Functional derivatives and local potentials

The variation of an orbital functional induced by infinitesimal orbital variations is 

If this becomes a density functional F,[p] for ground states, the density functional variation is 

Consistency between these variations of a density functional and of the equivalent orbital functional implies the chain rule = ./ (r), regardless of any 

An orbital functional derivative in general defines a linear operator such that 7 = vp pi. If the locality hypothesis were valid, then 4 = vF(r), and the implied local potential function could be computed directly from the sum rule, 

This formula was used by Slater [385] to define an effective local exchange potential. The generally unsatisfactory results obtained in calculations with this potential indicate that the locality hypothesis fails for the density functional derivative of the exchange energy Ex [294], 

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