Semilocal exchange-correlation potentials for solid-state calculations: Current status and future directions

Ahmad W. Huran; Fabien Tran; Jan Doumont; Leila Kalantari; Miguel A. L. Marques; Peter Blaha

REVIEW 2 major objections 5 minor 300 references

Semilocal exchange-correlation potentials for solid-state calculations: Current status and future directions

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A fast DFT potential predicts solid band gaps as well as costly hybrids.

desk verdict A useful, honest review of semilocal band-gap methods; the strong claim for mBJLDA rests on an in-sample 76-solid set, while the independent 472-solid set only shows parity with HSE06. read the letter →

arxiv 1908.01820 v2 pith:KBUHUDY7 submitted 2019-08-05 cond-mat.mtrl-sci

Fabien Tran , Jan Doumont , Leila Kalantari , Ahmad W. Huran , Miguel A. L. Marques , Peter Blaha This is my paper · ORCID

classification cond-mat.mtrl-sci PACS 71.15.Mb71.20.-b71.15.-m

keywords semilocalexchange-correlationpotentialsmBJLDAfundamentalbandgapKohn-ShamDFTderivativediscontinuityhybridfunctionalsmeta-GGAsolid-statebenchmarks

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper surveys the fastest class of density-functional approximations for solids — semilocal exchange-correlation potentials — and asks how accurately they reproduce the fundamental band gap, the energy needed to create a separated electron-hole pair. Drawing on benchmarks of 76 and 472 solids, its central conclusion is that the modified Becke-Johnson potential (mBJLDA) is on average the most accurate semilocal method and is at least as accurate as the much more expensive screened hybrid HSE06. Only dielectric-dependent hybrids and GW methods — many-body perturbation approaches — are consistently more accurate. The paper also shows where mBJLDA fails, namely Cu1+ compounds, ZnO, lead-halide perovskites, and 4f/5f oxides, and it identifies the two ingredients that make the potential work: the kinetic-energy density and a cell-averaged density-gradient measure. It argues that the formally sound next step is a non-multiplicative meta-GGA or GLLB-SC-style potential that carries an explicit derivative discontinuity.

What carries the argument

The central object is the mBJLDA exchange potential, defined as $v^{\mathrm{mBJ}}_{x,\sigma}(\mathbf{r}) = c\,v^{\mathrm{BR}}_{x,\sigma}(\mathbf{r}) + (3c-2)\,\frac{1}{\pi}\sqrt{\frac{5}{6}}\sqrt{t_\sigma(\mathbf{r})/\rho_\sigma(\mathbf{r})}$, with $c = \alpha + \beta\sqrt{g}$, $\alpha = -0.012$, $\beta = 1.023\,\mathrm{bohr}^{1/2}$, and $g$ the cell average of $|\nabla\rho_\sigma|/\rho_\sigma$. The Becke-Roussel term $v^{\mathrm{BR}}$ approximates the Slater exchange potential, while the $\sqrt{t_\sigma/\rho_\sigma}$ term supplies orbital-angular-momentum sensitivity, and the global factor $c$ is what makes the gap come out near experiment. The GLLB-SC potential plays a supporting role in the argument as the formally preferable multiplicative alternative because its orbital-energy dependence yields a calculable derivative discontinuity.

What would settle it

Take a set of about 100 solids not used in the original alpha/beta fit, measure their fundamental gaps experimentally, and compare mBJLDA against HSE06 mean absolute errors; if mBJLDA's error is clearly worse than HSE06's, or if the c(g) fit fails on a new family as it already does for lead-halide perovskites, the at-least-as-accurate claim is refuted.

Watch

Extended reading notes

Core claim

The authors claim that among semilocal methods, mBJLDA is the most accurate potential for fundamental band gaps. In their 76-solid benchmark it has the lowest mean absolute error (0.47 eV, 15% mean absolute relative error), slightly better than HSE06; in the 472-solid set it ties with AK13, HLE16, and HSE06 at about 0.5 eV and 30-35% mean absolute relative error. Crucially, it is the only semilocal method that handles antiferromagnetic transition-metal oxides well, reaching hybrid-level accuracy for those gaps and magnetic moments. The success is traced to the potential's dependence on the kinetic-energy density $t_\sigma$ and on $g$, the cell average of $|\nabla\rho_\sigma|/\rho_\sigma$: the former lets the potential act differently on orbitals of different angular character, opening d-d gaps; the latter tunes the overall strength $c = \alpha + \beta\sqrt{g}$ and correlates with the value that would reproduce the experimental gap. The paper is explicit that this accuracy comes at a formal price: because mBJLDA is a multiplicative potential with no derivative discontinuity, comparing its eigenvalue gap to the experimental fundamental gap is not in principle correct, and the fitted parameters $\alpha$ and $\beta$ inherit that limitation.

Load-bearing premise

The whole ranking assumes that comparing a Kohn-Sham eigenvalue gap with the experimental fundamental gap is meaningful, even though the paper concedes that for a multiplicative potential like mBJLDA this comparison is in principle invalid because the derivative discontinuity is missing.

Editorial extensions

If this is right

For routine band-gap screening of semiconductors and insulators, mBJLDA offers hybrid-level accuracy at a small fraction of the computational cost, making scans over thousands of candidate materials practical.
In antiferromagnetic transition-metal oxides, mBJLDA is the semilocal method of choice: other semilocal potentials fail badly, while mBJLDA matches hybrid accuracy for gaps and magnetic moments.
Because its eigenvalue gap omits the derivative discontinuity, mBJLDA-based gaps should not be interpreted as fundamental gaps in a formally exact sense; the agreement with experiment is the result of empirical fitting.
mBJLDA's bands are too narrow and its electron density is not particularly accurate, so bandwidths, effective masses, and density-sensitive properties like electric-field gradients should be taken from other methods or treated with caution.
The two mBJLDA ingredients — kinetic-energy density and a cell-averaged inhomogeneity measure — are identified as the key building blocks for future fast potentials that include a derivative discontinuity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If mBJLDA's accuracy is as robust as the benchmarks suggest, then a purely local or semilocal functional that omits a cell-averaged inhomogeneity term is unlikely to reproduce its success on antiferromagnetic oxides; this is a testable constraint for meta-GGA design.
The fitted relation $c(\sqrt{g})$ means every new materials family is, in effect, a prediction of the original fit; the documented failures on lead-halide perovskites and 4f/5f oxides imply the fitted domain is narrower than all solids, and any large screening using mBJLDA should validate against a few higher-level references per family.
A natural extension that the paper does not pursue is to replace the global cell average $g$ with a position-dependent local average for surfaces, interfaces, and molecules; the correlation visible in the paper's c-versus-g plot suggests such a locality-preserving variant could retain much of the gap accuracy while extending applicability.
Combining mBJLDA's two ingredients with the GLLB-SC derivative-discontinuity framework, as the paper suggests, would yield a potential that is both fast and formally grounded; the test of that program is whether the fitted $c$ retains its clean correlation once the discontinuity term is added.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a useful review, not a breakthrough. The authors summarize the evidence for mBJLDA as the best fast semilocal potential for band gaps, add a few new benchmark numbers, and are refreshingly honest about the formal problems with their own method.

What's actually new: WIEN2k values for EV93PW91, AK13, and GLLB-SC on the 472-solid set; a small table on CeO2/Ce2O3/UO2; and the analysis of how the unit-cell average g controls the gap in TM oxides vs metals. The review of other properties is well done—bandwidth, magnetism, electron density all get a fair treatment, and the message is balanced: mBJLDA wins for gaps, loses for bandwidths, and is mediocre for densities.

Now the soft spots. The central summary claim—'mBJLDA is on average the most accurate'—rests mainly on the 76-solid benchmark, which substantially overlaps the original fitting set for alpha and beta (Fig. 4). On that set, comparing a fitted potential against unfitted hybrids is not an even playing field. The more independent 472-solid set in Table II shows mBJLDA, HSE06, and AK13 all around MAE 0.5 eV and MARE 30-35%; the paper itself says they are 'equally well.' So the strong version of the claim doesn't hold. The weaker version—mBJLDA is competitive with HSE06 and much cheaper—is well supported and practically useful.

The formal concern about comparing multiplicative KS gaps with experiment is explicitly acknowledged in Sec. II D, so it is not a hidden flaw. It does mean the fitted parameters are pragmatic rather than theoretically grounded, and the authors say as much. The new numbers in Tables III and VI are plausible but lack error bars and fine computational details; that's a minor issue for a review.

I would send this to review. It deserves referee time. The requested revision should be light: soften the summary to 'competitive with HSE06,' and add a sentence quantifying the overlap between the 76-solid set and the original fitting set. That would make the paper's claims more robust.

Referee Report

2 major / 5 minor

Summary. The manuscript is a review of semilocal exchange-correlation potentials for calculating band structures of solids, with a strong focus on the modified Becke-Johnson (mBJLDA) potential. It reviews the formal distinction between multiplicative and non-multiplicative potentials, discusses the role of the derivative discontinuity, and compiles benchmark results from a 76-solid and a 472-solid test set. The paper also presents new WIEN2k results for rare-earth and actinide oxides (Table III) and an analysis of the role of the density-gradient average g in the mBJLDA potential (Table VI). The central conclusions are that mBJLDA is on average the most accurate semilocal method for the fundamental band gap and that it is at least as accurate as hybrid functionals such as HSE06, while its accuracy for other properties such as bandwidths and electron densities is limited. The paper ends with an outlook discussing possible routes toward improved potentials that incorporate a derivative discontinuity.

Significance. If the headline claim is accepted, the paper would establish that a cheap semilocal potential can replace much more expensive GW or hybrid calculations for band gaps of many solids, which would be practically important. The review is comprehensive and useful as a status report, and it includes original benchmarks: WIEN2k data for EV93PW91, AK13, and GLLB-SC on the 472-solid set, f-electron oxide calculations, and the g-dependence analysis in Table VI. The authors are also candid about formal limitations: Section II D clearly states that comparing multiplicative-potential eigenvalue gaps with experimental gaps is not in principle correct, and Section V concedes that mBJLDA gives poor bandwidths and densities. The main weakness is that the strongest claim about mBJLDA's superiority rests on benchmarks that overlap with its fitting set, while the larger independent benchmark shows parity rather than superiority.

major comments (2)

[Sec. III A 1, Table I; Sec. IV] The central claim that mBJLDA is 'on average the most accurate' semilocal method for the fundamental band gap is not supported by the out-of-sample evidence. The headline numbers in Table I (mBJLDA MAE 0.47 eV vs. HSE06 0.82 eV and GLLB-SC 0.64 eV) come from the 76-solid benchmark, which overlaps substantially with the fitting set used to determine alpha and beta in Eq. (16) (Ref. 20; Fig. 4 shows the fit for those solids). The larger, more independent 472-solid benchmark in Table II shows mBJLDA and HSE06 both at MAE = 0.5 eV, with AK13 and HLE16 close behind. The paper should either quantify the overlap between the 76-solid set and the original fitting set, or present the 472-solid result as the primary evidence and state the conclusion as 'competitive with HSE06 and among the most accurate semilocal methods' rather than 'the most accurate'.
[Sec. II D and Sec. IV] The central claim in Sec. IV that 'mBJLDA is at least as accurate as hybrid functionals like HSE06' should be accompanied by the caveat, already acknowledged in Sec. II D, that this is a pragmatic comparison of a multiplicative Kohn-Sham eigenvalue gap with experiment, not a formally justified one. Because the derivative discontinuity is missing, mBJLDA's E_KS^g is not the fundamental gap in the exact KS sense; the paper states this in Sec. II D, but the abstract and the summary bullet points do not carry the caveat. Please add a sentence to the abstract or conclusions clarifying that the claim is empirical and specific to the benchmark sets considered.

minor comments (5)

[Sec. III A 2 and Table II] The WIEN2k calculations for EV93PW91, AK13, and GLLB-SC on the 472-solid set are new, but the text gives no computational parameters; please provide k-point sampling, muffin-tin radii, and convergence criteria in the supplementary material, since the VASP-WIEN2k agreement is not uniform for large-gap systems.
[Table VI] The definition of g_fixed is not explicit; please state which reference value of g from which metal or oxide was used for each row and report the g_fixed values themselves.
[Table III] The experimental references are single values with no uncertainty estimates, and several of these systems have a range of reported gaps; please include error bars or multiple references and state the spin-orbit treatment for UO2 explicitly.
[Fig. 4] The caption should explicitly define copt and gopt and state that two solids (FeO and ZnO) are excluded from the fit; currently the reader must infer this from the text in Sec. V.
[Sec. II D] The bullet on LDA/GGA states 'in Ref. 131 it is shown that Δxc = Eg − E_KS^g = 0 in solids'; this should be qualified as referring to the thermodynamic limit for LDA/GGA to avoid a universal reading.

Circularity Check

1 steps flagged · score 5.0 of 10

mBJLDA's 'most accurate' claim on the 76-solid set is partly in-sample because α and β were fitted to band gaps; the out-of-sample 472-solid set shows parity with HSE06, softening but not eliminating the claim.

fitted input called prediction [Sec. II B 2 (Eq. 16); Sec. III A 1 (Table I, Figs. 1-2, 4); Sec. IV]
"The values of α and β in Eq. (16) were determined to be α = −0.012 (dimensionless) and β = 1.023 bohr 1/2 by minimizing the mean absolute error of the band gap for a group of solids. ... For the more accurate methods, the ranking depends on whether the MAE or the MARE is considered, however for both quantities mBJLDA is the most accurate method with MAE = 0.47 eV and MARE = 15%."

Equation (16) defines c = α + β√g with α and β 'determined ... by minimizing the mean absolute error of the band gap for a group of solids.' The benchmark in Sec. III A 1 then evaluates mBJLDA on the 76-solid set built in the authors' previous works (Refs. 29 and 32); Fig. 4 shows that many of the original fitting solids (Ne, Ar, Kr, Xe, C, Si, Ge, LiF, LiCl, MgO, ScN, MnO, NiO, SiC, BN, GaN, GaAs, AlP, ZnS, CdS, AlN) are part of this benchmark. Presenting mBJLDA's MAE = 0.47 eV on this partly in-sample set as evidence that 'mBJLDA is the most accurate method' compares a gap-fitted potential with non-fitted methods (HSE06, GLLB-SC).

full rationale

The paper is a review/benchmark, not a derivation, and much of its content is self-contained against external benchmarks. The mBJLDA parameters were indeed fitted to band-gap data (Eq. 16), so the 76-solid benchmark, which overlaps the original fitting solids (see Fig. 4), is partly in-sample; this is a genuine fitted-input-called-prediction issue for the claim that mBJLDA is 'on average the most accurate' semilocal method. However, the paper also reports a larger 472-solid out-of-sample benchmark (Table II) in which mBJLDA is tied with HSE06 and AK13, and it cites independent studies (Lee et al., Nakano and Sakai, Jiang) that support mBJLDA's gap accuracy on other solids. Thus the weaker claim, 'at least as accurate as HSE06,' has independent support, while the stronger 'most accurate' claim is inflated by in-sample evaluation. The formal objection that a multiplicative KS potential's eigenvalue gap should not be directly compared with the experimental gap is explicitly acknowledged in Sec. II D and V, so it is an honest limitation rather than a hidden circular step. On balance, the central 'most accurate' claim is partially circular (score 5), but the paper is not globally circular and contains substantial independent evidence.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim that mBJLDA is the most accurate fast method for band gaps rests on a fitted empirical potential and on benchmarks that compare Kohn-Sham gaps directly to experiment. The free parameters and assumptions listed above are the main external inputs that the reader is asked to accept.

free parameters (3)

mBJLDA alpha = -0.012 (dimensionless)
Fitted in Ref. 20 to minimize band gap error for a set of solids; used in Eq. (16).
mBJLDA beta = 1.023 bohr^{1/2}
Fitted in Ref. 20 along with alpha; controls c(g) in Eq. (16).
PBE+U U and J = U = 5 eV, J = 0.5 eV
Chosen for the f-electron oxides in Table III; values are typical from literature (fully localized limit).

assumptions (4)

domain assumption Validity of Kohn-Sham DFT as the computational framework.
The entire review operates within KS-DFT; the accuracy rankings are for KS-DFT approximations.
domain assumption Experimental band gaps used as reference are accurate.
The benchmark MAE/MARE values depend on experimental references from Refs. 28, 35, 36, 103, 145-151; systematic errors in those values would change rankings.
ad hoc to paper Direct comparison of Kohn-Sham eigenvalue gaps to experimental fundamental gaps is a meaningful ranking criterion.
The paper concedes in Sec. II D that this is formally incorrect for multiplicative potentials, yet the central claim relies on such comparisons.
domain assumption WIEN2k and VASP results are consistent enough to be combined in the 472-solids benchmark.
Table II mixes WIEN2k (EV93PW91, AK13, GLLB-SC) and VASP results; the paper notes discrepancies for large-gap solids but says they do not affect conclusions.

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Cite this review

Pith. "Pith review of Semilocal exchange-correlation potentials for solid-state calculations: Current status and future directions." pith.science (2026). https://pith.science/paper/KBUHUDY7

@misc{pith2026190801820,
  author       = {Pith},
  title        = {Pith review of: Semilocal exchange-correlation potentials for solid-state calculations: Current status and future directions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KBUHUDY7}},
  note         = {Machine review of arXiv:1908.01820}
}

abstract

Kohn-Sham (KS) density functional theory (DFT) is a very efficient method for calculating various properties of solids as, for instance, the total energy, the electron density, or the electronic band structure. The KS-DFT method leads to rather fast calculations, however the accuracy depends crucially on the chosen approximation for the exchange and correlation (xc) functional $E_{\text{xc}}$ and/or potential $v_{\text{xc}}$. Here, an overview of xc methods to calculate the electronic band structure is given, with the focus on the so-called semilocal methods that are the fastest in KS-DFT and allow to treat systems containing up to thousands of atoms. Among them, there is the modified Becke-Johnson potential that is widely used to calculate the fundamental band gap of semiconductors and insulators. The accuracy for other properties like the magnetic moment or the electron density, that are also determined directly by $v_{\text{xc}}$, is also discussed.

Figures

Figures reproduced from arXiv: 1908.01820 by the authors.

**Figure 3.** FIG. 3. Calculated versus experimental fundamental band ga [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 2.** FIG. 2. Calculated versus experimental fundamental band ga [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Plot of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

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2014

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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