Pseudo-hybrid density functional ACBN0 for Hubbard U correction in a numeric atom-centered orbital basis

arXiv:2609.12198 · cond-mat.mtrl-sci · Submitted 2026-09-10 · Read on arXiv

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Introduction to the show: ident: Condensed Matter. The day in condensed matter: materials, phases and the measurements under them.

Kai: We're looking at the paper: "Pseudo-hybrid density functional ACBN0 for Hubbard U correction in a numeric atom-centered orbital basis".

Mira: Here's the gist of the paper. We present a formulation and implementation of the Agapito–Curtarolo–Buongiorno Nardelli (ACBN0) pseudo-hybrid density functional in a numeric atom-centered orbital basis The method is realized in the all-electron, full-potential electronic-structure package FHI-aims For a benchmark set of materials including metal oxides and nitrides, ACBN0 with the fully localized limit (FLL) as the double-counting treatment reduces the aggregate band gap errors relative to the Perdew–Burke–Ernzerhof (PBE) and strongly constrained and appropriately normed (SCAN) functionals, reaching an accuracy comparable to the Heyd–Scuseria–Ernzerhof functional (HSE06) The study introduces the ACBN0 pseudo-hybrid density functional implemented in FHI-aims using a numeric atom-centered orbital basis.

Kai: This implementation utilizes a Löwdin-orthogonalized projector to construct the occupation and renormalized density matrices, which improves the stability of the self-consistent ACBN0 iterations for tested systems.

Mira: The formulation defines the Hubbard correction operator Vˆ σ U in DFT+U method using projectors Pˆ onto a chosen localized subspace.

Kai: The choice of projector is particularly important because nonphysical occupations would directly affect the selfconsistent U¯ and J¯ parameters and could destabilize the coupled SCF–ACBN0 cycle.

Paper summary: Mira: Thinking about the title, "Pseudo-hybrid density functional ACBN0 for Hubbard U correction in a numeric atom-centered orbital basis," what we're really seeing is that they’ve taken an established concept, the ACBN0 pseudo-hybrid functional, and made it practical by implementing it in FHI-aims with a numeric atom-centered orbital basis.

Kai: What that means for us is that we have a concrete way to test how this method behaves in practice on materials like metal oxides and nitrides, using all-electron calculations. They’ve shown this formulation can get band gap errors down to levels comparable to the HSE06 functional when using the fully localized limit of ACBN0.

Lev: From an engineering view, it confirms that we can build a system that implements this without resorting to external DFT+U loops, updating everything within a single SCF calculation. That simplifies the computational pipeline considerably if you're trying to move toward more integrated methods.

Mira: The implications are that for anyone working on DFT+U corrections, they need to be aware that the optimal parent functional depends on the material itself, and they have to pay close attention to how they construct that Hubbard subspace.

Kai: So, this paper gives us a toolset—a formulation and implementation—that we can use as a baseline for testing how robust it is across different materials, moving beyond just theoretical comparisons.

Conclusion: Mira: So we're wrapping up on this ACBN0 formulation in FHI-aims using a numeric atom-centered basis, and I want to talk about what that actually means for the field of materials science.

Kai: Yeah, it’s about taking this pseudo-hybrid density functional idea and making it runnable on real hardware instead of just sitting in a theoretical box.

Lev: It means they’ve baked the Hubbard U correction right into the self-consistent calculation, so you don't have to run a separate DFT+U loop on top of the main energy calculation.

Mira: Right, and they show that when you use this formulation with the fully localized limit—the FLL—the errors in predicting band gaps compared to experimental data drop down close to what we see with HSE06 functionals.

Kai: That's significant because we know HSE06 is computationally heavy, so getting results comparable to it using a different approach is a real win for practical applications.

Lev: I mean, the complexity of the ACBN0 iteration itself has its own cost, but if you're only doing one SCF cycle and you get that level of accuracy, that overhead might actually be manageable on certain systems.

Mira: The authors point out that what makes this work well across different materials is how they adjust the Hubbard subspace, because the best parent functional changes depending on whether you’re looking at a metal oxide or a nitride.

Kai: That points to the fact that it isn't one single magic formula; it requires tailoring your setup based on the specific crystal structure you're simulating.

Lev: From an engineering standpoint, that means we can't just use one fixed parameter set for everything; we have to build a procedure for selecting those parameters before we even start the simulation.

Mira: Exactly, and they also looked at surface stuff, like the oxygen evolution reaction on a nickel oxide surface, showing that this localized basis method still holds up well in lower-dimensional systems.

Kai: So it’s not just about bulk crystals; it’s showing versatility for surfaces too, which is crucial when you're trying to model things like battery materials or catalysis.

Lev: It gives us a solid piece of software to test on, which is better than just seeing a paper and hoping the implementation actually works in practice.

Center for Energy Science and Technology, Skolkovo Institute of Science and Technology

cond-mat.mtrl-sci

Submitted: 2026-09-10

Updated: 2026-10-29

Comments: 33 pages, 8 figures

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 79/100

The gist: We present a formulation and implementation of the Agapito–Curtarolo–Buongiorno Nardelli (ACBN0) pseudo-hybrid density functional in a numeric atom-centered orbital basis The method is realized

Key concepts

ACBN0 Pseudo-hybrid Density Functional: This is a new method for calculating electronic properties in materials. It combines elements of standard Density Functional Theory (DFT) with a Hubbard correction term, which helps improve the accuracy of predicting things like band gaps and magnetic moments in solids.
Numeric Atom-Centered Orbital Basis: Instead of using traditional basis sets, this method uses a specific numerical approach where orbitals are centered on individual atoms. This setup is used to construct occupation and density matrices, which helps stabilize the calculation process for the ACBN0 method.
Hubbard Correction Treatment (FLL): The ACBN0 method allows different ways to handle 'double-counting' errors in DFT calculations. The Fully Localized Limit (FLL) is one such treatment that reduces aggregate band gap errors compared to simpler functionals like PBE, bringing the results closer to more accurate reference data.
DFT+U Method: DFT+U is a technique used in DFT calculations to correct for strong electron correlation effects, particularly in materials with localized electrons. ACBN0 is an implementation of this method that uses projectors onto a chosen localized subspace to determine the Hubbard parameters.

Terminology

Localization — 16× in this paper · explained in 2 episodes
Supporting Information — 10× in this paper
Accuracy — 8× in this paper · explained in 1 episode
Benchmark — 5× in this paper · explained in 2 episodes
Polarization — 5× in this paper · explained in 2 episodes
computational cost — 5× in this paper · explained in 2 episodes
Ab Initio — 3× in this paper · explained in 1 episode
Core — 2× in this paper · explained in 2 episodes
Exchange Interaction — 2× in this paper · explained in 2 episodes
ACBN0 Pseudo-hybrid Density Functional — 1× in this paper · explained in 1 episode
Consistency — 1× in this paper · explained in 3 episodes
Mott Insulator — 1× in this paper · explained in 2 episodes
Orthogonal Projection — 1× in this paper · explained in 2 episodes
Robustness — 1× in this paper · explained in 6 episodes
Self-Consistency — 1× in this paper · explained in 3 episodes
Structural Properties — 1× in this paper · explained in 2 episodes
Symmetry Breaking — 1× in this paper · explained in 4 episodes

Summary

We present a formulation and implementation of the Agapito–Curtarolo–Buongiorno Nardelli (ACBN0) pseudo-hybrid density functional in a numeric atom-centered orbital basis The method is realized in the all-electron, full-potential electronic-structure package FHI-aims For a benchmark set of materials including metal oxides and nitrides, ACBN0 with the fully localized limit (FLL) as the double-counting treatment reduces the aggregate band gap errors relative to the Perdew–Burke–Ernzerhof (PBE) and strongly constrained and appropriately normed (SCAN) functionals, reaching an accuracy comparable to the Heyd–Scuseria–Ernzerhof functional (HSE06)

Methodology Overview

The study introduces the ACBN0 pseudo-hybrid density functional implemented in FHI-aims using a numeric atom-centered orbital basis. This implementation utilizes a Löwdin-orthogonalized projector to construct the occupation and renormalized density matrices, which improves the stability of the self-consistent ACBN0 iterations for tested systems. The formulation defines the Hubbard correction operator Vˆ σ U in DFT+U method using projectors Pˆ onto a chosen localized subspace. The choice of projector is particularly important because nonphysical occupations would directly affect the selfconsistent U¯ and J¯ parameters and could destabilize the coupled SCF–ACBN0 cycle.

Hubbard Correction Treatments

The ACBN0 method allows for different double-counting treatments, including the fully localized limit (FLL), around-mean-field (AMF), and Petukhov mixing. The Hubbard parameter in ACBN0, U I,n,l = U¯ I,n,l−J¯I,n,l > is determined through electron-repulsion integrals and renormalized density matrices. The self-consistent ACBN0 implementation follows a specific sequence of steps involving evaluating electron-repulsion integrals and solving the Kohn–Sham equations iteratively.

Performance on Bulk Materials

For six bulk transition metal oxides, ACBN0@PBE with Petukhov mixing required up to 1.49 times the PBE time per iteration, while an HSE06 iteration was approximately 13–33 times as expensive as the corresponding ACBN0@PBE iteration with Petukhov mixing. The ACBN0 results substantially reduce the band gap errors of PBE and SCAN relative to experiment and bring their magnetic moments closer to both experimental and HSE06 reference values. Specifically, ACBN0@SCAN–FLL gives the best result among the tested ACBN0 combinations, with the lowest experimental band gap MAPE of 21.3%, which is comparable to the HSE06 reference.

Surface and Reaction Energetics

The applicability of the localized-basis implementation to low-dimensional systems is demonstrated by calculations of adsorption and the oxygen evolution reaction (OER) overpotential on a β-NiOOH(001) surface. The mean absolute deviation from HSE06/“tight” for uncorrected rSCAN/“tight” OER steps was reduced to 0.107 eV with ACBN0@rSCAN with the “light” settings. The converged Hubbard parameters in Supporting Information Table S11 show how the self-consistently evaluated U values change among the clean slab, adsorbate-covered slabs, and molecular references.

Conclusions and Future Work

The present work establishes a practical all-electron, fullpotential formulation of ACBN0 in a localized orbital basis and provides a foundation for its further development. The comparison across the 3d transition metal series shows that the optimal parent functional depends on the material. Developing robust procedures for constructing an appropriate Hubbard subspace is therefore an important task for DFT+U in general and for selfconsistent ACBN0 in particular. The surface calculations also reveal the sensitivity of the meta-GGA functional rSCAN, and consequently ACBN0@rSCAN, to the numerical settings and magnetic initialization.

Supporting Information Details

The computational study consists of two parts: tests for bulk semiconductors and insulators, and an examination of adsorption and OER energetics on the β-NiOOH(001) surface. The FHI-aims “light” and “tight” settings denote complete sets of numerical defaults that differ in their integration grids and, most importantly for the present comparison, in the number of numeric atom-centered basis functions. The bulk tests comprise transition metal, alkaline earth, and post-transition metal oxides and nitrides. The surface test was performed for OER energetics using a stoichiometric monolayer slab model of the β-NiOOH(001) surface, which has previously been examined in first-principles studies of nickel oxyhydroxides.

References

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[3] J. Heyd, G. E. Scuseria, M. Ernzerhof, Hybrid functionals based on a screened coulomb potential, The Journal of chemical physics 118 (18) (2003) 8207–8215

[4] J. Paier, M. Marsman, K. Hummer, G. Kresse, I. C. Gerber, J. G. Ángyán, Screened hybrid density functionals applied to solids, The Journal of chemical physics 124 (15) (2006) 8207–8215

[7] K. J. May, A. M. Kolpak, Improved description of perovskite oxide crystal structure and electronic properties using selfconsistent hubbard corrections from ACBN0, Physical Review B 101 (2020) 165117

[8] D. S. Lambert, D. D. O’Regan, Use of DFT+u+j with linear response parameters to predict non-magnetic oxide band gaps with hybrid-functional accuracy, Physical Review Research 5 (1) (2023) 013160

[9] M. Aykol, C. Wolverton, Local environment dependent GGA+u method for accurate thermochemistry of transition metal compounds, Physical Review B 90 (11) (2014) 115105

[33] V. Blum, R. Gehrke, F. Hanke, P. Havu, V. Havu, X. Ren, K. Reuter, M. Scheffler, Ab initio molecular simulations with numeric atom-centered orbitals

[34] A. C. Ihrig, J. Wieferink, I. Y. Zhang, M. Ropo, X. Ren, P. Rinke, M. Scheffler, V Blum, Accurate localized resolution of identity approach for linear-scaling hybrid density functionals and for many-body perturbation theory

[35] M. J. Eslamibidgoli, A Groß, M Eikerling, Surface configuration and wettability of nickel(oxy)hydroxides: A first-principles investigation, Physical Chemistry Chemical Physics 19 (34) (2017) 22659–22669

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[38]

Improvements for AI systems

Improve accuracy of band gap predictions for transition metal oxides by using ACBN0@SCAN–FLL with light numerical settings, achieving MAPEs of 21.3% and 25.4%, which is comparable to the HSE06 reference for the test set.
Enhance prediction of magnetic moments in antiferromagnetic oxides by using ACBN0@SCAN–FLL, which reduces experimental MAPE from 37.3% with PBE to 15.2–17.8%, comparable to HSE06 values and showing that FLL gives the lowest errors among the tested ACBN0 combinations.
Develop a method for OER overpotential estimation on low-dimensional surfaces by using ACBN0@rSCAN with tight numerical settings, which shifts estimated overpotential toward HSE06 and PBE0 reference results.
Enable more stable and accurate calculations of electronic structure in low-dimensional systems by utilizing the Löwdin projector instead of the Mulliken construction, as Löwdin orthogonalization is applied to the complete finite basis at each k point.
Improve computational efficiency for correlated materials by comparing ACBN0@PBE with Petukhov mixing to HSE06 iterations, noting that an HSE06 iteration is approximately 13–33 times as expensive as the corresponding ACBN0@PBE iteration with Petukhov mixing.

Abstract

We present a formulation and implementation of the Agapito-Curtarolo-Buongiorno Nardelli (ACBN0) pseudo-hybrid density functional in a numeric atom-centered orbital basis. The method is realized in the all-electron, full-potential electronic-structure package FHI-aims. The implementation uses a Löwdin-orthogonalized projector, which improved the stability of the self-consistent ACBN0 iterations for the tested systems. For a benchmark set of materials including metal oxides and nitrides, ACBN0 with the fully localized limit (FLL) as the double-counting treatment reduces the aggregate band gap errors relative to the Perdew-Burke-Ernzerhof (PBE) and strongly constrained and appropriately normed (SCAN) functionals, reaching an accuracy comparable to the Heyd-Scuseria-Ernzerhof functional (HSE06). The computational cost of a single ACBN0 self-consistent-field iteration is comparable to that of the parent functional and substantially lower than that of HSE06. The applicability of the localizedbasis implementation to low-dimensional systems is demonstrated by calculations of adsorption and the oxygen evolution reaction (OER) overpotential on a β-NiOOH(001) surface. On structures relaxed with regularized SCAN (rSCAN), single-point ACBN0 calculations based on rSCAN (ACBN0@rSCAN) with the "tight" numerical settings reduce the mean absolute deviation of the reaction-step energies from HSE06 and shift the estimated overpotential toward the HSE06 and Perdew-Burke-Ernzerhof hybrid (PBE0) reference results.