Aufbau derived from a unified treatment of occupation numbersin Hartree-Fock, Kohn-Sham, and natural orbital theories with theKarush-Kuhn-Tucker conditions for the inequality constraints n i ≤ 1 and n i ≥ 0

In the major independent particle models of electronic structure theory-Hartree-Fock, Kohn-Sham (KS), and natural orbital (NO) theories-occupations are constrained to 0 and 1 or to the interval [0,1]. We carry out a constrained optimization of the orbitals and occupation numbers with application of the usual equality constraints ∑ i ∞ n i = N and ⟨ ϕ i | ϕ j ⟩ = δ i j . The occupation number optimization is carried out, allowing for fractional occupations, with the inequality constraints n i ≥ 0 and n i ≤ 1 with the Karush-Kuhn-Tucker method. This leads in all cases to an orbital energy spectrum with (only for NO and KS) possibly fractionally occupied degenerate levels at energy equal to the Lagrange multiplier ϵ for the first equality constraint, completely occupied levels at lower energies and completely unoccupied levels at higher energies. Aufbau thus follows in all cases directly from this general derivation.