  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.
