Comparing Hubbard parameters from linear-response theory and a Hartree-Fock-based approach

Abstract

Density-functional theory with on-site U and inter-site V Hubbard corrections (DFT+U+V) is widely used to predict properties of transition-metal and rare-earth compounds, but its accuracy depends critically on how these parameters are obtained. While they can be determined empirically, first-principles approaches offer better consistency, though their results can differ and a systematic comparison is lacking. Here, we compare two widely used approaches, linear-response theory (LRT) and the Hartree-Fock-based pseudohybrid functional formalism, applied to representative oxides (MnO, NiO, CoO, FeO, BaTiO₃, ZnO, and ZrO₂). For partially occupied transition-metal d states, both methods yield comparable U values, but they differ significantly for nearly empty or fully filled d shells. For O-2p states, LRT systematically predicts large U values (~ 10eV), whereas the pseudohybrid approach gives system-dependent values. Even larger differences appear for the inter-site V: the former yields small values (< 1eV), while the latter gives larger values (~ 3eV) due to its dependence on charge redistribution. We further show that the pseudopotential choice, which defines Hubbard projectors, affects the parameters and their agreement. Overall, while parallels between these two methods exist, they rely on different assumptions, leading to variations in predictions of material properties.

Introduction

Density-functional theory (DFT)^1,2 underpins modern first-principles simulations in physics, chemistry, and materials science. Despite its remarkable success, progress hinges on the development of increasingly accurate exchange-correlation (xc) functionals. Standard approximations, such as the local spin-density approximation (LSDA) and spin-polarized generalized-gradient approximation (σ-GGA), are invaluable for many systems, yet they perform poorly for transition-metal (TM) and rare-earth (RE) compounds, where localized d and f states suffer from large self-interaction errors (SIEs)^3,4. To address these limitations, several beyond-standard approaches have been introduced. Among the most prominent are Hubbard-corrected DFT (DFT+U^5–7 and its extension DFT+U+V⁸), meta-GGA functionals (e.g., SCAN⁹ and its variants^10,11), and hybrid functionals (e.g., PBE0¹² and HSE06^13,14). Combinations such as SCAN+U have also been explored^15–18. Each strategy embodies a distinct philosophy: DFT+U selectively corrects partially filled orbitals (typically d and f)^19,20; meta-GGA incorporates the kinetic-energy density and satisfies a broad set of exact constraints (17 in the case of SCAN); hybrid functionals mix a fraction of Fock exchange with (semi-)local functionals. While these methods improve predictive power, they each come with intrinsic trade-offs and limitations²¹. Among them, Hubbard-corrected DFT has become especially popular for its ability to enhance accuracy at modest computational cost²², provided reliable Hubbard parameters are available. These can be obtained either semi-empirically (by fitting to experimental data or higher-level calculations such as hybrids or GW²³) or through systematic calibration. Examples include tuning U to reproduce binary metal formation energies^24,25 or to match properties such as band gaps, magnetic moments, lattice parameters, and enthalpies^26–28.

While semi-empirical fitting of Hubbard parameters can be effective, it becomes impractical in materials discovery where experimental data are unavailable or when multiple target properties must be simultaneously reproduced, sometimes requiring advanced optimization schemes such as Bayesian approaches^29,30. A natural alternative is to compute Hubbard parameters from first principles using methods such as constrained DFT (cDFT)^31–39, Hartree-Fock-based schemes^40–45, or the constrained random-phase approximation (cRPA)^46–49. However, predicted values can differ substantially among these approaches^22,49,50, and systematic comparisons of their theoretical foundations remain scarce, although notable progress is beginning to emerge⁵¹. Moreover, the dependence of U on factors such as the choice of Hubbard projectors, pseudopotentials, oxidation state of TM ions, underlying exchange-correlation functional, local coordination environment, ligand species, and crystal structure further complicates a direct comparison of U values obtained using different computational methods and setups^{20,39,51–60}. Therefore, further work similar to ref. ⁵¹ is needed to systematically compare and benchmark different first-principles approaches for computing Hubbard parameters using identical computational setups (The selection of a specific first-principles method to compute U can be influenced by the intended applications and the properties of interest. Notably, cRPA is the preferred method for determining U in the context of subsequent DFT plus dynamical mean field theory (DFT+DMFT) calculations⁶¹).

Here, we present a comparative study of two such first-principles approaches. The first is based on cDFT formulated within linear-response theory (LRT)⁶², which determines on-site U by enforcing piecewise linearity of the total energy with respect to the occupation of localized orbitals (Hubbard manifold), thereby reducing SIEs⁶³ and improving predictive accuracy. Standard LRT requires supercell calculations, but its reformulation within density-functional perturbation theory (DFPT)^64,65 replaces these with monochromatic reciprocal-space perturbations in a unit cell. This substantially reduces computational overhead and has enabled applications to increasingly complex systems^66–73. The second approach for computing U is the Hartree-Fock-based Agapito, Curtarolo, and Buongiorno Nardelli (ACBN0) functional⁴³ and its extension to inter-site V dubbed as eACBN0^44,45. This method derives Hubbard parameters directly within self-consistent-field (SCF) cycles from an approximate form of the renormalized Hartree-Fock interaction energy. Using projections of Kohn-Sham (KS) states onto localized orbitals, U and V are updated at each iteration of the SCF cycle with negligible additional cost relative to standard DFT. This method has been shown to capture electronic structures and dynamical properties with good accuracy^74,75, and it holds promise for accurate evaluations of electron-phonon interactions^76,77.

Results

Formalism: the DFT+U+V approach

In DFT+U+V^8,22, a corrective Hubbard energy E_U+V is added to the standard DFT energy E_DFT:

$$E_{\mathrm{DFT}+U+V}=E_{\mathrm{DFT}}+E_{U+V}.$$ 1

Unlike DFT+U, which only accounts for on-site interactions scaled by U, DFT+U+V also includes inter-site interactions between an atom and its neighbors, scaled by V. In the simplified rotationally-invariant formulation⁷, the extended Hubbard correction reads⁶⁵:

$$\begin{array}{ccc}\hfill E_{U+V}& \hfill =\hfill & \frac{1}{2}\sum _I\sum _{\sigma m{m}^{\prime }}{U}^I\left({\delta }_{m{m}^{\prime }}-n_{m{m}^{\prime }}^{II\sigma }\right)n_{m^{\prime }m}^{II\sigma }\hfill \\ \hfill & \hfill \hfill & -\frac{1}{2}\sum _I\sum _{J\left(J\ne I\right)}^{*}\sum _{\sigma m{m}^{\prime }}{V}^{IJ}{n}_{m{m}^{\prime }}^{IJ\sigma }{n}_{m^{\prime }m}^{JI\sigma },\hfill \end{array}$$ 2

where I and J index atomic sites, m and $m\prime$ denote magnetic quantum numbers, and U^I and V^IJ are the effective on-site and inter-site Hubbard parameters, respectively (here and everywhere in the following, U^I denotes the difference between the screened Coulomb interaction parameter and the Hund’s exchange interaction parameter). The asterisk denotes that, for each atom I, the sum over J includes all neighbors up to a defined cutoff distance. The generalized occupation matrices $n_{m{m}^{\prime }}^{IJ\sigma }$ are obtained by projecting KS states ${\psi }_{v,\mathbf{k}}^{\sigma }$ onto localized orbitals ${\varphi }_m^I\left(\mathbf{r}\right)$:

$$n_{m{m}^{\prime }}^{IJ\sigma }=\sum _{v,\mathbf{k}}{f}_{v,\mathbf{k}}^{\sigma }⟨{\psi }_{v,\mathbf{k}}^{\sigma }\mid {\varphi }_{m^{\prime }}^J⟩⟨{\varphi }_m^I\mid {\psi }_{v,\mathbf{k}}^{\sigma }⟩,$$ 3

where v and σ label bands and spin, k runs over the first Brillouin zone (BZ), and $f_{v,\mathbf{k}}^{\sigma }$ are the KS occupations. In Eq. (3), m ∈ I and $m\prime \in J$, meaning that m runs over the magnetic quantum numbers associated with a fixed principal quantum number n and orbital angular momentum l of atom I, and analogously $m\prime$ for atom J. The corresponding generalized KS equation can be written as⁸,

$$\left({\widehat{\mathcal{H}}}_{\mathrm{KS}}^{\sigma }+{\widehat{\mathcal{V}}}_{U+V}^{\sigma }\right)∣{\psi }_{v,\mathbf{k}}^{\sigma }⟩={\epsilon }_{v,\mathbf{k}}^{\sigma }∣{\psi }_{v,\mathbf{k}}^{\sigma }⟩,$$ 4

where ${\widehat{\mathcal{H}}}_{\mathrm{KS}}^{\sigma }$ is the KS Hamiltonian of the standard DFT, ${\epsilon }_{v,\mathbf{k}}^{\sigma }$ is the v-th band energy at momentum k and spin σ, and ${\widehat{\mathcal{V}}}_{U+V}^{\sigma }$ is the Hubbard potential that is derived from the Hubbard energy (2) and which reads⁶⁵:

$$\begin{array}{ccc}\hfill {\widehat{\mathcal{V}}}_{U+V}^{\sigma }& =& \sum _I\sum _{m{m}^{\prime }}\frac{U^I}{2}\left({\delta }_{m{m}^{\prime }}-2n_{m{m}^{\prime }}^{II\sigma }\right)∣{\varphi }_m^I⟩⟨{\varphi }_{m^{\prime }}^I∣\hfill \\ \hfill & & -\sum _I\sum _{J(J\ne I)}^{*}\sum _{m{m}^{\prime }}{V}^{IJ}{n}_{m{m}^{\prime }}^{IJ\sigma }∣{\varphi }_m^I⟩⟨{\varphi }_{m^{\prime }}^J∣.\hfill \end{array}$$ 5

In Eqs. (2) and (5), the effective on-site U^I and inter-site V^IJ terms act in opposition when both parameters are positive: the on-site term favors electron localization on atomic sites and suppresses hybridization, whereas the inter-site term enhances hybridization with neighboring atoms. Accurate first-principles evaluation of U^I and V^IJ is therefore crucial to correctly describe the balance between localization and hybridization.

It is important to note that the computed values of U^I and V^IJ depend on the choice of Hubbard projector functions ${\varphi }_m^I\left(\mathbf{r}\right)$ used to define the occupation matrix in Eq. (3) and the Hubbard potential in Eq. (5)^44,52,53. In the following, we briefly describe how Hubbard parameters are defined and computed within LRT and eACBN0. For both methods, we employ the same projector functions, namely atomic orbitals orthogonalized using the Löwdin method^78,79, to ensure a fair and consistent comparison of the computed U^I and V^IJ values.

Within the LRT framework, the effective Hubbard parameters are defined as^8,54,62:

$$U^I={\left({\chi }_0^{-1}-{\chi }^{-1}\right)}_{II},$$ 6

and

$$V^{IJ}={\left({\chi }_0^{-1}-{\chi }^{-1}\right)}_{IJ},$$ 7

where χ₀ and χ are the bare and self-consistent response matrices, describing how the occupation matrices respond to localized electronic perturbations. The self-consistent response matrix is given by

$${\chi }_{IJ}=\sum _{m\sigma }\frac{d{n}_{mm}^{I\sigma }}{d{\alpha }^J},$$ 8

where $n_{m{m}^{\prime }}^{I\sigma }\equiv n_{m{m}^{\prime }}^{II\sigma }$ denotes the on-site occupation matrix, and α^J is the strength of the perturbation on the Jth site. The bare response χ₀ is defined analogously but evaluated before the self-consistent adjustment of the Hartree and xc potentials.

DFPT allows this calculation to be performed without supercells, by reformulating LRT in terms of monochromatic reciprocal-space perturbations in the unit cell. This significantly reduces computational cost^64,65. The response of the occupation matrix is then computed as:

$$\frac{d{n}_{m{m}^{\prime }}^{I\sigma }}{d{\alpha }^J}=\frac{1}{N_{\mathbf{q}}}\sum _{\mathbf{q}}^{N_{\mathbf{q}}}{e}^{i\mathbf{q}\cdot \left({\mathbf{R}}_l-{\mathbf{R}}_{l^{\prime }}\right)}{\mathrm{\Delta }}_{\mathbf{q}}^{s^{\prime }}{n}_{m{m}^{\prime }}^{s\sigma },$$ 9

where q is the wavevector of the monochromatic perturbation, N_q is the total number of perturbations, and ${\mathrm{\Delta }}_{\mathbf{q}}^{s^{\prime }}{n}_{m{m}^{\prime }}^{s\sigma }$ is the lattice-periodic response of the occupation matrices to a q-specific perturbation. Here, I ≡ (l, s) and $J\equiv (l^{\prime },s^{\prime })$, with s and $s^{\prime }$ indexing atoms within the unit cell, while l and $l^{\prime }$ indexing unit cells; R_l and ${\mathbf{R}}_{l^{\prime }}$ are the corresponding Bravais lattice vectors. The quantities $\mathrm{\Delta }{\mathbf{q}}^{\prime }{n}_{m{m}^{\prime }}^{s\sigma }$ are obtained from the response KS wavefunctions, which are computed by solving q-specific Sternheimer equations. Further details on the DFPT implementation can be found in refs. ^64,65. The efficiency of DFPT for calculating U^I and V^IJ has been demonstrated in numerous studies, with several recent highlights^54,80–84.

In the eACBN0 method, the effective Hubbard parameters are defined by equating the corrective Hubbard energy [Eq. (2)] with the Hartree-Fock interaction energy^43,44. This leads to the following expressions:

$$\begin{array}{c}{U}^I=\frac{{\sum }_{\left\{m\right\},\sigma {\sigma }^{\prime }}{\overline{P}}_{m{m}^{\prime }}^{II,\sigma }{\overline{P}}_{m^{″}{m}^{″\prime }}^{II,\sigma \prime }\left(m{m}^{\prime }\mid m^{″}{m}^{″\prime }\right)}{{\sum }_{m{m}^{\prime },\sigma }{n}_{mm}^{II,\sigma }{n}_{m^{\prime }{m}^{\prime }}^{II,-\sigma }+{\sum }_{m\ne m^{\prime },\sigma }{n}_{mm}^{II,\sigma }{n}_{m^{\prime }{m}^{\prime }}^{II,\sigma }}\hfill \\ -\frac{{\sum }_{\left\{m\right\},\sigma }{\overline{P}}_{m{m}^{\prime }}^{II,\sigma }{\overline{P}}_{m^{″}{m}^{″\prime }}^{II,\sigma }\left(m{m}^{″\prime }\mid m^{″}{m}^{\prime }\right)}{{\sum }_{m\ne m^{\prime },\sigma }{n}_{mm}^{II,\sigma }{n}_{m^{\prime }{m}^{\prime }}^{II,\sigma }},\hfill \end{array}$$ 10

and

$$V^{IJ}=\frac{1}{2}\frac{{\sum }_{m{m}^{\prime },\sigma {\sigma }^{\prime }}\left[{\overline{P}}_{mm}^{II,\sigma }{\overline{P}}_{m^{\prime }{m}^{\prime }}^{JJ,\sigma \prime }-{\delta }_{\sigma {\sigma }^{\prime }}{\overline{P}}_{m{m}^{\prime }}^{IJ,\sigma }{\overline{P}}_{m^{\prime }m}^{JI,\sigma \prime }\right]\left(mm\mid m^{\prime }{m}^{\prime }\right)}{{\sum }_{m{m}^{\prime },\sigma {\sigma }^{\prime }}\left(n_{mm}^{II,\sigma }{n}_{m^{\prime }{m}^{\prime }}^{JJ,{\sigma }^{\prime }}-{\delta }_{\sigma {\sigma }^{\prime }}{n}_{m{m}^{\prime }}^{IJ,\sigma }{n}_{m^{\prime }m}^{JI,{\sigma }^{\prime }}\right)},$$ 11

where $\left(m{m}^{\prime }\mid m^{″}{m}^{″\prime }\right)$ represents the bare Coulomb interaction between electrons with $m,m^{\prime }$ of atom I and $m^{″},m^{″\prime }$ of atom J. The renormalized density matrix ${\overline{P}}_{m{m}^{\prime }}^{IJ,\sigma }$ is defined as

$${\overline{P}}_{m{m}^{\prime }}^{IJ,\sigma }=\sum _{v,\mathbf{k}}{f}_{v,\mathbf{k}}^{\sigma }{\overline{N}}_{v,\mathbf{k}}^{IJ,\sigma }⟨{\psi }_{v,\mathbf{k}}^{\sigma }\mid {\varphi }_{m^{\prime }}^J⟩⟨{\varphi }_m^I\mid {\psi }_{v,\mathbf{k}}^{\sigma }⟩,$$ 12

where ${\overline{N}}_{v,\mathbf{k}}^{IJ,\sigma }$ is the renormalized occupation number defined as

$$\begin{array}{ccc}\hfill {\overline{N}}_{v,\mathbf{k}}^{IJ,\sigma }& =& \sum _{\left\{I\right\}}\sum _m\langle {\psi }_{v,\mathbf{k}}^{\sigma }\mid {\varphi }_m^I\rangle \langle {\varphi }_m^I\mid {\psi }_{v,\mathbf{k}}^{\sigma }\rangle \hfill \\ \hfill & & +\sum _{\left\{J\right\}}\sum _{m^{\prime }}\langle {\psi }_{v,\mathbf{k}}^{\sigma }\mid {\varphi }_{\prime }^J\rangle \langle {\varphi }_{\prime }^J\mid {\psi }_{v,\mathbf{k}}^{\sigma }\rangle ,\hfill \end{array}$$ 13

where m ∈ I and $m^{\mathrm{\prime }}\in J$, and the summations ∑_{I} and ∑_{J} mean that we sum up contributions coming from all equivalent atoms of the same type. For the case of U^I-only calculations (I = J), the second term in Eq. (13) is dropped⁴⁴, so ${\overline{N}}_{v,\mathbf{k}}^{II,\sigma }$ reduces to the Mulliken charge of atom I in state ${\psi }_{v,\mathbf{k}}^{\sigma }\left(\mathbf{r}\right)$. Moreover, ${\overline{N}}_{v,\mathbf{k}}^{IJ,\sigma }$ can be viewed as a position-dependent mixing parameter within the Hartree-Fock approximation, encoding the effective on-site and inter-site interactions in the localized subspace. It is important to note that for a fixed ground-state density, the U^I values computed in ACBN0 remain unchanged upon inclusion of inter-site V^IJ terms in eACBN0. However, in self-consistent implementations of eACBN0, a coupling between U^I and V^IJ emerges because the ground state is obtained under the influence of both on-site and inter-site terms in the Hubbard potential (5), with self-consistency enforced for both the electron density and the Hubbard parameters. Further details on the eACBN0 formalism can be found in refs. ^43–45.

Finally, we comment on the normalization entering Eq. (13). Upon enlarging the unit cell, the number of equivalent atomic orbital projectors included in the sums increases. However, each KS wavefunction is normalized to unity within the unit cell, and Eq. (13) is defined at the level of individual KS states (there is no summation over v and k). As a result, when the cell is enlarged, each KS state delocalizes over a larger number of equivalent atoms, reducing its projection on each atom, while the sum over all equivalent atoms increases proportionally. These two effects exactly compensate, so that the renormalized occupation number in Eq. (13) remains unchanged for the corresponding KS states. Consequently, the Hubbard parameters obtained from Eqs. (10) and (11) are invariant with respect to the choice of unit cell.

Overview of computational protocols

In this section, we present results for the computed on-site U and inter-site V Hubbard parameters across three classes of systems: (i) TM oxides with partially occupied d states (MnO, FeO, CoO, and NiO); (ii) compounds with nominally empty d states (ZrO₂ and BaTiO₃); and (iii) a compound with fully occupied d states (ZnO). These categories allow us to assess similarities and differences between LRT and eACBN0 under distinct electronic configurations.

Hubbard parameters can be computed at different levels of self-consistency. The simplest approach is a “one-shot” calculation, where the parameters are evaluated using the converged DFT charge density while keeping the geometry fixed. A more advanced “self-consistent” approach iteratively updates both the charge density and the corresponding U and V values, but still at a fixed geometry. Finally, in the “structurally self-consistent” scheme, structural relaxations (atomic positions and unit-cell parameters) are included in the self-consistent cycle^54,65. In the main text, we report results obtained with the “self-consistent” scheme, i.e., calculations in which the electronic degrees of freedom are converged together with the Hubbard parameters while the geometry remains fixed. In this approach, all U and V values are determined consistently with the self-consistent charge density^44,65, as illustrated in Fig. 1. This allows us to isolate purely electronic effects from structural ones and to assess the influence of extended Hubbard interactions in LRT and eACBN0 with high internal consistency. For completeness, in the Supplemental Information (SI) we also present results obtained using the “structurally self-consistent” scheme, in which the self-consistent cycle includes full structural optimization with updated Hubbard parameters and charge density (see Sec. S1 and S2 in the SI). In this case, the crystal structure, the Hubbard parameters, and the charge density are iterated to convergence, yielding mutually consistent electronic and crystal structures. All computational details are provided in the “Methods” section.

Fig. 1. Protocols for computing Hubbard parameters within the “self-consistent” scheme.

a LRT (DFPT)⁶⁵, and b eACBN0⁴⁴. In the LRT workflow, Hubbard parameters are evaluated self-consistently after the electronic charge density has been fully converged. In contrast, the eACBN0 scheme updates the charge density and the Hubbard parameters simultaneously until convergence is reached. In both workflows, atomic positions are fixed, and no structural relaxations are performed.

We note that results obtained using the “one-shot” scheme are not reported in this work. In this scheme, the occupation matrices of Eq. (3) and Kohn-Sham wavefunctions employed to compute the Hubbard parameters (via Eqs. (6) and (7) for LRT and Eqs. (10) and (11) for eACBN0) differ from those that are obtained during the final DFT+U+V ground-state SCF calculation, which is determined by the Hubbard energy [Eq. (2)] and the Hubbard potential [Eq. (5)]. This difference between the Hubbard parameters evaluation step and the subsequent materials-property calculations makes the “one-shot” scheme not consistent. Moreover, the eACBN0 method is never used in a “one-shot” form in practice. To avoid potential ambiguity for the reader, we therefore do not include “one-shot” results in this work.

Using the computed self-consistent Hubbard parameters, we perform DFT+U and DFT+U+V calculations to evaluate various material properties. The band gap serves as our primary benchmark for accuracy. The experimental band gaps used for comparison were taken from the literature and were obtained using different techniques depending on the material: X-ray emission spectroscopy (XES) and X-ray absorption spectroscopy (XAS) for ZnO⁸⁵, MnO, CoO, and NiO⁸⁶; photoemission spectroscopy (PES) and Bremsstrahlung isochromat spectroscopy (BIS) for MnO⁸⁷, CoO⁸⁸, and NiO⁸⁹; conductivity for MnO⁹⁰, CoO⁹¹, and NiO⁹²; ultraviolet photoelectron spectroscopy (UPS) and inverse photoemission spectroscopy (IPS) for ZrO₂⁹³; and optical absorption edge measurements for FeO⁹⁴ and BaTiO₃⁹⁵. We assume that excitonic effects are weak in BaTiO₃ due to the relatively large dielectric screening, so that the measured absorption edge closely approximates the fundamental band gap with minimal influence from excitonic states. In contrast, FeO exhibits substantially weaker dielectric screening than BaTiO₃, which in principle could lead to more pronounced electron-hole interactions. However, we make an assumption that the optical response in FeO is dominated by the valence-conduction interband excitations, so that well-defined excitonic features are not expected to be the primary determinant of the absorption edge. In the absence of reliable experimental determinations of the fundamental gap in BaTiO₃ and FeO, we therefore use the reported optical absorption edge as an approximate proxy, while noting that it may include many-body effects. For magnetic oxides, we also examine magnetic moments, while for BaTiO₃, we additionally analyze structural properties relevant to ferroelectricity. The results are systematically compared with experiments.

Local and semilocal XC functionals (LSDA, σ-GGA) notoriously underestimate band gaps in TM oxides (TMOs) due to strong SIEs, particularly on TM ions. DFT+U can alleviate this issue, but only when the correction is applied to states that define the gap edges (for insulators) or to states near the Fermi level (for spurious DFT metals)⁵³. The inter-site V correction can further improve band gaps in covalent systems, but again, only when applied to orbitals directly involved in gap formation. Prior work supports these points: ref. ⁵³ showed that DFT+U with LRT-derived U improves agreement with experimental band gaps compared to (semi-)local functionals, while DFT+U+V with LRT-derived parameters yields even better accuracy in TMOs with strong covalency^52,66,67,69. For p-block semiconductors without TM ions, ref. ⁴⁴ demonstrated that DFT+U+V with eACBN0-derived parameters accurately reproduces experimental band gaps. These insights provide the foundation for our analysis below.

MnO, FeO, CoO, and NiO

TMOs are well-known systems where SIEs are prominently manifested due to localized 3d orbitals. Standard DFT calculations at the LDA or GGA level typically exhibit considerable discrepancies compared to experimental measurements, motivating the use of the DFT+U or DFT+U+V method. Particularly, 3d electrons of TM ions exhibit significant spatial localization, which can be effectively treated using the Hubbard U corrections, while V might be necessary to describe their hybridization with ligands. Therefore, we select representative 3d TMOs (MnO, FeO, CoO, and NiO) as first targets and apply two Hubbard parameter calculation methods, LRT and eACBN0, to understand the trends in Hubbard parameters and their influence on materials properties such as band gaps and magnetic moments.

To compare the two methods under as equivalent conditions as possible, we perform LRT and eACBN0 calculations using the self-consistent scheme described in Fig. 1. The resulting Hubbard parameters are reported in Figs. 2 and 3. We first focus on the U values obtained within DFT+U, shown as unhatched bar plots in Fig. 2. We find that the U_TM values obtained from LRT and ACBN0 are in excellent agreement for MnO, CoO, and NiO, with differences not exceeding 0.3 eV. In contrast, a significantly larger discrepancy of about 1.5 eV is observed for FeO. To understand this behavior, it is important to consider the role of the pseudopotentials used in our calculations. Specifically, we employ pseudopotentials from the GBRV library (v1.5)⁹⁶, where the TM ions are represented in ionized configurations: 3d⁵4s^1.5 for Mn, 3d⁵4s² for Fe, 3d⁷4s¹ for Co, and 3d⁸4s⁰ for Ni. Notably, for Mn, Co, and Ni, the ionization affects the 4s shell relative to the neutral atom, whereas for Fe, it affects the 3d shell. This distinction is crucial because the computed U_TM values are sensitive to the choice of pseudopotentials when Hubbard projectors are constructed from (orthogonalized) atomic orbitals derived from them, as is the case here. In particular, previous studies have shown that, within LRT, variations in the ionization of the 3d states can lead to changes in U_TM of up to 2 eV for Fe²⁰. Therefore, to further investigate this effect, in Sec. S3 in the SI, we report results obtained using the PSlibrary (v0.3.1)⁹⁷, in which the TM pseudopotentials are not ionized. In this case, the agreement between LRT and ACBN0 for FeO improves significantly, with deviations reduced to about 0.3 eV. However, this improvement comes at the expense of a larger discrepancy for MnO, CoO, and NiO, reaching up to 1.8 eV for NiO. These observations highlight the importance of paying attention to pseudopotentials when comparing Hubbard parameters obtained from different methods. Overall, U_TM computed within LRT is considerably more sensitive to the choice of pseudopotentials (and thus to the definition of Hubbard projectors), whereas ACBN0 exhibits a much weaker dependence, with variations not exceeding 0.4 eV for the TMOs considered here. Returning to Fig. 2, and excluding the LRT result for FeO, a clear trend emerges: U_TM increases from MnO to NiO, following the increasing occupation of the 3d shell. A notable feature of the eACBN0 results is that the converged U_TM values closely track the nominal electron count of the divalent TM cations. This behavior is consistent with the eACBN0 formalism, in which the effective on-site interaction is constructed from renormalized occupations. Consequently, U_TM naturally reflects the filling of the TM-3d states.

Fig. 2. DFT+U and DFT+U+V self-consistent calculations for TMOs without on-site corrections on O-2p states (U_O), using the workflow described in Fig. 1.

a On-site Hubbard U_TM parameter for 3d orbitals of TM ions and inter-site Hubbard V_TM,O parameter between 3d orbitals of TM ions and 2p orbitals of oxygen, b band gap, and c magnetic moment. LRT and eACBN0 denote the method used to obtain the Hubbard parameters. The experimental band gaps and magnetic moments for MnO^{86,87,90,114,115}, FeO^94,116,117, CoO^{86,88,91,118,119}, and NiO^{86,89,92,114,115} are indicated by the dashed blue lines in panels (b, c), respectively.

Fig. 3. DFT+U and DFT+U+V self-consistent calculations for TMOs with on-site U_O corrections on O-2p states.

a On-site Hubbard U_TM and U_O parameters for 3d orbitals of TM ions and 2p orbitals of oxygen, respectively, and inter-site Hubbard V_TM,O parameter between 3d orbitals of TM ions and 2p orbitals of oxygen, b band gap, and c magnetic moment. LRT and eACBN0 denote the method used to obtain the Hubbard parameters. The experimental band gaps and magnetic moments for MnO^{86,87,90,114,115}, FeO^94,116,117, CoO^{86,88,91,118,119}, and NiO^{86,89,92,114,115} are indicated by the dashed blue lines in panels (b, c), respectively.

We now discuss the magnetic moments and band gaps obtained within the DFT+U framework, shown in Figs. 2(b, c). Starting with the magnetic moments, we observe a decrease with increasing 3d shell occupancy, i.e., when going from MnO to NiO, as expected. Since the values of U_TM obtained from LRT and ACBN0 are very similar for these systems (with the exception of FeO, as discussed above), the corresponding magnetic moments are also in close agreement. Notably, even for FeO, where the difference in U_TM between LRT and ACBN0 is relatively large, the computed magnetic moments remain very similar, indicating that they are not strongly sensitive to the value of U_TM. Overall, the calculated magnetic moments are in good agreement with available experimental data, with the largest discrepancy observed for CoO, of about 0.6 − 1.2 μ_B. Turning to the band gaps, we find that the predictions based on U_TM from LRT and ACBN0 are generally consistent, with the largest deviation again occurring for FeO, because it shows the largest difference in U_TM. The computed band gaps are in relatively good agreement with some experimental data for FeO, CoO, and NiO, while larger discrepancies are found for MnO, amounting to approximately 2 eV. Focusing on MnO, the lowest conduction band is dominated by Mn-4s character and is therefore largely insensitive to the Hubbard U_TM applied to Mn-3d states. The primary effect of U_TM is to increase the splitting between occupied and unoccupied Mn-3d states. This is confirmed by the partial density of states (PDOS), which shows that, with increasing U_TM, the occupied 3d states shift to lower energies, while the unoccupied 3d states (located above the conduction band minimum) shift to higher energies⁷⁵. This mechanism is general and also applies to other TM oxides such as FeO, CoO, and NiO, which exhibit a significant TM-4s contribution at the conduction band minimum at the Γ point. For these materials, although the energy splitting between occupied and unoccupied 3d states increases proportionally with U_TM, the resulting change in the fundamental band gap remains significantly smaller.

We now turn to the case where inter-site Hubbard interactions (V_TM,O) between TM-3d and O-2p states are included. Within LRT, the determination of V_TM,O requires oxygen to be treated as an additional perturbed subspace. As a result, the response matrices are modified, since additional screening channels involving O states are activated. In LRT, U_TM and U_O appear as diagonal elements, and V_TM,O as off-diagonal elements of the matrix obtained by inverting the interacting and non-interacting response matrices, χ and χ₀ [Eqs. (6) and (7)]. Once both TM and O sites are perturbed, the structure of the response matrices in DFT+U and DFT+U+V becomes effectively similar: both schemes capture the same screening pathways involving TM and O states. An important practical aspect is whether the on-site Hubbard term U_O should be retained in the self-consistent DFT+U and DFT+U+V cycles or excluded. In Fig. 2, we present results obtained when U_O is disregarded, while Fig. 3 shows the corresponding results when U_O is included. The choice of whether to include U_O is not unique in practical calculations. It depends on the material under study, the property of interest, and the specific method used to compute the Hubbard parameters^44,45,53,98.

When the inter-site Hubbard interactions (V_TM,O) between TM-3d and O-2p states are included, the differences between the two methods remain almost unchanged. In Fig. 2, the results with the V_TM,O term are illustrated with hatched patterns. We observe that the inclusion of inter-site interactions within both LRT and ACBN0 does not significantly modify the values of U_TM, as evidenced by the close agreement between the hatched and unhatched bars of the same color in Fig. 2(a). This weak sensitivity originates from the self-consistent determination of the Hubbard parameters: although the inclusion of V can modify the U_TM values at the initial stage of the calculation, these differences are largely reduced once the Hubbard parameters and the charge density are updated self-consistently. We note, however, that this behavior depends on the system. In other materials, the inclusion of inter-site interactions may lead to more pronounced changes in U_TM even within a self-consistent scheme (see e.g., Fig. 4). Next, LRT yields relatively small V_TM,O values (below 1 eV), resulting in negligible differences in band gaps and magnetic moments whether V_TM,O term is included or not in the total energy calculation. In contrast, eACBN0 consistently produces larger V_TM,O values (close to 3 eV) across all TMOs, resulting in more noticeable changes in band gaps compared to LRT, while the magnetic moments change by a small margin. This effect arises because the V_TM,O term facilitates electron delocalization into the overlapping regions between atomic pairs. This counteracts the strong localization favored by the U_TM term and thus reduces the local magnetic moment, as previously shown through charge density difference analysis in our earlier studies⁷⁵. Figure 3 extends our analysis by including a Hubbard correction on the O-2p states (U_O), computed with both LRT and eACBN0. Within LRT, the resulting U_O values are large (~ 10 eV), consistent with previous studies⁵³. This can be understood in terms of oxygen’s large chemical hardness, arising from its high ionization energy and negative electron affinity. Therefore, the O-2p subspace exhibits a weak occupation response to local perturbations. Within the LRT framework, this weak response naturally results in a large Hubbard U_O^59,80,99. Despite these large U_O values, the inter-site interaction V_TM,O remains small (below 1 eV), indicating that TM–O hybridization has only a minor effect on the electronic structure of these TMOs even when U_O is considered. Importantly, the value of U_TM can change when U_O is included in the calculation. Comparing Figs. 2 and 3, we observe a slight increase of U_TM for CoO and NiO (by 3–5%), a substantial increase for FeO (18%), and almost no change for MnO. Conversely, eACBN0 yields smaller U_O values than LRT ones, in the range of 2 − 3 eV. Similar to the case with the V_TM,O correction, the converged Hubbard parameters can vary depending on the Hubbard manifolds being considered. In this self-consistent cycle, the parameters from one iteration feed back into the next as a correction term, thereby influencing the KS states and the final converged values. Comparing Figs. 2 and 3 for eACBN0, the changes are likewise system dependent and can be small increases, decreases, or negligible variations, depending on whether the DFT+U or DFT+U+V scheme is employed.

Fig. 4. DFT+U and DFT+U+V self-consistent calculations for ZrO₂ without on-site corrections on O-2p states (U_O).

a On-site Hubbard U_Zr parameter for 4d orbitals of Zr ions and inter-site Hubbard V_Zr,O parameter between 4d orbitals of Zr ions and 2p orbitals of oxygen, and b band gap. The experimental band gap⁹³ is shown by the dashed blue line in panel (b).

The effects of V_TM,O and U_O on the magnetic moments are not very significant. The predicted magnetic moments for the TMOs are consistent regardless of computational schemes, and they agree well with the experimental range, irrespective of the inclusion of V_TM,O. Conversely, the influence of V_TM,O is pronounced on the band gap. For eACBN0, the inclusion of V_TM,O introduces a non-negligible difference for FeO, CoO, and NiO. Nevertheless, the inclusion of U_O has a greater overall impact on the band gap magnitude than V_TM,O does. Without U_O (Fig. 2), both methods tend to underestimate the band gaps of MnO and NiO, while results for FeO and CoO are generally in good agreement with experiment. Including U_O leads to a significant increase in the band gaps within LRT, whereas the corresponding increase within ACBN0 is more moderate (Fig. 3). For MnO, LRT yields a band gap that is close to the experimental values, while eACBN0 still underestimates it. In contrast, for FeO, the LRT band gap overshoots the experimental value by approximately 2eV, whereas ACBN0 remains in good agreement with experiment. For CoO, LRT overestimates the lower bound of the experimental band gap range but agrees well with the upper bound, while ACBN0 continues to match the lower bound. Finally, for NiO, LRT overestimates the experimental band gap by about 1 − 1.5 eV, whereas ACBN0 falls within the experimental range.

Finally, following the protocol in Fig. S1 in the SI⁶⁵, we compute the self-consistent Hubbard parameters and their corresponding optimized structural parameters using both LRT and eACBN0. The results are presented in Figs. S2 and S3, and their discussion is in the SI. For both methods, the structural parameters (lattice constants and rhombohedral angles) are generally in good agreement with available experimental data.

ZrO₂ and BaTiO₃

To further compare the LRT and eACBN0 methods, we examine ZrO₂ and BaTiO₃, where both Zr⁴⁺ and Ti⁴⁺ ions have nominal d⁰ electronic configurations. Since these materials are nonmagnetic, the accuracy of the predictions will be assessed based on the band gap as well as the structural properties in the case of BaTiO₃. Figure 4 shows the results for ZrO₂ without the U_O correction. In LRT, the U parameter for Zr-4d states (U_Zr) is ~ 3 eV, reflecting the finite response arising from the small but nonzero occupancy of Zr-4d states in the PBEsol ground state. In contrast, ACBN0, being more sensitive to the small residual occupancy of the nominal d⁰ states, yields a much smaller U_Zr value of ~ 0.3 eV, which is one order of magnitude smaller than the LRT value. Consequently, the very small U_Zr obtained from ACBN0 leads to a significantly smaller band-gap correction in DFT+U compared to that obtained using LRT [see Fig. 4(b)].

For the inter-site interactions between Zr-4d and O-2p states (V_Zr,O), LRT yields values below 1 eV, comparable to those obtained for the TMOs discussed in the previous section. Including these interactions in the DFT+U+V calculation increases the band gap by 7% relative to DFT+U with only U_Zr from LRT. In contrast, eACBN0 produces substantially larger V_Zr,O values (2.3-2.5 eV), leading to a 22% band-gap increase compared to DFT+U with only U_Zr, as shown in Fig. 4(b). Since V_Zr,O depends on the interatomic distances between Zr and its nearest-neighbor O atoms, a range of values is reported in Fig. 4(a) rather than a single number. Despite these V_Zr,O corrections, both methods still underestimate the experimental band gap by more than 1 eV⁹³.

Including U_O substantially increases the band gap, as shown in Fig. 5. Both methods (LRT and eACBN0) yield large U_O values in the range of 7–9 eV, with LRT producing the higher values and eACBN0 the lower ones. In the eACBN0 formalism, the two slightly inequivalent O sublattices in ZrO₂ are included in the same O-species renormalization factor, while the projections of the KS states onto the local atomic orbitals remain site-dependent and thus retain sensitivity to the distinct local environments. Incorporating these U_O corrections within either DFT+U or DFT+U+V consistently increases the band gap relative to calculations without U_O, bringing the results closer to the experimental value. Overall, LRT-based predictions (whether or not V_Zr,O is included) tend to overestimate the band gap, whereas eACBN0-based predictions underestimate it within DFT+U and slightly overestimate it within DFT+U+V.

Fig. 5. DFT+U and DFT+U+V self-consistent calculations for ZrO₂ with on-site U_O corrections on O-2p states.

a On-site Hubbard U_Zr and U_O parameters for 4d orbitals of Zr ions and 2p orbitals of oxygen, respectively, and inter-site Hubbard V_Zr,O parameter between 4d orbitals of Zr ions and 2p orbitals of oxygen, and b band gap. The experimental band gap⁹³ is shown by the dashed blue line in panel (b).

The fully relaxed structures of ZrO₂ are well described by both methods, as shown in Fig. S4, and they are only weakly affected by the inclusion of V_Zr,O and U_O. We also note that the method-dependent variations of the “structurally self-consistent” Hubbard parameters during structural relaxation follow the same trends discussed above for the “self-consistent” case at a fixed PBEsol geometry.

Next, we turn to BaTiO₃, which is similar to ZrO₂ in the sense that Ti⁴⁺ also has a nominal d⁰ configuration. In this case, we consider not only the band gap but also the crystal-structure properties, which are highly sensitive to Hubbard corrections^69,100. BaTiO₃ is a well-known ferroelectric material, exhibiting a rhombohedral distortion at low temperatures. However, standard DFT+U with sizable U corrections for Ti-3d states incorrectly stabilizes a centrosymmetric cubic structure, thereby suppressing its intrinsic polarization¹⁰¹. However, including the inter-site interactions (V_Ti,O) between Ti-3d and O-2p states preserves their hybridization, which is essential for maintaining the rhombohedral distortion and thus restoring the polarization^69,100.

In Fig. 6 we present the results obtained using the “self-consistent” scheme described in Fig. 1 without the U_O correction. Within LRT, the Hubbard parameter for the Ti-3d states (U_Ti) is ~ 6 eV, reflecting the finite response associated with the small but nonzero 3d occupancy in the PBEsol ground state, similarly to the behavior observed in ZrO₂. Since Ti-3d states are more localized than Zr-4d states, the resulting U_Ti is roughly twice as large as U_Zr in LRT. In contrast, ACBN0 yields a very small U_Ti of ~ 0.3 eV, identical to U_Zr, since the Ti-3d and Zr-4d occupancies are both very low. Thus, ACBN0 appears largely insensitive to the degree of d-state localization (3d vs. 4d), depending primarily on the actual orbital occupancies. As shown in Fig. 6, these differences lead to markedly different DFT+U outcomes: LRT-based DFT+U underestimates the band gap by ~ 13%, whereas ACBN0-based DFT+U underestimates it by ~ 31%. This picture changes substantially when V_Ti,O is included. The V_Ti,O values obtained from LRT and eACBN0 are similar in magnitude to those found for ZrO₂. With these inter-site interactions included, both methods yield band gaps in excellent agreement with experiment within DFT+U+V. Thus, an accurate band gap is achieved either through a combination of a large U_Ti and small V_Ti,O (as in LRT) or a small U_Ti and large V_Ti,O (as in eACBN0).

Fig. 6. DFT+U and DFT+U+V self-consistent calculations for BaTiO₃ without on-site corrections on O-2p states (U_O).

a On-site Hubbard U_Ti parameter for 3d orbitals of Ti ions and inter-site Hubbard V_Ti,O parameter between 3d orbitals of Ti ions and 2p orbitals of oxygen, and b band gap. The experimental band gap⁹⁵ is shown by the dashed blue line in panel (b).

As shown in Fig. 7(a), the U_O values obtained from LRT and eACBN0 are comparable to those for ZrO₂ (see Fig. 5(a)). Including these U_O corrections has a strong impact on the band gap of BaTiO₃. In particular, when U corrections are applied to both Ti-3d and O-2p states (and even when the inter-site term V_Ti,O is also included), both methods substantially overestimate the band gap (Fig. 7(b)). The only exception is the DFT+U calculation with ACBN0-derived U values, which still underestimates the band gap but by a smaller margin (7%).

Fig. 7. DFT+U and DFT+U+V self-consistent calculations for BaTiO₃ with on-site U_O corrections on O-2p states.

a On-site Hubbard U_Ti and U_O parameters for 3d orbitals of Ti ions and 2p orbitals of oxygen, respectively, and inter-site Hubbard V_Ti,O parameter between 3d orbitals of Ti ions and 2p orbitals of oxygen, and b band gap. The experimental band gap⁹⁵ is shown by the dashed blue line in panel (b).

Figure S5 in the SI shows the results of the “structurally self-consistent” calculations for BaTiO₃. The Hubbard parameters U_Ti and V_Ti,O are only slightly modified compared with those obtained using the PBEsol geometry (compare Fig. S5(c) with Fig. 6(a)). Without U_O, the two methods exhibit different structural trends: LRT-based DFT+U predicts a cubic phase (α = 90⁰), whereas ACBN0-based DFT+U retains the rhombohedral distortion (α < 90⁰), as shown in Fig. S5(b). When V_Ti,O is included, both methods capture the rhombohedral distortion, but their quantitative predictions differ. Specifically, DFT+U+V based on LRT parameters yields a lattice parameter, rhombohedral angle, and band gap in very good agreement with experiment, whereas DFT+U+V based on eACBN0 parameters shows significantly larger deviations for all three properties.

When U_O is included, the structural behavior changes, as shown in Figs. S5 (e)–(h). In this case, LRT fails to capture the ferroelectric nature of BaTiO₃: the large U_Ti and U_O values overly localize the Ti-3d and O-2p states, respectively, driving the system into a non-polar cubic phase even when the small inter-site interaction V_Ti,O is added (Fig. S5 (f)). A similar trend occurs in ACBN0-based DFT+U calculations, where the combined inclusion of small U_Ti and large U_O suppresses the polarization and restores cubic symmetry. However, unlike LRT, adding large V_Ti,O within the eACBN0 framework recovers the rhombohedral distortion and the associated polarization, consistent with previous findings¹⁰⁰. Nonetheless, the resulting lattice parameter and band gap remain substantially overestimated.

These results highlight that, although both approaches rely on similar Hubbard correction schemes, they lead to qualitatively different effects on structural and electronic properties of BaTiO₃. Namely, LRT yields the most accurate properties when U_Ti and V_Ti,O are included but U_O is excluded, whereas eACBN0 achieves its best accuracy when all correction terms are included.

ZnO

After examining TMOs with partially filled d states (from Mn²⁺ to Ni²⁺) and then early TM compounds with nominal d⁰ configurations (Ti⁴⁺ and Zr⁴⁺), we now turn to a material with a late d-block element: ZnO, where Zn²⁺ has a nominally filled d¹⁰ configuration. ZnO represents the opposite extreme of the d-electron occupancy spectrum relative to the d⁰ systems, offering a distinctly different test case for evaluating Hubbard parameters.

We first discuss the results for ZnO without the U_O correction. As shown in Fig. 8, the LRT method yields an unphysically large self-consistent U_Zn exceeding 100 eV. Even without performing the full self-consistent cycle (see Fig. 1), i.e., using the “one-shot” scheme, the LRT estimate already exceeds 25 eV. This well-known limitation of LRT arises from the fully occupied Zn-3d shell, which leads to spuriously large U values⁹⁹. The “structurally self-consistent” scheme produces a similarly large U_Zn, as shown in Fig. S6. It is important to emphasize that in ZnO, the valence-band maximum is primarily O-2p in character, while the conduction-band minimum is dominated by Zn-4s states. As a result, applying U_Zn to the Zn-3d states influences the band gap only indirectly. In fact, several studies have emphasized that applying a Hubbard correction to the Zn-4s subspace can be more effective than correcting only the deep Zn-3d states^102,103. Related evidence that SIEs in metal 4s orbitals can be non-negligible has also been reported in molecular systems¹⁰⁴. Indeed, LRT-based DFT+U increases the PBEsol band gap from ~ 0.7 eV to 1.94 eV, which remains well below the experimental value of 3.3 eV⁸⁵. In contrast, the ACBN0 approach also predicts a large but more moderate U_Zn of ~ 16 eV, resulting in a slightly smaller band gap of 1.52 eV, as shown in Fig. 8(b).

Fig. 8. DFT+U and DFT+U+V self-consistent calculations for ZnO without on-site corrections on O-2p states (U_O).

a On-site Hubbard U_Zn parameter for 3d orbitals of Zn ions and inter-site Hubbard V_Zn,O parameter between 3d orbitals of Zn ions and 2p orbitals of oxygen, and b band gap. The experimental band gap⁸⁵ is shown by the dashed blue lines in panel (b).

For the inter-site interactions between Zn-3d and O-2p states (V_Zn,O), LRT yields small negative values. These negative values likely originate from numerical instabilities during the inversion of response matrices whose elements are close to zero and, therefore, highly sensitive to numerical noise. In contrast, eACBN0 produces significantly larger and more stable V_Zn,O values of about 3 eV, consistently with the trends observed for the other materials discussed above. However, these inter-site corrections have no effect on the band gap, as shown in Fig. 8(b). This insensitivity arises because the band gap in ZnO is defined between O-2p (valence) and Zn-4s (conduction) states, while the Zn-3d states lie much deeper in energy. As a result, the V_Zn,O term mainly alters the hybridization between the O-2p and Zn-3d states but does not influence the band edges, leaving the band gap unchanged.

Including U_O in addition to U_Zn, as shown in Fig. 9, leads to markedly different outcomes. In LRT, the already unphysical U_Zn value combined with a large U_O ~ 11 eV results in a band gap that overestimates the experimental value by 44%. In contrast, ACBN0 yields still large but more moderate values, U_Zn ~ 15 eV and U_O ~ 7 eV, which together produce a band gap in excellent agreement with experiment. As before, adding the inter-site term V_Zn,O has no influence on the band gap for the reasons discussed above. These results show that the band gap is highly sensitive to U_O, since this correction directly shifts the O-2p states that form the valence-band maximum in ZnO.

Fig. 9. DFT+U and DFT+U+V self-consistent calculations for ZnO with on-site U_O corrections on O-2p states.

a On-site Hubbard U_Zn and U_O parameters for 3d orbitals of Zn ions and 2p orbitals of oxygen, respectively, and inter-site Hubbard V_Zn,O parameter between 3d orbitals of Zn ions and 2p orbitals of oxygen, and b band gap. The experimental band gap⁸⁵ is shown by the dashed blue lines in panel (b).

When structural relaxations are included, both the Hubbard parameters and the resulting band gaps follow similar trends as described above (see Fig. S6). We also note that, with these Hubbard corrections, both methods reproduce the experimental structural parameters very well (see Fig. S6 (a) and (d)).

Discussion

Based on the numerical results presented in the previous section, we now provide a discussion and interpretation of the observed trends in the behavior of the on-site U and inter-site V Hubbard parameters obtained from LRT and eACBN0. Despite the fundamentally different mathematical formulations and physical concepts underlying these two methods, there are cases where they yield comparable U values, as well as cases where the results diverge dramatically. By contrast, the inter-site V parameters systematically exhibit large discrepancies between the two approaches. In what follows, we analyze these similarities and differences in more detail.

We start with the on-site U_TM values. For TMOs with partially occupied 3d shells, LRT and eACBN0 predict similar U_TM values except U, for Fe in FeO that differs by 2–3 eV. This discrepancy can be largely attributed to the choice of pseudopotentials, which determine the Hubbard projectors and can significantly influence the computed U_TM values, particularly in LRT. As a result, the level of agreement between LRT and eACBN0 can vary depending on the degree of ionization and on the specific construction scheme of the pseudopotentials. However, when the 3d shell is nearly empty or fully occupied, the two methods behave in fundamentally different ways. In particular, for nominally empty 3d states, LRT still produces finite and sizable U_TM values in the range of 3-6 eV, while for fully occupied 3d states, it yields unphysically large values. The latter is a well-known limitation of the LRT framework⁹⁹: for filled shells, the occupation matrices exhibit vanishing variations under a local electronic perturbation, leading to nearly singular response matrices whose inversion generates spuriously large U_TM values. In ZnO, this behavior can also be interpreted as an indication that the Zn-3d states are not the most relevant ones for the band edge, since the conduction band minimum has predominantly Zn-4s character. Consequently, applying Hubbard corrections directly to the Zn-4s manifold has been suggested as a useful strategy to better target the states controlling the band edges in ZnO^102,103. The behavior of eACBN0 is markedly different: It produces U_TM values that are finite and in the range of physical values even for closed shells. The root cause of this behavior lies in the definition of the auxiliary objects—the renormalized occupation numbers [Eq. (13)] and renormalized density matrices [Eq. (12)]. These quantities are constructed to ensure that the sum of the Mulliken charge of each atom and the interatomic charge reproduces the correct total electronic population. As a consequence, the Hubbard parameters derived from eACBN0 naturally become very small for nearly empty shells, while for fully filled 3d shells they approach values of ~ 16 eV. This trend is not mirrored in LRT because of the different physical idea: U_TM is computed from the occupation-matrix variations in response to local electronic perturbations. For nearly empty shells, even an infinitesimal local perturbation can induce relatively large fractional changes in the occupation matrix, which translates into sizable U_TM values. Conversely, for fully occupied shells, the occupation matrix is extremely rigid, and the small changes captured by the linear-response equations produce nearly vanishing matrix elements. The inversion of these nearly singular matrices amplifies the response artificially, resulting in unphysically large U_TM values. Thus, while LRT and eACBN0 can yield comparable results for partially filled shells, their behavior diverges sharply at the two extremes of the d-electron occupation spectrum.

Regarding the on-site Hubbard U_O parameter for O-2p states, as obtained from LRT and eACBN0, we observe notable differences. On the one hand, LRT consistently predicts large U_O values, close to 10 eV, essentially independent of the specific compound under investigation. This is consistent with the intrinsically large chemical hardness of oxygen, which makes the O-2p subspace weakly respond to the local perturbations and relatively insensitive to the chemical environment, as discussed by Moore et al.⁵⁹. In contrast, the eACBN0 approach provides a much broader and more system-dependent distribution of U_O values, ranging from 2 to 8 eV. Specifically, U_O is about 2–3 eV for TMOs, but increases to 7–8 eV in ZnO, ZrO₂ and BaTiO₃. These results indicate that the U_O value does not follow the same trend as the U_TM values, which generally scale with the electron count within eACBN0. This behavior can be rationalized by considering that the degree of localization and hybridization of O-2p states varies strongly across materials. In some cases, such as in ZrO₂ and BaTiO₃, the oxygen states are more localized, resulting in larger U_O values. Additionally, it is important to account for the fact that O-2p states participate in significant charge redistribution and hybridization with TM-3d. This naturally calls for the inclusion of inter-site V terms in order to properly capture the covalency of such interactions, since purely on-site corrections are insufficient to accurately describe the electronic interactions.

Next, we turn to the inter-site V interactions between TM-d and O-p states in the studied materials. From our LRT calculations, we find that for all compounds except ZnO, V values are typically around 1 eV or smaller, while in ZnO, they are small and negative. Such values are characteristic of LRT: V corresponds to the off-diagonal elements of the response matrix, describing how the occupation on a given atom changes when a neighboring atom is perturbed. Since these inter-site responses are much weaker than the on-site ones (which define U), the resulting V is generally small. In ZnO, the small negative V originates from numerical instabilities caused by inverting nearly singular response matrices. In contrast, eACBN0 yields a markedly different picture: The inter-site V values are much larger, often comparable in magnitude to the on-site U. For instance, in TMOs with partially occupied 3d shells and in ZnO, V is 2–5 times smaller than the corresponding U_TM. Even more strikingly, in ZrO₂ and BaTiO₃, V exceeds the U_TM values of the nominally empty d states by a factor of ~ 10. This behavior stems from the fact that in eACBN0, the formation of a chemical bond involves charge redistribution between atoms, placing U and V on a more equal footing than in LRT. Indeed, Eq. (13) shows that V^IJ scales with the sum of Mulliken charges of atoms I and J, and is therefore not intrinsically smaller than its on-site counterpart. Moreover, since eACBN0 derives U and V from energy contributions that depend on the spatial distribution of electronic charges, bonding tends to lower on-site U while enhancing inter-site V. In this framework, the ratio U/V reflects the balance between atomic and bond-centered charges, and is thus directly tied to bond strength. Consequently, V remains sizable relative to U in all systems. By contrast, in LRT, the U and V parameters emerge from occupation matrix responses, where the dominant contribution is always local. As a result, U is much larger than V, with ratios often an order of magnitude.

Finally, we discuss the sensitivity of the band gaps to the Hubbard parameters obtained from LRT and eACBN0. For FeO and CoO, applying U on the TM ions alone yields band gaps in good agreement with experiment, whereas for MnO and NiO, the gaps are somewhat underestimated. In these cases, the inter-site V contribution is negligible. The inclusion of U on oxygen strongly affects the predicted gaps. LRT tends to overestimate the experimental values for FeO and NiO, while yielding good agreement for MnO and for the upper bound of the experimental band gap range in CoO. Conversely, eACBN0 shows good agreement with experimental values for FeO and NiO, as well as with the lower bound of the CoO band gap, while significantly underestimating the band gap of MnO. These trends directly reflect the substantial differences in the values of U on oxygen obtained from the two methods. For ZrO₂, applying U only on the TM ion underestimates the band gap for both LRT and eACBN0, and the inclusion of V has only a small effect. Adding U on oxygen generally increases the gap, but this often leads to an overestimation relative to the experiment. For BaTiO₃, applying U only on the TM ion while including V produces accurate band gaps for both methods, whereas adding U on oxygen significantly overestimates the gaps. For ZnO, applying U on TM ions (with or without V) underestimates the band gaps in both LRT and eACBN0. Including U on oxygen, however, overshoots the gaps in LRT while eACBN0 yields results in good agreement with experiment. Overall, these trends indicate that the predicted band gaps are most sensitive to the treatment of oxygen states, while inter-site V typically has a smaller effect. In particular, LRT tends to produce larger U_O values than eACBN0, often leading to overestimation of the gaps when oxygen is corrected. Conversely, eACBN0 often provides a more balanced description across different materials. These observations suggest that careful consideration of which Hubbard terms to include (especially on oxygen) can provide practical guidance for achieving accurate band gaps in TM compounds, highlighting a pathway for improved computational predictions in related materials.

In summary, we have carried out a systematic comparison of two widely used approaches, LRT and eACBN0, for computing on-site U and inter-site V Hubbard parameters across three classes of materials spanning nearly empty, partially filled, and fully occupied TM-d shells. These two methods rest on fundamentally different physical pictures and mathematical frameworks, yet our analysis reveals both points of agreement and distinct discrepancies. For TMOs with partially occupied d states, the two methods yield broadly consistent U values, whereas for nearly empty or fully filled d shells, the disagreement becomes substantial, reflecting intrinsic differences in how each method treats electronic interactions. For the O-2p states, LRT systematically predicts large U values independent of chemistry, while eACBN0 produces a much wider spread, sensitive to localization and hybridization with TM-d states. The contrast is even sharper for the inter-site V: LRT consistently yields small values (~ 1 eV or less), while eACBN0 predicts much larger values, often comparable to on-site U. These differences can be traced to the distinct mathematical assumptions underpinning the definitions of U and V, and to the different physical insights each method encodes—LRT focusing on response functions, eACBN0 on energetics and charge redistribution. Since the Hubbard parameters obtained from these approaches are ultimately used within the same DFT+U or DFT+U+V energy functional, they are often implicitly assumed to represent the same physical quantity. However, the differences observed here suggest that their interpretation may, at least in part, remain method dependent, particularly in how the screening processes are captured for a given electronic configuration. Overall, our study highlights that while certain parallels exist for the considered set of materials, the divergences between LRT and eACBN0 are pronounced and can affect the predicted material properties for specific cases. This work represents a first step toward a systematic understanding of how these methods compare and where they diverge. Bridging their conceptual and mathematical foundations and testing them across a broader range of materials and chemistry remains an important direction for future developments.

Methods

Computational details

All calculations were performed using the plane-wave pseudopotential method as implemented in the QUANTUM ESPRESSO (QE) distribution^105–107. For the xc functional, we adopted the σ-GGA with the PBEsol parametrization¹⁰⁸, and used pseudopotentials from the GBRV library (v1.5)⁹⁶. The KS wavefunctions and potentials were expanded in plane waves with kinetic-energy cutoffs of 65 and 780 Ry for the TMOs, and 60 and 600 Ry for ZnO, ZrO₂, and BaTiO₃. The first BZ was sampled using uniform Γ-centered k-point meshes: 10 × 10 × 10 for TMOs, 12 × 12 × 7 for ZnO, 7 × 7 × 6 for ZrO₂, and 8 × 8 × 8 for BaTiO₃. Spin-orbit coupling was neglected. Geometry optimizations were carried out using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm¹⁰⁹, with convergence criteria of 10⁻⁴ Ry for the total energy, 10⁻³ Ry/Bohr for forces, and 0.5 kbar for pressure in TMOs. For ZnO, ZrO₂, and BaTiO₃, criteria of 10⁻⁶ Ry for the total energy, 10⁻⁵ Ry/Bohr for forces, and 0.5 kbar for pressure were adopted. For antiferromagnetic TMOs, we estimated the magnetic moment per TM ion from the absolute magnetization at the end of the self-consistent calculations. Since the spin polarization is largely localized on the TM sites, we computed the moment per ion by dividing the converged absolute magnetization of the unit cell by the number of TM atoms.

The LRT Hubbard parameters were computed using DFPT^64,65 as implemented in the HP code¹¹⁰. Hubbard projectors were constructed from atomic orbitals orthogonalized using the Löwdin scheme⁷⁸. In this scheme, the Löwdin orthogonalization is applied to the full set of atomic orbitals provided by the pseudopotentials, rather than being restricted to the selected Hubbard manifolds. Uniform Γ-centered k- and q-point grids were employed, with sizes of 10 × 10 × 10 and 6 × 6 × 6 for TMOs, 12 × 12 × 7 and 6 × 6 × 3 for ZnO, 7 × 7 × 6 and 3 × 3 × 3 for ZrO₂, and 8 × 8 × 8 and 4 × 4 × 4 for BaTiO₃, respectively.

The eACBN0 Hubbard parameters were computed using our modified in-house QE package^44,74. For consistency with the LRT approach, the Hubbard projectors were constructed from atomic orbitals orthogonalized using the same Löwdin scheme⁷⁸. The same Γ-centered k-point meshes used for the electronic structure calculations were employed, and convergence thresholds of 10⁻¹⁰ and 10⁻¹⁵ Ry were applied for the total eACBN0 energy and SCF cycle, respectively.

The U and V parameters were obtained using two self-consistent protocols, summarized in Fig. 1 and Fig. S1 of the SI. In the main text, we report the values computed by iteratively evaluating U and V on the PBEsol-relaxed structures using the LRT and eACBN0 methods until convergence (the “self-consistent” scheme). In the case of ZnO, however, this iterative procedure was limited to two steps due to its tendency to diverge⁹⁹. For completeness, the SI also presents results from the “structurally self-consistent” scheme, in which DFT+U+V structural optimizations (including Hubbard contributions to forces and stresses¹¹¹) are performed, and the Hubbard parameters are recomputed until both the structure and the U and V values are converged. Moreover, for FeO and CoO, we employ occupation matrix control¹¹² by initializing the 3d occupation matrices with 3d⁶ and 3d⁷ configurations for Fe and Co, respectively. These occupations are appropriately distributed between spin-up and spin-down channels with integer eigenvalues. This procedure is necessary to ensure convergence to the correct insulating ground state⁶². Without it, the systems tend to converge to a metallic solution, leading to different values of the Hubbard parameters.

Author contributions

I.T. and Y.-W.S. led the project. W.Y. and I.T. developed methodology and software. W.Y., I.T., F.A., and Y.-W.S. contributed conceptualization and analysis. All authors wrote, reviewed and edited the manuscript and supplemented information.

Data availability

The data used to produce the results of this work is available in the Materials Cloud Archive [https://doi.org/10.24435/materialscloud:69-v6]¹¹³.

Code availability

Our modified in-house version of QUANTUM ESPRESSO, used for eACBN0 calculations, is available at https://github.com/KIAS-CMT/DFT-U-V. All LRT calculations, in contrast, were performed using the official QUANTUM ESPRESSO distribution, which can be accessed at https://www.quantum-espresso.org/.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41524-026-02123-9.