The last section is devoted to illustrative examples and discussion. 

First-principles methods are the most convincing choice for the study of bulk and surface materials. Although in general rather slow, DFT is the most prominent and most successful way nowadays to solve the many-body problem of quantum physics. It is based on the Hohenberg- Kohn theorem and the dominant role of the electronic density, as well as the Kohn-Sham method for finding the ground-state... 

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