Kohn–Sham inversion for open-shell systems

Methods based on density-functional theory usually treat open-shell atoms and molecules within the spin-unrestricted Kohn-Sham (KS) formalism, which breaks symmetries in real and spin space. Symmetry breaking is possible because the KS Hamiltonian operator does not need to exhibit the full symmetry of the physical Hamiltonian operator, but only the symmetry of the spin density, which is generally lower. Symmetry breaking leads to spin contamination and prevents a proper classification of the KS wave function with respect to the symmetries of the physical electron system. Formally well-justified variants of the KS formalism that restore symmetries in real space, in spin space, or in both have been introduced long ago, but have rarely been used in practice. Here, we introduce numerically stable KS inversion methods to construct reference KS potentials from reference spin-densities for all four possibilities to treat open shell systems, non-symmetrized, spin-symmetrized, space-symmetrized, and fully-symmetrized. The reference spin-densities are obtained by full configuration interaction and high-level coupled cluster methods for the considered atoms and diatomic molecules. The decomposition of the total energy in contributions such as the non-interacting kinetic, the exchange, and the correlation energy is different in the four KS formalisms. Reference values for these differences are provided for the considered atoms and molecules. All KS inversions, except the fully symmetrized one, lead in some cases to solutions violating the Aufbau principle. In the purely spin-symmetrized KS formalism, this represents a violation of the KS v-representability condition, i.e., no proper KS wave functions exist in those cases.