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 (KS) potential. We develop the fully general degenerate DFT perturbation theory. The requirement that a differentiable map connects the unperturbed state to the perturbed state defines the unitary transformations and changes in Fermi-level occupation numbers at each order. The resulting methodology is applied to the iron atom ground state, which displays 3$d$-4$s$ degeneracy, calculated with the VWN functional. By applying a perturbing quadrupole field, we demonstrate how the equations for the first-order density and first through third-order energies can be solved in a system where degeneracy appears both due to symmetry requirements and accidentally, between different representations of the symmetry group.