A recently proposed variation principle [N. I. Gidopoulos, Phys. Rev. A 83, 040502(R) (2011)] for the determination of Kohn–Sham effective potentials is examined and extended to arbitrary electron-interaction strengths and to mixed states. Comparisons are drawn with Lieb’s convex-conjugate functional, which allows for the determination of a potential associated with a given electron density by maximization, yielding the Kohn–Sham potential for a non-interacting system. The mathematical structure of the two functionals is shown to be intrinsically related; the variation principle put forward by Gidopoulos may be expressed in terms of the Lieb functional. The equivalence between the information obtained from the two approaches is illustrated numerically by their implementation in a common framework.

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