Energetics of the coupled electronic-structural transition in the rare-earth nickelates

Rare-earth nickelates exhibit a metal-insulator transition accompanied by a structural distortion that breaks the symmetry between formerly equivalent Ni sites. The quantitative theoretical description of this coupled electronic-structural instability is extremely challenging. Here, we address this issue by simultaneously taking into account both structural and electronic degrees of freedom using a charge self-consistent combination of density functional theory and dynamical mean-field theory, together with screened interaction parameters obtained from the constrained random phase approximation. Our total energy calculations show that the coupling to an electronic instability towards a charge disproportionated insulating state is crucial to stabilize the structural distortion, leading to a clear first order character of the coupled transition. The decreasing octahedral rotations across the series suppress this electronic instability and simultaneously increase the screening of the effective Coulomb interaction, thus weakening the correlation effects responsible for the metal-insulator transition. Our approach allows to obtain accurate values for the structural distortion and thus facilitates a comprehensive understanding, both qualitatively and quantitatively, of the complex interplay between structural properties and electronic correlation effects across the nickelate series.

Rare earth nickelates, RNiO3, exhibit a metal-insulator transition (MIT) accompanied by a breathing mode distortion of the octahedral network which breaks the symmetry between formerly equivalent Ni sites. The quantitative theoretical description of this coupled electronic-structural instability, driven by electronic correlation effects, is extremely challenging. Here, we address this issue by simultaneously taking into account both structural and electronic degrees of freedom using a self-consistent combination of density functional theory and dynamical mean-field theory (DFT+DMFT), together with interaction parameters obtained from the constrained random phase approximation. Our total energy calculations allow to obtain accurate values for the breathing mode distortion across the series and elucidate the coupling between the structural distortion and the electronic instability towards a charge disproportionated insulating state. Our analysis shows that the MIT is crucial to stabilize the structural distortion, leading to a clear first order character of the coupled transition. Decreasing octahedral rotations across the series suppress the electronic instability and simultaneously increase the screening of the effective Coulomb interaction, thus weakening the correlation effects responsible for the MIT. Our results enable a comprehensive understanding, both qualitatively and quantitatively, of the complex interplay between structural properties and electronic correlation effects across the nickelate series, and pave the way for future studies of similar materials systems.
Complex transition metal oxides exhibit a variety of phenomena, such as, e.g., multiferroicity 1 , non-Fermi liquid behavior 2 , high-temperature superconductivity 3 , or metal-insulator transitions 4 , which are not only very intriguing, but are also of high interest for future technological applications 5-7 . However, the quantitative predictive description of these materials and their properties represents a major challenge for modern computational materials science, due to the importance of electronic correlation effects as well as due to the intimate coupling between electronic, magnetic, and structural degrees of freedom. 4, 8 .
An example, which has received considerable attention recently, is the family of rare-earth nickelates, RNiO 3 , with R=La-Lu and Y, which exhibit a rich phase diagram that is highly tunable by strain, doping, and electromagnetic fields (see, e.g., . All members of the nickelate series (except LaNiO 3 ) exhibit a metalinsulator transition (MIT) as a function of temperature, which is accompanied by a structural distortion that lowers the space group symmetry from orthorhombic P bnm, where all Ni sites are symmetry-equivalent, to monoclinic P 2 1 /n, with two inequivalent types of Ni sites [15][16][17][18] .
The structural distortion results in a three-dimensional checkerboard-like arrangement of long bond (LB) and short bond (SB) oxygen octahedra surrounding the two inequivalent Ni sites (see Fig. 2a), with a volume difference between LB and SB NiO 6 octahedra of ∼ 12 % in LuNiO 3 17,18 . This distortion corresponds to a zoneboundary breathing mode of the octahedral network with symmetry label R + 1 19 . In addition, all systems exhibit antiferromagnetic (AFM) order at low temperatures. 9,20,21 For R from Lu to Sm, the AFM transition occurs at lower temperatures than the MIT, whereas for R=Nd and Pr, the magnetic transition coincides with the MIT. AFM order in LaNiO 3 was only reported recently 21 and is still under discussion 22 .
Due to challenges in synthesis, experimental data on the bulk materials is relatively sparse, and in particular information on the temperature dependence of structural parameters is still lacking. Quantitative predictive calculations are therefore highly valuable to gain a better understanding of this system. As shown recently, 23-27 density functional theory plus dynamical mean field theory (DFT+DMFT) 28 allows to describe the complex physics found in nickelates, resulting in the classification of the MIT as site-selective Mott transition 23,26 . However, so far, the capability of DFT+DMFT to address structural properties is not well established, even though promising results have been achieved in previous work. 24,25,27 Thereby, these studies either employed simplified interpolation procedures between different structures, fixed lattice parameters to experimental data, and/or used ad hoc values for the interaction parameters.
Here, we show that accurate structural parameters across the whole series can be obtained fully ab initio, using charge self-consistent (CSC) DFT+DMFT 29 with interaction parameters obtained within the constrained random phase approximation 30 . To achieve this, we use a symmetry-based mode decomposition, which allows to obtain all structural degrees of freedom for which correlation effects are not crucial from standard DFT calculations, while the important breathing mode distortion is then obtained from DFT+DMFT total energy calculations. Our analysis of the structural energetics leads to a transparent and physically sound picture of the MIT in the nickelates, which is determined by the interplay between electronic correlations and structural distortions, where the octahedral rotations control both the strength of the electronic instability as well as the magnitude of the screened interaction parameters across the series.

Relaxation of P bnm structures and definition of correlated subspace
All systems are fully relaxed within the hightemperature P bnm space group using non-spinpolarized DFT calculations 31 . We then use symmetry-based mode decomposition 32 to analyze the relaxed P bnm structures and quantify the amplitudes of the various distortion modes. The mode decomposition allows for a clear conceptional distinction between different structural degrees of freedom. As shown in our previous work 33 , all structural parameters of these P bnm structures, in particular the amplitudes of the octahedral tilt modes (with symmetry labels R + 4 and M + 3 , see also Ref. 19) are obtained in excellent agreement with available experimental data. This is very important, since the octahedral tilt modes in turn have a strong influence on the energetics of the R + 1 breathing mode distortion, as reported recently 34 . For further details on the DFT results and our distortion mode analysis we refer to our previous work 33 .
Next, we construct a suitable low energy electronic subspace, for which the electron-electron interaction is then treated within DMFT. Here, we follow the ideas of Subedi et al. 26 , and construct Wannier functions only for a minimal set of bands with predominant Ni-e g character around the Fermi level, which in all cases (except LaNiO 3 ) is well separated from other bands at lower and higher energies. The Wannier functions are then used as localized basis orbitals to construct the effective impurity problems for our CSC DFT+DMFT calculations, where the LB and SB Ni sites are treated as two separate impurity problems (even for zero R + 1 amplitude) coupled through the DFT+DMFT self-consistency loop, and the system is constrained to remain paramagnetic. More details on the construction of the Wannier functions and the technical aspects of our CSC DFT+DMFT calculations can be found in the "Methods" section.
We first establish the main overall effect of the interaction parameters U and J on the electronic properties of LuNiO 3 within the high symmetry P bnm structure, i.e. R + 1 = 0.0Å. The resulting phase diagram is presented in Fig. 1. Analogously to Ref. 26, we can identify three distinct phases: First, a standard Mott-insulating phase for large U values, with vanishing spectral weight around the Fermi level, A(ω = 0) = 0, and equal occupation of all Ni sites. Second, another insulating phase for moderate U values of around 2 eV to 3.5 eV and relatively large J ( 0.4 eV), which is characterized by a strong difference in total occupation of the Wannier functions centered on LB and SB Ni sites, respectively (n LB ≥ 1.5 and n SB ≤ 0.5). We denote this phase as charge disproportionated insulating (CDI) phase 37 . Third, a metallic phase for small U values in between the two insulating regions, with equal occupation on all Ni sites, n SB ≈ n LB ≈ 1.0, and non-vanishing spectral weight at the Fermi level, A(ω = 0) > 0.
The CDI phase has been identified as the insulating low-temperature phase of nickelates in Ref. 26 38 , where it has also been shown that the strong charge disproportionation is linked to the MIT. We note that the Wannier basis within our low energy subspace, while being centered on the Ni sites with strong e g character, also exhibits strong tails on the O ligands, and thus the corresponding charge is distributed over the central Ni atom and the surrounding O atoms. The strong charge disproportionation found within our chosen basis set is thus fully consistent with the observation that the integrated charge around the two different Ni atoms differs only marginally 23 . Alternatively, within a negative charge transfer picture, the MIT can also be described, using a more atomic-like basis, as ( One should also note that the CDI phase appears even though all Ni sites are structurally equivalent (R + 1 = 0 in Fig. 1), which indicates an electronic instability towards spontaneous charge disproportionation. This has already been found in Ref. 26. However, compared to the non-CSC calculations of Ref. 26, within our CSC DFT+DMFT calculations the CDI phase appears at significantly lower J and a more confined U range. This is consistent with non-CSC DFT+DMFT calculations pre- sented in Ref. 36, where the (effective) inter-site Coulomb interaction has been included on the Hartree level, and was shown to significantly reduce the value of J necessary to stabilize the CDI phase. We note that within our CSC DFT+DMFT calculations, the change in the inter-site Hartree interaction due to the charge disproportionation is accounted for within the DFT part, as the charge density is updated in each CSC loop.
Next, we investigate how the electronic instability corresponding to the CDI phase couples to the structural R + 1 breathing mode distortion. For this, we vary only the R + 1 amplitude, while keeping all other structural parameters fixed to the fully relaxed (within nonmagnetic DFT) P bnm structures, and calculate (U, J) phase diagrams for different values of the R + 1 amplitude. We do this for both LuNiO 3 and PrNiO 3 , i.e., for the two compounds with the smallest and largest rare earth cations within the series that exhibit the MIT. The (U, J) range of the CDI phase for a given R + 1 amplitude is then extracted by interpolating the convex hull of the phase boundary using cubic splines (similar to the red line in Fig. 1). The results are summarized in Fig. 2b.
In both cases, R=Lu and R=Pr, the R + 1 amplitude couples strongly to the CDI state, and increases the corresponding area within the (U, J) phase diagam. In particular, the minimal J required to stabilize the CDI phase is significantly lowered. Furthermore, also for R=Pr, there is a spontaneous instability towards the formation of a CDI state, but the corresponding (U, J) range is noticeably smaller than for R=Lu. However, the initial increase of this area with increasing R + 1 amplitude is even stronger than for R=Lu. In addition, the minimal U required to stabilize the CDI phase for a given R + 1 amplitude is slighty higher for R=Pr than for R=Lu. We note that, since the R ions do not contribute notice-ably to any electronic states close to the Fermi level, the differences between the two materials are mainly due to the different underlying P bnm structures, i.e., the weaker octahedral tilts in PrNiO 3 compared to LuNiO 3 . This increases the electronic bandwidth, which opposes the tendency towards charge disproportionation.

Calculation of interaction parameters
So far we have varied U and J in order to obtain the general structure of the phase diagram. Next, we calculate U and J corresponding to our correlated subspace for all systems across the series to see where in these phase diagrams the real materials are located. We use the constrained random phase approximation (cRPA) 30 to extract the interaction parameters (U, J) within the Hubbard-Kanamori parameterization from the static value of the frequency-dependent screened interaction matrix, W r (ω = 0), as described, e.g., in Ref. 42.
The results of these cRPA calculations are shown in Fig. 3 as a function of the R cation and the corresponding R + 4 amplitude, i.e., the main octahedral tilt mode in the P bnm structure. The effective interaction parameters U corresponding to our e g correlated subspace are strongly screened compared to the bare interaction parameters V . For LuNiO 3 , we obtain V = 13.91 eV and U = 1.85 eV, while J = 0.42 eV with a corresponding bare value of 0.65 eV. This is in good agreement with Ref. 36, which obtained U = 1.83 eV and J = 0.37 eV using the experimental P 2 1 /n structure. Furthermore, both U and J decrease monotonically across the series (for decreasing R + 4 amplitude), leading to an additional reduction of U by 25% in LaNiO 3 compared to LuNiO 3 .

Lu Er Dy Gd Sm
Pr La This decrease is also observed in the ratio U/V , indicating that it is due to an even stronger screening for R=La compared to R=Lu.
Our calculated (U, J) parameters for R=Lu and R=Pr are also marked in the corresponding phase diagrams in Fig. 2. It is apparent, that for R=Lu the calculated cRPA values are well within the stability region of the CDI phase, even for a relatively small R + 1 amplitude of 0.02Å. In contrast, for R=Pr, the values are outside the CDI phase even for R + 1 amplitudes larger than the one experimentally observed. Thus, at their respective experimental breathing mode amplitudes, our calculations predict a paramagnetic CDI state for LuNiO 3 but not for PrNiO 3 .

Lattice energetics
So far, we have been addressing the stability of the CDI phase for a given (fixed) R + 1 amplitude. Now, we will address the stability of the R + 1 mode itself and calculate its amplitude across the series using total energy calculations within CSC DFT+DMFT. Again, the symmetrybased mode decomposition allows us to systematically vary only the R + 1 mode, while keeping all other structural parameters fixed to the values obtained from the nonmagnetic DFT calculations. Thus, in contrast to interpolation procedures such as the ones used in Refs. 25 or 27, our approach automatically excludes any additional energy contributions related to other structural distor- tions, in particular due to simultaneous changes in the octahedral tilt modes. Fig. 4 shows the total energy and the spectral weight around the Fermi level,Ā(ω = 0), as a function of the R + 1 amplitude for LuNiO 3 , calculated using different values for (U, J). First, we focus on the results obtained using our cRPA calculated values (J = 0.42 eV, U = 1.85 eV, orange crosses). It can be seen, that the energy indeed exhibits a minimum for an R + 1 amplitude very close to the experimental value. Furthermore, as seen fromĀ(ω = 0), the system undergoes a MIT transition for increasing R + 1 amplitiude and is clearly insulating in the region around the energy minimum. Thus, our CSC DFT+DMFT calculations together with the calculated cRPA interaction parameters correctly predict the CDI ground state for LuNiO 3 , and furthermore result in a breathing mode amplitude that is in excellent agreement with experimental data.
To see how subtle changes in (U, J) influence the energetics of the system, we also perform calculations using the cRPA values obtained in Ref. 36 (J = 0.37 eV, U = 1.83 eV, red diagonal crosses). In this case, we obtain a more shallow energy minimum at a slightly reduced amplitude of R + 1 = 0.06Å. This reduction is mainly caused by the slightly smaller J. Moving the values of (U, J) even closer to the boundary of the stability region of the CDI phase for the experimental R + 1 amplitude, cf. amplitude. Nevertheless, a kink in the total energy is clearly visible at the R + 1 amplitude for which the system becomes insulating, indicating the strong coupling between the structural distortion and the MIT. A similar kink can also be recognized (for rather small R + 1 amplitide) in the total energy obtained for J = 0.37 eV and U =1.83 eV, resulting in an additional local energy minimum at R + 1 = 0, indicating a first order structural transition. In addition, we also perform calculations where (U, J) are increased by 10 % compared to our cRPA values (J = 0.47 eV, U = 2.04 eV, red circles), which leads to a deeper energy minium and an R + 1 amplitude in near perfect agreement with experiment R + 1 = 0.075Å 18 . Finally, we examine how the energetics of the R + 1 mode varies across the nickelate series, by comparing the two end members LuNiO 3 and PrNiO 3 , as well as SmNiO 3 , which is the compound with the largest R cation in the series that still exhibits a paramagnetic CDI state. In each case we use (U, J) values that are increased by 10 % relative to the corresponding cRPA values. The use of such slightly increased interaction parameters is motivated by the observation that the U values obtained from the static limit of the (frequency-dependent) screened cRPA interaction are often too small to reproduce experimental data for various materials 28,[43][44][45] . The results are depicted in Fig. 5.
For LuNiO 3 (blue circles), we obtain an energy minimum exactly at the experimentally observed amplitude of R + 1 = 0.075Å 18 . For SmNiO 3 (purple triangles), we obtain a much more shallow minimum at R + 1 = 0.06Å, which corresponds to a reduction by ≈ 20 % compared to LuNiO 3 . Unfortunately, structural refinements for SmNiO 3 are only available within the P bnm space group, and thus no information on the R + 1 amplitude exists 46 .
However, the reduction of the R + 1 amplitude from R=Lu to R=Sm is much more pronounced compared to previous DFT+U calculations with AFM order 33 , where the reduction is only about 8 %.
For PrNiO 3 (green squares), no stable R + 1 amplitude is obtained within our paramagnetic DFT+DMFT calculations, but a kink marking the MIT is still visible at R + 1 = 0.06Å. This is also in agreement with the experimental observation that no paramagnetic CDI phase occurs in PrNiO 3 9 . Furthermore, it was recently demonstrated using DFT+DMFT calculations that for NdNiO 3 the CDI state becomes only favorable in the antiferromagnetically ordered state 27 . Our results indicate that this also holds for PrNiO 3 , while in SmNiO 3 a stable R + 1 amplitude can be found even in the paramagnetic case. Thus, the phase boundaries across the series are correctly described within the DFT+DMFT approach. The conclusion that AFM order is necessary to induce a breathing mode distortion for R=Nd and R=Pr has also been drawn from recent DFT+U calculations 34 . We further note that, considering the (U, J) phase diagrams for PrNiO 3 in Fig. 2, a U of up to 2.5 or even 3 eV would be required to put PrNiO 3 well within the CDI phase region at its experimental R + 1 amplitude, which appears necessary to obtain a stable R + 1 amplitude. However, such a large U seems highly unrealistic considering the calculated cRPA values.

DISCUSSION
In summary, we have shown that accurate structural parameters can be obtained across the nickelate series using CSC DFT+DMFT with interaction parameters obtained using the cRPA, i.e., without ad hoc assumptions regarding the strength of the Hubbard interaction or fixing some structural parameters to experimental data. Thereby, the symmetry-based mode decomposition allows to selectively relax specific degrees of freedom for which electronic correlation effects are crucial, while keeping the required computational effort manageable.
Moreover, our calculations show that the MIT, which is related to an electronic instability towards spontaneous charge disproportionation, leads to a significant restructuring of the energy landscape, indicated by a kink in the calculated total energy. This creates a minimum at a finite R + 1 amplitude (for approporiate (U, J)), and suggests a first order character of the coupled structural and electronic transition, consistent with experimental observations that both MIT and structural distortion appear at the same temperature.
The strength of the electronic instability towards spontaneous charge disproportionation and thus the stability range of the CDI phase, is strongly affected by the amplitude of the octahedral rotations, varying across the series. In addition, the octahedral rotations also influence the screening of the effective interaction parameters, disfavoring the CDI state for larger R cations. As a re-sult, magnetic order appears to be crucial to stabilize the breathing mode distortion for both R=Nd and Pr.
We note that, to arrive at a fully coherent picture, it is crucial to treat both structural and electronic degrees of freedom on equal footing. For example, even though a CDI state can be obtained for PrNiO 3 for fixed R + 1 amplitude > 0.06Å, our calculations show that this is indeed energetically unstable.
Furthermore, our calculations show that the use of a minimal correlated subspace not only incorporates the essential physics, but even provides quantitatively accurate structural parameters across the nickelate series. We note that the use of such a reduced correlated subspace can be advantageous, since it not only allows to reduce the computational effort (due to less degrees of freedom), but also because the double counting problem is typically less severe if the O-p dominated bands are not included in the energy window of the correlated subspace. 47,48 In the present case, the resulting more extended Wannier functions, which also incorporate the hybridization with the surrounding ligands, also provide a rather intuitive picture of the underlying charge disproportionation.
Finally, our study represents the successful application of a combination of several state-of-the-art methods that allows to tackle other open issues related to the entanglement of structural and electronic properties in correlated materials, such as Jahn-Teller and Peierls instabilities, charge density wave, or polarons.

ACKNOWLEDGMENTS
We are indebted to Oleg Peil and Antoine Georges for helpful discussions. This work was supported by ETH Zurich and the Swiss National Science Foundation through grant No. 200021-143265 and through NCCR-MARVEL. Calculations have been performed on the clusters "Mönch" and "Piz Daint", both hosted by the Swiss National Supercomputing Centre, and the "Euler" cluster of ETH Zurich.

DFT calculations
All DFT calculations are performed using the projector augmented wave (PAW) method 49 implemented in the "Vienna Ab initio Simulation Package"(VASP) 50-52 and the exchange correlation functional according to Perdew, Burke, and Ernzerhof 53 . For Ni, the 3p semi-core states are included as valence electrons in the PAW potential. For the rare-earth atoms, we use PAW potentials corresponding to a 3+ valence state with f -electrons frozen into the core and, depending on the rare-earth cation, the corresponding 5p and 5s states are also included as valence electrons. A k-point mesh with 10 × 10 × 8 grid points along the three reciprocal lattice directions is used and a plane wave energy cut-off of 550 eV is chosen for the 20 atom P bnm unit cell. The structures are fully relaxed, both internal parameters and lattice parameters, until the forces acting on all atoms are smaller than 10 −4 eV/Å.
Distortion mode analysis For the symmetrybased mode decomposition 32 we use the software ISODISTORT 54 . Thereby, the atomic positions within a distorted low-symmetry crystal structure, r dist i , are written in terms of the positions in a corresponding non-distorted high-symmetry reference structure, r 0 i , plus a certain number of independent distortion modes, described by orthonormal displacement vectors, d im , and corresponding amplitudes, A m : (1) The distortion modes of main interest here are the outof-phase and in-phase tilts of the oxygen octahedra, R + 4 and M + 3 , for characterization of the high-temperature P bnm structure, and the R + 1 breathing mode distortion within the low-temperature P 2 1 /n structure. A more detailed description for nickelates can be found, e.g., in Refs. 19 and 33.
DMFT calculations The Wannier functions for our CSC DFT+DMFT calculations are constructed via projections on local Ni e g orbitals as described in Ref. 55 and 56, using the TRIQS/DFTTools software package. 57,58 The effective impurity problems within the DMFT loop are solved with the TRIQS/cthyb continuoustime hybridization-expansion solver 59 , including all offdiagonal spin-flip and pair-hopping terms of the interacting Hubbard-Kanamori Hamiltonian. 42 The LB and SB Ni sites are treated as two separate impurity problems (even for zero R + 1 amplitude), where the number of electrons per two Ni sites is fixed to 2, but the occupation of each individual Ni site can vary during the calculation (while the solution is constrained to remain paramagnetic).
The fully-localized limit 60 is used to correct for the double-counting (DC) in the parametrization given in Ref. 61: where n α is the occupation of Ni site α, obtained in the DMFT loop, and the averaged Coulomb interaction is defined asŪ = 3U − 5J/3. Note, that in our Wannier basis the occupations change quite drastically from the original DFT occupations and the choice of the DC flavor can therefore influence the outcome. However, with respect to the lattice energetics we found no difference in the physics of the system when changing the DC scheme or using fixed DFT occupation numbers for the calculation of the DC correction. If the DFT occupations are used instead of the DMFT occupations, larger interaction parameters are required to obtain the same predicted R + 1 amplitude. However, we note that the DFT occupations have no clear physical meaning within CSC DFT+DMFT.
The spectral weight around the Fremi level,Ā(ω = 0), is obtained from the imaginary time Green's function: 62 For T = 0 (β → ∞),Ā is identical to the spectral function at ω = 0. For finite temperatures, it represents a weighted average around ω = 0 with a width of ∼ k B T 62 . The total energy is calculated as described in Ref. 28: The first term is the DFT total energy, the second term subtracts the band energy of the Ni e g dominated bands (index λ), the third term evaluates the kinetic energy within the correlated subspace via the lattice Green's function, the fourth term adds the interaction energy, where we use the Galitskii-Migdal formula 63,64 , and the last term subtracts the DC energy. To ensure good accuracy of the total energy, we represent both G imp and Σ imp in the Legendre basis 65 and obtain thus smooth high-frequency tails and consistent Hartree shifts. Moreover, we sample the total energy over a minimum of additional 60 converged DMFT iterations after the CSC DFT+DMFT loop is converged. Convergence is reached when the standard error of the Ni site occupation of the last 10 DFT+DMFT loops is smaller than 1.5 × 10 −3 . That way we achieve an accuracy in the total energy of < 5 meV. All DMFT calculation are performed for β = 40 eV −1 , which corresponds to a temperature of 290 K.
cRPA calculations We use the cRPA method as implemented in the VASP code 66 to extract interaction parameters for our correlated subspace. These calculations are done for the relaxed P bnm structures 33 . We follow the ideas given in the paper of Subedi et al. 26 and construct maximally localized Wannier functions (MLWFs) for the Ni-e g dominated bands around the Fermi level using the wannier90 package 67 . Since the corresponding bands are isolated from other bands at higher and lower energies, no disentanglement procedure is needed, except for LaNiO 3 , for which we ensured that the resulting Wannier functions are well converged and have a very similar spread as for all other compounds of the series.
The constrained polarization, P r , and the static limit of the screened interaction matrix, W r (ω = 0), are calculated using a 5 × 5 × 3 k-point mesh, a plane wave energy cut-off of E cut = 600 eV, and 576 bands. Furthermore, we divide the polarization into a contribution involving only transitions within the effective "e g " correlated subspace and the rest, P = P eg + P r . We then extract effective values for the Hubbard-Kanamori interaction parameters (U, J) from W r (ω = 0) as described in Ref. 42. Our procedure is analogous to the calculation of effective interaction parameters for LuNiO 3 in Ref. 36.
It should be noted that the MLWFs used for the cRPA calculations are not completely identical to the projected Wannier functions used as basis for the correlated subspace within our DMFT calculations. However, test calculations for the case of LuNiO 3 showed only minor differences between the hopping parameters corresponding to the MLWFs and the ones corresponding to the Wannier functions generated by the projection scheme implemeted in VASP. Furthermore, we did not find a noticeable difference between the screened (U, J) values calculated for the MLWFs and the ones calculated for the initial guesses for these Wannier functions, i.e., before the spread minimization, which are also defined from orthogonalized projections on atomic-like orbitals. We thus conclude that the two sets of Wannier functions are indeed very similar, and that the cRPA values of (U, J) obtained for the MLWFs are also representative for the Wannier basis used in our DMFT calculations.
Additionally, we point out that, in contrast to what was found in Ref. 36, we obtained only negligible differences in the interaction parameters obtained for the relaxed P bnm structure and the ones obtained for the experimental low-temperature P 2 1 /n structure for LuNiO 3 18 . Moreover, the difference in the intra-orbital U matrix elements between the d z 2 and the d x 2 −y 2 orbitals is negligible small, ∼ 0.01 eV, in our calculations and all given interaction parameter values are averaged therefore over all e g orbitals.

AUTHOR CONTRIBUTIONS
A.H. performed and analyzed all DFT and DMFT calculations. The cRPA calculations were done by A.H. with the help of P.L. and supervised by C.F. The whole project was initiated by C.E. The initial manuscript was written by A.H. and C.E. All authors discussed the results at different stages of the work and contributed to the final manuscript.