Performance of the r²SCAN functional in transition metal oxides

We used the Vienna ab-initio simulation package (VASP 6.2.1)[48, 49, 50] for all the spin-polarized DFT calculations, where the frozen-core PBE-based projector augmented wave (PAW)[51] potentials employed were identical to previous work.[37, 38] The plane waves for each system were expanded up to a kinetic energy of 520 eV, with each structure converged until the total energy differences and atomic forces became <0.01 meV and <$|0.01|$ eV/Å, respectively. We adopted a $\Gamma$-centered Monkhorst-Pack[52] grid with a density of 48 $k$-points per Å for all systems. The conjugate gradient algorithm was used to relax the structures (i.e., cell shapes, volumes, and ionic positions), without preserving any underlying symmetry. An ‘accurate’ level of precision was maintained while projecting the wavefunctions in the reciprocal space. The Fermi surface of each system was integrated with a Gaussian smearing of partial occupancies, with a width of 0.05 eV. In terms of DFT+U calculations, we used the Dudarev framework[53] for adding a effective U correction on the $d$ orbitals of TM atoms. All U values used in SCAN+U calculations were taken from previous work (see Table S1 of the Supporting Information –SI).[37, 38] Since we used different computing systems to perform our structure relaxations for different systems, we normalized the computational time with the number of cores used in each calculation to compare the computational efficiency of the different XC functionals considered.

For calculating band gaps, GGA functionals typically use the Kohn Sham potential as a multiplicative term, which typically underestimates the band gap of solids even at the SCAN level.[54, 55] Here, we use the generalized Kohn Sham technique to determine the band gaps by calculating the density of states (DOS) for all systems considered. For each DOS calculation, we used the optimized structure and the initial charge density from a previous structure relaxation. Subsequently, we introduced a set of zero-weighted $k$-points, corresponding to a density of 96 $k$-points per Å, where the $k$-points that were used for the structure relaxation retained their original weights (as determined by VASP). Finally, we performed a single-SCF calculation where the DOS was sampled between energies of -20 to 20 eV in steps of 0.005 eV.

We considered the binary oxides of each TM, i.e., Ti, V, Cr, Mn, Fe, Co, Ni, and Cu with different oxidation states, similar to previous studies.[37, 38] The main criteria in selection of these metal oxides are the availability of reliable thermodynamic data (i.e., formation energies[56, 57, 58]) and the experimentally-determined ground-state structures that are compiled in the inorganic crystal structure database (ICSD)[59] Note that the structures from the ICSD were the initial structures in all our DFT structure relaxations, including the systems used as transferability checks. In the case of Ni oxides, we chose NiO and LiNiO₂ (similar to previous work,[38]), as reliable thermodynamic data is not available for higher-oxidation-state binary Ni oxides (e.g., Ni₂O₃ and NiO₂). The TM in all oxides, except select Co and Ni compounds, was initialized in its high-spin configuration (e.g., high-spin configuration of Fe³⁺ consists of five unpaired $d$ electrons). A detailed description of all structures utilised in this work is provided in the SI, under the ’Crystal Structures’ section, with the magnetic configurations depicted in Figure S1.

The magnetic configuration of each TMO considered (see Figure S1) was initialized to its appropriate (in several cases, experimentally-known) ground state configuration during the structural relaxation. For example, we considered the ferromagnetic (FM) ground state configuration for CrO₂ and VO₂, given that CrO₂ is metallic[60] and VO₂ undergoes a metal-to-insulator transition (MIT) below 341 K.[61] The rocksalt (RS) TMOs, namely, VO, MnO, FeO, CoO, and NiO were initialized with their experimentally-known type-II antiferromagnetic (AFM) configuration.[62, 63, 64, 65, 66, 67] Each Ni’s spin in NiO was initialized with two unpaired $d$ electrons (i.e., its high-spin configuration). In CuO, we arranged the magnetic moments of Cu²⁺ antiferromagnetically along the Cu-O-Cu chains in the [10$\bar{1}$] direction.[68, 69]

We initialized $\alpha$-Mn₂O₃ (bixbyite structure) in a FM configuration as this configuration was found to be the most stable in previous work.[37] AFM configurations were utilized for rutile-MnO₂,[70], and the other TM₂O₃ oxides, namely, V₂O₃, Fe₂O₃, Ti₂O₃, and Cr₂O₃. Note that V₂O₃ becomes AFM below its MIT temperature,[71, 72, 73] while Fe₂O₃ displays an AFM configuration with the magnetic moment of Fe alternating every two consecutive layers along the $c$-axis.[74] Cr₂O₃ and Ti₂O₃ exhibit $\uparrow\downarrow\uparrow\downarrow$ and $\uparrow\downarrow\downarrow\uparrow$ magnetic configurations, respectively, on the TM centers along the $a$-axis.[75, 76]

In case of spinels, we used different ferrimagnetic (FIM) configurations, as per experimental observations. For example, spinel-Fe₃O₄ contains both Fe³⁺ and Fe²⁺, with up-spin Fe³⁺ occupying tetrahedral sites and down-spin Fe³⁺ occupying half the octahedral sites. The remaining octahedral sites in Fe₃O₄ are occupied by up-spin Fe²⁺.[77, 78] In Co₃O₄, no-spin Co³⁺ occupies octahedral sites, while high-spin Co²⁺ (three unpaired $d$ electrons) occupies tetrahedral sites in an AFM configuration.[79, 80, 81] For Mn₃O₄, we adopted the ”FIM6” configuration, as this was found to be the ground state in previous work.[37] TiO₂, CrO₃, and V₂O₅ are diamagnetic, since they contain TMs with empty 3$d$ orbitals. Similarly, Cu₂O is diamagnetic owing to the completely-filled 3d orbitals of Cu.

We determined the required U value, with r²SCAN, for each binary TMO oxidation reaction (e.g., Ti${}^{3+}\rightarrow$ Ti⁴⁺ in 2Ti₂O₃ + O${}_{2}\rightarrow$ 4TiO₂) by comparing the experimental enthalpy (per mole of O₂) with the calculated (r²SCAN+U) value that minimizes the error against the experimental value. Note that U = 0 eV in our data simply reflects a r²SCAN calculation. In order to obtain the experimental oxidation enthalpy, standard enthalpy of formation for all the considered TMOs were taken from the Wagman and/or Kubaschewski tables,[56, 57] thus ignoring the $p-V$ and entropic contributions, similar to previous works.[37, 38, 82] The overall optimal U value for each TM was obtained by taking the average of the required U for each of the available oxidation reactions. In the case of Ni oxides, oxidation of NiO to LiNiO₂ by 2Li₂O + 4NiO + O₂ $\rightarrow$ 4LiNiO₂ was considered as a proxy for the Ni${}^{2+}\rightarrow$ Ni³⁺ oxidation reaction.[38]

Figure 1 displays the variation of the enthalpy of different oxidation reactions among binary TMOs, as a function of applied U in the r²SCAN+U framework, for all TMs considered except Cr and Cu. Solid lines in each panel of Figure 1 represent DFT-calculated oxidation enthalpies, with each color corresponding to different oxidation reactions for the TM. For instance in V oxides (Figure 1b), the solid black line corresponds to the oxidation reaction, VO $\rightarrow$ V₂O₃, while the solid red and green lines indicate V₂O${}_{3}\rightarrow$ VO₂ and VO${}_{2}\rightarrow$ V₂O₅, respectively. Similarly, the experimental enthalpy of each oxidation reaction is represented by dashed horizontal line of the same color. For example, the black dashed line in Figure 1b indicates the experimental oxidation enthalpy (-7.36 eV) of VO $\rightarrow$ V₂O₃. Also, dotted vertical line of a given color highlights the required U value to minimize the error between DFT-calculated and experimental value for the oxidation reaction enthalpy indicated by the same color. The dotted blue line in each panel signifies the overall optimal U for the TM that is averaged across all available oxidation reactions.

We report an optimal U value of 2.3, 1.0, 1.8, 3.1, 1.8, and 2.1 eV, respectively, for Ti, V, Mn, Fe, Co, and Ni oxides, within the r²SCAN+U framework (Figure 1). Notably, the optimal U obtained with r²SCAN is less than that reported previously for SCAN functional (Table S1) for all 3d TMs considered (except V and Fe), which can be attributed to better accuracy of r²SCAN compared to SCAN, as observed in non-TMOs.[36] For V oxides, the required U value for VO₂ $\rightarrow$ V₂O₅, V₂O₃ $\rightarrow$ VO₂, VO $\rightarrow$ V₂O₃ is 0.0, 0.7, and 2.2 eV, respectively. Thus, the optimal U value for V is 1.0 eV (average of the three required U values), which is identical to the U correction required with SCAN.[38] The decreasing required U with increasing oxidation state of V in V oxides is expected due to the decrease in the strength of exchange interactions among the d electrons as oxidation state increases. In the case of Fe, FeO $\rightarrow$ Fe₂O₃ and FeO $\rightarrow$ Fe₃O₄ reactions require a U of 2.9 and 3.3 eV, respectively, resulting in an optimal U of 3.1 eV, which is also identical to the optimal U with SCAN.[37] Moreover, we obtain the highest optimal U of 3.1 eV for Fe, among all TMs considered in this work, which is consistent with the fact that Fe³⁺ has the highest number of unpaired $d$ electrons resulting in the strongest exchange interactions.

For Ti and Ni, we observe a marginal improvement in the U-value for r²SCAN when compared to SCAN. Specifically, we obtain an optimal U of 2.3 eV and 2.1 eV for Ti and Ni, respectively, versus 2.5 eV for both elements with SCAN. We find an optimal U value of 1.8 eV for both Mn (2.7 eV with SCAN) and Co (3.0 eV with SCAN). In Mn-oxides, the required U for the oxidation of Mn₂O${}_{3}\rightarrow$ MnO₂, and MnO $\rightarrow$ Mn₂O₃ are 1.5 and 2.1 eV, respectively. The optimal U for Mn is transferable to other Mn oxides as well, indicated by the robust agreement between r²SCAN+U-calculated and experimental oxidation enthalpy for MnO $\rightarrow$ Mn₃O₄ (green lines in Figure 1c).

For Cr and Cu oxides, we obtain reasonable agreement with experimental data without a U correction (Figure S2), similar to our observation with SCAN.[38] In fact, for Cu, introducing U-correction worsens the error in the calculated oxidation enthalpy for Cu₂O $\rightarrow$ CuO versus experiment, similar to our observation with SCAN(+U) as well, which can be attributed to PAW potentials derived at the PBE-level.[38] However, the magnitude of error (versus experiment) is smaller with r²SCAN ($\approx$13.1%) than with SCAN ($\approx$25.7%). In case of Cr, the oxidation reaction of CrO${}_{2}\rightarrow$ CrO₃ requires U $\sim$ 0.9 eV, but introducing a U correction worsens any agreement with experiment for Cr₂O${}_{3}\rightarrow$ CrO₂ (where required U = 0 eV). Thus, the optimal U for Cr oxides is 0.45 eV (<0.5 eV), which only provides a marginal improvement in describing oxidation enthalpies. Hence, we recommend using only r²SCAN for calculating any Cr oxide framework.

All r²SCAN(+U) and SCAN(+U) calculated lattice parameters, on-site magnetic moments, and band gaps for each TMO are tabulated in Table S2. Additionally, the calculated lattice volumes by the four XC functionals are plotted against experimental data in Figure 2a for all oxides. Generally, both SCAN (green squares in Figure 2a) and r²SCAN (blue symbols) offer $<$ 2.8% deviation from the experimental lattice parameters for all the TMOs considered, except VO, FeO, CuO, and LiNiO₂, indicating robust agreement with experiments for both functionals. In VO, SCAN and r²SCAN overestimate (by $\sim$8%) the experimental lattice constants, while the deviation in FeO and CuO is $\sim$3-4% and $\sim$8-10%, respectively. In LiNiO₂, SCAN’s $\beta$ angle evaluation is $\sim$4.1% different from experiment.

Notably, SCAN and r²SCAN do show qualitative differences in their calculated lattice parameters (when compared against experiments) across TMOs. For instance, both functionals overestimate the experimental lattice constants in TiO₂, Ti₂O₃, and VO, while they underestimate in CrO₂, CrO₃, MnO₂, and Fe₃O₄. There are also examples (MnO and Mn₂O₃) where SCAN underestimates the experimental lattice constants while r²SCAN overestimates. Overall, there are cases where SCAN’s errors in lattice parameter estimations are lower versus experiments (e.g., Cr₂O₃, CoO), r²SCAN’s errors are lower (e.g., CrO₂, CrO₃, MnO₂, Fe₃O₄), and both functionals exhibit identical errors (e.g., TiO₂, Co₃O₄, NiO, Cu₂O), signifying that both functionals offer similar performance in terms of geometrical properties.

Comparing r²SCAN and SCAN, we find that r²SCAN’s lattice constants are generally larger than SCAN across TMOs (e.g., Ti₂O₃, Cr₂O₃, CrO₃, VO₂, etc.). As a range, r²SCAN estimates lattice constants that are a maximum of $\sim$1.5% larger than SCAN (in CrO₃) and a minimum of $\sim$0.1% larger than SCAN (in Mn₂O₃). Having said that, there are instances where r²SCAN’s lattice constant evaluations are lower than SCAN (VO, CoO, CuO, and LiNiO₂) and cases where both functionals are identical (TiO₂, Co₃O₄, NiO, and Cu₂O). In specific TMOs, SCAN and r²SCAN calculate an identical (individual) lattice constant, while the other lattice constants with r²SCAN are larger than SCAN. For example, $a$ and $c$ lattice constants with r²SCAN are higher than SCAN in V₂O₅ while both functionals estimate $b$ = 3.55 Å.

On introducing the optimal U correction, an increase in the value of calculated lattice constants is obtained for both SCAN and r²SCAN functionals for all TMOs. The lattice constants computed by r²SCAN+U (yellow symbols in Figure 2a) is up to 1.3% higher than r²SCAN, except FeO ($\sim$4.2% higher). Similar to the comparison of r²SCAN vs. SCAN, there are systems where r²SCAN+U predicts larger, smaller, and identical lattice constants compared to SCAN+U (red triangles). For example, r²SCAN+U calculates larger lattice constants than SCAN+U in VO₂, V₂O₅, MnO, Mn₂O₃ and Fe₃O₄ (maximum of $\sim$0.5% higher in V₂O₅), while for Ti₂O₃, CoO and NiO, r²SCAN+U’s estimations are smaller than SCAN+U (maximum deviation of $\sim$2.1% in Ti₂O₃). Both SCAN+U and r²SCAN+U functionals evaluate identical lattice parameters for TiO₂, Co₃O₄ and LiNiO₂.

Overall, lattice constants calculated by SCAN+U and r²SCAN+U deviate $<\sim$3.3% from experiments for all TMOs, except VO and VO₂ where deviations of $\sim$8.5% and $\sim$4.6% are observed, respectively. Adding U improves the agreement with experiment for both SCAN and r²SCAN in Co₃O₄, while r²SCAN+U gives the best estimate of the lattice parameters in FeO ($<1$% deviation vs. experiments) compared to SCAN, SCAN+U and r²SCAN. Notably, all functionals break the rocksalt symmetry of VO, MnO, and FeO, while the cubic symmetry of Fe₃O₄ is retained only by SCAN. In Ti₂O₃, the hexagonal symmetry is broken by SCAN but the symmetry is preserved by the other frameworks. In summary, we find that the differences in lattice parameter estimations to be minimal across the four functionals on average, with notable exceptions of a few systems.

On-site magnetic moments of the TMOs (Figure 2c and Table S2) computed by SCAN and r²SCAN generally underestimate experimental values, with the exception of MnO₂, Mn₂O₃, CrO₂, and VO₂. Note that larger magnetic moments typically indicate stronger localization of d electrons. Comparing r²SCAN and SCAN calculations, we find that r²SCAN typically estimates smaller magnetic moments than SCAN but with several exceptions, such as MnO, MnO₂, Mn₂O₃, Cr₂O₃, and VO₂. Thus, on average, SCAN’s magnetic moment predictions are in better agreement with experiments. However, in terms of magnitude, moments predicted by r²SCAN deviate by $<$ 3% from SCAN’s estimates, except CuO ($\sim$ 6.8% deviation), CrO₂ ($\sim 3.5$%), and MnO₂ ($\sim 3.5$%), highlighting that the differences in the predictions are marginal.

Adding optimal U to both SCAN and r²SCAN increases the magnitude of the calculated on-site magnetic moments for all TMOs (except VO₂, which is predicted to be metallic by all functionals), consistent with the expectation that the U correction facilitates d electron localization. r²SCAN+U-calculated data are similar to the corresponding SCAN+U values ($<2.3$% variation), except LiNiO₂ ($\sim$6.3% variation), and Ti₂O₃ ($\sim$3.8%). Similar to r²SCAN versus SCAN, r²SCAN+U estimates smaller magnetic moments than SCAN+U, with notable exceptions being VO₂, Mn₂O₃, MnO₂ and FeO. Overall, we observe the accuracy in calculated on-site magnetic moments versus experiments to follow the order SCAN+U$>$ r²SCAN+U $>$ SCAN $>$ r²SCAN for several TMOs. However, there are specific cases where specific XC frameworks offer better accuracy in calculating magnetic moments, such as SCAN in CrO₂, Mn₂O₃, MnO₂, Fe₃O₄ and CuO, r²SCAN in Mn₃O₄ and Cr₂O₃, and r²SCAN+U in V₂O₃. Given the numerically marginal deviations in calculated magnetic moments across the XC frameworks ($\sim$10% deviation), we expect an increase/decrease in accuracy to be marginal amongst the XC frameworks considered.

The differences between calculated and experimental band gaps of all TMOs considered are visualized as violin plots for SCAN (green violin), SCAN+$U$ (red), r²SCAN (blue), and r²SCAN+$U$ in Figure 2b. The top and bottom ends of the individual violins mark the highest and lowest differences in the respective calculated data. Note that the mean values (white empty circles) are similar for SCAN and r²SCAN, and in turn are lower than their U-corrected versions. In other words, addition of the $U$-correction reduces the error of calculated band gaps compared to experimental values, which is expected given that semi-local DFT typically underestimates band gaps. Also, we find that SCAN+$U$ displays the lowest mean band gap difference among the XC functionals considered, indicating that on-average SCAN+$U$ provides better computed band gaps.

We present calculated electronic DOS of select TMOs, namely CoO (panels a and b), V₂O₃ (c and d), and Mn₂O₃ (e and f), in Figure 3, to illustrate qualitative trends in computed band gaps. The DOS for the remaining TMOs, calculated by the four XC frameworks, are compiled in Figures S3-S19 of the SI. In each DOS panel, solid orange and solid green lines correspond to the 2p-states of O and the 3d-states of the TM, respectively. Dashed black lines represent Fermi levels in metallic compounds. Dotted vertical lines represent valence and conduction band edges in semiconducting/insulating compounds, with the band gaps indicated by the text annotation near the conduction band minimum (CBM). The zero of the energy scale is set to the valence band maximum (VBM) for TMOs with a band gap and to the Fermi level in metallic TMOs.

We observe that r²SCAN generally calculates a smaller band gap than SCAN for most TMOs (maximum of $\sim$66% lower in MnO₂, see Table S2), as illustrated by the case of CoO in panels a and b of Figure 3. Notable exceptions do exist to this observation, such as V₂O₅ ($\sim$1.7% larger), CrO₃ ($\sim$3.2%), MnO ($\sim$4.3%), and Fe₂O₃ ($\sim$1.7%), where r²SCAN calculated band gaps are marginally larger than SCAN. Both SCAN and r²SCAN incorrectly describe the ground state electronic configuration of narrow band gap TMOs (i.e., experimental band gaps $<1$ eV), including Ti₂O₃ (Figure S4), V₂O₃(Figure 3c and S3c), VO₂ (Figure S7) and Fe₃O₄ (Figure S15) to be metallic, with the exception of MnO₂ where both SCAN and r²SCAN estimate a narrow gap (Figures S12a and S12c). Additionally, both functionals also calculate the wrong electronic structure in the case of a non-narrow-gap semiconductor, Mn₂O₃ (Figure S3), which exhibits an experimental gap of 1.2-1.3 eV.[83, 84] However, SCAN and r²SCAN qualitatively describe the right electronic structure in the case of wide band gap TMOs such as FeO (Figure S13), Fe₂O₃ (Figure S14), and NiO (Figure S17), with a significant quantitative underestimation of the experimental gaps. In any case, the differences in electronic structure predictions between SCAN and r²SCAN in TMOs are minimal, with SCAN being marginally better in accuracy.

Introducing a U correction to SCAN and r²SCAN widens or opens the band gap, especially in narrow band gap TMOs, as illustrated by the case of V₂O₃ (panels c and d in Figure 3). The opening of band gap with U correction is expected since localization of d electrons, which form the VBM and/or CBM in 3d-TMOs, is faciliated with U addition, in turn resulting in a larger gap. However, in the case of VO₂ (Figure S7), adding U does not capture the MIT that occurs at low temperatures ($<341$ K[61]) with either SCAN or r²SCAN, causing the erroneous prediction of metallic behavior. Generally, SCAN+U calculates a larger band gap than r²SCAN+U (Table S2), as highlighted by the case of Mn₂O₃ (panels e and f in Figure 3). In fact, SCAN+U is the only framework (among those considered) to estimate a band gap in Mn₂O₃, which is consistent with experiment. Moreover, SCAN+U’s evaluations of larger band gaps results in better (poorer) quantitative agreement with experiments in wide (narrow) gap materials, such as MnO and FeO (V₂O₃ and MnO₂).

Note that SCAN+U and r²SCAN+U do underestimate the experimental band gaps, similar to SCAN and r²SCAN, in wide gap TMOs. The only exception to this observation is CoO, where SCAN+U overestimates the band gap versus experiment (Figure S3a and Table S2), as also observed in our previous work.[38] In select TMOs, including Fe₂O₃ and V₂O₅, r²SCAN+U’s band gap is larger than SCAN+U, but the magnitude of difference ($\leq 0.2$ eV) is meagre. Thus, for electronic structure predictions, we expect SCAN+U to provide the best qualitative and quantitative band gaps across TMOs, among the functionals considered here, especially for wide gap semiconductors/insulators. However, the qualitative trends provided by r²SCAN+U are quite robust as well and in small gap semiconductors ($<1$ eV gap), r²SCAN+U’s quantitative accuracy is often better than SCAN+U.

To examine the transferability of the optimal U values determined in this work (with r²SCAN), to oxide systems not used for obtaining the values, we perform checks on systems with different oxidation state and/or coordination environment for each TM. We compare calculated values against available experimental data, such as structural, electronic, magnetic, and/or electrochemical properties. Specifically, we choose Ba₂TiO₄ as a check for Ti, BiVO₄ for V, K₃MnO₄, K₂MnO₄, and Mn₂O₇ for Mn, SrFeO₃ for Fe, LiCoO₂-CoO₂ for Co, and LiNiO₂-NiO₂ for Ni. Data related to transferability checks are compiled in Figure 4, Table 1, and Table S3.

In the case of Ba₂TiO₄, we compare the calculated lattice parameters with experimental values (see Table S3 and lattice voliume differences plotted in Figure 4). Ba₂TiO₄ crystallizes in a monoclinic structure (space group P2₁/n) at low temperatures, where the unit cell is composed of four formula units.[85, 86] Ti atoms are present in distorted tetrahedra composed of neighbouring oxygen atoms (TiO₄) within the Ba₂TiO₄ lattice, which is different from the octahedral environments sampled in TiO₂ and Ti₂O₃. Upon structure relaxation, we observe that both r²2SCAN and r²2SCAN+U functionals marginally overestimate (by $\sim$2%) experimental lattice parameters (Figure  4 and Table S3). Similar to trends observed in Table S2, adding U to r²SCAN increases the calculated lattice parameters in Ba₂TiO₄ (by $\sim$0.03 Å), thereby marginally reducing the agreement with experiment.

We benchmark both structural and electronic properties of BiVO₄ as a transferability check for V-based systems. Note that BiVO₄ transforms from tetragonal (I41/a) to a monoclinic (I2/b) ‘scheelite’ phase below $\sim$ 528 K,[87, 88] which is a reversible second order ferroelastic transition driven by soft optical phonon modes. The BiVO₄ unit cell possesses four formula units, with tetrahedrally coordinated V ions, which is different from the coordination environments of V in VO, V₂O₃, VO₂, and V₂O₅. Importantly, monoclinic-BiVO₄ spontaneously transforms to the tetragonal structure upon structure relaxation with r²2SCAN and r²2SCAN+U, similar to the observation by Liu et al [87] with GGA and hybrid functionals. Thus, neither r²2SCAN nor r²2SCAN+U predict the correct ground state structure. Additionally, BiVO₄ possess a band gap of 2.4–2.48 eV[89] and is a candidate photocatalyst.[87] Both r²2SCAN and r²SCAN+U provide similar band gap predictions (2.01-1.98 eV), which is in good qualitative agreement with experiment. Surprisingly, r²SCAN+U evaluates a marginally lower band gap than r²SCAN (see panels a and b in Figure 4). However, both r²2SCAN and r²SCAN+U predict similar states occupying the valence band (O_p) and conduction band (V_d) edges.

The rationale behind the choice of K₃MnO₄, K₂MnO₄, and Mn₂O₇ as checks for Mn-based systems is to explore the higher, unsampled oxidation states of Mn, namely +5, +6, and +7 in K₃MnO₄, K₂MnO₄, and Mn₂O₇, respectively. Also, Mn resides in tetrahedral coordination in these compounds, which is different from the octahedral coordination observed in MnO, Mn₂O₃, and MnO₂. Although Mn²⁺ resides in tetrahedral sites in spinel-Mn₃O₄, we had not used in the spinel structure to obtain our optimal U. We benchmark the calculated lattice parameters versus experiments for all Mn-based transferability checks.

Mn₂O₇ is a volatile liquid at 298 K and solidifies to a monoclinic crystal structure (P2₁/c) below $\sim$ 279 K, with the unit cell consisting of 8 formula units of corner sharing tetrahedral MnO₄ pairs.[90, 91] Upon structural relaxation, both r²SCAN and r²SCAN+U underestimate the lattice constants of monoclinic-Mn₂O₇ by $\sim$1-3% (Figure  4 and Table S3). In the case of K₃MnO₄, the tetragonal symmetry (I$\overline{4}$2m)[92] is broken with r²SCAN functional resulting in an orthorhombic structure, while the symmetry is preserved by r²SCAN+U (see Figure  4 and Table S3). Nonetheless, both r²SCAN and r²SCAN+U significantly underestimate the $c$ parameter (by $\sim$ 13.5%) and overestimate the $a$ or $b$ parameter ($\sim$ 10.2%). K₂MnO₄ is an orthorhombic crystal (Pnma) with four formula units per unit cell.[93] Here, r²SCAN and r²SCAN+U predict identical lattice parameters, which marginally underestimate experimental values (by $\sim$ 0.4-1%, see Figure  4 and Table S3).

The choice of SrFeO₃, a cubic perovskite, as a check for Fe is largely motivated by the 4+ oxidation state exhibited by Fe in the structure, which is not sampled in FeO, Fe₂O₃, or Fe₃O₄. Both r²SCAN and r²SCAN+U preserve the cubic symmetry during structure relaxation, with r²SCAN+U’s lattice parameters identical to experiments and r²SCAN’s parameters being a slight underestimation ($\sim$ 0.5%, see Figure  4 and Table S3). In terms of magnetic configuration of Fe in SrFeO₃, Takeda et al.[94] reported a helical spin structure via their neutron diffraction experiments, with competing FM and AFM interactions. However, Shein et al.[95] found a FM metallic state to be the ground state of SrFeO₃, over a wide range of pressures, based on their first principles calculations, which they attributed to stronger FM than AFM interactions. We considered a FM configuration of Fe atoms in the SrFeO₃ unit cell, and the on-site magnetic moments on Fe calculated by both r²SCAN (3.375 $\mu_{B}$, Table 1) and r²SCAN+U (3.819 $\mu_{B}$) overestimate the experimental value (2.7$\pm$0.4 $\mu_{B}$[94]). However, our calculated magnetic moments do indicate a localization of $\sim$4 electrons on the $d$ orbitals of Fe, consistent with its +4 oxidation state.

We choose CoO₂ (R$\overline{3}$m or ‘O3‘ polymorph[96]), and NiO₂ (P1m1 or ‘O1’[97]), both layered structures, as transferability checks for Co and Ni, respectively, owing to the unsampled 4+ oxidation states of each TM. In terms of experimental property to benchmark, we choose the average Li intercalation voltage in these structures, i.e., LiCoO₂-CoO₂, and LiNiO₂-NiO₂ pairs, since they have been measured with high precision. The reader is referred to previous works on calculating and benchmarking average ‘topotactic’ intercalation voltages.[98, 99] r²SCAN underestimates the experimental average voltage[96, 100, 101, 102, 103, 99] in LiNiO₂-NiO₂ (by $\sim$ 8%), while it overestimates the average voltage in LiCoO₂-CoO₂ (by $\sim$ 1.7%), similar to trends observed with SCAN.[99] The addition of U to r²SCAN leads to an improvement in agreement with the experimental voltage in the Ni-system (deviation of $\sim$ 1.8%), while it worsens the agreement in the Co-system (deviation of $\sim$ 4.4%). Nevertheless, r²SCAN+U does overestimate the average voltage in both Co and Ni systems, similar to the behavior of SCAN+U.[99]