Within the context of a density matrix approach based on the 1-matrix, where pure-state N-representability conditions are explicitly incorporated, the Schrodinger equation may be exactly reduced to a set of 2m coupled equations (m is the number of single particle orbitals) that give solutions for both the ground state and excited states are discussed. These coupled equations given via realization of the two-step variational procedure are shown to yield under appropriate simplifications the usual Kohn–Sham equations. The relevance of these novel equations, in relation to molecular physics and condensed matter theory calculations, is discussed. The chapter also provides a review of Hohenberg–Kohn theorems and illustrates their relation to the two-step variational principle of Levy and Lieb. It describes the constrained variational procedure that is used in the determination of the auxiliary universal functional and analyzes in this respect the importance of N-representability conditions. The connections between the system of coupled orbital equations which result from this formalism and those of Kohn-Sham are also discussed.