Foundation for the {\Delta}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 $\Delta SCF$ 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 $n_{s}$ and the potential $w_{s}(\mathbf{r})$ (excited-state potential-functional theory, $n$PFT), (2) the noninteracting wavefunction $\Phi$ ($\Phi$-functional theory, $\Phi$FT), or (3) the noninteracting one-electron reduced density matrix $\gamma_{s}(\mathbf{r},\mathbf{r}')$ (density-matrix-functional theory, $\gamma_{s}$FT). 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 \textit{same} universal functional, regardless of whether $\left(n_{s},w_{s}(\boldsymbol{r})\right)$, $\Phi$, or $\gamma_{s}(\mathbf{r},\mathbf{r}')$ 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 $\Delta SCF$ method.