# Performance of the r2SCAN functional in transition metal oxides

S. Swathilakshmi  Department of Materials Engineering, Indian Institute of
Science, Bengaluru, 560012, India  Reshma Devi  Department of Materials
Engineering, Indian Institute of Science, Bengaluru, 560012, India
Gopalakrishnan Sai Gautam  Department of Materials Engineering, Indian
Institute of Science, Bengaluru, 560012, India  Email<EMAIL_ADDRESS>
###### Abstract

We assess the accuracy and computational efficiency of the recently developed
meta-generalized gradient approximation (metaGGA) functional, the restored
regularized strongly constrained and appropriately normed (r2SCAN), in
transition metal oxide (TMO) systems and compare its performance against SCAN.
Specifically, we benchmark the r2SCAN-calculated oxidation enthalpies, lattice
parameters, on-site magnetic moments, and band gaps of binary 3d TMOs against
the SCAN-calculated and experimental values. Additionally, we evaluate the
optimal Hubbard _U_ correction required for each transition metal (TM) to
improve the accuracy of the r2SCAN functional, based on experimental oxidation
enthalpies, and verify the transferability of the _U_ values by comparing
against experimental properties on other TM-containing oxides. Notably,
including the U-correction to r2SCAN increases the lattice parameters, on-site
magnetic moments and band gaps of TMOs, apart from an improved description of
the ground state electronic state in narrow band gap TMOs. The r2SCAN and
r2SCAN+U calculated oxidation enthalpies follow the qualitative trends of SCAN
and SCAN+_U_ , with r2SCAN and r2SCAN+U predicting marginally larger lattice
parameters, smaller magnetic moments, and lower band gaps compared to SCAN and
SCAN+U, respectively. We observe that the overall computational time (i.e.,
for all ionic+electronic steps) required for r2SCAN(+U) to be lower than
SCAN(+U). Thus, the r2SCAN(+U) framework can offer a reasonably accurate
description of the ground state properties of TMOs with better computational
efficiency than SCAN(+U).

##  1 Introduction

Density functional theory (DFT[1]) calculations are the bedrock of modern
computational materials science in terms of predicting thermodynamic and
kinetic properties, with such property predictions being put to use in
subsequent materials discovery[2, 3, 4, 5, 6, 7] and understanding underlying
physical phenomena.[8, 9, 10, 11, 12] In recent years, machine learning has
been used to augment DFT in property predictions, thereby reducing
computational cost and accelerating materials discovery.[13, 14, 15, 16, 17]
Note that a key approximation within DFT is the exchange-correlation (XC)
functional, the exact form of which is unknown. However, several
approximations for the XC functional have been proposed over the years, which
can be categorized into different classes depending on the degree of
sophistication and accuracy, and visually represented as rungs on the Jacob’s
ladder.[1, 2, 18, 19] As with most computational tools, the higher the
accuracy (higher up Jacob’s ladder) higher is the computational cost.

Most DFT calculations for “large” solid systems (10s to 100s of atoms) are
performed using the Perdew-Burke-Ernzerhof (PBE) parameterization of the
generalized gradient approximation (GGA) XC functional,[20] as it offers fair
accuracy at reasonable computational cost for a wide variety of materials.[21,
22, 23] Specifically, GGAs include the local electron density as well as the
gradient of the electron density in describing the XC. As a semilocal
functional of electron density, PBE captures short range interactions but
fails to capture medium and long-range dispersions and also exhibits large
electronic self-interaction errors (SIEs), especially in highly correlated
systems.[24, 25] Also, PBE typically underestimates the formation energies[26,
27] and semiconductor band gaps of crystalline solids,[28, 26] while
overestimating their lattice volumes.[26, 29]

As we move higher in the Jacob’s ladder,[19] we obtain metaGGA functionals,
which may account for medium range dispersions and exhibit lower SIEs. Some
metaGGAs consider orbital kinetic energy density in addition to the local
electron density and its gradient, such as the recently developed strongly
constrained and appropriately normed (SCAN[30]) functional, which offers
better numerical accuracy than PBE and satisfies all 17 known constraints for
a XC functional (namely, 6 for exchange, 6 for correlation, and 5 for both).
The iso-orbital indicator ($\alpha$), which includes the kinetic energy
density in SCAN, distinguishes various bonding environments in a given
material and consequently improves the accuracy of SCAN over GGA. However,
SCAN suffers from numerical instability during self-consistent-field (SCF)
calculations[31] wherein denser $k$-grids (than PBE) are required for accurate
and consistent predictions.[32, 33, 31] Thus it is computationally expensive
(per SCF step) compared to PBE.[21]

To overcome the numerical instability and reduce the computational cost of
SCAN, Bartok and Yates[34] developed regularized SCAN (rSCAN), which satisfies
13 out of the 17 known constraints. The authors replaced the non-analytical
switching $\alpha$ interpolation function in SCAN with a simple polynomial
function, which improves computational speed.[35] However, subsequent
investigations showed a significant drop in numerical accuracy with rSCAN
(compared to SCAN), which is attributed to the failure of the polynomial
$\alpha$ function to fully recover the uniform gas limit.[32, 31]
Subsequently, Furness et al.[32] introduced the restored regularized SCAN (or
r2SCAN), wherein the constraints broken by rSCAN were restored except the
fourth order gradient expansion constraint for exchange (or GE4X). Furness et
al. claimed that the new r2SCAN functional combines the numerical accuracy of
SCAN and computational speed of rSCAN as the smooth polynomial $\alpha$
function of rSCAN is modified to satisfy the uniform gas limit in r2SCAN.[32]
Recently, Kingsbury et al.[36] demonstrated that r2SCAN functional indeed
delivers robust numerical accuracy (i.e., similar to SCAN) and better
computational performance (faster and numerically stable) by comparing r2SCAN
and SCAN for solids using a high-throughput computational workflow.
Specifically, the authors[36] reported that while r2SCAN predicts a smaller
band gap (for most of the strongly-bound materials) and larger lattice volumes
than SCAN, the mean atomization error with r2SCAN is $\sim$15-20% lower for
most solids. However, the performance of r2SCAN in correlated electron
systems, i.e., transition metal oxides (TMOs) containing open-shell d
electrons, remains to be seen and forms the main focus of this work.

Despite the accuracy of SCAN, it still has shortcomings in TMOs, which can be
mitigated by adding an on-site Hubbard U correction term for the transition
metal (TM) under consideration.[37, 38] This approach is similar to the one
followed to mitigate the SIEs of PBE in TMOs.[39, 40] However, the magnitude
of the U correction required is not known a priori, and there are both theory-
based approaches such as density functional perturbation theory,[41] linear
response theory,[42, 43, 44] embedded Hartree-Fock method,[45, 46] and machine
learning based Bayesian optimisation,[47] and experimental-data-based
approaches to identify the appropriate U values. For example, Gautam et
al.[37, 38] used the experimental oxidation enthalpies among binary TMOs to
identify optimal U values across various oxidation states of 3d TMs. A similar
experimental-data-based Hubbard U correction scheme can be developed in
conjunction with r2SCAN as well, resulting in a r2SCAN+U framework, in case
r2SCAN exhibits similar SIEs as SCAN in TMOs. We explore the usefulness of
such a r2SCAN+U framework also in this work.

Here, we verify the numerical accuracy and computational efficiency of the
r2SCAN and r2SCAN+U frameworks in comparison to SCAN and SCAN+U, respectively,
in describing material properties such as lattice parameters, on-site magnetic
moments, and band gaps of binary 3d TMOs, including Ti, V, Cr, Mn, Fe, Co, Ni,
and Cu. As necessary, we evaluate the optimal Hubbard U correction with r2SCAN
for each TM by using the experimental-data-based approach employed in previous
works.[37, 38] We find that r2SCAN predicts marginally larger lattice
constants and smaller on-site magnetic moments than SCAN for most of the TMOs
considered. On addition of the U-correction to both SCAN and r2SCAN, we
observe an increase in the calculated lattice constants, on-site magnetic
moments and band gaps. In the case of narrow band gap TMOs, SCAN+U and
r2SCAN+U generally estimate a non-zero band gap, with r2SCAN+U’s band gap in
better agreement with experiments. Also, we perform transferability checks for
the optimal _U_ values derived in this work for each TM, by benchmarking
various properties in oxides that were not used in obtaining the _U_ values.
Finally, we compare the computational performance of r2SCAN/r2SCAN+U relative
to SCAN/SCAN+U to explore the accuracy-cost trade-off. We report that
r2SCAN/r2SCAN+U is computationally less expensive than SCAN and SCAN+_U_ ,
when all required ionic and electronic steps are taken into account for
convergence during structure relaxations. We hope that our work will provide a
foundational basis for further studies on understanding material behavior and
computationally discovering new materials in the near future.

##  2 Methods

###  2.1 Computational Methods

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.

###  2.2 Structures and magnetic configurations

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.

###  2.3 Determining $U$

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]

##  3 Results

###  3.1 Oxidation energetics

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.

Figure 1: Calculated oxidation enthalpy versus the magnitude of U correction
within r2SCAN+U framework for (a) Ti, (b) V, (c) Mn, (d) Fe, (e) Co, and (f)
Ni oxides. Solid, dashed, and dotted lines of a given color indicate
calculated, experimental, and required U values for a given oxidation
reaction. Optimal U for each TM is indicated by the dotted blue line in each
panel.

###  3.2 Lattice parameters

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.

Figure 2: (a) Comparison of calculated and experimental lattice volume (in Å3)
of all TMOs considered. (b) Violin plot capturing the difference between the
experimental and computed band gap (in eV) across TMO systems using the four
XC frameworks. The empty circle and horizontal line in the inner box plot
corresponds to the mean and median of the calculated band gaps, respectively.
(c) Heat map representation of the differences between the experimental and
calculated on-site magnetic moments (in $\mu_{B}$) using the four XC
functionals and across all TMOs. A value of zero indicates perfect
consistency, while red (blue) colors indicate overestimation (underestimation)
of magnetic moments. Hatched boxes either correspond to experimentally
undetermined magnetic moments (VO) or calculations not executed with _U_
frameworks (Cr and Cu oxides).

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.

###  3.3 On-site magnetic moments

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.

###  3.4 Band gaps

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.

Figure 3: DOS for CoO calculated using (a) SCAN and (b) r2SCAN, DOS for V2O3
computed using (c) r2SCAN and (d) r2SCAN+U, and DOS for Mn2O3 estimated using
(e) SCAN+U and (f) r2SCAN+U.

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.

###  3.5 Transferability checks

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 (Op) and conduction band (Vd) edges.

Figure 4: DOS for BiVO4 calculated using (a) r2SCAN and (b) r2SCAN+U. (c)
Difference between experimental and calculated lattice volumes (using r2SCAN
and r2SCAN+_U_), plotted as a heatmap, for various systems. Red (blue) squares
indicate overestimated (underestimated) calculated lattice volumes versus
experiment.

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]

Table 1: Voltage and magnetic moments calculated by r2SCAN, and r2SCAN+U compared against experimental values (denoted by ‘Expt.’). The U values used with r2SCAN+U are the corresponding optimal U values obtained for each TM (from Figure 1). Composition | Source | Voltage | Magnetic moment
---|---|---|---
(space group) |  | (V) | ($\mu_{B}$)
LiCoO2-CoO2 | Expt. | 4.05 | -
(R$\bar{3}$m) | r2SCAN | 4.12 | -
| r2SCAN+U | 4.23 | -
LiNiO2-NiO2 | Expt. | 3.85 | -
(P1m1) | r2SCAN | 3.54 | -
| r2SCAN+U | 3.92 | -
SrFeO3 | Expt. | - | 2.7$\pm$0.4
(Pm$\bar{3}$m) | r2SCAN | - | 3.375
| r2SCAN+U | - | 3.819

##  4 Discussion

In this work, we evaluated the performance of the r2SCAN functional among
binary TMOs consisting of 3d-TMs by calculating the oxidation enthalpies,
lattice parameters, on-site magnetic moments, and band gaps. Additionally, for
each TM-O2 system considered, we calculated the optimal Hubbard-U corrections
to be used in a r2SCAN+_U_ framework, based on experimental oxidation
enthalpies. Although theoretical approaches exist to derive U values,[45, 46,
42, 44, 43, 41, 47] using oxidation enthalpies nominally gives an “average”
correction that is suitable across several oxidation states of a given TM.
Specifically, our optimal U values are 2.3, 1.0, 1.8, 3.1, 1.8, and 2.1 eV for
Ti, V, Mn, Fe, Co, and Ni, respectively, while we don’t deem a U correction
necessary for Cr and Cu oxides. Interestingly, the optimal U corrections
needed with r2SCAN are lower in magnitude compared to SCAN for Ti, Mn, Co, and
Ni oxides (while the corrections are identical for V and Fe oxides),
indicating that r2SCAN exhibits lower errors with oxidation enthalpies and
possibly lower SIEs than SCAN. However, this is not reflected in other
physical properties. On an average, we find the accuracy, versus experimental
values, to be similar for r2SCAN compared to SCAN, and for r2SCAN+U compared
to SCAN+U, respectively, in lattice parameter, on-site magnetic moment, and
band gap evaluations as seen in Figure 2.

Figure 5: (a) Overall computational time (electronic+ionic steps) (b)
computational time per ionic step and (c) computational time per electronic
loop taken for each TM-O2 binary system with SCAN+U, r2SCAN, and r2SCAN+U
frameworks relative to SCAN. Values greater (smaller) than 1 in each panel
indicates that a given calculation is slower (faster) than SCAN.

The general trends in lattice parameter, magnetic moment, and band gap
predictions, across the XC frameworks considered, can be summarized as
follows. We observe that r2SCAN generates larger lattice constants than SCAN
and on addition of the U correction to both functionals, the lattice constants
further increase. Thus, in systems where SCAN underestimates experimental
lattice constants (e.g., CrO2, CrO3, MnO2), shifting to r2SCAN improves
agreement (e.g., error in r2SCAN in CrO3 is 0.8% versus 2.3% with SCAN). Also,
there are instances where the ground state symmetry of the TMO is not
preserved by some or all of the XC frameworks considered (i.e., in VO, MnO,
FeO, Fe3O4, and Ti2O3), highlighting systematic issues in the XC treatment
across the four frameworks considered. The calculated on-site magnetic moments
by r2SCAN (and r2SCAN+U) are marginally lower than SCAN (SCAN+U), with the U
correction nominally increasing the calculated moments calculated by r2SCAN
and SCAN. However, calculated magnetic moments across the four XC frameworks
differ by $<10$% (except LiNiO2), signifying marginal differences in accuracy.
Both SCAN and r2SCAN underestimate band gaps across all TMOs (except MnO2),
with band gaps calculated by r2SCAN typically being lower than SCAN, and
adding the U opens/widens the gap. Thus, SCAN+U offers the best quantitative
accuracy versus experimental band gaps, especially for wide gap
semiconductors. Note that the qualitative trends from r2SCAN+U are consistent
with the trends exhibited by SCAN+U and should be reliable in electronic
structure predictions in other TM-based oxide systems.

r2SCAN adopts the smooth polynomial interpolation function of rSCAN to
maintain numerical stability during SCF calculations. Additionally, the
reformed gradient expansion for correlation introduced in r2SCAN (partially)
negates the error introduced to the slowly varying density by the non-
vanishing interpolation function,[32] which largely accounts for the observed
variation in accuracy of r2SCAN versus SCAN. Based on our data, we observe
that r2SCAN is not systematically more accurate than SCAN across all TMOs and
for all property predictions. For example, we have lower optimal U values
indicating lower SIEs with r2SCAN versus SCAN, but also lower on-site magnetic
moments (except Mn and Cr oxides) signifying poorer d-electron localization
with r2SCAN. Further, the smaller band gaps with r2SCAN (versus SCAN) may be
caused by the residual SIEs, resulting in an underestimation of the CBM across
TMOs. Hence, usage of r2SCAN(+U) in TM-based systems must be done with care
and efforts should be made to benchmark as many available experimental
properties as possible before performing “true” computational predictions.

We considered the transferability of the _U_ values estimated in this work,
with r2SCAN, by examining systems for each TM with oxidation states and/or
coordination environments not sampled while calculating the optimal _U_. In
general, we find that r2SCAN or its Hubbard _U_ corrected version estimate
similar lattice parameters and hence yield similar accuracies on structural
properties. Analogously, the calculated on-site magnetic moments in SrFeO3 and
the band gaps in BiVO4 are similar between r2SCAN and r2SCAN+_U_. In case of
electrochemical properties, we do find tangible variations in the calculated
average voltages of r2SCAN and r2SCAN+_U_ , with r2SCAN+_U_ exhibiting overall
lower errors across the Co and Ni systems. Thus, we find the optimal _U_
values obtained in this work to be transferable across oxide frameworks not
sampled _a priori_. Nevertheless, more benchmarking studies to compare the
performance of r2SCAN+_U_ with r2SCAN (and experiments) will help in
quantifying the reliability and errors associated with using r2SCAN+_U_.

Given that r2SCAN(+U) is not systematically more or less accurate than
SCAN(+U), the computational performance and numerical stability of r2SCAN(+U)
is critical in determining its utility in property predictions across
materials. Thus, we have quantified the computational time of r2SCAN(+U) and
SCAN+U relative to SCAN for each TM-O2 system considered in Figure S1.
Specifically, panels a, b, and c of Figure 5 plot the overall
(electronic+ionic steps), per ionic step, and per electronic step
computational time, respectively, taken by the SCAN+U (blue bars), r2SCAN
(red), and r2SCAN+U (yellow) frameworks, relative to the computational time
taken by the SCAN functional (dotted black lines), for each TM-based set of
oxides. Details on calculating the computational times used by the functionals
is described in the ‘Computational time’ section of the SI. Note that our
objective is not to provide a rigorous quantification of computational
resources required for each XC framework, but to provide a qualitative
understanding of the relative computational costs across the frameworks
considered.

For each electronic step, r2SCAN(+U) is typically faster than SCAN (Figure 5),
signifying better numerical stability than SCAN, with Mn, Ni, and Cu oxides
being marginal exceptions. In contrast, on a per-ionic step basis, r2SCAN and
r2SCAN+U is slower than SCAN, by $\sim$1.05-1.78$\times$ and
$\sim$1.1-1.31$\times$, respectively, highlighting that r2SCAN(+U) takes more
electronic steps to converge per ionic step. Importantly, the overall
computational time (ionic+electronic steps, Figure 5) required for structural
relaxation of TMOs using r2SCAN and r2SCAN+U is lower than SCAN, by
$\sim$12.1-61.2% and $\sim$1.9-34.5%, respectively, except in Fe oxides,
indicating that r2SCAN(+U) takes lower number of ionic steps to converge,
which possibly indicates a better description of atomic forces. The higher
overall computation time in Fe oxides with r2SCAN(+U) than SCAN is primarily
due to the difficulty in converging Fe3O4 with r2SCAN(+U). Comparing r2SCAN
and r2SCAN+U, we find that r2SCAN+U takes a higher overall computational time
to converge, except in Fe and Ni oxides. Thus, we expect r2SCAN(+U) to provide
good utility in property predictions in TM-containing systems given its better
computational performance and reasonable accuracy compared to SCAN(+U).

##  5 Conclusion

3d-TMs and their compound phases find applications in several fields such as
energy storage, solar cells, catalysts, thermochemical water splitting, etc.,
and it is imperative to predict their properties such as lattice constants,
magnetic moments, reaction enthalpies, and band gaps accurately using DFT-
based techniques for designing better materials. Recently, the r2SCAN metaGGA
XC functional was proposed to exhibit the accuracy of its predecessor, SCAN,
and the computational performance of rSCAN in main-group compounds, but the
accuracy of r2SCAN was not rigorously tested on TM-based systems. Here, we
assessed the numerical accuracy and computational performance of r2SCAN in
binary 3 _d_ -TMOs, in calculating the lattice parameters, on-site magnetic
moments, binary oxidation enthalpies, and band gaps against experimental data.
Notably, we observed that r2SCAN exhibited similar qualitative trends as that
of SCAN, with marginally larger estimations of lattice parameters than SCAN,
while the on-site magnetic moments and band gap calculations are marginally
smaller than SCAN. While both r2SCAN and SCAN underestimated the band gaps in
wide gap TMOs, with SCAN offering slightly better accuracy, they failed to
predict the correct ground state electronic configurations of narrow band gap
TMOs (e.g., Mn2O3).

On analysing the addition of Hubbard U-correction to improve the accuracy of
the r2SCAN functional, we observed that a lower optimal U value, based on
experimental oxidation enthalpies, was required in a r2SCAN+_U_ framework for
Ti, Mn, Co and Ni oxides, when compared to a SCAN+_U_ framework. The optimal
_U_ values were identical in both r2SCAN+_U_ and SCAN+_U_ frameworks for V and
Fe oxides, while we did not observe the need for a _U_ correction in Cr and Cu
oxides with r2SCAN, similar to SCAN. Moreover, introducing the U-correction to
SCAN and r2SCAN increased the calculated lattice parameters, on-site magnetic
moments and the band gaps of the TMOs.

r2SCAN+U and SCAN+U successfully opened a band gap for narrow gap TMOs (except
VO2 and Mn2O3 with r2SCAN+U). Upon testing the optimal _U_ values with
r2SCAN+_U_ on oxides with different oxidation states and/or coordination
environments, we found that the _U_ values derived in this work are in general
transferable to other TM-containing oxides as well. Furthermore, we observed
that r2SCAN(+U) took less overall computational time (ionic+electronic steps)
to converge when compared to SCAN, which indicated that r2SCAN(+U) was
computationally more efficient than SCAN(+U). Since r2SCAN+_U_ offers a
reasonably accurate prediction of material properties at a lower computational
expense than SCAN+_U_ , we observe that r2SCAN+_U_ can be used in high-
throughput materials discovery, after adequate benchmarking tests are done in
each new chemical space explored.

### Acknowledgments

G.S.G. acknowledges the Indian Institute of Science (IISc) Seed Grant,
SG/MHRD/20/0020 and SR/MHRD/20/0013 and the Science and Engineering Research
Board (SERB) of the Department of Science and Technology, Government of India,
under sanction numbers SRG/2021/000201 and IPA/2021/000007 for financial
support. R.D. thanks the Ministry of Human Resource Development, Government of
India, for financial assistance. S.S. acknowledges financial support from SERB
under IPA/2021/000007. All the authors acknowledge the computational resources
provided by the Supercomputer Education and Research Centre, IISc, for
enabling some of the density functional theory calculations showcased in this
work.

### Author Contributions

G.S.G. envisioned and designed the work. S.S. and R.D. performed the
calculations. All authors contributed in data analysis and writing the paper.

### Conflicts of Interest

The authors declare no competing financial or non-financial interests.

### Availability of data

The data that support the findings of this study are openly available at
https://github.com/sai-mat-group/r2SCAN-U-benchmarking.

### Supplementary Materials

Electronic Supporting Information is available online at , with details on the
crystal structures used for calculations, oxidation energetics of Cr and Cu
oxides, densities of states of all systems not showcased in the main text, and
details on computational time calculations.

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