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# As shown in Figs a and b with the SO

As shown in Figs. 3(a) and (b), with the SO effect, the absolute values of the VDEs change by at most 0.19eV. Table 1 also shows that the observed peak splitting energies in the Sm-Ho complexes were reproduced even without the SO effect. Considering its significant importance in the electronic structures of general Ln systems, the effect of the SO interaction on the VDE appears too small. The reason may be found in the approximate Cy3 TSA expressions for the initial anion and the relevant final neutral states that are connected in the PES. See Supporting Information (S3) of Ref. [5]. By treating L and S as the good quantum numbers, the approximate SO coupled wave function for the anion complex is expanded as follows,where the expansion coefficients are approximated by the Clebsch-Gordan (CG) coefficients; ; then, the energy is expressed as the sum of the spin-free and SO energies,
Note that the spin-free energy has a form of the weighted average energy of the multiplet levels with the squared CG coefficients as the weighting factors. The connected neutral states are similarly expanded considering the coupling between Ln and ligand parts as follows,with , thus their energies are
These spin-free energies also have a form of the weighted average of the low- and high-spin multiplet levels and with the same weighting factors. in Eqs. (5), (7) is the common ground state SOC energy of Ln3+,where the SOC parameter λ is used for ease of discussion. Now, the SO effect on the VDE values is clearly small because the initial and final states in the PES must have the identical ground 2+1 multiplet for the Ln portions as Eq. (2) suggests. Thus, the SO effect induces only a parallel shift for the two states.
Eqs. (6), (7) can also be used to explain the notable difference between MCQDPT2 and SO-MCQDPT2 results. As explained, the X peak splittings occur when the neutral state responsible for the X peak has additional second-order stabilization energy,
This occurrence and magnitude critically depend on the S′ and values of the neutral complexes as fully discussed [5]. As long as this perturbation can be treated as a second-order energy, it can be added into the spin-free energies and in Eq. (7). Then, with the SO effect, the stabilization energy is ‘diluted’ with the squared CG coefficients as the weighting factors and results in a slight decrease in splitting size (Table 1) and a smoothed change in the X peaks (Fig. 3(a)) compared to those without the SO effect. In addition, the mixing of different spin states with the CG coefficient in Eq. (6) decreases the relative intensity of the X′ peaks in the SO-MCQDPT2 spectra. Since the relative intensities of the X and X′ peaks without the SO interaction are dominated by the spin multiplicity of the corresponding neutral states, with the SO interaction a portion of the intensity of the X′ peaks that correspond to the high spin states moves to the X peaks because of the spin mixing, thus I(X′)/I(X) decreases.

Conclusion
The SO effects on the anion photoelectron spectra of Ln(COT)2− (Ln=Ce-Yb) were systematically investigated by the SO-MCQDPT2 method. Although the SO interaction has a significant effect on the electronic structures of the Ln portion, it is qualitatively insignificant for the VDEs and magnitudes of the energy splittings as long as the detachment occurs from the ligand COT. On the other hand, the inclusion of the SO effect improves the splitting magnitudes and the relative intensities of the X and X′ peaks. Therefore, the SO effect is essential for explaining the detailed structures of the experimental spectra.

Acknowledgments
The authors gratefully acknowledge the helpful discussions with Prof. A. Nakajima, his laboratory members and Mr. Tomohide Masuda. This work was supported by JSPS KAKENHI Grant Number 16K05668. The computations were partly performed using the computer facilities at the Research Center for Computational Science, Okazaki National Institutes.