Density Functional Theory (DFT) maps the interacting many-electron problem onto a system
of non-interacting electrons in an effective potential. The Kohn-Sham equations are:
[−½∇² + V_ext + V_Hartree + V_xc]ψᵢ = εᵢψᵢ.
The exchange-correlation (XC) functional encodes all many-body effects.
LDA uses the homogeneous electron gas result: ε_x = −(3/4)(3n/π)^(1/3).
GGA-PBE adds gradient corrections. Shown here: self-consistent radial DFT for atom Z
(simplified hydrogen-like basis for illustration).