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arxiv: 2604.25706 · v1 · submitted 2026-04-28 · ❄️ cond-mat.str-el · cond-mat.mtrl-sci· physics.chem-ph· physics.comp-ph

Validity of DFT+U band gaps in all its known functional forms

Andrew C. Burgess , David D. O'Regan This is my paper

Pith reviewed 2026-05-07 14:45 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mtrl-sciphysics.chem-phphysics.comp-ph
keywords DFT+Uband gapfundamental gapHubbard correctionKohn-Sham eigenvaluesperiodic systemsdensity functional theory
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0 comments X

The pith

The DFT+U eigenvalue gap equals the fundamental gap from total-energy differences in pristine periodic systems.

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

The paper proves that for every known DFT+U functional form the gap between the highest occupied and lowest unoccupied eigenvalues matches the fundamental gap obtained by subtracting total energies of the N+1 and N-1 electron systems. This equality is shown analytically to hold when the crystal is perfectly periodic and k-point sampling is fully converged. The result justifies the routine use of the simpler eigenvalue gap for band-gap estimates in transition-metal and lanthanide solids. The equality survives pseudopotentials, PAW potentials, hybrid functionals, and any level of subspace projection, yet it fails in defective supercells or isolated molecules. The authors survey all published DFT+U variants and illustrate the behavior in the hydrogen lattice Mott-Hubbard limit.

Core claim

In all known DFT+U functionals the difference between the highest occupied and lowest unoccupied Kohn-Sham eigenvalues equals the total-energy difference between the N+1 and N-1 electron systems for pristine periodic solids with converged k-point sampling. This equality is shown to survive the use of pseudopotentials, projector augmented waves, and hybrid functionals, and to hold independently of the subspace projection level for the Hubbard correction.

What carries the argument

The analytic verification that the DFT+U eigenspectrum gap matches the total-energy fundamental gap, performed separately for each published functional form including Dudarev, Liechtenstein, and hybrid variants.

If this is right

  • The eigenvalue gap can be used directly to estimate band gaps in perfect crystals for any published DFT+U form.
  • Band-gap validity extends to hybrid DFT+U and to calculations that employ pseudopotentials or PAW potentials.
  • The equality does not hold for defective supercells or isolated molecules, so total-energy differences must be used instead.
  • Total energies and gaps remain well-defined and consistent with the eigenvalue spectrum in the hydrogen lattice Mott-Hubbard limit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that careful k-point convergence is essential for reliable DFT+U gap predictions in solids.
  • In materials containing defects or dopants, practitioners may need to switch to total-energy difference methods for accurate gap levels.
  • The unification across all DFT+U variants may guide the design of new corrective functionals that preserve gap validity under periodic boundary conditions.

Load-bearing premise

The system must be a pristine periodic crystal with fully converged k-point sampling; any defect, finite-size effect, or incomplete k-point convergence that breaks periodicity can make the eigenvalue gap differ from the total-energy gap.

What would settle it

A numerical calculation in a pristine periodic system with converged k-points in which the DFT+U eigenvalue gap and the total-energy difference gap differ by more than the numerical precision would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.25706 by Andrew C. Burgess, David D. O'Regan.

Figure 1
Figure 1. Figure 1: FIG. 1. The number of Google Scholar hits [ view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. DFT+ view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Correction to the bandgap of the non-spin polarized view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Dudarev’s DFT+ view at source ↗
read the original abstract

The Density Functional Theory plus Hubbard $U$ (DFT+$U$) technique is one of the most widely used tools by condensed matter physicists and solid state chemists for the simulation of transition-metal and lanthanide bearing crystals, and increasingly of much more diverse chemistries. Although often synonymous with the corrective functionals of Dudarev et al. and Liechtenstein et al., there exists a wide variety of DFT+$U$-type functionals ready to be utilized, and no doubt yet to be developed. Since the earliest days, the gap in the DFT+$U$ single-particle eigenspectrum has been associated with the fundamental band gap, and the method has typically found more success for spectra than for total-energy derived properties. There has been some doubt, however, as to the conceptual validity of this association. Here, extending findings from recent years regarding local and semi-local functionals, we prove that the DFT+$U$ eigenspectrum gap is indeed valid, in the sense that it matches its own fundamental gap calculated using total-energy differences. This is true for pristine periodic systems with converged $k$-point sampling but not, however, for defective ones or isolated systems. We show that bandgap validity for solids holds in the presence of pseudopotentials and PAW potentials, when using hybrid functionals, and in DFT+$U$(+$J$) irrespective of the level of subspace projection onto the band-edge states. We survey every DFT+$U$-type functional known to have been published to date, within a unified notation. We verify analytically under which conditions the eigenvalue gap equals its fundamental gap for each functional, and analyze its effect on total energies and gaps for the hydrogen lattice in the Mott-Hubbard limit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript provides an analytical proof that, for every published DFT+U functional form, the single-particle eigenvalue gap equals the fundamental gap obtained from total-energy differences. This equality holds for pristine periodic systems with converged k-point sampling but fails for defective or isolated systems. The derivation covers pseudopotentials/PAW, hybrid functionals, and DFT+U(+J) irrespective of subspace projection, and is illustrated with the hydrogen lattice in the Mott-Hubbard limit.

Significance. If the analytical verification is correct, the result is significant: it removes a long-standing conceptual uncertainty about the validity of DFT+U band gaps in solids, extending earlier results for local functionals to the entire family of DFT+U corrections. The unified notation and exhaustive survey of all known functional forms, together with the explicit delimitation of the regime of validity, supply a clear and reusable foundation for the method. The analytical (rather than numerical) character of the proof and the explicit treatment of the exceptions constitute genuine strengths.

minor comments (2)
  1. §2 (unified notation): the mapping between the various published DFT+U forms and the general expression in Eq. (X) could be tabulated for immediate reference.
  2. The hydrogen-lattice numerical test in §5 would benefit from an explicit statement of the k-point mesh and U value at which the equality was verified to machine precision.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, their accurate summary of its scope and conclusions, and their recommendation to accept. We are pleased that the referee recognizes the significance of the analytical proof and the unified treatment of all DFT+U functional forms.

Circularity Check

0 steps flagged

No significant circularity: analytical verification of gap equality

full rationale

The paper derives an analytical proof that the DFT+U single-particle eigenvalue gap equals the fundamental gap (from total-energy differences) for all published DFT+U functional forms, but only in the explicitly stated regime of pristine periodic systems with converged k-point sampling. This is verified across pseudopotentials, PAW, hybrids, and DFT+U(+J) variants irrespective of subspace projection. The derivation uses unified notation to check conditions for equality and tests the hydrogen lattice in the Mott-Hubbard limit. No load-bearing step reduces to a self-definition, fitted input renamed as prediction, or self-citation chain; the central result is an independent analytical identity within the delimited scope, with explicit negation for defects and isolated systems. The claim is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard mathematical properties of DFT functionals and the specific algebraic structure of each DFT+U correction; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The DFT+U functionals satisfy variational and self-consistency properties that permit the eigenvalue gap to equal the total-energy difference gap under periodic boundary conditions with converged k-point sampling.
    This assumption is required to extend the equality to all functional forms, pseudopotentials, PAW, and hybrids as stated.

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