Regarding the excitation energies, an average error of 0.079/0.088 eV and r 2 value of 0.99/0.97 are obtained by eo-TDA calculations based on the ground-state density matrix for OLU - and pHBDI -, respectively, as shown in Figs. and. For these solvated chromophores, eo-TDA(P 0 ) is thus more accurate than TDA/TIP3P and TDA/EFP, which is not surprising because of an improved description of frontier orbitals. [Such an improved accuracy come with a much higher computational cost, though, due to the need of performing first full-system DFT and then eo-TDA calculations with orbitals extending beyond the impurity.] The excitation energy errors of eo-TDA calculations could be further lowered to 0.041 and 0.066 eV (Figs. and), when the eo density matrix is augmented with the contribution from only one virtual orbital. However, these errors decay rather slowly with the further addition of virtual orbitals, reaching only 0.034 and 0.035 eV with the eo-TDA(P +20 ) calculations (Figs. and). This is expected because TDDFT and TDA calculations often involve excitations into higher virtuals with small amplitudes but nevertheless non-negligible perturbations to the excitation energies. As shown in Figs. and, at the limit where a maximum number of virtuals (i.e. the number of nearly zero singular values of P AE 0 ) are included, the net errors are reduced to 0.016 and 0.014 eV for OLU -and pHBDI -, respectively.