Electronic Structure of Mononuclear Cu-based Molecule from Density-Functional Theory with Self-Interaction Correction

We investigate the electronic structure of a planar mononuclear Cu-based molecule [Cu(C$_6$H$_4$S$_2$)$_2$]$^z$ in two oxidation states ($z$$=$$-2$, $-$1) using density-functional theory (DFT) with Fermi-L\"owdin orbital (FLO) self-interaction correction (SIC). The dianionic Cu-based molecule was proposed to be a promising qubit candidate. Self-interaction error within approximate DFT functionals renders severe delocalization of electron and spin densities arising from 3$d$ orbitals. The FLO-SIC method relies on optimization of Fermi-L\"owdin orbital descriptors (FODs) with which localized occupied orbitals are constructed to create the SIC potentials. Starting with many initial sets of FODs, we employ a frozen-density loop algorithm within the FLO-SIC method to study the Cu-based molecule. We find that the electronic structure of the molecule remains unchanged despite somewhat different final FOD configurations. In the dianionic state (spin $S=1/2$), FLO-SIC spin density originates from the Cu $d$ and S $p$ orbitals with an approximate ratio of 2:1, in quantitative agreement with multireference calculations, while in the case of SIC-free DFT, the orbital ratio is reversed. Overall, FLO-SIC lowers the energies of the occupied orbitals and in particular the 3$d$ orbitals unhybridized with the ligands significantly, which substantially increases the energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) compared to SIC-free DFT results. The FLO-SIC HOMO-LUMO gap of the dianionic state is larger than that of the monoionic state, which is consistent with experiment. Our results suggest a positive outlook of the FLO-SIC method in the description of magnetic exchange coupling within 3$d$-element based systems.