By Capelle K. A
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Additional resources for Bird's-Eye View of Density-Functional Theory
It has a spatially varying density n(r), and it would clearly be useful to also include information on the rate of this 45 variation in the functional. A first attempt at doing this were the so-called ‘gradient-expansion approximations’ (GEA). , to the LDA. A famous example is the lowest-order gradient correction to the Thomas-Fermi approximation for Ts [n], Ts [n] ≈ TsW [n] = TsLDA [n] + h ¯2 8m d3 r |∇n(r)|2 . 48 Similarly, in Ex [n] ≈ ExGEA(2) [n] = ExLDA [n] − 10q 2 432π(3π 2)1/3 d3 r |∇n(r)|2 n(r)4/3 (87) the second term on the right-hand side is the lowest-order gradient correction49 to ExLDA [n].
55) is that the unknown functional Exc [n] is typically much smaller than the known terms Ts , UH and V . One can thus hope that reasonably simple approximations for Exc [n] provide useful results for E[n]. Some successful approximations are discussed in Sec. 5. Exact properties, such as the sum rule d3 r ′ nxc (r, r′) = −1, described in the preceding section, are most valuable guides in the construction of approximations to Exc [n]. Among the known properties of this functional are the coordinate scaling conditions first obtained by Levy and Perdew  Ex [nλ ] = λEx [n] 30 (59) Ec [nλ ] > λEc [n] Ec [nλ ] < λEc [n] for λ > 1 for λ < 1, (60) (61) where nλ (r) = λ3 n(λr) is a scaled density integrating to total particle number N.
For pure illustrative purposes only, Table 1 contains ground-state energies of the Ar atom, obtained with several of the methods discussed previously in this chapter. Footnote 7 contains additional information on the performance of DFT for larger systems. 50 The PBE GGA  and the TPSS MGGA  (see below) may be partial exceptions [99, 100] because they work reasonably well near the equilibrium distance of the van der Waals bond, but they recover only the short-range behaviour and do not describe correctly the long-range asymptotic regime of the van der Waals interaction.
Bird's-Eye View of Density-Functional Theory by Capelle K. A