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Higuchi and K. Higuchi, Phys. Rev. B 69, 035113 (2004)] enables us to choose arbitrary quantities as the basic variables of density functional theory. In this paper, we apply it to several cases. In the case in which the occupation matrix of localized orbitals is chosen as a basic variable, we can obtain the single-particle equation which is equivalent to that of the LDA+U method. The theory also leads to the Hartree-Fock-Kohn-Sham equation by letting the exchange energy be a basic variable. 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