Aufbau derived from a unified treatment of occupation numbers in Hartree–Fock, Kohn–Sham, and natural orbital theories with the Karush–Kuhn–Tucker conditions for the inequality constraints ni≤1 and ni

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∞ ni=N and ⟨ϕi|ϕj⟩=δij. The occupation number optimization is carried out, allowing for fractional occupations, with the inequality constraints ni≥0 and ni≤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.