Starting point is the energy expectation value as a functional of the one-particle density matrix γ and the two-particle density cumulant λ2. We decompose γ into a best idempotent approximation κ and a correction τ, that is entirely expressible in terms of λ2. So we get the energy E as a functional of κ and λ2, which can be varied independently. Approximate n-representability conditions, derived by perturbation theory are imposed on the variation of λ2. A nonlinear system of equations satisfied by λ2 is derived, the linearized version of which turns out to be equivalent to the coupled electron-pair approximation, variant zero. The start for κ is Hartree-Fock, but κ is then updated to become the best idempotent approximation of γ. Relations to density matrix functional theory and Kohn-Sham type density functional theory are discussed.