where E i is usually the ground state and E f corresponds to a non-Aufbau Kohn-Sham determinant with an electron excited into the virtual space and therefore a hole below the highest-occupied MO (HOMO). Various algorithms have been developed to relax non-Aufbau MOs while avoiding variational collapse. The maximum overlap method (MOM) often works well for the lowest excited state, and has previously been applied to core → LUMO excitations. In our experience, however, more sophisticated methods are often required to converge higher-lying excited states. B. Slater-Type Methods. The use of a fractionalelectron SCF calculation as the basis for computing excitation or ionization energies originated with Slater. Fractional-electron calculations are now widely used to diagnose and correct problems with delocalization error and self-interaction in DFT, but have also been used in conjunction with correlated wave function models. The concept originates in the Slater-Janak theorem, which states that SCF eigenvalues ε i are derivatives of the SCF energy with respect to orbital occupancy, ε i = ∂E/∂n i , which holds for all of the occupied orbitals including lower-lying ones. As a result, the occupied level ε i (1/2) that is computed with n i = 1/2 electrons in occupied MO ψ i approximates the ∆SCF ionization energy. This fact forms the basis of the STM, which has long been applied to both core-level electron binding energies and core-level excitation energies. For K-edge XPS, this method exhibits an accuracy of ∼ 0.5 eV at the Hartree-Fock level but is significantly worse at DFT levels of theory. However, reasonable accuracy can be recovered using DFT via generalized (G)STM approaches that require more than one fractional-electron SCF calculation per ionization energy. In the present work, we examine core-level excitation (rather than ionization), using eigenvalue differences as in eq. 1. In the original STM, transition energies ∆E are computed according to the formula