Semilocal functionals GEA, GGA and beyond

In the LDA one exploits knowledge of the density at point r. Any real system is spatially inhomogeneous, i.e., it has a spatially varying density n(r), and it would clearly be useful to also include information on the rate of this 

This second term on the right-hand side is called the Weizsacker term.48 Similarly, in 

In this situation it was a major breakthrough when it was realized, in the early eighties, that instead of power-series-like systematic gradient expansions one could experiment with more general functions of n(r) and Vn(r), which need not proceed order by order. Such functionals, of the general form 

Different GGAs differ in the choice of the function f(n,Vn). Note that this makes different GGAs much more different from each other than the different parametrizations of the LDA essentially there is only one correct 

48If one adds this term to the Thomas-Fermi expression (35) one obtains the so-called Thomas-Fermi-Weizsacker approximation to E[n. In a systematic gradient expansion the 8 in the denominator is replaced by a 72 [5, 6]. 

In the LDA one exploits knowledge of the density at point r. Any real system is spatially inhomogeneous, i.e. it has a spatially varying density (r), and it would clearly be useful to also include information on the rate of this variation in the functional. A first attempt at doing this was the so-called GEAs. In this class of approximation one tries to systematically calculate gradient corrections of the form V (r), V (r)p, V n(r), etc. to the LDA. A famous example is the lowest-order gradient correction to the Thomas-Fermi approximation for Ts n], 

Remarkably, the form of this term is fully determined already by dimensional analysis. In I fin, V p) the func- 

Different GGAs differ in the choice of the function /(n, V ). Note that this makes different GGAs much more different from each other than the different parameterizations of the LDA essentially there is only one correct expression for ejj (n), and the various parameterizations of the LDA i09, iii j.g jngj-giy different ways of writing it. On the other hand, depending on the method of construction employed for obtaining /(n, Vn) one can obtain very different GGAs. 

Nowadays, the most popular (and most reliable) GGAs are PBE (denoting the parameter-free functional proposed in 1996 by Perdew, Burke and Ernzerhof ) and B88LYP (denoting the combination of Becke s 1988 one-parameter exchange functional B88 ° with 

---