Foundation for the {Δ}SCF Approach in Density Functional Theory

We extend ground-state density-functional theory to excited states and provide the theoretical formulation for the widely used Δ SCF method for calculating excited-state energies and densities. As the electron density alone is insufficient to characterize excited states, we formulate excited-state theory using the defining variables of a noninteracting reference system, namely (1) the excitation quantum number nₛ and the potential wₛ(r) (excited-state potential-functional theory, nPFT), (2) the noninteracting wavefunction Φ (Φ-functional theory, ΦFT), or (3) the noninteracting one-electron reduced density matrix γₛ(r,r') (density-matrix-functional theory, γₛFT). We show the equivalence of these three sets of variables and their corresponding energy functionals. Importantly, the ground and excited-state exchange-correlation energy use the same universal functional, regardless of whether (nₛ,wₛ(r)), Φ, or γₛ(r,r') is selected as the fundamental descriptor of the system. We derive the excited-state (generalized) Kohn-Sham equations. The minimum of all three functionals is the ground-state energy and, for ground states, they are all equivalent to the Hohenberg-Kohn-Sham method. The other stationary points of the functionals provide the excited-state energies and electron densities, establishing the foundation for the Δ SCF method.