General degeneracy in density functional perturbation theory

Degenerate perturbation theory from quantum mechanics is inadequate in density functional theory (DFT) because of nonlinearity in the Kohn-Sham potential. Herein, we develop the fully general perturbation theory for open-shell, degenerate systems in Kohn-Sham DFT, without assuming the presence of symmetry or equal occupation of degenerate orbitals. To demonstrate the resulting methodology, we apply it to the iron atom in the central field approximation, perturbed by an electric quadrupole. This system was chosen because it displays both symmetry required degeneracy, between the five 3\textit{d} orbitals, as well as accidental degeneracy, between the 3\textit{d} and 4\textit{s} orbitals. The quadrupole potential couples the degenerate 3\textit{d} and 4\textit{s} states, serving as an example of the most general perturbation.