Perdew-Wang formula

The real-space cutoff of the GEA hole provides a powerful nonempirical way to construct GGA s. Since exchange and correlation should be treated in a balanced way, there was a need to extend the 1986 real-space cutoff construction [13] from exchange to correlation with the help of a second cutoff radius Wc(r) chosen to satisfy (1.97). Without accurate formulas for the correlation hole of the uniform electron gas, this extension had to wait until 1991, when it led to the Perdew-Wang 1991 (PW91) [18,79] GGA for Exc- For most practical purposes, PW91 is equivalent to the Perdew-Burke-... 

Several gradient corrected exchange [28-30] and correlation [31-34] functionals have been proposed. In our study we use different combinations of those of Becke [28] and Perdew and Wang [30] for the exchange and the functionals proposed by Perdew [31] and Proynov [34] for the correlation. Recently, Becke [29] has introduced the so-called hybrid functional that is based on the "adiabatic connection" formula [35], Because of its reliability, largely validated in the literature [36-38], we employ in our work also the B3LYP functional which uses the Becke gradient... 

The analytical form for the correlation energy of a uniform electron gas, which is purely dynamical correlation, has been derived in the high and low density limits. For intermediate densities, the correlation energy has been determined to a high precision by quantum Monte Carlo methods (Section 4.16). In order to use these results in DFT calculations, it is desirable to have a suitable analytic interpolation formula, and such formulas have been constructed by Vosko, Wilk and Nusair (VWN) and by Perdew and Wang (PW), and are considered to be accurate fits. The VWN parameterization is given in eq. (6.36), where a slightly different spin-polarization function has been used. 

The last formula is circumvented to the high-density total correlation density approaches rooting at their turn on the Thomas-Fermi atomic theory. Very interesting, the relation (4.477) may be seen as an atomic reflection of the (solid state) high-density regime (r < 1) given by Perdew et al. (Perdew, 1986 Wang Perdew, 1989 Seidl et al., 1999 Perdew et al., 1996) ... 

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