$f$-electron and magnetic ordering effects in nickelates LaNiO$_2$ and NdNiO$_2$: remarkable role of the cuprate-like $3d_{x^2-y^2}$ band

Recent discovery of superconductivity in the doped infinite-layer nickelates has renewed interest in understanding the nature of high-temperature superconductivity more generally. The low-energy electronic structure of the parent compound NdNiO$_{2}$, the role of electronic correlations in driving superconductivity, and the possible relationship betweeen the cuprates and the nickelates are still open questions. Here, by comparing LaNiO$_2$ and NdNiO$_2$ systematically within a parameter-free density functional framework, all-electron first-principles framework, we reveal the role Nd 4$f$ electrons in shaping the ground state of pristine NdNiO$_2$. Strong similarities are found between the electronic structures of LaNiO$_2$ and NdNiO$_2$, except for the effects of the 4$f$-electrons. Hybridization between the Nd 4$f$ and Ni 3$d$ orbitals is shown to significantly modify the Fermi surfaces of various magnetic states. In contrast, the competition between the magnetically ordered phases depends mainly on the gaps in the Ni $d_{x2-y2}$ band, so that the ground state in LaNiO$_2$ and NdNiO$_2$ turns out to be striking similarity to that of the cuprates. The $d-p$ band-splitting is found to be much larger while the intralayer 3$d$ ion-exchange coupling is smaller in the nickelates compared to the cuprates. Our estimated value of the on-site Hubbard $U$ is similar to that in the cuprates, but the value of the Hund's coupling $J_H$ is found to be sensitive to the Nd magnetic moment. The exchange coupling $J$ in NdNiO$_2$ is only half as large as in the curpates, which may explain why $T_c$ in the nickelates is half as large as the cuprates.


Introduction
Since the discovery of high-T c superconductivity (HTSC) in the lanthanum-based cuprates in 1986 [1], understanding the mechanism of HTSC has drawn intense interest [2][3][4][5][6]. Despite vigorous efforts, however, many unanswered questions still remain and a clear consensus on the underlying mechanism of HTSC has not been reached. A promising route in this connection is to find superconducting analogs of the cuprates which could provide new clues to the origin of HTSC. One such candidate materials is the perovskite nickel oxides. Specifically, the infinite-layer NdNiO 2 compound holds great promise since it exhibits an intrinsic 3d 9 filling much like the cuprates, although challenges of crystal growth have presented problems for undertaking systematic investigations of this material.
Recently, superconductivity in the hole-doped infinitelayer nickelate NdNiO 2 at 9∼15K has been reported in thin film samples grown on SrTiO 3 [7][8][9][10], although The bulk of these studies focus on the NiO 2 plane, finding differences in quantities such as the d − p orbital splitting as compared to the cuprates [18] in accord with experimental reports [33]. However, by focusing on the NiO 2 plane neglects the effects of the f -electrons on the electronic and magnetic structure, despite the presence of superconductivity in Nd and Pr based compounds but not in La [7,21]. The active role of the f -electrons is also suggested by a Kondo-like logarithmic temperature dependence of the resistivity and the Hall coefficient at low temperatures [7], and other recent experiments demonstrating strong similarities between the infinitelayer nickelates and rare earth intermetallics [33], although a recent study attributes this strange behavior to the Nd 5d orbitals [15]. A few electronic structure studies utilizing the DFT+U [16] and Heyd-Scuseria-Ernzerhof (HSE) hybrid functional [24] approaches have considered the f -electrons and find significant hybridization between the Nd 4f and Ni 3d orbitals near the chemical potential along with a possible ferromagnetic order. But, these calculations neglected the effect of spin-orbit coupling (SOC) crucial to capture the correct f -band splittings and required the introduction and fine tuning of external ad hoc parameters such as the Hubbard U and the exact-exchange admixture, which limit their predictive power [34].
In this article, we present a systematic study of the electronic and magnetic structures of both LaNiO 2 and NdNiO 2 using the strongly-constrained-appropriatelynormed (SCAN) density functional [35] with spin-orbit coupling to examine effects of the f -electron physics. The SCAN functional has a proven track record of accurately modeling many correlated materials families including the cuprates [36][37][38][39][40], iridates [41], and ABO 3 materials [42]. In particular, SCAN accurately predicts the f -band splitting in SmB 6 in good accord with experimental values [43]. We consider several magnetic phases, whose energy and ordering are found to be quite similar for LaNiO 2 and NdNiO 2 , reflecting their sensitivity to the opening of magnetic gaps in the Ni d x2−y2 band. Dispersion of this band is quite similar to the corresponding band in cuprates, as is the order of the resulting magnetic phases, with an antiferromagnetic state having the lowest energy. In line with this, the estimated values of the Hubbard U are close to those commonly found in the cuprates, while Hund J H varies for different Nd magnetic sublattice. In contrast, we find the the intralayer nearestneighbor exchange coupling J approximately one half as large as that in La 2 CuO 4 [36]. Lastly, the 4f -electrons play an important role in modifying the Fermi surfaces. We find the charge transfer energy between Ni 3d and O 2p orbitals is large and does not change much with magnetic order. These latter findings suggest distinct physics in nickelates, which could be confirmed by further experiments.

Results and Discussion
Crystal and magnetic structures: Figure 1 shows the crystal structure of LaNiO 2 and NdNiO 2 in the P 4/mmm symmetry [22], where NiO 2 planes are sandwiched together with La or Nd spacer layers. In the NiO 2 planes the Ni sites are surrounded by four O atoms in square-planar coordination. While the non-magnetic (NM) and ferromagnetic (FM) phases can be calculated in the primitive cell [Figs. 1 (a) and 1 (b)], the remaining three antiferromagnetic (AFM) phases require distinct supercells. Specifically, we use a √ 2× √ 2×1, √ 2× √ 2×2 , and 1 × 1 × 2 supercell for the C-type AFM (C-AFM), G-type AFM (G-AFM), and A-type AFM (A-AFM) orders, respectively, as shown in Figs. 1 (c)-(e). In the C-AFM phase, the intralayer coupling in both Nd and Ni layers is AFM, whereas the interlayer is FM coupled. In the G-AFM phase both the intra-and interlayer coupling are AFM. In contrast, the A-AFM phase displays an intralayer FM coupling with an AFM interlayer coupling for both Nd and Ni sublattices. We note that the coupling between Ni and Nd nearest neighbors is frustrated in the A-AFM and G-AFM configurations.  Table I gives our theoretically predicted total energies, lattice constants, and spin magnetic moments for various magnetic phases of LaNiO 2 and NdNiO 2 . For LaNiO 2 , the C-AFM phase is the most stable, with the G-AFM, A-AFM, and NM phases lying at higher energies. Upon optimizing the crystal structure we find the predicted lattice parameters for the magnetic phases in good accord with the experimental values, while those from the NM phase are quite further away. Moreover, the lattice constant and energy of the FM (C-AFM) phase are almost same as in the A-AFM (G-AFM) phase, suggesting that interlayer coupling in LaNiO 2 is very weak. Finally, our theoretically predicted local nickel magnetic moment is ∼1.0 µ B irrespective of the magnetic configuration.
Our results are in contrast to those found with GGA+U [18,44] (U = 3) which predict a much smaller energy separation between the magnetic configurations along with reduced magnetic moments of ∼0.7 µ B and ∼0.5 µ B in the C-AFM and FM phases, respectively. The reduced moment values could possibly be due to SOC effects being neglected, but this result is still surprising since a significant Hubbard U was introduced on the Ni sites. We now compare our LaNiO 2 results to the corresponding properties of NdNiO 2 (see Table I). For both compounds the C-AFM configuration is the ground state, with almost the same energy separation with the FM phase. However, the energy differences between the other AFM phases are much larger in NdNiO 2 than those in LaNiO 2 , with the NM phase lying at 3840 meV/f.u. The larger energy differences in NdNiO 2 between the FM (G-AFM ) and A-AFM (C-AFM ) phases suggests that the interlayer exchange coupling is strongly affected by the presence of the local magnetic moment of Nd. Furthermore, like LaNiO 2 , the lattice constants of NdNiO 2 in the C-and G-AFM phases are extremely close to the experimental values compared to those obtained in the NM, FM, and A-AFM arrangements. The stabilization of C-AFM order over that of G-AFM appears to be due to the frustration between Nd and Ni magnetic moments, clearly illustrating the importance of the Nd f -electrons in the calculation. Table I shows the theoretically predicted local magnetic moments of Nd and Ni to be ∼3.0 µ B and ∼1.0 µ B , respectively. This suggests that the electron configuration of Nd and Ni are [Xe] 4f 3 and 3d 9 , respectively. By breaking the Ni magnetic moment down into the various orbital contributions, we find 0.75 µ B and 0.25 µ B on the d x 2 −y 2 and d z 2 orbitals, respectively, where the t 2g states have negligible moments. A number of theoretical studies in the literature have reported magnetic ordering and the associated local moments [16,27,44,46] with strong spin fluctuations possibly playing a key role in cooper pairing [31,46,47]. Specifically, the GGA finds the local moment on Ni to be significantly reduced, yielding ∼0.52 and ∼0.35 µ B in the FM and AFM phases, respectively [16]. Where the SCAN values are only recovered when a large Hubbard U of 8 (5) eV for Nd (Ni) is assumed [16]. Moreover, a previous calculation utilizing the SCAN functional [25] surprisingly finds reduced Ni magnetic moment values of 0.76 µ B [25], but these calculations neglect Nd 4f electrons and SOC effects. Lastly, the HSE06 hybrid functional finds stabilized moments of ∼3.03 µ B and ∼0.89 µ B on Nd and Ni sites, respectively, for the FM phase [24], similar to the SCAN values. Overall, the SCAN predictions are closely aligned with the expected d and f -filling of NdNiO 2 without any fine-tuning.
Electronic structure of NM Phase: Figures 2 (a) and 2 (b) show the theoretically obtained band structures and density of states (DOS) for LaNiO 2 and NdNiO 2 in the non-magnetic (NM) phase, with the various orbitalresolved atomic site projections overlaid. For LaNiO 2 , Fig. 2 (a), we find two distinct bands crossing the Fermi level: one of nearly pure Ni-3d x 2 −y 2 character, and the other composed of Ni (3d z 2 , 3d xy/yz ) and La 5d orbitals. The latter band produces a 3D spherical electron-like Fermi surface at Γ and A [ Fig. 2 (c)], whereas the former generates a large slightly warped quasi-2D cylindrical Fermi surface similar to the cuprates. These findings are consistent with previous LDA [18], GGA [24], DFT+U [44], and LDA+DMFT [30] studies. We note that the 3d x 2 −y 2 band bears a striking resemblance to the corresponding band in cuprates [36], except for a shift of the VHS energy between Γ and Z planes in the Brillouin zone. The shifting of the VHS from below to above the Fermi level along k z is directly reflected in the find all the Ni bands to be relatively unchanged, except for significant hybridization between the Nd 4f -electrons and Ni 3d x 2 −y 2 orbitals. Interestingly, this similarity between the Ni 3d dispersions in LaNiO 2 and NdNiO 2 persists across all magnetic phases studied [Figs. [3][4][5][6]. The significant hybridization between Ni 3d and Nd 4f levels can give rise to self-doping effects and induce Kondo physics [23,[25][26][27]29]. Such hybridizations also radically alter the Fermi surfaces. Figure 2 (d) displays a doublegoblet-like hole pocket along the Γ-Z direction, with a very narrow stem at Γ. Moreover, a large and complex hole Fermi surface appears surrounding Γ near the k z = 0 plane, along with the formation of electron pockets near M . We further note that the narrow neck of the goblet Fermi surface at Γ suggests that the system is close to a Fermi surface topological transition where the goblet splits into Fermi pockets centered on the Z point [ Fig. 2 (d)].
Electronic structure of C-AFM Phase: Figures 3 (a) and 3 (b) present the unfolded theoretical electronic structure of LaNiO 2 and NdNiO 2 in the C-AFM phase. Similar to the cuprates, the C-AFM order is stabilized by opening a 2 eV band gap in the d x 2 −y 2 dominated band. However, unlike the cuprates, 5d and 4f states fill the gap maintaining the metallic nature of these compounds. For example, the states near the Fermi level in LaNiO 2 are mainly governed by the Ni 3d z 2 and La 5d orbitals [ Fig. 3 (a)]. Moreover, an extremely flat band is found pinned near the Fermi level along the Z −R−A−Z line, originating from the La 5d and Ni 3d z 2 hybridized orbitals. A similar flat band is found in NdNiO 2 , along with a second flat band along Z − R − A, stemming from Nd 4f Ni 3d xy/yz hybridization. These flat band features have also been observed by Choi et. al [46] where a large U was used to push the Nd 4f states away from the Fermi level. These flat-band features also produce highly anisotropic Fermi surfaces near the k z =π/c plane for both LaNiO 2 and NdNiO 2 [ Fig. 3 (c) and (d)]. For NdNiO 2 there are two additional heavy electron pockets ellipsoidal in shape [ Fig. 3(d)], which are induced by strong Nd 4f and Ni 3d z 2 hybridization.
Electronic structure of G-AFM Phase: While the C-AFM phase is found to be the ground state, it is important to study other low-lying phases in correlated quantum materials that could contribute to various intertwined orders [36,37,[41][42][43]. Figure 4 (a) and 4(b) show the electronic structures of LaNiO 2 and NdNiO 2 in the G-AFM phase. The G-AFM magnetic phase exhibits AFM coupling between the intra-and interlayer magnetic sites, in contrast to C-AFM where the interlayer coupling is FM. Interestingly, here we find no f -bands near the Fermi level for NdNiO 2 , making the Fermi surfaces in LaNiO 2 and NdNiO 2 equivalent. While both G-AFM and C-AFM phases are dominated by the splitting of the d x2−y2 -band, and both have a region of suppressed DOS within ∼0.7 eV of the Fermi level, they differ in that a 3d z 2 band is above the low DOS region in the C-AFM phase, but below it in the G-AFM phase, causing the Fermi energy to shift by ∼0.7 eV.
Electronic structure of FM Phase: Figures 5 (a) and 5 (b) present the electronic structure of LaNiO 2 and NdNiO 2 in the FM phase. Although the spin moments of Nd and Ni were initialized in the same direction, the system selfconsists into a ferrimagnetic configuration [ Fig. 1  (b)]. Compared with the NM phase [ Fig. 2], the Ni 3d x 2 −y 2 and 3d z 2 bands are spin split, due to the spinpolarization in FM phase. In this case the Fermi surface is composed of a Ni 3d x 2 −y 2 hole pocket (red) at the M point and an electron pocket from the hybridization between Ni 3d z 2 and Nd 4f orbitals at Γ point (blue) [ Fig. 5 (b)]. Interestingly, the majority spins in the Ni 3d x 2 −y 2 and 3d z 2 bands point in opposite directions for a given atom. Figs. 5 (c) and (d) show that the Fermi surfaces of these two compounds are quite similar, except that the Γ point electron pocket in Fig. 5 (c) has grown 'propellers' in Fig. 5 (d), which is due to hybridization between Ni 3d z 2 and Nd 4f orbitals. We also find a 2D Fermi sheet centered at the M point extending in the k z direction, produced by hybridization between the Ni 3d xy/yz and Nd 4f orbitals. Finally, there is an Acentered hole pocket generated by the Ni 3d x 2 −y 2 band.
Electronic structure of A-AFM Phase: cally ordered, the resulting band-splitting is quite similar to the FM phase [ Fig. 5]. Figure 6 (a), shows that the bands near the Fermi level in LaNiO 2 are mainly of Ni 3d x 2 −y 2 , and hybridized Ni 3d z 2 and La 5d character. However, in NdNiO 2 , we find the main lowlying states near the Fermi level to originate from Nd 4f states hybridizing with Ni 3d x 2 −y 2 and 3d z 2 orbitals. Notably, around -1 eV we see a strong mixing between the Ni 3d x 2 −y 2 and 3d z 2 orbitals in both LaNiO 2 and NdNiO 2 along Γ − X and R − A directions, which are absent in the NM and FM phases. Such a strong 'orbitalmixing' effect could make the physics in the nickelates quite different from the cuprates. along the z-axis due to the AFM stacking of adjacent FM layers. This effect leads to the appearance of two pockets near the M point in Γ-plane. (ii) The Γ − Z direction is more reminiscent of the NM case, with a Ni 3d z 2 electron pocket at the Γ point in LaNiO 2 , while in NdNiO 2 , the goblet Fermi surface of the NM phase has split into Z-centered pockets. The blue color of these features (low Fermi velocity) suggests strong f -electron mixing.
f -electron dispersions: Based on the preceding comparison of LaNiO 2 and NdNiO 2 , we find the stabilization of the various magnetic phases of NdNiO 2 to be mainly driven by Ni d-electrons, with the f -electrons playing only a minor role. In contrast, the Fermi surfaces are strongly affected by the Nd 4f -electrons, exhibiting strong mixing with Ni 3d orbitals. While the f -electrons form one large cluster in the NM phase, they split into three subbands once the Nd atoms become polarized with slight shifts depending on the magnetic order consistent with our recent work on SmB 6 [43]. Table II gives the calculated ∆ dp , ∆ eg , U, and J H for the various magnetic phases of LaNiO 2 and NdNiO 2 , along with the corresponding values for the cuprates for comparison. The values of ∆ dp for LaNiO 2 phases range from 1.58 (FM) to 2.08 eV (G-AFM ), while the corresponding values for NdNiO 2 span 1.82 to 2.84 eV. To illustrate the charge-transfer gap further and compare to C-AFM of the infinite-layer cuprate CaCuO 2 the partial density of states for the Ni (Cu) 3d and O 2p orbitals is shown in Figs. 7 (a) and (c). Here, the O 2p band-center is clearly lower than the center of gravity of the Ni 3d states by ∼2 eV in NdNiO 2 , whereas the O 2p levels are strongly hybridized with Cu 3d x 2 −y 2 orbitals near the Fermi level in CaCuO 2 . To quantify this, we find ∆ dp for CaCuO 2 in the C-AFM phase to be 0.19 eV, which is significantly smaller than nickel-based compounds. Additionally, we estimated ∆ dp for the single-layer La 2 CuO 4 in the G-AFM phase to be 0.6 eV, still much smaller than nickelates. Based on these results, LaNiO 2 and NdNiO 2 are closer to the Mott-Hubbard limit rather than chargetransfer case based on the ZaanenSawatzkyAllen classification scheme [48]. Interestingly, values of ∆ eg for the various magnetic phases of LaNiO 2 and NdNiO 2 are very close to 2 eV. For example, the ∆ eg for LaNiO 2 in the NM phase is 1.93 eV. Our estimate of ∆ eg is consistent with the value of 1.95 eV obtained in Ref. 18. The similarity of ∆ eg values across the infinite-layer nickelates suggests that the Nd 4f electrons play a very limited role in splitting the Ni 3d levels. Additionally, we estimate the value of ∆ eg for C-AFM CaCuO 2 and G-AFM La 2 CuO 4 to be close to those obtained from the nickelates [ Table II]. This information also can be read from Figs. 7 (b) and (d), where e g orbitals are quite splitting for both C-AFM NdNiO 2 and CaCuO 2 .

Comparisons with cuprates:
To gauge the strength of correlations on the nickel site we calculate the Hubbard U and Hund's coupling J H for LaNiO 2 and NdNiO 2 in various magnetic arrangements according to Eq.(1)- (3). We find the on-site potential U for LaNiO 2 and NdNiO 2 to be almost the same and very close to values obtained for CaCuO 2 and La 2 CuO 4 , suggesting strong electron interactions in nickelates. Note that these estimated U values are also consistent with recent works [23,49]. The behavior of J H is more subtle. In general, J H for nickelates and cuprates are very similar, whereas J H s for NdNiO 2 are larger in the A-AFM and G-AFM phases, suggesting that J H is highly sensitive to the interlayer coupling.
TABLE II. Comparison of properties of different phases of nickelates and cuprates. ∆ dp and ∆e g represent the splitting of the metallic (Ni and Cu) 3d and O 2p bands, and the splitting of tradition metal ions (Ni and Cu) eg bands, respectively. U and JH are on-site Hubbard potential and Hunds coupling, respectively.

LaNiO2
Phases ∆ dp (eV) ∆e g (eV) U (eV) JH (eV) NM Despite there being no clear theoretical description of the mechanism of HTSC, the view that spin-fluctuations play a central role in determining the physical properties of the cuprates has been gaining increasing support. Furthermore, in this picture, the exchange coupling strength is a good descriptor of the robustness of superconductivity.
In order to determine the strength of the exchange coupling, we map the total energies of the AFM and FM phases onto those of the nearest-neighbor spin-1/2 Heisenberg Hamiltonian in the mean-field approximation [36,50]. The Heisenberg Hamiltonian gives a reason-able description of the low-lying excitations for La 2 CuO 4 , and thus a good estimate of the Heisenberg exchange parameter J [51], and we expect this also to be the case for the infinite-layer nickelates. In the mean-field limit, the difference in total energies of the FM and AFM phases is where N is the total number of magnetic moments, S is the spin on each site, and Z is the coordination number. Since the in-plane interactions within the Ni-O planes in La(Nd)NiO 2 are much stronger than the interplanar interactions, we can take Z = 4. Since we normalize to one formula unit, N = 1. Using the total energies for the FM and C-AFM states obtained from our firstprinciples computations we find J to be 62 and 65 meV for LaNiO 2 and NdNiO 2 , respectively. These exchange parameters are half as large as those in La 2 CuO 4 [36] and larger than those estimated in Ref. [17]. The small J is consistent with the finding that the AFM gap is twice as large as in curpates, which in turn may be related to the larger value of Ni magnetic moments, which may be related to smaller Ni-d and O-p hybridization (larger ∆ dp ).

LaNiO2
NdNiO2 La2CuO4 J (meV) 62 65 138 [36] Superconductivity in the cuprates evolves out of a Mott insulator [3], whereas in the iron pnictides superconductivity emerges out of a metallic state [52] with strong local magnetic fluctuations. Is magnetic order necessary for d-electron high-T c superconductivity? The new Ni-based superconductors appear to be a counterexample. However, both Ni and Nd sites typically display significant magnetic moments with considerable evidence of magnetic fluctuations or short-range order [31,46]. Moreover, the undoped nickelates are not ordinary metals, but weak insulators [8,10]. In our previous SCANbased studies of other correlated materials, we found many low-energy magnetic phases indicative of prominent magnetic fluctuations [36,37,43]. In the nickelates, our study of the various AFM orders finds ∼0.7 eV pseudogap (regions of low DOS) near the Fermi level, which could explain the weak insulating behavior. In Figure 8, we compare the AFM gaps in the ground state structures of the cuprates (a) and the nickelates (b). We find the gap is about twice as large in the nickelates as compared to the cuprates [36], which may explain why J is only half as large in the former.
Hence, we find great similarity between the cuprates and nickelates, both in the dispersion of the NM d x 2 −y 2 band, and in the resulting magnetic orders, with the felectrons playing little role in the magnetic transitions despite significantly modifying the Fermi surfaces. In principle, the fact that J is only half as large in nickelates as in cuprates could tell us something about why the T c dome is only half as large in nickelates. On the other hand, so far superconductivity has been found in two rare-earth substituted nickelates [7,21], but not the parent La-based compound, suggesting a more significant role for f -electrons. An interesting possibility is that the f -electrons could lead to heavy-fermion physics (flat bands) not present in cuprates.
Conclusion: We examine in-depth the role of felectrons and magnetic ordering effects in LaNiO 2 and NdNiO 2 within a parameter-free, all-electron firstprinciples framework. The magnetic orders in the nickelates are found to be very similar to those in the cuprates in that the transitions are driven by the gapping of the d x 2 −y 2 band. We find a reduced J value in the nickelate compared to the cuprates, however, which could explain the weaker superconductivity in the nickelates. While the 4f electrons play little role in the nickelate magnetism, they substantially modify Fermi surfaces for various mag-netic states. Our study further reveals the importance of fluctuating magnetic order in correlated materials [37].

Methods
All calculations were performed by using the pseudopotential projector-augmented wave method [53] as implemented in the Vienna ab initio simulation package (VASP) [54,55]. A high-energy cutoff of 520 eV was used to truncate the plane-wave basis set. The exchangecorrelation effects were treated using the SCAN [35] meta-GGA scheme. Spin-orbit coupling effects were included self-consistently. The crystal structures and ionic positions were fully optimized with a force convergence criterion of 0.01 eV/Å for each atom along with a total energy tolerance of 10 −5 eV. The Fermi surface was obtained with the FermiSurfer code [56]. The unfolded band structures including orbital characters are extracted from the supercell pseudo-wavefunction calculation [57], which has been implemented based on the VaspBandUnfolding code [58].
To facilitate comparison with the cuprates, we calculated two quantities: (1) charge-transfer energies between the Ni 3d and O 2p orbitals ∆ dp = ε d −ε p and (2) the energy splitting of the two Ni e g orbitals ∆ eg = ε x 2 −y 2 −ε z 2 . Here, ε i refers to the band centers of the corresponding orbital i. Following previous works, [18,59] the band centers are defined as ε i = gi(ε)εdε gi(ε)dε , where g i (ε) refers to the partial-density-of-states (PDOS) associated with orbital i. The integration range for ∆ dp is set to cover the full bonding and antibonding bands [59], whereas ∆ eg is obtained from an integral over the antibonding bands alone, using an energy window of -3.5 to 2 eV and -4 to 4 eV for the NM and magnetic phases, respectively.
To estimate the on-site Hubbard potential U and the Hunds coupling J H , we follow the method developed by Lane et al. [36]. Using the site-projected orbital-resolved partial DOS g µσ , we determine the average spin-splitting of the µ levels as follows: where N ↑ (N ↓ ) is the occupation of the spin-up (down) d x 2 −y 2 orbital and the integration is over the full bandwidth W .

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.