Classification of Exchange-Correlation Functionals

In the previous chapter, the fundamentals of the Kohn-Sham method and its derivative theories have been explained without referring to the specific forms of exchange-correlation functionals used. The Kohn-Sham method is an established quantum theory based not only on electron density but also on a rigorous exchange-correlation functional. It is therefore difficult to assess the reliability of the Kohn-Sham method and its derivative theories without specifically considering the exchange-correlation functional used. In this chapter, let us examine the exchange-correlation functionals that have thus far been developed, with their features and problems. 

As shown in Eq.(4.6), the exchange-correlation functional is the only part that is approximated in the Kohn-Sham equation. Therefore, the reliability of the Kohn-Sham method depends on the validity of this approximated functional. A variety of exchange-correlation functionals have thus far been developed on the basis of different physical models. In Fig. 5.1, exchange-correlation functionals are classified based on their characteristics. 

Tsuneda, Density Functional Theory in Quantum Chemistry, DOI 10.1007/978-4-431-54825-6 5, Springer Japan 2014 

Although there are exchange-correlation functionals that do not fall into this classification, they are mostly functionals correcting the above functionals for some sort of physical factor. Such physical factors for functionals are introduced in Chap. 6. In this chapter, let us focus on functionals conforming to a standard paradigm in which LDA and GGA functionals are expanded by additional terms. These standard functionals are often arranged hierarchically, as in Jacob s ladder ( the ladder to Heaven in the Old Testament, Book of Genesis, Fig. 5.2) (Perdew et al. 2006). 

In past developments of exchange-correlation functionals, the following two criteria have been emphasized  

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