Density functional derivative Frechet, local potential

This argument shows that the locality hypothesis fails for more than two electrons because the assumed Frechet derivative must be generalized to a Gateaux derivative, equivalent in the context of OEL equations to a linear operator that acts on orbital wave functions. The conclusion is that the use by Kohn and Sham of Schrodinger s operator t is variationally correct, but no equivalent Thomas-Fermi theory exists for more than two electrons. Empirical evidence (atomic shell structure, chemical binding) supports the Kohn-Sham choice of the nonlocal kinetic energy operator, in comparison with Thomas-Fermi theory [288]. A further implication is that if an explicit approximate local density functional Exc is postulated, as in the local-density approximation (LDA) [205], the resulting Kohn-Sham theory is variation-ally correct. Typically, for Exc = f exc(p)p d3r, the density functional derivative is a Frechet derivative, the local potential function vxc = exc + p dexc/dp. 

When Exc p is specified, the relevant ground-state density for Hohenberg-Kohn theory is p0, computed using the equivalent orbital functional Exc in the OEL equahons, (Q — e,-)local potential w(r) in the corresponding KS equahons is determined by the KSC by minimizing T for p = p0. Assuming the locality hypothesis, that w — v is the Frechet derivative of the model ground-state functional h p — Ts[p, this implies that w = vh + vxc + v is a sum of local potentials. If i>xc in the OEL equahons was equivalent to a local potential vxc(r), the KS and OEL equations would produce the same model wave function. 

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