The analog of Koopmans’ theorem in spin-density functional theory

For spin-unrestricted Kohn–Sham (KS) calculations on systems with an open shell ground state with total spin quantum number S, we offer the analog of the Koopmans’-type relation between orbital energies and ionization energies familiar from the Hartree–Fock model. When (case I) the lowest ion state has spin $\text{S−1/2}$ (typically when the neutral molecule has a (less than) half filled open shell), the orbital energy of the highest occupied orbital ${\mathrm{(\phi }}_H\mathrm{),}$ belonging to the open shell with majority spin (α) electrons, is equal to the ionization energy to this lowest ion state with spin $\text{S−1/2:}$ ${\epsilon }_H^{\alpha }{\mathrm{=-I}}^{\text{S−1/2}}{\mathrm{(\phi }}_H^{\text{−1}}\mathrm{).}$ For lower (doubly occupied) orbitals the ionization ${\phi }_H^{\text{−1}}$ leaves an unpaired electron that can couple to the open shell to $\text{S±1/2}$ states: ${\epsilon }_i^{\beta }{\mathrm{\approx -I}}^{\text{S+1/2}}{\mathrm{(\phi }}_i^{\text{−1}})$ (exact identity for $\text{i=H−1),}$ ${\epsilon }_i^{\alpha }{\text{≈−{[2S/(2S+1)]I}}^{\text{S−1/2}}{\mathrm{(\phi }}_i^{\text{−1}}{\text{)+[1/(2S+1)]I}}^{\text{S+1/2}}{\mathrm{(\phi }}_i^{\text{−1}}\mathrm{)\},}$ reducing to a simple average in the case of a doublet ground state (single electron outside closed shells). When the lowest ion state has spin $\text{S+1/2}$ (case II; typically for more than half filled open shells): ${\epsilon }_H^{\alpha }{\mathrm{=\epsilon }}_H^{\beta }{\mathrm{=-I}}^{\text{S+1/2}}{\mathrm{(\phi }}_H^{\text{−1}}\mathrm{);}$ for $\mathrm{i<H,}$ ${\epsilon }_i^{\beta }{\mathrm{\approx -I}}^{\text{S+1/2}}{\mathrm{(\phi }}_i^{\text{−1}}\mathrm{),}$ ${\epsilon }_i^{\alpha }{\text{≈−{[2S/(2S+1)]I}}^{\text{S−1/2}}{\mathrm{(\phi }}_i^{\text{−1}}{\text{)+[1/(2S+1)]I}}^{\text{S+1/2}}{\mathrm{(\phi }}_i^{\text{−1}}\mathrm{)\}.}$ A physical basis is thus provided for the KS orbital energies also in the spin unrestricted case and an explanation is given for the common observation in approximate Kohn–Sham calculations of more negative majority spin (α) levels ${\epsilon }_i^{\alpha }$ for $\mathrm{i<H,}$ than minority spin levels ${\epsilon }_i^{\beta }.$