Gedanken Densities and Lower Bounds in Density Functional Theory

A gedanken density is not a real one but one imagined in the construction of density functional approximations. The uniform electron gas is the original gedanken density, but we will be concerned here with two others: (1) the groundstate density of one electron in the presence of a nonuniform periodic potential , in which the reduced density gradient s = |∇n| /[2(3π2)1/3n4/3diverges almost everywhere as the volume tends to infinity. This density was used in the construction [1] of a generalized gradient approximation (GGA): To satisfy the general Lieb-Oxford lower bound [2] on the exchangecorrelation energy for all possible densities, the exchange enhancement factor Fx ≡ eapprox x /eunif x in the large-s limit for a spin-unpolarized density must be less than or equal to 1.804. (2) a two-electron spherical ground-state density in which s takes the same arbitrary positive value wherever the density is non-zero [3]. This density can be used to show that, to satisfy the tight Lieb-Oxford bound on the exchange energy of a two-electron density for every possible such density, Fx for such a density (and probably for every density) must be less than 1.174. The local spin density approximation (LSDA) for exchange (Fx = 1) satisfies this tight bound, but standard GGA’s and meta-GGA’s do not. A talk by Jianwei Sun will present what may be the first beyond-LSDA approximation to satisfy this strong new constraint.