To determine β, we use the same data set of 20 higherlying excitation energies that were to obtain error statistics in Table . Fitted values of β, along with errors resulting from this approach, are summarized in Table . All functionals except BH&HLYP and SRC1-r1 afford considerable improvement over the unshifted XTPM approach. (For SRC1-r1, the improvement is almost 1 eV yet the MAE itself remains large, whereas for BH&HLYP the improvement is only 0.25 eV and the error also remains large.) Using B3LYP, the shifted XTPM approach yields a MAE of 0.3 eV, as compared to 1.8 eV without the shift. Thus, the shifted-XPTM approach based on B3LYP is more accurate than ∆SCF with the same functional, for which the MAE is 1.2 eV. In conjunction with other functionals (including SCAN, SCAN0, CAM-B3LYP, LRC-ωPBE, LRC-ωPBEh, and ω97X-V), the shifted-XTPM procedure is also a significant improvement with respect to either ∆SCF or fractional-electron procedures without the shift. In contrast, for BH&HLYP and SRC1-r1 we do not find any value β > 0 that can improve over unshifted XTPM results with the same functional. This is probably due to the large fraction of exact exchange, which may lead to over-localization.