This suggests a value of β ≈ 2, although we treat β as a fitting parameter. Note that both ε i (0) and ε i (1/2) are already required for a STM calculation. The value ε i (0) comes from the integer-occupancy SCF calculation, and orbitals obtained from that calculation serve as a starting point for the for the fractional-occupancy SCF calculation that is used to obtain ε i (1/2). The correction δ i can be understood to eliminate differential selfinteraction between the initial state and the core-ionized state that remains in the Kohn-Sham eigenvalues, while at the same time compensating for higher-order terms in the Taylor expansion of Eq. ( ), which are omitted in the conventional STM. That the value of δ i must be elementspecific reflects the localized atomic nature of the core hole, whose radial size (and therefore the magnitude of its self-interaction) depends strongly on atomic number. Errors resulting from of this approach, for the CORE65 data set and using best-fit values of β, are summarized in Table . (See Table for the full set of results.) The SCAN functional with β = 3.2 yields considerable improvement over unshifted STM-SCAN results, achieving a MAE of 0.15 eV that is smaller than the ∆SCF error (0.19 eV) for the same functional. The shift δ i improves the results significantly, for all functionals except HFX, although best-fit value of β differ considerably from one functional to the next. Especially notable are BH&HLYP, where the MAE is reduced to 0.5 eV (better than the corresponding ∆SCF error) and SRC1-r1, for which the shifted STM error is 0.3 eV whereas the ∆SCF error is 1.5 eV. On the other hand, the SRC1-r1 functional requires a rather large fitting parameter (β = 15.2) in order to achieve this result. Safer bets are the shifted STM-SCAN and shifted STM-B3LYP methods, for which absolute CEBEs are plotted versus experiment in Figs. and A survey of the errors obtained using several of the best methods (including ∆SCF, GSTM, and empirically shifted STM with various functionals) is presented in Fig. , alongside results from several variants the GW method that have recently been tested using the same data set. The shifted-STM approach achieves a MAE of 0.14 eV in conjunction with any of several different functionals: SCAN0, B3LYP, LRC-ωPBE, and LRC-ωPBEh. This is actually slightly smaller than the MAE obtained using a variety of (considerably more expensive) GW methods. The latter include the nonself-consistent variant G 0 W 0 @PBEh(α), whose MAE is 0.33 eV, the "eigenvalue self-consistent" evGW 0 @PBE approach (MAE = 0.30 eV), and GW with Hedin shift, G ∆H W 0 @PBE, whose MAE is 0.25 eV. Moreover, these GW methods are not free from empiricism. For example, G 0 W 0 @PBEh(α) uses a fraction of HFX (α = 0.45) that has been adjusted in order to minimize errors with respect to experimental CEBEs. GSTM methods based on the SCAN0, B3LYP, and ωB9X-V also yield similar accuracy as compared to the GW methods.