# Moreover the vibrational contributions provide a plausible e

Moreover, the vibrational contributions provide a plausible explanation for the missing X′ peaks in the experimental PES for the Er and Tm complexes. The current SO-MCQDPT2 (Fig. 1(k) and (l)) and MCQDPT2 II (Fig. 2(k) and (l)) calculations suggest the appearance of peak splittings for these complexes. Because these theoretical X′ peaks appear at 0.15–0.19eV higher GSK503 structure than the X peaks and are relatively weak, the experimental X′ peaks of electronic origin might have been buried in the vibrational subpeaks. Particularly, the X′ peaks in the SO-MCQDPT2 spectra have even smaller intensities than that of MCQDPT2 II, and these peaks are more easily hidden by the vibrational structure. Indeed, compared to the vibrational structure in the PES of the Lu (f14) complex (Fig. 1 of Ref. [7]), those of the Er and Tm complexes are obviously complicated. A notable difference with/without SO effect appears in the Nd complex (Table 1). The MCQDPT2 method shows that the X peak that was derived from the lowest multiplet level with =3 of this anion complex does not split, whereas that derived from the first excited multiplet level with =4 has a 0.238eV splitting (Table 3 of Ref. [5]). The current SO-MCQDPT2 method indicates that the X peak has a 0.127eV splitting. Our analysis shows that the SO effect couples these two lowest initial anion states, and the peak splitting appears. Although the splitting over 0.1eV is not observed in the experimental Nd PES (Fig. 1 of Ref. [7]), small splitting is observed in the main X peak. Fig. 3 compares (a) the VDEs of the X peaks by the SO-MCQDPT2 method with (b) those by the MCQDPT2 II method and (c) the experimental values. The change rates of these two types of peaks were calculated and added in the figures. For the SO-MCQDPT2 (MCQDPT2 II) results, the change rate of the highest VDEs from Ce to Yb complexes is −0.0130 (−0.0135)eV, and that of the lowest VDEs from Pm to Yb complexes is 0.0060 (0.0125)eV. For the experimental values, the change rate of the higher VDEs from La to Ho complexes was −0.0203eV, and that of the lower VDEs from Sm to Lu complexes is −0.0058eV. Although the SO-MCQDPT2 VDEs are still 0.76–0.83eV smaller than the experimental absolute values (this error can be reduced by using larger basis sets for the COT part. See Fig. S2 and Table 1 in Supplementary material for the basis set dependence of the VDEs and splitting magnitudes), the Ln dependence of the X and X′ peaks is generally better reproduced by the SO-MCQDPT2 results than the MCQDPT2 II results. In particular, the decreasing trend of the higher VDEs and the flatness of the lower VDEs across the Ln series are reproduced well in the SO-MCQDPT2 VDEs. Since the higher VDEs for the X′ peaks do not include the configuration interaction, Koopmans’ theorem can be applied, and their Ln dependence is easily found by approximating . In fact, the decreasing trend was interpreted as a result of the destabilizing HOMO energies across the Ln series [5]. However, the lower VDEs for the X peaks are considered to be stabilized by the additional configuration interaction between the 4fe2u3 and 4f−1e2u4 configurations, which is caused by the spin-free ligand field perturbation . Therefore, their VDEs are symbolically expressed by adding the second-order perturbation energy as follows,where the energy denominator . The difference in the lower and higher VDEs is just the magnitude of the perturbation energy (the second term on the right hand side of Eq. (3)). Therefore, with the decreasing trend of , the notably weak Ln dependence in VDE(X) suggests that the absolute value of the second term in Eq. (3) also decreases across the Ln series at almost the identical rate. This decreasing trend of the perturbation energy was previously discussed by relating the 4f orbital size contraction across the Ln series to the numerator of the second term in Eq. (3) and the fourth ionization energy of Ln to the energy denominator [5].