Fig. 4: Microscopic origin of anomalous properties of Pr⁴⁺ and a universal model for f¹ single-ions.

a Schematic of p and 4f energy levels for Pr⁴⁺ and Ce³⁺. t_pfπ (t_pf(π+σ)) is the hopping integral between p and ²T_2u (²T_1u) orbitals. The corresponding pf charge transfer gap is indicated with Δ_pf,4+ < Δ_pf,3+. b Sketch of the hopping processes between occupied f_ζ orbitals mediated by the π interacting 2p orbitals analogous to t_2g − p − t_2g hopping in d⁵ systems. c Sketch of the hopping processes between occupied f_ζ and unoccupied f_α orbitals mediated by the π + σ interacting 2p orbitals analogous to t_2g − p − e_g hopping in d⁵ systems. d–g Probability density of the ground state KD in ideal Γ₇, 2-Pr, 1-Pr, and 0-Pr, respectively and shows the impact of mixing excited states in to the original Γ₇ doublet. h Schematic of the splitting of f orbitals as a function of CF (Δ and θ) relative to SOC (ζ). The value of ξ for Pr⁴⁺ was calculated from 0-Pr, and the values for Ce³⁺, U⁵⁺, and Np⁶⁺ were obtained from ref. ⁴⁹. Using this as a universal model for f¹ ions, Pr⁴⁺ is categorized together with the actinides, where the traditional Ln³⁺ picture breaks down. The figure also shows the evolution of the shape of the Γ₇ KD as a function of \({B}_{4}^{0}\) in the \({\hat{{{{{{{{\mathcal{H}}}}}}}}}}_{CF}^{{O}_{h}}={B}_{4}^{0}{\hat{O}}_{4}^{0}+{B}_{4}^{4}{\hat{O}}_{4}^{4}\), where \({B}_{4}^{4}=5{B}_{4}^{0}\). Increasing \({B}_{4}^{0}\) from ~0 (ζ_SOC > > Δ_CF) yielding an almost perfect Γ₇ KD (leftmost figure) to ~2000 (ζ_SOC < < Δ_CF, rightmost figure) and the resultant drastic changes of the nature of the KD. The original nature of the Γ₇ KD is retained until the eigenvalue of the Γ₈ ≈ 75 meV where J_eff = 1/2 limit still applies.