Does the Harris energy functional possess a local maximum at the ground-state density?

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Abstract
We prove that for any system, and within the framework of any reasonable local approximation to the Hohenberg-Kohn-Sham exchange-correlation energy functional, the Harris energy functional posseses either a saddle point or a local minimum at a self-consistent ground-state charge density. This invalidates the common suggestion that this functional has a local maximum at this charge density. Some numerical results in support of our arguments are presented.