Extended Data Fig. 5: Theoretical energy diagrams.

a, Splitting of the NiS₆ cluster mutiplet with parameters shown in Extended Data Table 1 as a function of hopping strength. Here r is the scale parameter of hopping strengths like V_pdσ = −0.74r eV and V_pdπ = 0.4r eV. Magenta, green, light blue and red lines represent the energy levels of singlet, doublet, triplet and nonuplet states, respectively. Dashed orange and solid grey lines refer to the energy levels of charge-transfer states which are not coupled with d⁸ multiplets. Because the multiplets shown by grey lines are weakly coupled with d¹⁰L² multiplets, their energy is slightly lower than that of multiplets shown by dashed orange lines. The irreducible representations for r = 0 and r = 1 are given on the top and bottom axes. b, Schematic diagram of the nature of the bonding between d⁸ and d⁹L¹ multiplets. Dotted red and dashed blue lines represent respectively the pdσ- and pdπ-type bonding between d⁸ and d⁹L¹ multiplets.