Wigner-Seitz truncated TDDFT approach for the calculation of exciton binding energies in solids

A. Ayuela; A. Leonardo; M. Arruabarrena

arxiv: 2303.13389 · v1 · submitted 2023-03-23 · ❄️ cond-mat.mtrl-sci

Wigner-Seitz truncated TDDFT approach for the calculation of exciton binding energies in solids

M. Arruabarrena , A. Leonardo , A. Ayuela This is my paper

Pith reviewed 2026-05-24 09:59 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords TDDFTexciton binding energiesWigner-Seitz truncationCoulomb singularityLRC kernelshybrid TDDFTsolidsoptical properties

0 comments

The pith

Numerical treatment of the long-range Coulomb singular term limits accurate exciton binding energies in TDDFT for solids

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

The paper establishes that challenges in obtaining both accurate optical spectra and precise exciton binding energies with long-range corrected kernels in TDDFT stem from difficulties in numerically handling the long-range Coulomb singular term. Applying a Wigner-Seitz truncated kernel to both pure-TDDFT and hybrid approaches shows these issues persist across methods. A sympathetic reader cares because TDDFT offers a lower-cost alternative to many-body perturbation theory for optical properties in solids, and clarifying this numerical bottleneck could improve its reliability for excitonic effects.

Core claim

The central claim on the paper's own terms is that the key to the observed discrepancies resides in the numerical treatment of the long-range Coulomb singular term, and that computing this term presents technical difficulties that are hard to overcome in both the pure-TDDFT and hybrid approaches when using the Wigner-Seitz truncated kernel, pointing to the need for a better description of the electron-hole interaction.

What carries the argument

Wigner-Seitz truncated kernel, applied to isolate and study the effect of the long-range Coulomb singular term within the exchange-correlation kernel of TDDFT.

Load-bearing premise

The observed technical difficulties originate primarily from the numerical handling of the Coulomb singularity rather than from limitations inherent to the LRC or hybrid kernels themselves.

What would settle it

A recalculation of the same systems using an alternative numerical scheme or truncation that fully eliminates the singularity-handling issues, followed by checking whether the exciton binding energies then match experiment while spectra remain good.

Figures

Figures reproduced from arXiv: 2303.13389 by A. Ayuela, A. Leonardo, M. Arruabarrena.

**Figure 2.** Figure 2: shows the obtained α values that brought the binding energies close to experiments for a set of semiconductors and insulators in the following three cases: a) setting Fxc(G = 0) = 0 in Eq.[4], i.e., neglecting the indetermination; b) only considering the Fxc(G = 0) term different from zero and neglecting the all the other terms of the summatory (LRC Head-only); and c) full solution (LRC Diagonal). Moreov… view at source ↗

**Figure 3.** Figure 3: Calculated exciton binding energies of GaAs with [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

read the original abstract

Time-Dependent Density Functional Theory (TDDFT) has been currently established as a computationally cheaper, yet effective, alternative to the Many-Body Perturbation Theory (MBPT) for calculating the optical properties of solids. Within the Linear Response formalism, the optical absorption spectra are in good agreement with experiments, as well as the direct determination of the exciton binding energies. However, the family of exchange-correlation kernels known as long-range corrected (LRC) kernels that correctly capture excitonic features have difficulties simultaneously producing good-looking spectra and accurate exciton binding energies. More recently, this discrepancy has been partially overcome by a hybrid-TDDFT approach. We show that the key resides in the numerical treatment of the long-range Coulomb singular term. We carefully study the effect of this term, both in the pure-TDDFT and hybrid approach using a Wigner-Seitz truncated kernel. We find that computing this term presents technical difficulties that are hard to overcome in both approaches, and that points to the need for a better description of the electron-hole interaction.

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.

Desk Editor's Note private letter to a colleague

The paper flags real numerical headaches with the Coulomb singularity under Wigner-Seitz truncation in both LRC and hybrid TDDFT, but stops short of showing that fixing the numerics alone would fix the binding energies.

read the letter

The main thing to know is that this work isolates the long-range Coulomb term via Wigner-Seitz truncation and reports that it creates stubborn technical problems for getting both decent spectra and accurate exciton binding energies in solids, whether using pure LRC kernels or the hybrid-TDDFT route. That observation is the concrete contribution. It is not a new kernel or a new derivation; it is a targeted numerical check on an existing difficulty that the abstract already flags in the literature. The authors do a service by showing the issue appears in both approaches rather than blaming one kernel family. That narrows where the problem likely sits. The soft spot is exactly the one the stress-test note flags. The paper observes the difficulties but does not run the obvious control: keep the identical kernel, swap in a demonstrably more accurate treatment of the singular term, and check whether binding energies then come out right. Without that step the claim that the numerics are the key obstacle remains plausible but unproven. The rest of the manuscript would need to supply error bars, convergence tests, or direct comparisons to make the case tighter. This is the sort of focused methods paper that people who actually implement TDDFT for real materials will want to read. It is not ready for a broad audience yet, but it is worth a referee's time because the numerical point is practical and the field needs clearer diagnostics on why LRC-type kernels keep underperforming on binding energies. I would send it out for review with a request for the missing control calculation.

Referee Report

2 major / 0 minor

Summary. The manuscript investigates TDDFT calculations of exciton binding energies in solids, focusing on long-range corrected (LRC) kernels and a hybrid-TDDFT approach. It claims that discrepancies between accurate optical spectra and binding energies arise primarily from numerical difficulties in treating the long-range Coulomb singular term, which are studied via a Wigner-Seitz truncated kernel in both pure-TDDFT and hybrid settings; the authors conclude that these difficulties are hard to overcome and point to the need for an improved description of the electron-hole interaction.

Significance. If the attribution of discrepancies to numerical treatment of the Coulomb term holds after controlled testing, the work would usefully highlight a practical computational bottleneck in TDDFT for excitons and could guide refinements in truncation schemes or kernel implementations. The paper reports a computational observation on technical challenges rather than a parameter-free derivation or falsifiable prediction, limiting its immediate impact on the field.

major comments (2)

[Abstract] Abstract: the central claim that 'computing this term presents technical difficulties that are hard to overcome in both approaches' and that this 'points to the need for a better description of the electron-hole interaction' is not supported by a controlled comparison isolating the numerical handling of the singularity from limitations in the LRC or hybrid kernel forms themselves. No test is shown in which an alternative high-accuracy treatment of the same singular term (with identical kernel) simultaneously recovers accurate binding energies and spectra.
[Abstract] Abstract (paragraph on hybrid-TDDFT and Wigner-Seitz study): the assumption that observed technical difficulties originate primarily from the numerical handling of the Coulomb singularity (rather than from the kernels) lacks direct evidence such as error analysis, convergence metrics, or side-by-side comparisons with non-truncated high-precision treatments of the same term.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments on our abstract. We address each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that 'computing this term presents technical difficulties that are hard to overcome in both approaches' and that this 'points to the need for a better description of the electron-hole interaction' is not supported by a controlled comparison isolating the numerical handling of the singularity from limitations in the LRC or hybrid kernel forms themselves. No test is shown in which an alternative high-accuracy treatment of the same singular term (with identical kernel) simultaneously recovers accurate binding energies and spectra.

Authors: Our calculations apply the Wigner-Seitz truncation to the long-range Coulomb term within both LRC kernels and the hybrid-TDDFT framework, and we report that accurate exciton binding energies remain difficult to obtain simultaneously with good spectra. This observation underpins the abstract statement. We agree that the study does not include a side-by-side comparison against an alternative high-accuracy treatment of the identical singular term, so the numerical origin is not isolated from possible kernel limitations. We will revise the abstract to state the findings more narrowly as results obtained with the truncation scheme. revision: yes
Referee: [Abstract] Abstract (paragraph on hybrid-TDDFT and Wigner-Seitz study): the assumption that observed technical difficulties originate primarily from the numerical handling of the Coulomb singularity (rather than from the kernels) lacks direct evidence such as error analysis, convergence metrics, or side-by-side comparisons with non-truncated high-precision treatments of the same term.

Authors: The manuscript shows that the same truncation produces persistent difficulties in both the pure-TDDFT and hybrid settings. While this consistency is the evidence we present, we do not supply quantitative error analysis, convergence plots, or comparisons with non-truncated high-precision implementations of the singular term. We will therefore revise the abstract paragraph to avoid implying that the numerical handling has been shown to be the primary source and to indicate that further work on improved treatments would be valuable. revision: yes

Circularity Check

0 steps flagged

No circularity; computational observations are self-contained without fitted predictions or load-bearing self-citations

full rationale

The paper reports numerical results from applying a Wigner-Seitz truncated kernel to study the long-range Coulomb term in TDDFT and hybrid-TDDFT calculations of exciton binding energies. No derivation reduces a claimed prediction to a fitted parameter by construction, no uniqueness theorem is imported via self-citation, and no ansatz is smuggled through prior work. The central observation—that technical difficulties arise in handling the singular term—follows directly from the reported computations rather than from re-labeling inputs as outputs. The study is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the work rests on the standard linear-response TDDFT framework and the validity of Wigner-Seitz truncation for periodic systems. No free parameters, new entities, or ad-hoc axioms are mentioned.

axioms (1)

domain assumption Linear-response TDDFT with LRC kernels is applicable to optical absorption and exciton binding in solids
Invoked as the established starting point for the discrepancy discussed.

pith-pipeline@v0.9.0 · 5721 in / 1231 out tokens · 22919 ms · 2026-05-24T09:59:29.218769+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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However, in comparison with TDDFT, the q = G =0 term for the c = c′ and v = v′ matrix elements now diverges as 1 /02

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[1] [1]

However, in comparison with TDDFT, the q = G =0 term for the c = c′ and v = v′ matrix elements now diverges as 1 /02

Wigner-Seitz truncation of the kernel In the SXX approach, a correct treatment for the Coulomb term is crucial. However, in comparison with TDDFT, the q = G =0 term for the c = c′ and v = v′ matrix elements now diverges as 1 /02. The use of real-space truncation methods to handle numerically the Coulomb interaction have been previously studied to cal- cul...

work page 2050

[2] [2]

Onida, L

G. Onida, L. Reining, and A. Rubio, Electronic exci- tations: density-functional versus many-body green’s- function approaches, Rev. Mod. Phys. 74, 601 (2002)

work page 2002

[3] [3]

Botti, A

S. Botti, A. Schindlmayr, R. D. Sole, and L. Reining, Time-dependent density-functional theory for extended systems, Rep. Prog. Phys. 70, 357 (2007)

work page 2007

[4] [4]

Sharma, J

S. Sharma, J. K. Dewhurst, A. Sanna, and E. K. U. Gross, Bootstrap approximation for the exchange- correlation kernel of time-dependent density-functional theory, Phys. Rev. Lett. 107, 186401 (2011)

work page 2011

[5] [5]

Rigamonti, S

S. Rigamonti, S. Botti, V. Veniard, C. Draxl, L. Reining, and F. Sottile, Estimating Excitonic Eﬀects in the Ab- sorption Spectra of Solids: Problems and Insight from a Guided Iteration Scheme, Phys. Rev. Lett. 114, 146402 (2015)

work page 2015

[6] [6]

C. A. Ullrich and Z.-h. Yang, Excitons in Time- Dependent Density-Functional Theory (Springer, Cham,

work page

[7] [7]

Yang and C

Z.-h. Yang and C. A. Ullrich, Direct calculation of exciton binding energies with time-dependent density-functional theory, Phys. Rev. B 87, 195204 (2013)

work page 2013

[8] [8]

Byun and C

Y.-M. Byun and C. A. Ullrich, Assessment of long-range- corrected exchange-correlation kernels for solids: Accu- rate exciton binding energies via an empirically scaled bootstrap kernel, Phys. Rev. B 95, 205136 (2017)

work page 2017

[9] [9]

M. E. CASIDA, Time-Dependent Density Functional Re- sponse Theory for Molecules (1995) pp. 155–192

work page 1995

[10] [10]

Turkowski, a

V. Turkowski, a. Leonardo, and C. a. Ullrich, Time- dependent density-functional approach for exciton bind- ing energies, Phys. Rev. B 79, 1 (2009)

work page 2009

[11] [11]

Resta, Quantum-Mechanical Position Operator in Ex- tended Systems, Phys

R. Resta, Quantum-Mechanical Position Operator in Ex- tended Systems, Phys. Rev. Lett. 80, 1800 (1998)

work page 1998

[12] [12]

B. Gu, N. H. Kwong, and R. Binder, Relation between the interband dipole and momentum matrix elements in semiconductors, Phys. Rev. B 87, 125301 (2013)

work page 2013

[13] [13]

Z.-h. Yang, F. Sottile, and C. A. Ullrich, Simple screened exact-exchange approach for excitonic properties in solids, Phys. Rev. B 92, 035202 (2015)

work page 2015

[14] [14]

Y.-M. Byun, J. Sun, and C. A. Ullrich, Time-dependent density-functional theory for periodic solids: assess- ment of excitonic exchange–correlation kernels, Electron. Struct. 2, 023002 (2020)

work page 2020

[15] [15]

J. Sun, J. Yang, and C. A. Ullrich, Low-cost alternatives to the bethe-salpeter equation: Towards simple hybrid functionals for excitonic eﬀects in solids, Phys. Rev. Res. 2, 013091 (2020)

work page 2020

[16] [16]

Sun and C

J. Sun and C. A. Ullrich, Optical properties of CsCu 2X3 (X = Cl, Br, andI): A comparative study between hybrid time-dependent density-functional theory and the Bethe- Salpeter equation, Phys. Rev. Mater. 4, 095402 (2020)

work page 2020

[17] [17]

R. M. Martin, L. Reining, and D. M. Ceperley, Inter- acting Electrons: Theory and Computational Approaches (Cambridge University Press, 2016)

work page 2016

[18] [18]

Botti, F

S. Botti, F. Sottile, N. Vast, V. Olevano, L. Reining, H.-C. Weissker, A. Rubio, G. Onida, R. Del Sole, and R. W. Godby, Long-range contribution to the exchange- correlation kernel of time-dependent density functional theory, Phys. Rev. B 69, 155112 (2004)

work page 2004

[19] [19]

Rigamonti, S

S. Rigamonti, S. Botti, V. Veniard, C. Draxl, L. Rein- ing, and F. Sottile, Estimating excitonic eﬀects in the absorption spectra of solids: Problems and insight from a guided iteration scheme, Phys. Rev. Lett. 114, 146402 (2015)

work page 2015

[20] [20]

Baroni and R

S. Baroni and R. Resta, Ab initio calculation of the macroscopic dielectric constant in silicon, Phys. Rev. B 33, 7017 (1986)

work page 1986

[21] [21]

M. L. Cohen and T. K. Bergstresser, Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zinc-blende structures, Phys. Rev. 141, 789 (1966)

work page 1966

[22] [22]

Giannozzi, S

P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ- cioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Sc...

work page 2009

[23] [23]

Deslippe, G

J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, BerkeleyGW: A massively parallel computer package for the calculation of the quasi- particle and optical properties of materials and nanos- tructures, Comput. Phys. Commun. 183, 1269 (2012)

work page 2012

[24] [24]

Kim and A

Y.-H. Kim and A. G¨ orling, Excitonic optical spectrum of semiconductors obtained by time-dependent density- functional theory with the exact-exchange kernel, Phys. Rev. Lett. 89, 096402 (2002)

work page 2002

[25] [25]

Kim and A

Y.-H. Kim and A. G¨ orling, Exact kohn-sham exchange kernel for insulators and its long-wavelength behavior, Phys. Rev. B 66, 035114 (2002)

work page 2002

[26] [26]

Sundararaman and T

R. Sundararaman and T. A. Arias, Regularization of the coulomb singularity in exact exchange by wigner-seitz truncated interactions: Towards chemical accuracy in 7 nontrivial systems, Phys. Rev. B 87, 165122 (2013)

work page 2013

[27] [27]

Parenteau, C

M. Parenteau, C. Carlone, and S. M. Khanna, Damage coeﬃcient associated with free exciton lifetime in GaAs irradiated with neutrons and electrons, J. Appl. Phys. 71, 3747 (1992)

work page 1992

[28] [28]

Levinshtein, S

M. Levinshtein, S. Rumyantsev, and M. S. Shur, Prop- erties of advanced semiconductor materials : GaN, AlN, InN, BN, SiC, SiGe (2001) pp. 1–30

work page 2001

[29] [29]

Feneberg, M

M. Feneberg, M. R¨ oppischer, C. Cobet, N. Esser, J. Sch¨ ormann, T. Schupp, D. J. As, F. H¨ orich, J. Bl¨ asing, A. Krost, and R. Goldhahn, Optical properties of cubic GaN from 1 to 20 eV, Phys. Rev. B 85, 155207 (2012)

work page 2012

[30] [30]

D. J. As, F. Schmilgus, C. Wang, B. Sch¨ ottker, D. Schikora, and K. Lischka, The near band edge pho- toluminescence of cubic gan epilayers, Appl. Phys. Lett. 70, 1311 (1997)

work page 1997

[31] [31]

Jakobson, V

M. Jakobson, V. Kagan, R. Seisyan, and E. Goncharova, Optical properties of “pure” CdS and metal-insulator- semiconductor structures on CdS at electrical operation, J. Cryst. Growth 138, 225 (1994)

work page 1994

[32] [32]

Voigt, F

J. Voigt, F. Spiegelberg, and M. Senoner, Band param- eters of CdS and CdSe single crystals determined from optical exciton spectra, Phys. Status Solidi B 91, 189 (1979)

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