Chemical design of electronic and magnetic energy scales of tetravalent praseodymium materials

Abstract

Lanthanides in the trivalent oxidation state are typically described using an ionic picture that leads to localized magnetic moments. The hierarchical energy scales associated with trivalent lanthanides produce desirable properties for e.g., molecular magnetism, quantum materials, and quantum transduction. Here, we show that this traditional ionic paradigm breaks down for praseodymium in the tetravalent oxidation state. Synthetic, spectroscopic, and theoretical tools deployed on several solid-state Pr⁴⁺-oxides uncover the unusual participation of 4f orbitals in bonding and the anomalous hybridization of the 4f¹ configuration with ligand valence electrons, analogous to transition metals. The competition between crystal-field and spin-orbit-coupling interactions fundamentally transforms the spin-orbital magnetism of Pr⁴⁺, which departs from the J_eff = 1/2 limit and resembles that of high-valent actinides. Our results show that Pr⁴⁺ ions are in a class on their own, where the hierarchy of single-ion energy scales can be tailored to explore new correlated phenomena in quantum materials.

Introduction

The electronic structure of lanthanide and actinide materials inherits on-site correlations and unquenched orbital degrees of freedom from atomic f-electron states. In the most stable trivalent oxidation state (Ln³⁺), the core-like 4f orbitals are usually only weakly perturbed by the crystalline environment¹. Yet, the energetic splitting of the otherwise 2J + 1 fold-degenerate (free-ion) ground-state yields rich physics and applications. For example, Ln³⁺ insulators can host anisotropic magnetic moments with effective spin-1/2 character (J_eff = 1/2) that are promising to stabilize entangled states in quantum magnets^2,3. Metallic 4f and 5f systems also display a wealth of quantum phenomena rooted in the hybridization between localized f-electrons and conduction d-electrons such as the Kondo effect, valence fluctuations, correlated insulators, and unconventional superconductivity^4,5.

In rare instances, Ce, Pr, Tb, (and less definitively Nd, and Dy) ions exist in a high, formally tetravalent, oxidation state, i.e., Ce⁴⁺ (4f⁰), Pr⁴⁺ (4f¹), and Tb⁴⁺ (4f⁷)^6,7. Although covalent metal-ligand interactions involving the 4f shell are generally weak in Ln³⁺ systems^8,9, this paradigm breaks down for Ln⁴⁺ as 4f orbitals directly participate in bonding and hybridize with the valence orbitals of the ligands (e.g., the 2p states for oxygen) analogous to transition metals. A high oxidation state and strong 4f covalency are expected to significantly impact the redox, electronic, and magnetic properties of these systems, but, surprisingly, only a few Ln⁴⁺ insulators have been studied in detail to date^10,11. As Ce⁴⁺ is nominally non-magnetic and Tb⁴⁺ has a half-filled 4f shell, the one-electron 4f¹ configuration of Pr⁴⁺ makes it unique to search for new quantum phenomena at the nexus of strong electronic correlations, quantum magnetism, and spin-orbital entanglement.

The emergence of an insulating state in PrBa₂Cu₃O_6+δ (PBCO)—a compound obtained by substituting Y by Pr in the high-T_c superconductor YBa₂Cu₃O_6+δ (YBCO), and valence fluctuations driven metal-insulator transitions in Pr-containing complex oxides—epitomizes such anomalous behavior. In PBCO, the significant Pr-4f/O-2p covalency (Fehrenbacher-Rice hybridization) drives a mixed-valent state for Pr ions that competes with Cu-3d/O-2p hybridization (Zhang-Rice) and dramatically suppresses superconductivity in favor of local magnetism¹². In Pr containing complex oxides like (Pr_1−yY_y)_1−xCa_xCoO_3−δ and Pr_1−xSr_xCoO₃, valence transition from Pr⁴⁺ to Pr³⁺ drives a spin state/metal-insulator transition, making them attractive for oxide-based electronics^13,14,15. This observation stimulated early experimental and theoretical work to understand the magnetism of cubic Pr⁴⁺ oxides such as PrO₂ and BaPrO₃^10,11. More recently, the edge-sharing PrO₆ octahedral in Na₂PrO₃ have attracted attention to stabilize antiferromagnetic Kitaev interactions between J_eff = 1/2 moments¹⁶. But much like in PrO₂¹¹, the hallmark of Na₂PrO₃ magnetism is the unusually large crystal field (CF) splitting Δ_CF ≥ 230 meV that competes with spin-orbit coupling (SOC) ζ_SOC ≈ 100 meV¹⁷. The competition between CF and SOC yields drastically different single-ion and exchange properties than expected in the Δ_CF ≪ ζ_SOC limit where J_eff = 1/2 moments usually form, as illustrated in Fig. 1a. The most noticeable consequences for Na₂PrO₃ are the low effective magnetic moment with g ≈ 1 and the surprisingly large J_ex ≈ 1 meV exchange interactions¹⁷. The precise mechanisms that endow Pr⁴⁺ ions with these unusual properties are poorly understood.

Fig. 1: Competing interactions in Pr⁴⁺ oxides.

a Hierarchy of single-ion energy scales for Pr⁴⁺ ions in octahedral oxygen environments starting from a ²F, (S = 1/2, L = 3) free-ion state. For spin-orbit coupling (SOC) as the dominant energy scale (J_eff = 1/2 limit, left, brown), the low-symmetry crystal field (CF) lifts the degeneracy of the ground-state ²F_5/2 and excited ²F_7/2 multiplets into seven Kramers doublets (KDs, with selected radial squared wave-functions represented). For approximate O_h symmetry, the Γ₇ doublet, in the $\left|J,{m}_{J}\right\rangle$ basis, is given by ${\Gamma }_{7}^{\pm }=\sin \theta \left|\frac{5}{2},\pm \frac{5}{2}\right\rangle+\cos \theta \left|\frac{5}{2},\mp \frac{3}{2}\right\rangle$, where ${\sin }^{2}\theta \, \approx \, 1/6$ (See SI). For CF as the dominant energy scale (S_eff = 1/2 limit, right, blue), SOC and distortion from O_h symmetry lift the ²A_2u ground state and the triply degenerate ²T_2u and ²T_1u excited states into seven KDs. The ground-state doublet is given in the $\left|{m}_{l},{m}_{s}\right\rangle$ basis by $\left|\pm \right\rangle=A\,\left|\mp 3,\pm \frac{1}{2}\right\rangle -\,B\,\left|\mp 2,\mp \frac{1}{2}\right\rangle \,+\,C\,\left|\pm 1,\pm \frac{1}{2}\right\rangle -\,D\,\left|\pm 2,\mp \frac{1}{2}\right\rangle$, where ${({A}^{2}/{B}^{2})}^{{\Gamma }_{7}} \, \approx \, 2.6$ and ${({C}^{2}/{D}^{2})}^{{\Gamma }_{7}} \, \approx \, 0.33$ for the $\left|{\Gamma }_{7}\right\rangle$ doublet⁶² (See SI). In the intermediate limit, the competition between Δ_CF and ζ_SOC scales in Pr⁴⁺ yields seven KD (indicated by the mix of brown/blue lines) with magnetic properties that are distinct from the J_eff = 1/2 and S_eff = 1/2 limits, such as large magnetic super-exchange achievable by tuning the ligand field. b–d Crystal structure, magnetic lattice dimensionality, and Pr⁴⁺ coordination environment for the oxides studied in this work: Na₂PrO₃ (2-Pr), Sr₂PrO₄ (1-Pr), and Li₈PrO₆ (0-Pr), respectively.

In this work, we focus on the microscopic mechanisms that underpin the electronic and magnetic behavior of Pr⁴⁺ materials comprising octahedral [PrO₆]⁸⁻ units. We examine a series of insulating oxides with decreasing order of lattice dimensionality: quasi-2D layers in Na₂PrO₃ (2-Pr, Fig. 1b, ref. ¹⁸) and quasi-1D chains in Sr₂PrO₄ (1-Pr, Fig. 1c, ref. ¹⁹) to understand magnetic exchange and the role of Pr-4f/O-2p hybridization, and quasi-isolated “0D” PrO₆ octahedra in Li₈PrO₆ (0-Pr, Fig. 1d, ref. ²⁰) to understand the intrinsic behavior of the [PrO₆]⁸⁻ moiety without the complication of magnetic exchange interactions. Using inelastic neutron scattering (INS) and infrared magnetospectroscopy (IRMS), we demonstrate that the magnetic ground-state of Pr⁴⁺ ions systematically deviates from the J_eff = 1/2 limit and can be understood from an intermediate coupling regime where significant admixture of nominally excited J-states enter the ground-state wavefunction. X-ray magnetic circular dichroism (XMCD) at the Pr-M_4,5 edge strengthens this picture and elucidates the mechanism behind the low effective magnetic moments of Pr⁴⁺ ions. Oxygen K-edge and Pr M_4,5-edge x-ray absorption spectroscopy (XAS) evidences Pr-4f/O-2p hybridization with a degree of Pr-O covalency influenced by the symmetry of the [PrO₆]⁸⁻ moiety. These results are supported and explained by first-principles calculations. Taken together, this study establishes Pr⁴⁺ ions as an important building block to design quantum magnetic behavior distinct from trivalent lanthanides. Furthermore, the study demonstrates that the competition between energy scales in the [PrO₆]⁸⁻ unit is reminiscent of 4d and 5d transition metals²¹, and can serve as a simplification of 5f¹ actinide systems for which Δ_CF, ζ_SOC and on-site Hubbard interaction U strongly compete^22,23.

Results

Crystalline powder samples of 0-Pr, 1-Pr, and 2-Pr were synthesized using solid-state reactions and phase purity was confirmed using powder X-ray diffraction (See Methods and Supplementary Methods 1,2). These materials incorporate low symmetry, but close-to-octahedral [PrO₆]⁸⁻ units with D_2d symmetry in 2-Pr (C_2/c space group, Fig. 1b), C_2h in 1-Pr (Pbam space group, Fig. 1c), and S₆ in 0-Pr ($R\overline{3}$ space group, Fig. 1d). 0-Pr contains spatially isolated PrO₆ octahedral with the nearest-neighbor Pr-Pr distance of d ≈ 5.6 Å, which is significantly longer than d ≈ 3.5 Å in 1-Pr and 2-Pr, and sufficient to effectively magnetically isolate the [PrO₆]⁸⁻ units.

Broadband INS measurements were used to probe the CF states accessible to the dipole selection rule. Given that Pr⁴⁺ is a 4f¹ Kramers ion (isoelectronic to Ce³⁺), the standard approach starts from a ²F free-ion manifold split by SOC into J = 5/2 and J = 7/2 multiplets. For a six-oxygen environment with O_h symmetry, the CF further splits the ²F_5/2 multiplet into a doublet ground-state (Γ₇) and an excited quartet (Γ₈), and the ²F_7/2 multiplet into two doublets (${\Gamma }_{7}^{{\prime} }$ and Γ₆) and a quartet (${\Gamma }_{8}^{{\prime} }$). Any deviation from O_h symmetry, as is the case for our materials (see Fig. 1), splits the quartets and yields seven Kramers doublets (KDs). Thus, the magnetic properties of Pr⁴⁺ ions in the hypothetical Δ_CF ≪ ζ_SOC limit are dominated by the Γ₇ doublet ground-state, which is spanned by pseudospin variable $\left|\pm \right\rangle$ associated with an effective angular momentum J_eff = 1/2. The wave function of the Γ₇ doublet is well-known¹⁶ and can be written in either $\left|J,{m}_{J}\right\rangle$ or $\left|{m}_{l},{m}_{s}\right\rangle$ basis (see Fig. 1a and SI).

However, as ζ_SOC ≈ Δ_CF for Pr⁴⁺, the simple J_eff = 1/2 picture breaks down. Indeed, INS on 0-Pr readily reveals an intense magnetic signal at ${E}_{1}^{{{{{{{{\bf{0-Pr}}}}}}}}}=274(1)$ meV which we assign to the first CF excitation, see Fig. 2a. This energy is 2.5 times larger than reported for isoelectronic Ce³⁺ in KCeO₂^24,25, and to the best of our knowledge, this is the largest first CF excited state observed for a lanthanide ion. Given the uncommonly large Δ_CF, modeling the single-ion properties of 0-Pr requires an intermediate coupling approach¹¹ that uses the complete set of 14 $\left|{m}_{l},{m}_{s}\right\rangle$ basis states and diagonalizes the single-ion CF Hamiltonian ${\hat{{{{{{{{\mathcal{H}}}}}}}}}}_{{{{{{{{\rm{CF}}}}}}}}}^{\Pr }={B}_{2}^{0}{\hat{O}}_{2}^{0}+{B}_{4}^{0}{\hat{O}}_{4}^{0}+{B}_{4}^{4}{\hat{O}}_{4}^{4}+{B}_{6}^{0}{\hat{O}}_{6}^{0}+{B}_{6}^{4}{\hat{O}}_{6}^{4}$ for a fixed value of the spin-orbit interaction ζ_SOC (see Methods and Supplementary Note 1 for definitions). The above single-ion CF Hamiltonian is written in a truncated symmetry basis to avoid over-parametrization and treat all materials on equal footing (see Methods). Irrespective, it is impossible to constrain the parameters of ${\hat{{{{{{{{\mathcal{H}}}}}}}}}}_{{{{{{{{\rm{CF}}}}}}}}}^{{{{{{{{\bf{0-Pr}}}}}}}}}$ solely using the one observed excitation. Thus 1-Pr is examined first because the PrO₆ octahedra further depart from ideal symmetry and is likely to present a richer spectrum in INS.

Fig. 2: Crystal field excitations and magnetic properties in Pr⁴⁺ oxides.

a, d Energy-dependent neutron-scattering intensity at low temperature integrated into the range 6 < Q < 7 Å⁻¹ for several neutron energies and for 1-Pr (Sr₂PrO₄) and 0-Pr (Li₈PrO₆), respectively. b, e Magnetic susceptibility (χ(T)) and inverse susceptibility (χ(T)⁻¹) data measured under μ₀H = 3 T. Also plotted is a CF model and a Curie–Weiss analysis in the temperature range 4 < T < 40 K that yields ${\Theta }_{{{{{{{{\rm{CW}}}}}}}}}^{{{{{{{{\rm{1-Pr}}}}}}}}}=-7.52(2)$ K. The CF model calculations were carried out in Stevens operator formalism using the PYCRYSTALFIELD package⁵⁸ with 14 $\left|{m}_{l},{m}_{s}\right\rangle$ basis states to account for intermediate coupling. The inset shows macroscopic magnetic behavior under an applied field of μ₀H = 0.1 T. 1-Pr exhibits an AFM order with a pronounced peak in χ(T) with no splitting between ZFC and FC. c Normalized IR transmission spectra as a function of applied magnetic field for 0-Pr. The blue color indicates the area with intense CF transitions, while yellow corresponds to the flat line. The experimental data were taken at 5 K and normalized to the average spectra as described in Methods. f Isothermal magnetization M(H) at T = 50 K for 0-Pr and 1-Pr plotted together with the CF model. T = 50 K was chosen so that 1-Pr is well above the ordering temperature and free from short-range correlations.

Unlike 0-Pr, INS results on 1-Pr reveal three magnetic excitations at ${E}_{1}^{{{{{{{{\bf{1-Pr}}}}}}}}}=168(1)$ meV, ${E}_{2}^{{{{{{{{\bf{1-Pr}}}}}}}}}=335(1)$ meV, and ${E}_{3}^{{{{{{{{\bf{1-Pr}}}}}}}}}=387(1)$ meV (Fig. 2d); more states than available in the sole J = 5/2 manifold. Although 1-Pr exhibits an antiferromagnetic transition at T_N = 3.0 K with a pronounced peak in χ(T) (Fig. 2e), the magnetic susceptibility at μ₀H = 3 T above T > 40 K can be used to further constrain the parameters of the single-ion CF Hamiltonian. To proceed, ${\hat{{{{{{{{\mathcal{H}}}}}}}}}}_{{{{{{{{\rm{CF}}}}}}}}}^{\Pr }$ is diagonalized with fixed ζ_SOC = 112 meV (free-ion value) and the CF parameters fit to the observed INS energies and magnetic susceptibility data (see Methods and Supplementary Note 1). This yields a model Hamiltonian that reproduces the isothermal magnetization at T = 50 K (Fig. 2f) and predicts an unusually small powder-averaged g factor ${g}_{{{{{{{{\rm{CF}}}}}}}}}^{{{{{{{{\rm{avg}}}}}}}},{{{{{{{\bf{1-Pr}}}}}}}}}=1.13$ and an effective moment ${\mu }_{{{{{{{{\rm{CF}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}},{{{{{{{\bf{1-Pr}}}}}}}}} \, \approx \, 1\,{\mu }_{{{{{{{{\rm{B}}}}}}}}}$/Pr comparable to the value extracted from a Curie–Weiss fit ${\mu }_{{{{{{{{\rm{CW}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}},{{{{{{{\bf{1-Pr}}}}}}}}}=1.13(1)\,{\mu }_{{{{{{{{\rm{B}}}}}}}}}$/Pr (Fig. 2e).

Having established an approach to model the single-ion Hamiltonian for Pr⁴⁺, 0-Pr is examined and employ IRMS measurements conducted up to 17.5 T. The normalized IR spectra reveal three field-dependent features around ${E}_{1}^{{{{{{{{\bf{0-Pr}}}}}}}}}=267$ meV, ${E}_{2}^{{{{{{{{\bf{0-Pr}}}}}}}}}=270$ meV, and ${E}_{4}^{{{{{{{{\bf{0-Pr}}}}}}}}}=670$ meV (Fig. 2c) that can be associated with magnetic dipole allowed CF transitions from the ground-state doublet. The distinct features at ${E}_{1}^{{{{{{{{\bf{0-Pr}}}}}}}}}$ and ${E}_{2}^{{{{{{{{\bf{0-Pr}}}}}}}}}$—resolved due to the excellent energy resolution of IRMS—correspond to the sole transition observed in INS. The first excited level in 0-Pr is thus a quasi-degenerate quartet (Γ₈-like) split by the weak distortion of the PrO₆ octahedra from an ideal O_h symmetry. This model is fully supported by ab-initio calculations (multireference CASPT2 + SOC, Methods and Supplementary Note 2), which predicts the quasi-degeneracy of ${E}_{2}^{{{{{{{{\bf{0-Pr}}}}}}}}}$ and ${E}_{2}^{{{{{{{{\bf{0-Pr}}}}}}}}}$ at 241 and 246 meV, respectively. The 670 meV transition is likely weak in INS and masked by the strong background (recoil intensity observed from hydroxide impurities, <5% in starting materials, see Supplementary Methods 1). The ab-initio calculations assign it as the fourth ground to excited state transition, and further predict a third (IR inactive)²⁶ transition at 396 meV with ²T_2u + ²A_2u origin. The parameters of ${\hat{{{{{{{{\mathcal{H}}}}}}}}}}_{{{{{{{{\rm{CF}}}}}}}}}^{{{{{{{{\bf{1-Pr}}}}}}}}}$ are fitted using the same procedure as for 0-Pr (Fig. 2b). The resulting model yields ${g}_{{{{{{{{\rm{CF}}}}}}}}}^{{{{{{{{\rm{av}}}}}}}},{{{{{{{\bf{0-Pr}}}}}}}}}$ = 0.94, in good agreement with the isothermal magnetization (Fig. 2f), and ${\mu }_{{{{{{{{\rm{CF}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}},{{{{{{{\bf{0-Pr}}}}}}}}}=0.81{\mu }_{{{{{{{{\rm{B}}}}}}}}}$/Pr consistent with ${\mu }_{{{{{{{{\rm{CW}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}},{{{{{{{\bf{0-Pr}}}}}}}}}=0.86(1)\,{\mu }_{{{{{{{{\rm{B}}}}}}}}}$/Pr (Fig. 2b) and first-principles calculations (See Discussion). Analysis of the INS spectrum of 2-Pr leads to similar conclusions¹⁷.

Analysis of the single-ion physics of these three materials therefore suggests that the ground-state wavefunction of Pr⁴⁺ systematically deviates from the Γ₇ doublet expected in the J_eff = 1/2 limit. For example, the ratio of $\left|{m}_{l}=\mp 3,{m}_{s}=\pm 1/2\right\rangle$ to $\left|{m}_{l}=\pm 2,{m}_{s}=\pm 1/2\right\rangle$ basis states (parametrized by (A²/B²), see Fig. 1 caption and Methods for Definition and Supplementary Tables 5–7 for full wavefunction) changes from 2.6 for the $\left|{\Gamma }_{7}\right\rangle$ doublet to 0.53, 2.13, and 1.51 for the ground-state doublet of 0-Pr, 1-Pr, and 2-Pr, respectively. When recast in the $\left|J,{m}_{J}\right\rangle$ basis, our analysis suggests that intermediate coupling mixes $\left|J=7/2,{m}_{J}=\pm 3/2\right\rangle$ and $\left|J=7/2,{m}_{J}=\pm 5/2\right\rangle$ states into the ground doublet, leading to an increased $|J,{m}_{J}=\pm 5/2\rangle$ character for the ground-state wavefunction (see Supplementary Note 1).

Probing the density of electronic states around the 4f level, XAS (see Methods) provides definitive spectroscopic evidence of this hypothesis and further elucidates the origin of the large Δ_CF. The M₅ (3d_5/2 → 4f_7/2 and 4f_5/2) and M₄ (3d_3/2 → 4f_5/2) edges for 2-Pr and 0-Pr are shown in Fig. 3a, b. The splitting between the Pr M₅ edge at 931 eV and M₄ edge at 951 eV originates from SOC within the 3d core-holes. The M_5,4 edges for both 0-Pr and 2-Pr show intense main peaks (labeled A and B) followed by higher-energy satellites (${A}^{{\prime} }$ and ${B}^{{\prime} }$) raised by ≈5 eV, and a smeared shoulder (labeled B^S) starting ≈3 eV below the main peaks. For a 4f¹ ion with a ground-state wavefunction described by a pure J-multiplet, isotropic M_5,4 edges are expected²⁷ as predicted by the Wigner–Eckart theorem (Fig. S4). Thus, the complex spectral lineshapes observed in 0-Pr and 2-Pr, which resemble previous observations for PrO₂²⁸ (see Supplementary Figs. 4,5), are direct evidence for the mixing of ²F_7/2 and ²F_5/2 multiplets in the ground and excited states of our compounds. This analysis is corroborated by the branching ratio (BR) ${I}_{{M}_{5}}/({I}_{{M}_{5}}+{I}_{{M}_{4}})$, evaluated from the total spectral weight under all M₅ or M₄ peaks, with values of 0.445(17) (2-Pr), 0.443(10) (0-Pr), and 0.453(6) (PrO₂). These values are less than ≈0.50 reported for ionic Ce³⁺ systems^27,28 (a BR of ≈0.5 also applies for an ionic Pr⁴⁺ system with no hybridization, as shown in Fig. S4).

Fig. 3: Fingerprints of Pr-4f/O-2p hybridization from X-ray scattering spectra.

a, b Isotropic XAS (top) and XMCD (bottom) spectra at the Pr M₅ and M₄ edges for 2-Pr (left) and 0-Pr (right), respectively measured using the Scanning Transmission X-Ray Microscope (STXM) mode (XAS, μ₀H = 0 T and T = 300 K) and the total electron tield (TEY) mode (XMCD, μ₀H = 5 T and T = 20 K). For the XAS spectra, first-principle calculations (CI under AIM framework) are shown as orange sticks with Gaussian and Lorentzian broadening. For the XMCD spectra, the integration range for the sum-rule analysis is shown as a purple shaded region. c, d Isotropic XAS spectra at the oxygen K edge for 2-Pr (left) and 0-Pr (right), both measured in STXM mode (T = 300 K). The peak corresponding to Pr-4f/O-2p hybridization is shown in yellow. For comparison, reference data on PrO₂²⁸ is shown in Table. S2 and Fig. S4 and S8.

The complex M_5,4 lineshapes and multiplet mixing in 4f¹ systems have previously been ascribed to electronic hybridization, i.e., covalent bonding, between 4f and O-2p states^27,29. In the charge-transfer limit (Zaanen–Sawatzky–Allen (ZSA) scheme³⁰), the electronic ground-state of Pr⁴⁺ is a superposition $|{\psi }_{g}\rangle=\sin \theta|4{f}^{1}\rangle+\cos \theta|4{f}^{2}\underline{\nu }\rangle$ where $\underline{\nu }$ is a hole in the O 2p band. Configuration-interaction (CI) calculations using the Anderson-impurity model (AIM)^31,32 were carried out to understand the impact of 4f covalency on the M_5,4 XAS spectra of the subject compounds. Within this framework, the initial and final states include the combination of $|3{d}^{10}4{f}^{1}\rangle$ & $|3{d}^{10}4{f}^{2}\underline{\nu }\rangle$, and $|3{d}^{9}4{f}^{2}\rangle$ & $|3{d}^{9}4{f}^{3}\underline{\nu }\rangle$, respectively (see Methods). In the limit of vanishing hybridization (V → 0), the energy difference between initial configurations is ΔE_g = 2.0 eV (2-Pr) and 3.0 eV (0-Pr), and between final states is ΔE_f = 0.5 eV (2-Pr) and 1.5 eV (0-Pr). Calculations for a realistic hybridization agree well with the experimental data (Fig. 3a, b) and allows to estimate the fraction of 4f¹ and 4f² configurations in $|{\psi }_{g}\rangle$ to be 69–31% for 2-Pr, and 75–25% in 0-Pr; i.e., 0-Pr is the least hybridized system. The estimated contributions to $|{\psi }_{g}\rangle$ in 2-Pr are similar to the values extracted for PrO₂ (see Fig. S5)²⁸. The smaller hybridization in 0-Pr relative to both 2-Pr and PrO₂ is evident theoretically from the increased ΔE_g and experimentally from the more dominant main peak at both edges. Due to the comparable energy scales of ΔE_g ≈ V_g, Pr⁴⁺ oxide systems are thus strongly correlated insulators in the charge-transfer limit (U_ff ≫ ΔE_g) and require a quantum many-body description for the ground state. Indeed, these systems behave similar to CeO₂ and the spectral features are describing ground and excited state charge transfer³³.

The weak magnetic moment observed for the Pr⁴⁺ ion can also be understood directly from XAS. The XMCD spectra at the Pr M_5,4 edge (See Methods) reveal sizeable dichroism (Fig. 3a (2-Pr) and 3b (0-Pr)). Quantitative analysis using sum rules (see Supplementary Note 1) allows for the extraction of the orbital μ_o = −〈L_z〉μ_B, spin (μ_s = −2〈S_z〉μ_B, and magnetic dipole 〈T_z〉 contributions to the total moment μ_t)³⁴ (see Methods, note that 〈T_z〉 ≠ 0 reflects the departure from spherical symmetry for μ_s)). Applying the orbital sum-rule yields μ_o = 0.34(5)μ_B (2-Pr) and 0.33(7)μ_B (0-Pr). The measured absolute macroscopic magnetization yields μ_t = 0.028(4)μ_B (2-Pr) and 0.044(3)μ_B (0-Pr) at μ₀H = 5 T and T = 20 K, and in turn yields μ_s = − 0.31(9)μ_B (2-Pr) and − 0.29(10)μ_B (0-Pr) based on μ_total = μ_spin + μ_orbital. These values correspond to ∣〈T_z〉/〈S_z〉∣ = 0.40(8) (2-Pr) and 0.38(1) (0-Pr) and ∣〈L_z〉/〈S_z〉∣ = 2.17(6) (2-Pr) and 2.29(9) (0-Pr); the latter is significantly lower than $|\langle {L}_{z}^{4{f}^{1}}\rangle /\langle {S}_{z}^{4{f}^{1}}\rangle |=8$ expected for a free 4f¹ ion³⁵ but resembles 5f–c hybridized uranium systems^36,37 and is usually attributed to 4f electron delocalization in lanthanides. Despite a low bulk magnetic moment, XMCD data reveals the existence of sizable spin and orbital moments with a reduced ∣〈L_z〉/〈S_z〉∣ ratio that provides a fingerprint for Pr-4f/O-2p hybridization in Pr⁴⁺ systems.

Finally, to gain ligand-based information about Pr-4f/O-2p hybridization, O K-edge XAS³⁸ were acquired. The spectra for 2-Pr and 0-Pr, Fig. 3c,d, reveal strong features in the 532.8 to 536 eV range attributed to excitations from the 1s shells of the ligand to states with Pr-5d and O-2p character. These features are a measure of the 5d-covalency of the Pr–O bond and show that nominally unoccupied 5d orbitals take part in covalent bonding³⁹. The splitting of the 5d states is estimated to be 3.67(11) eV in 2-Pr and 3.61(4) eV in 0-Pr and compares well with the value calculated for PrO₂ (≈3.6 eV, see Fig. S3)²⁸. Contributions from Pr-6sp states cannot be entirely neglected in the 5d-driven region⁴⁰. More subtle features common to both 2-Pr and 0-Pr include pre-5d-edge peaks at near ≈529 and ≈530.7 eV (labeled as 1s → 4f). These pre-edge peaks are a signature of strong Pr-4f/O-2p hybridization in the ground state ($|{\psi }_{g}\rangle$) because they reflect transitions from the O 1s-core states to 2p-hole states of the oxygen in the narrow 4f-dominated bands. These low-energy pre-edge features are characteristic of Ln⁴⁺ ions; if at all present in spectra of Ln³⁺ systems³⁹, they are quite weak. The integrated intensities of the 1s → 4f peaks is 3.4(1) and 2.5(1) larger for PrO₂ and 2-Pr, respectively, than for 0-Pr. This result indicates that 0-Pr has the least Pr-4f/O-2p hybridization in good accord with the Pr M_5,4 edge spectra. Overall, the presence of pre-edge features in the O K-edge XAS spectra confirms 4f covalency in the Pr-O bond and strongly indicates ligand holes induced by Pr-4f/O-2p hybridization⁴⁰.

X-ray absorption provides unambiguous evidence for the presence of 4f − 2p hybridization. If the $|4{f}^{2}\underline{\nu }\rangle$ character is localized (as in a truly multiconfigurational ground state), it would be reflected as CF excitations corresponding to Pr³⁺ (usually ≤50 meV)². It is evident that such CF transitions from localized Pr³⁺ are not observed in both neutron and IR spectra. The alternative and the most reasonable explanation is that the $|4{f}^{2}\underline{\nu }\rangle$ character is delocalized in the Pr-O bond as is the case for charge-transfer insulator like nickelates⁴¹ and cuprates⁴². This framework is inconsistent with ${\hat{{{{{{{{\mathcal{H}}}}}}}}}}_{{{{{{{{\rm{CF}}}}}}}}}^{\Pr }$ described earlier, where the basis states are pure 4f functions. The hybridized orbitals are analogous to molecular orbitals with significant ligand contribution. Therefore, we use a modified ${\hat{{{{{{{{\mathcal{H}}}}}}}}}}_{{{{{{{{\rm{CF}}}}}}}}}^{hyb}={\kappa }^{2}\,{B}_{2}^{0}{\hat{O}}_{2}^{0}+{\kappa }^{4}\,{B}_{4}^{0}{\hat{O}}_{4}^{0}+{\kappa }^{4}\,{B}_{4}^{4}{\hat{O}}_{4}^{4}+{\kappa }^{6}\,{B}_{6}^{0}{\hat{O}}_{6}^{0}+{\kappa }^{6}\,{B}_{6}^{4}{\hat{O}}_{6}^{4}$ by including an orbital reduction factor^43,44 (κ = 0.9) which accounts for metal-ligand hybridization and yields similar results to the original model (see Supplementary Note 1).

Discussion

Taken together, the present experiments point at Pr-4f/O-2p hybridization as the essential microscopic mechanism behind the unusual electronic and magnetic properties of Pr⁴⁺ systems. A qualitative understanding of Pr-O bonding is enabled by ab-initio calculations (CASPT2/CASSCF+SOC, see Methods and Supplementary Note 2). Considering an isolated [PrO₆]⁸⁻ fragment with perfect O_h symmetry, the 4f atomic orbitals (AOs) can be easily described in the $\left|{m}_{l},{m}_{s}\right\rangle$ basis. Here, the CF splitting leads to three spin states: a ²A_2u ground-state, and two excited ²T_1u and ²T_2u triply-degenerate states (Fig. 1a). The ²A_2u state has a δ-symmetry with respect to surrounding oxygens and thus remains strictly non-bonding. In contrast, the 4f AOs underlying the ²T_2u and ²T_1u states overlap with oxygen’s 2p atomic orbitals leading to bonding and anti-bonding molecular orbitals (MOs) with π and σ + π character, respectively, about the Pr–O axes (See Fig. S14). When SOC is turned on, Table S10, the ground state corresponds to the admixture 58%²A_2u + 42%²T_2u (what identifies with the E_5/2u term in the O_h double-group symmetry). Departing from O_h symmetry – as relevant for the PrO₆ distorted-octahedra of 0-Pr and 2-Pr—lifts the degeneracies of the ²T_1u and ²T_2u excited states (Tables S11, S12). However, regardless of distortion, the ground state in both 2-Pr and 0-Pr remains solely an admixture of ²A_2u and ²T_2u states.

The spectroscopy and thermomagnetic measurements are well explained by this model and calculations. For instance, the calculated ∣〈L_z〉/〈S_z〉∣ ≈ 1.8 (See Table S12) is consistent with the XMCD data, and the small Pr⁴⁺ magnetic moment can be attributed to self-compensating spin and orbital moments combined with an unusually small ∣〈L_z〉/〈S_z〉∣ that signals a strong reduction of the orbital character in the original Γ₇ ground-state doublet. This framework naturally explains the O K edge spectra of 2-Pr and 0-Pr through ligand holes induced by the formation of hybridized T_1u2p_σ + T_2u2p_σ+π states. The model also explains why the largest hybridization is observed for PrO₂: the eight-, rather than six-, oxygen coordination environment allows the Pr 4f a_2u orbital to covalently interact with the O 2p orbitals with σ symmetry, thereby exhibiting enhanced 4f-2p hybridization²⁸. The difference in 4f-2p hybridization between 2-Pr and 0-Pr likely comes from different point-group symmetries for the PrO₆ unit and the overall symmetry of the material. It is clear that 4f-2p hybridization can strongly influence single-ion energy scales such as the CF - this phenomenon is directly analogous to the behavior of d-block metals.

Beyond single-ion properties, Pr-4f/O-2p hybridization leads to unusually large two-ion magnetic exchange interactions. For example, J = 1.2 meV has been reported by some of us on 2-Pr¹⁷; a value 2.5 times larger than the typical scale of J ≈ 0.4 meV observed for 4f¹ or 4f¹³ systems such as KCeO₂ and NaYbO₂^24,45. The Weiss constant of 1-Pr of around ∣Θ_CW∣ = 7 K is also large, especially considering the quasi-1D nature of this system. MO theory can be used to understand these exchange interactions, as shown in Fig. 4a. In the charge transfer limit (U_ff ≫ ΔE_g), the nearest-neighbor exchange interaction scales as ${t}_{pf}^{4}/\Delta {E}_{pf}^{3}$, where t_pf is the hopping integral between 4f and 2p orbitals, and ΔE_pf is their energy difference (i.e., the ligand to metal charge transfer (LMCT)). The enhancement of magnetic exchange in Ln⁴⁺ compounds is likely primarily driven by the reduction of the charge transfer energy ΔE_pf, as evidenced by e.g., calculations of [CeCl₆]³⁻ (>5 eV) and [CeCl₆]²⁻ (~3.3 eV)⁴⁶. Large t_pf hopping and small Δ_pf implies a large ligand-hole character, consistent with our O K-edge spectra. First-principles calculations on a binuclear [Pr₂O₁₀]¹²⁻ embedded cluster model for 2-Pr (see Supplementary Note 2) quantitatively confirm this picture. In the S_eff = 1/2 limit, the spin-singlet minus spin-triplet energy, which identifies with the Heisenberg exchange interaction, yields J = 4.2 meV (See Table S13, Fig. S16). Upon including SOC, the energy splitting among the four lowest levels is ≈1.5 meV. To the extent that this energy separation approximates J, it is in good agreement with the measured value for 2-Pr (1.2 meV). Therefore exchange interactions in Pr⁴⁺ materials may change by an order of magnitude (0.3 to 4.2 meV) under changes of the ligand field (Fig. 1a). Similar effects have been observed in high-valent actinides, including U⁵⁺ and Np⁶⁺⁴⁷. Inspecting hopping pathways is also informative to comment on the Kitaev (AFM/FM) interactions proposed for 2-Pr through the T_2u − p − T_1u pathway (Fig. 4b) analogous to the t_2g − p − e_g process in d⁵ systems¹⁶. While this contribution is small in d⁵ systems due to the large t_2g to e_g separation, it is proposed to be larger in 4f¹-systems owing to the small CF energy scale¹⁶. However, as we have demonstrated, Pr⁴⁺ systems exhibit a very large CF splitting (or in other words, the J_eff = 1/2 limit is not adequate), making the T_2u − p − T_1u pathway energetically less favorable⁴⁸ than the T_2u − p − T_2u pathway (Fig. 4c) responsible for the large Heisenberg AFM interaction.

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.

Finally, and very importantly, the competition between CF and SOC energy scale in Pr⁴⁺ systems resembles that in high-valent actinide systems such as U⁵⁺ and Np⁶⁺ for which a CF energy scale as large as ≈800 meV is possible^49,50. The chemistry and physics of high-valent actinides is further complicated by an extra competition between Coulomb repulsion, CF, and SOC^51,52 leading to the dual nature of 5f electrons⁵³. In order to develop a universal description for f¹ single-ions, we argue that Pr⁴⁺ systems can facilitate the study of the delicate balance of various competing interactions in the absence of competing Coulomb repulsion. To showcase that idea, we use the model established by ref. ⁵⁰ where, the CF transitions for a f¹ ion in an ideal O_h symmetry can be written as a function of two CF parameters Δ and θ, and ζ_SOC, as shown in Fig. 4f. Using this framework and our experimentally determined values for 0-Pr, we calculate the parameter $\xi=\frac{\Delta+\theta }{7/2\zeta+\Delta+\theta } \, \approx \, 0.62$ for Pr⁴⁺. When compared with other f¹ ions, including Ce³⁺, U⁵⁺, and Np⁶⁺ (see Fig. 4h and refs. ^49,54), it is evident that Pr⁴⁺ lies closer to U⁵⁺ than Ce³⁺. Qualitatively, the observed trend can be generalized by a simple f-orbital bonding picture that puts hybridization with the ligands np-orbitals as the key microscopic phenomenon leading to an enhanced CF energy scale.

In summary, our work elucidated the mechanisms behind the anomalously large CF energy scale in Pr⁴⁺ systems and discussed how exotic magnetic and electronic properties emerge as a result. The covalent character of the Pr–O bond plays a key role in determining the single-ion and macroscopic physics in Pr⁴⁺ compounds, similar to familiar systems such as cuprates and nickelates. It is in sharp contrast to Ln³⁺ systems for which, conventionally, the metal-ligand bond is described using an ionic picture. While the focus of this study has been oxides, Pr⁴⁺ materials do exist as fluorides which require rigorous synthetic conditions using pure F₂ gas. The ability to stabilize Pr in the unusually high 4+ oxidation state demonstrates that there is a rich chemical space still to be explored beyond fluorides and oxides for designing new quantum materials, including mixed-anion materials. Our results define a strategy to design and control quantum materials by tuning the Pr–O covalency through site symmetry and ligand identity by moving to softer donors like S or Se. The inadequacy of the J_eff = 1/2 limit shows us how to change the fabric of spin-orbit entangled single-ion states to stabilize exotic exchange Hamiltonians or to develop universal models to understand the physics of high-valent actinides. Pr⁴⁺-based systems offer the rare possibility to tune the hierarchy of single-ion energy scales, as well as the p and f hole density, which may be harnessed to design new correlated phenomena in quantum materials.

Methods

Material synthesis

Na₂LnO₃ (1-Ln, Ln = Ce, Pr) were synthesized using established procedures¹⁸. Polycrystalline powder samples of Sr₂LnO₄ (2-Ln, Ln = Ce, Pr) and Li₈LnO₆ (3-Ln, Ln = Pr, Tb) were synthesized using traditional solid-state methods. The samples were fired under a flow of O₂ in a tube furnace. The firing was performed at 1100 °C for 24 h. The samples were taken out of the quartz tubes in air and placed into the antechamber of the glovebox as quickly as possible in order to minimize contact with the ambient atmosphere (See SI for details).

Experimental characterizations

PXRD

Laboratory powder X-ray diffraction (PXRD) was collected on a PANalytical X’Pert PRO Alpha-1 diffractometer with Cu Kα source in reflection geometry equipped with a fixed divergence slit of 1/8”, a convergence slit of 1/4” and a working radius of 240 mm. The samples were homogenized by finely grinding them inside the glovebox using an agate mortar for about ~15 min. To avoid the exposure of the sample to atmospheric air, the PANalytical domed sample holder is equipped with stainless steel base and a polycarbonate dome with a 70% X-ray transmission. A 2θ range of 5−85^∘ was used with a scan speed of 5 s and a step size of 0.1.

Physical property measurements

The d.c. magnetic susceptibility measurements and isothermal magnetization measurements were made using a Quantum Design MPMS3. The sample was sealed in a plastic capsule on a low-background brass holder.

Neutron-scattering measurements

Inelastic neutron-scattering measurements were carried out on ~8 g of polycrystalline samples of 1-Pr, 1-Ce, 0-Pr, and 0-Tb on the time-of-flight fine-resolution Fermi chopper spectrometer SEQUOIA, at the Spallation Neutron Source, ORNL^55,56. The powder samples were enclosed in standard 15-mm diameter cylindrical annular aluminum cans (with 2 mm sample gap) under one atmosphere of ⁴He at room temperature. All four samples were cooled using a closed-cycle refrigerator reaching a base temperature of T = 5 K. Measurements were carried out using incident neutron energies E_i = 300, 500, 700 meV at T = 5 K. Background and sample holder contributions were measured using empty can measurements. The lattice phonon contributions for 0-Pr and 1-Pr were subtracted by measuring the analogous 0-Tb and 1-Ce, respectively.

STXM O K edge XAS

STXM methodology was similar to that discussed previously²⁸. In an argon-filled glovebox, samples for STXM measurement were prepared by pulverizing the powder compounds and transferring particles to Si₃N₄ windows. Second windows were placed over the samples to sandwich the particles, and the windows were sealed together with Hardman Double/Bubble epoxy. Single-energy images and O K-edge XAS spectra were acquired using the STXM instrument at the Canadian light source (CLS) spectromicroscopy beamline 10ID-1, operating in decay mode (250 to 150 mA, in a ~0.5 atm He-filled chamber) at a base temperature of T = 300 K. The beamline uses photons from an elliptically polarizing undulator that delivers photons in the 130 to 2700 eV energy range to an entrance slit-less plane-grating monochromator. The maximum energy resolution E/ΔE was previously determined to be better than 7500, which is consistent with the observed standard deviation for spectral transitions of ±0.1 eV determined from the comparison of spectral features over multiple particles and beam runs. For these measurements, the X-ray beam was focused with a zone plate onto the sample, and the transmitted light was detected. The spot size and spectral resolution were determined from the characteristics of the 35 nm zone plate. Images at single energy were obtained by raster-scanning the sample and collecting transmitted monochromatic light as a function of the sample position. Spectra at particular regions of interest on the sample image were extracted from the “stack”, which is a collection of images recorded at multiple, closely spaced photon energies across the absorption edge. Dwell times used to acquire an image at a single photon energy were 2 ms per pixel and spectra were obtained using circularly polarized radiation. The incident beam intensity was measured through the sample-free region of the Si₃N₄ windows. In order to ensure that the spectra were in the linear regime of Beer–Lambert’s law, particles with an absorption of less than 1.5 OD were used. High-quality spectra were obtained by averaging measurements from multiple independent particles, samples, and beam runs.

STXM Pr M _5,4 edge XAS

Measurements at the Pr M_5,4-edges were conducted using the STXM instrument at the Canadian Light Source (CLS) spectromicroscopy beamline 10ID-1, operating in top-off mode (250 mA, in a ~0.5 atm He-filled chamber) at a base temperature of T = 300 K. The sample preparation and data acquisition methodology is the same as described above for the O K-edge measurements.

Pr M _5,4 edge XMCD

The XAS and XMCD measurements at Pr M_5,4-edges were conducted at beamline 4-ID-C of the Advanced Photon Source located at Argonne National Laboratory. XAS and XMCD spectra were collected simultaneously using total electron yield (TEY) and total fluorescence yield (TFY), respectively, with circularly polarized X-rays in a near normal (80^∘) configuration using a cryostat reaching a base temperature of T = 20 K. The applied field was along the beam direction and it defines the positive Z direction. The data was obtained at both a zero field and an applied field of μ_OH = ±5 T. The XMCD spectra were obtained point by point by subtracting right from left circular polarized XAS data. Measurements were taken for both positive and negative applied field directions, and then a difference of these two spectra XMCD = $\frac{1}{2}$ [XMCD(H_z > 0) - XMCD(H_z < 0)] was taken to eliminate polarization-dependent systematic errors. The TFY XAS data is identical to the STXM data described above. However, the TFY XMCD signal is weak and distorted by self-absorption effects. The TEY XAS data is similar to STXM data as well, except the high energy satellite peaks at both M_5,4 edges are weak and not as pronounced. Furthermore, the low-energy shoulder at the M₄ edge is more pronounced in TEY XAS than in both TFY and STXM. For discussions in the main text regarding M_5,4 edge isotropic XAS spectra, only the STXM data is discussed as it minimizes error due to self-absorption, saturation, and surface contamination. However, for our discussions with XMCD, we use the data collected in TEY mode. As noted in Fig. 3, the isotropic XAS was in the top panel and is measured in STXM mode, while the XMCD was at the bottom panel and is measured in TEY.

Infrared magnetospectroscopy

Broadband IR measurements were performed in the Voigt transmission configuration using a Bruker 80v Fourier-transform IR spectrometer. The incident IR light from a global source was guided to the top of the probe inside an evacuated beamline and then delivered to the bottom of the probe through brass light pipes. The sample was located in the middle of two confocal 90^o off-axis parabolic mirrors mounted at the bottom of the probe. While the first mirror focuses the IR radiation on the sample, the second mirror collimates the radiation to the short light pipe with a 4K composite Si bolometer at the end. About 25 mg of the powder sample was mixed with KBr inside a glovebox in a 1:1 ratio. The resulting mixture was pressed into 3 mm pellets and was secured by a thin polypropylene adhesive film, and mounted on the brass plate with a clear aperture of 3 mm. The sample was placed at the center of a μ₀H = 17.5 T vertical bore superconducting magnet in a helium exchange gas environment, providing the sample temperature of about 5.5 K. IR transmission spectra were collected for 3 min at a fixed magnetic field, changing with 1 T step. All spectra obtained at different magnetic fields were normalized to the same reference spectrum, which is their mean, computed after removing the outlier points at each frequency. Such normalization flattens those spectral features independent of magnetic fields and highlights those absorption peaks that shift as the magnetic field rises.

Crystal field modeling of inelastic neutron scattering (INS)

CF modeling was carried out using the truncated CF Hamiltonian ${\hat{{{{{{{{\mathcal{H}}}}}}}}}}_{{{{{{{{\rm{CF}}}}}}}}}={B}_{2}^{0}{\hat{O}}_{2}^{0}+{B}_{4}^{0}{\hat{O}}_{4}^{0}+{B}_{4}^{4}{\hat{O}}_{4}^{4}+{B}_{6}^{0}{\hat{O}}_{6}^{0}+{B}_{6}^{4}{\hat{O}}_{6}^{4}$ where ${B}_{n}^{m}$ are the second, fourth, and sixth-order terms and ${\hat{O}}_{m}^{n}$ are the corresponding Stevens operator equivalents⁵⁷ for all three materials studied here. The Stevens operators are expressed in terms of L and L_z. Although the true symmetries of Pr⁴⁺ in each system require more parameters based on point-group symmetry, any mixing induced by these parameters would not induce any further loss of degeneracy and hence we choose to parameterize their effects ${B}_{n}^{0}$ and ${B}_{n}^{4}$ parameters. This approach was taken in order to minimize over-parametrization while fitting to experimental data. All Hamiltonian diagonalizations were performed using the PYCRYSTALFIELD package⁵⁸. Fitting was carried out to a combination of eigen-energies extracted experimentally from INS and IRMS and to the temperature-dependent susceptibility data over T > 40 K in order to avoid short-range correlations present at lower temperatures. The final fit results are provided in Supplementary Table 3. The CF models were validated by calculating the isothermal magnetization at T = 50 K. The model calculation of g values for the ground state wavefunction was compared to experimentally determined values from Curie–Weiss fits and first-principles calculations. See Supplementary Note 1 for a detailed description of the fitting procedure. The f electron density plots were obtained using QUANTY⁵⁹ and plotted using Wolfram-Mathematica⁶⁰.

The Γ₇ KD in the Δ_CF < < ζ_SOC limit is written in the $\left|J,{m}_{J}\right\rangle$ basis as $\sin \theta \left|\frac{5}{2},\pm \frac{5}{2}\right\rangle+\cos \theta \left|\frac{5}{2},\mp \frac{3}{2}\right\rangle$, where $\sin {\theta }^{2} \sim 1/6$. The same Γ₇ KD can be written in the $\left|{m}_{l},{m}_{s}\right\rangle$ basis as $A\,\left|\mp 3,\pm \frac{1}{2}\right\rangle -\,B\,\left|\mp 2,\mp \frac{1}{2}\right\rangle \,+\,C\,\left|\pm 1,\pm \frac{1}{2}\right\rangle -\,D\,\left|\pm 2,\mp \frac{1}{2}\right\rangle$, where A = 0.352, B = 0.215, C = 0.454, D = 0.79, yielding $\alpha=\frac{{A}^{2}+{B}^{2}}{{C}^{2}+{D}^{2}} \, \approx \, 0.18$. The first two components of the Γ₇ KD in $\left|{m}_{l},{m}_{s}\right\rangle$ (m_l = ∓3, ∓2) map onto a linear combination of the $\left|\frac{5}{2},\pm \frac{5}{2}\right\rangle,\left|\frac{7}{2},\pm \frac{5}{2}\right\rangle$, states in $\left|J,{m}_{J}\right\rangle$ basis, while the last components (m_l = ±1, ±2) map onto $\left|\frac{5}{2},\pm \frac{3}{2}\right\rangle,\left|\frac{7}{2},\pm \frac{3}{2}\right\rangle$ states. For the Γ₇ KD, given that Δ_CF < < ζ_SOC, the contributions from the $J=\frac{7}{2}$ SOC manifold are negligible. As Δ_CF ~ ζ_SOC, non-negligible contributions from the $J=\frac{7}{2}$ SOC manifold enter the ground-state wavefunction making it impossible to deconvolute the individual contributions from each SOC manifold. Therefore, a better description of mixing can be obtained by looking at the ratios $\frac{{A}^{2}}{{B}^{2}}$ and $\frac{{C}^{2}}{{D}^{2}}$. Within this framework, irrespective of the symmetry at the metal center, for a six-coordinate system, the ground state wavefunction is always a linear combination of m_l = ±1, ±2, ∓3 states. This derives from the ²A_2u + ²T_2u symmetry (in O_h notation) as described in the main text and predicted by first-principles calculations. The introduction of intermediate coupling, changes only the relative mixtures of m_l = ±1, ±2, ∓3 states and does not introduce any new admixture into the ground state wavefunction. The relative change in a mixture can be viewed as introducing $\left|\frac{7}{2},\pm \frac{5}{2}\right\rangle$ and $\left.\frac{7}{2},\pm \frac{3}{2}\right\rangle$ states and increasing the amount of $\left|J,\pm \frac{5}{2}\right\rangle$ character in the ground state. This is clearly evident in the toy model established in Supplementary Note 1.

Multiplet modeling of X-ray absorption spectroscopy (XAS)

Multiplet calculations were implemented using the original code written by Cowan (ref. ³²) and further developed by de Groot based on AIM. The multi-electron configuration in the ground and the final states was implemented using a charge-transfer methodology analogous to nickelates and cuprates. For all calculations, a Gaussian broadening of 0.45 eV was applied to account for instrumental broadening and Lorentzian broadening of 0.3 and 0.6 eV were applied to the M₅ and M₄ edges, respectively. The model parameters had the following values for both 1-Pr and 3-Pr: U_ff ~14.1 eV, U_fc ~8.5 eV, ${\zeta }_{SOC}^{4f} \sim 0.12$ eV, and ${\zeta }_{SOC}^{3d} \sim 7.1$ eV where U_ff and U_fc are the 4f − 4f Coulomb interaction and the core-hole potential acting on the 4f electron, respectively. In the limit of vanishing V → 0, the difference between the two configurations in the ground state was ΔE_g = ϵ_f − ϵ_n = 2.0 eV (1-Pr) and 3.0 eV (3-Pr), and ΔE_f = ϵ_f − ϵ_n + U_ff − U_fc = 0.5 eV (1-Pr) and 1.5 eV (3-Pr), where ϵ_f and ϵ_n are one-electron energies of Pr 4f and O 2p levels and V is the hybridization energy between atomic like localized 4f states and delocalized O 2p states which determines the mixing between the multi-electron configurations. Hybridization energy in the ground state (V_g) was set to 1.4 eV (1-Pr and 3-Pr), and final state (V_f) was set to 1.4 eV (1-Pr) and 1.8 eV (3-Pr).

Ab-initio calculations

Without symmetry restrictions, wavefunction theory (WFT) calculations were performed within a commonly applied two-step spin-orbit coupled configuration-interaction formalism using OpenMolcas⁶¹. WFT calculations (See SI for more details) were performed on an isolated PrO${}_{6}^{8-}$ cluster and on embedded cluster models for Li₈PrO₆ and Na₂PrO₃. Additional calculations were performed with a binuclear Pr₂O${}_{10}^{12-}$ embedded cluster model of Na₂PrO₃. All geometries were extracted from the crystal structures and used without further optimization. A 40 Å sphere of atoms was generated from the crystal structure. An outer 32 Å sphere contained Pr⁴⁺, O²⁻, Na⁺/Li⁺ embedding point charges; the inner 8 Å sphere contained the PrO${}_{6}^{8-}$ ion treated quantum-mechanically surrounded by Pr⁴⁺, O²⁻, Na⁺/Li⁺ pseudocharges represented by ab-initio model potentials (AIMPs). All atoms treated quantum-mechanically were modeled with all-electron atomic natural orbital relativistically contracted basis sets of valence triple-ζ quality (ANO-RCC-VTZP).