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Optical excitations in nanographenes from the Bethe-Salpeter equation and time-dependent density functional theory: absorption spectra and spatial descriptors

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Pith's one-line read This paper shows that a many-body GW-BSE calculation reproduces the two lowest absorption peaks of 7-armchair nanographenes within 0.05 eV of experiment after a quadratic length extrapolation, and that the size of the lowest bright excitati

desk verdict A credible CP2K GW-BSE implementation backed by a strong cross-code benchmark; the sub-0.1 eV agreement with experiment is plausible but rests on a long extrapolation and should be read as suggestive, not definitive. read the letter →

arxiv 2510.25658 v2 pith:PHXUQY32 submitted 2025-10-29 physics.comp-ph physics.chem-ph

Maximilian Graml , Jan Wilhelm This is my paper · ORCID
classification physics.comp-phphysics.chem-ph
keywords Bethe-SalpeterequationGWapproximationnanographenesgraphenenanoribbonsopticalabsorptionspectraexcitonstime-dependentdensityfunctionaltheoryspatialdescriptors
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

The paper establishes that a many-body GW-BSE calculation (BSE@evGW0@PBE) can quantitatively reproduce the optical absorption of finite nanographene ribbons and, at the same time, predict the spatial extent of the bound electron-hole pair. After validating the new CP2K implementation against an established code on Thiel's molecular set (mean error under 3 meV), the authors compute spectra for ribbons of increasing length, extrapolate the two lowest bright peaks to the experimental ~20 nm length, and find agreement within 0.05 eV. They also compute the spatial descriptors of the lowest excitation and find that its size saturates at about 7.6 Å for ribbons longer than a few nanometers, indicating a bound exciton with intrinsic correlation length. A systematic TDDFT comparison shows that functionals with 0-65% exact exchange cannot simultaneously reproduce the BSE spectral shape and the exciton size. The work thus argues that many-body methods are needed for accurate excited-state description in nanoscale carbon nanostructures.

What carries the argument

The central machinery is the Bethe-Salpeter equation built on eigenvalue-self-consistent GW quasiparticle energies (BSE@evGW0@PBE), which treats the electron-hole pair via the statically screened Coulomb interaction W0(ω=0). The spatial analysis uses the size of the excitation des = sqrt(⟨|r_e - r_h|²⟩), decomposed into electron-hole separation, electron and hole spreads, and a correlation coefficient. The finite-size analysis uses a quadratic fit Ω(p)(L)=a+b/L+c/L², motivated by a one-dimensional free-electron model, to extrapolate peak positions from L=4-13 to the experimental length L≈20 nm.

What would settle it

Compute BSE@evGW0@PBE absorption peaks for a nanographene with L=15 or L=20 (or use a more rigorous extrapolation with more data points) and check whether the peak positions follow the fitted quadratic curve; alternatively, measure the two experimental peak positions for ribbons of several well-characterized lengths to test the extrapolation.

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Extended reading notes

Core claim

The central result is that the BSE@evGW0@PBE approach, implemented in the CP2K code, reproduces the two lowest longitudinal absorption peaks of 7-armchair nanographenes within 0.05 eV of experimental peak positions after a quadratic 1/L extrapolation to the experimental length, and that the spatial size of the lowest bright excitation converges to ~7.6 Å with increasing ribbon length. The paper further shows that TDDFT with any tested functional (PBE, BLYP, PBE0, B3LYP, HSE06, CAM-B3LYP, and PBEh with 0-100% exact exchange) fails to reproduce both the full spectral shape and the exciton size, highlighting the need for explicit electron-hole interaction and screening.

Load-bearing premise

The agreement with experiment rests on the quadratic 1/L extrapolation of the two peak positions from short ribbons (L=4-13) to the experimental length of about 20 nm; if the true length dependence deviates from this functional form at longer lengths, the claimed 0.05 eV agreement would not hold.

Editorial extensions

If this is right

  • The same GW-BSE workflow can now be applied to other finite nanostructures (nanoribbons, quantum dots, molecular aggregates) in CP2K, providing optical spectra and exciton sizes without empirical tuning.
  • The saturation of the exciton size at ~7.6 Å in 7-armchair ribbons implies that optical properties of longer ribbons are governed by a local exciton with well-defined radius, not by ribbon length.
  • TDDFT with standard hybrid functionals, even at 25-65% exact exchange, is insufficient for predicting both spectral shape and exciton spatial extent; quantitative exciton properties in nanographenes require many-body methods.
  • The quadratic 1/L extrapolation connects computationally accessible finite flakes to experimentally synthesized ~20 nm ribbons, enabling quantitative comparison with experiment.

Reading between the lines

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

  • If the quadratic 1/L scaling holds beyond L=13, a direct BSE calculation at L=15 or L=20 should land on the fitted curve; a deviation would indicate the extrapolated agreement is fortuitous.
  • The predicted exciton radius could be measured experimentally (e.g., via two-photon absorption or scanning tunneling spectroscopy), offering a direct test of the ~7.6 Å value.
  • The descriptor framework introduced here (size of excitation, correlation coefficient) could serve as a quantitative benchmark for judging improved TDDFT kernels, such as range-separated hybrids with optimally tuned screening.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The authors report an implementation of GW-BSE in the CP2K code and validate it against FHI-aims on Thiel's set of 28 organic molecules, obtaining a mean absolute error of 2.7 meV with different analytic-continuation settings (4.7 meV with matched settings). They then apply BSE@evGW0@PBE to finite 7-armchair nanographenes of increasing length (L=4–13), extract the two lowest longitudinal absorption peaks, and extrapolate quadratically in 1/L to the experimental length of about 20 nm. The extrapolated values (2.09 eV and 2.26 eV) agree with experiment (2.05 eV and 2.31 eV) to within 0.05 eV. The paper also computes spatial descriptors of the lowest bright excitation, finding that its longitudinal size converges to about 7.6 Å, and compares with TDDFT using several exchange–correlation functionals, concluding that none reproduce both the BSE size and the full spectral shape.

Significance. If the finite-size extrapolation is robust, the paper is a valuable contribution: it makes available an open-source GW-BSE implementation in a widely used code, backed by a strong cross-code validation (MAE < 5 meV) and reproducible data (GitHub/Zenodo). The 0.05 eV agreement with experiment would be an encouraging demonstration of BSE for finite nanostructures. The spatial-descriptor analysis provides a compact way to compare TDDFT and BSE excited-state character, and the finding that common hybrid functionals fail to simultaneously reproduce the size and spectral shape of the exciton is instructive. The claimed convergence of the exciton size to about 7.6 Å is an interesting falsifiable prediction. However, the central quantitative claim hinges on an extrapolation that is currently not sufficiently scrutinized, which tempers the significance until that point is addressed.

major comments (3)
  1. [Sec. III, Fig. 4 and Table I] The central claim of 0.05 eV agreement with experiment rests on the quadratic 1/L extrapolation of Ω^(p)(L) from L=4–13 to L≈46 (~20 nm). The extrapolation extends more than three times beyond the largest computed system, yet the paper provides no confidence intervals on a,b,c, no residual analysis, and no test of the assumed functional form. For p=2, b/L contributes 68 meV at L=46; a modest error in b would shift the extrapolated value by tens of meV. The BSE energy cutoff (E_empty=10 eV) already shifts peak positions by up to 45 meV at L=4 (Table II), an error comparable to the claimed agreement. Please report fit uncertainties, test sensitivity to excluding small-L points and to alternate 1/L forms, and preferably compute at least one longer system. If the extrapolated values shift by more than ~50 meV, the 'excellent agreement' wording should be softened.
  2. [App. B, Table II] All production BSE calculations use E_empty^cut=10 eV (App. D). Table II shows that this cutoff shifts the first three peak positions by 35–45 meV for L=4 relative to the fully converged result, and Table III shows a 38 meV shift and 2.3% change in d_exc for the strongest excitation. This error is comparable to the claimed 0.05 eV agreement with experiment and is not included in the error budget of the extrapolated values. The authors should quantify the cutoff convergence at larger L (e.g., L=8,13) and report whether the extrapolated peak positions are stable with respect to cutoff corrections.
  3. [Sec. III, Fig. 5] The experimental spectra (Ref. 118) are measured for nanographenes on noble-metal surfaces, while the calculations treat free-standing molecules. Substrate screening and image-charge effects can shift excitation energies by tens of meV and are not discussed. Without an estimate of the substrate-induced shift, the agreement to 0.05 eV is not fully interpretable. Please discuss this effect and, if possible, provide a rough estimate (e.g., via a dielectric screening model or literature values for similar GNRs on metal surfaces).
minor comments (6)
  1. [Abstract / App. E] The abstract and Sec. II state a MAE 'below 3 meV', but this is for different analytic-continuation settings in CP2K (128 Padé, 500 grid) and FHI-aims (16 Padé, 100 grid). With matched settings the MAE is 4.7 meV (App. E, Fig. 10). Please report both values together to avoid overstatement.
  2. [Sec. III / Sec. VI] The direction nomenclature is inconsistent: Sec. III says 'seven carbon atoms in zigzag direction and ... L in armchair direction', while Sec. VI says the width is along the armchair direction and the length along the zigzag direction. The latter is the correct description of a 7-armchair nanographene; please fix the earlier sentence.
  3. [Fig. 4] The axis label '1/∞' is unconventional; consider labeling the limit as 0.
  4. [Eq. (24) / Fig. 5] The frequency-independent constant C in Eq. (24) is not determined, so the comparison of line shapes in Fig. 5 is qualitative. This should be stated explicitly.
  5. [App. G] The main text quotes a converged size of ~7.6 Å from TDA. App. G shows that solving the full BSE (ABBA) changes d_exc by up to 0.32 Å at large L; this uncertainty should be reflected in the quoted value.
  6. [Fig. 4 caption] Typo: 'We fit the the obtained peak frequencies' should read 'We fit the obtained peak frequencies'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: experimental agreement is a genuine extrapolation, implementation is benchmarked against an independent code, and self-citations are not load-bearing.

full rationale

The paper's central claims do not reduce to their own inputs. The CP2K GW-BSE implementation is validated against FHI-aims, an independent code, for Thiel's set (Sec. II, Fig. 2), giving an external benchmark rather than a self-referential check. The comparison with experiment is a genuine extrapolation: the peak positions Ω^(p)(L) are computed for L=4–13, fitted to Ω^(p)(L)=a^(p)+b^(p)/L+c^(p)/L², and only then evaluated at the experimental length L≈20 nm (Sec. III, Fig. 4, Table I). The fit parameters a,b,c are determined from the authors' own computed data, not from the experimental peak positions, so the agreement with Ω_exp=2.05 eV and 2.31 eV is not forced by construction. The use of this quadratic form is an extrapolation ansatz and is a robustness limitation, but it is not circularity. The spatial descriptors in Secs. IV–V are standard exciton-size definitions (Refs. 55,56,59) evaluated from BSE eigenvectors; the claimed ~7.6 Å saturation is a computed result, not an input. TDDFT comparisons likewise use independently defined functionals and spectra. Self-citations to the authors' prior GW methodology (Refs. 90,91,120) occur, but the methodological choices they support are benchmarked externally and are not used to forbid alternatives. No load-bearing derivation step is equivalent by construction to its input.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities. The main free parameters are the coefficients of the 1/L fit used to compare computed peak positions to experiment; these are fitted to computed data, not to the experimental target, so they do not create circularity but they do add uncertainty to the extrapolation. The key modeling assumptions are the isolated-molecule model for an on-substrate experiment and the validity of the 1/L² extrapolation.

free parameters (7)
  • Fit coefficient a(p=1) for Ω(1)(L) = 2.08 eV
    Quadratic 1/L fit to computed peak positions for L=4..13 (Table I).
  • Fit coefficient b(p=1) for Ω(1)(L) = 0.39 eV
    Same fit.
  • Fit coefficient c(p=1) for Ω(1)(L) = 3.96 eV
    Same fit.
  • Fit coefficient a(p=2) for Ω(2)(L) = 2.19 eV
    Quadratic 1/L fit to computed peak positions for L=4..13 (Table I).
  • Fit coefficient b(p=2) for Ω(2)(L) = 3.13 eV
    Same fit.
  • Fit coefficient c(p=2) for Ω(2)(L) = 3.97 eV
    Same fit.
  • Broadening η = 0.05 eV (Figs. 3,9), 0.3 eV (Fig. 5)
    Chosen by hand; affects peak heights but not peak positions; standard practice.
assumptions (4)
  • domain assumption The GW approximation (Σ=iGW) and BSE with static screening provide a sufficiently accurate description of the optical excitations of these nanographenes.
    The paper uses the standard BSE@evGW0@PBE workflow (Eqs. 3-12) and does not compare against higher-level methods (e.g., CCSD or high-resolution gas-phase spectra) for the nanographenes themselves.
  • domain assumption The Tamm-Dancoff approximation (TDA) yields negligible deviations for the computed observables.
    The authors verify in App. G that TDA vs full BSE changes energies by ≤0.22 eV and sizes by ≤0.32 Å, so this assumption is partially tested; however, the main results in Sec. V are quoted in TDA.
  • ad hoc to paper The 1/L quadratic extrapolation model is valid over the range from L=4 to L→∞.
    Motivated by the free-electron model for conjugated polymers (Refs. 122-123), but the coefficients are fitted to computed data and no independent check of the functional form at larger L is given.
  • domain assumption The isolated-molecule model is representative of the experimental nanographenes measured on a metal surface.
    Experimental spectra (Ref. 118) are taken on Au(111); the calculations are for free-standing molecules. The paper does not discuss image-charge screening, charge transfer, or substrate-induced broadening.

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Pith. "Pith review of Optical excitations in nanographenes from the Bethe-Salpeter equation and time-dependent density functional theory: absorption spectra and spatial descriptors." pith.science (2026). https://pith.science/paper/PHXUQY32

@misc{pith2026251025658,
  author       = {Pith},
  title        = {Pith review of: Optical excitations in nanographenes from the Bethe-Salpeter equation and time-dependent density functional theory: absorption spectra and spatial descriptors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PHXUQY32}},
  note         = {Machine review of arXiv:2510.25658}
}
abstract

The GW plus Bethe-Salpeter equation (GW-BSE) formalism is a well-established approach for calculating excitation energies and optical spectra of molecules, nanostructures, and crystalline materials. We implement GW-BSE in the CP2K code and validate the implementation for a standard organic molecular test set, obtaining excellent agreement with reference data, with a mean absolute error in excitation energies below 3 meV. We then study optical spectra of nanographenes of increasing length, showing excellent agreement with experiment. We further compute the size of the excitation of the lowest optically active excitation which converges to about 7.6 $\r{A}$ with increasing length. Comparison with time-dependent density functional theory using functionals of varying exact-exchange fraction shows that none reproduce both the size of the excitation and optical spectra of GW-BSE, underscoring the need for many-body methods for accurate description of electronic excitations in nanostructures.

Figures

Figures reproduced from arXiv: 2510.25658 by the authors.

Figure 1
Figure 1. Workflow of a GW-BSE calculation within this work, where we employ BSE@evGW0@PBE. A KS–DFT calculation provides the KS energies ε KS p and orbitals ψ KS p as input. In the evGW0 scheme, the quasiparticle energies are iterated (bold arrows) in a self-consistent loop, starting from G = G0 and using W0 from the DFT starting point. This yields the evGW0 quasiparticle energies ε evGW0 p [Eq. (6)]. BSE matrices A, B [Eq. … view at source ↗
Figure 2
Figure 2. Absolute error |Ω (n) CP2K − Ω (n) aims| of BSE excitation energies Ω(n) computed from CP2K and FHI-aims by solving Eq. (8) with BSE@evGW0@PBE. The mean absolute error (23) over the ten low￾est excitation energies across all molecules is only 2.7 meV. For a benchmark on the impact of parameters of GW calculations on BSE excitation energies, we refer to App. E. III. Absorption spectrum of nanographenes from BSE and c… view at source ↗
Figure 4
Figure 4. Excitation frequencies Ω(p=1,2) of the first two dominant peaks (p = 1, 2) in Im αxx (L) (ω) for L ∈ [4, 13] (see example spectra in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (9 more)
Figure 3
Figure 3. Figure 3: Optical absorption spectrum from the imaginary part of the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]
Figure 5
Figure 5. Figure 5: Optical absorption spectrum (a) reproduced from exper [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Spatial properties of the first bright peak [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 8
Figure 8. Figure 8: Optical absorption spectrum for GW-BSE (black) and TDDFT with two different hybrid functionals PBE0 (green) and B3LYP (beige) for the nanographene with L = 13, computed from Eq. (20). Dashed lines denote the contribution of individual excita￾tions n to Im αxx(ω = Ω(n) …
Figure 7
Figure 7. Figure 7: Excitation energy Ω(p=1) and size of the excitation d (x,p=1) exc computed from BSE@evGW0 and TDDFT (both in TDA; effect of TDA investigated in App. G) employing different ex￾change–correlation functionals (exact exchange fraction shown in the legend). (a) Peak frequen…
Figure 9
Figure 9. Figure 9: Optical absorption spectra computed from [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: Absolute error |Ω (n) CP2K − Ω (n) aims| of BSE excitation en￾ergies Ω(n) computed from CP2K and FHI-aims by solving Eq. (8) with BSE@evGW0@PBE. The analytical continuation for the GW￾modules in both codes was applied with 16 parameters of the Pade´ function and 100 p…
Figure 11
Figure 11. Figure 11: Excitation energy Ω(p=1) and size of the excitation d (x,p=1) exc computed from BSE@evGW0 via Eq. (8) (yellow) and Eq. (18) (blue). (a) Peak frequency of the first prominent peak in the optical absorption spectrum Ω(p=1). (b) Longitudinal size of the excitation for th…
Figure 13
Figure 13. Figure 13: Excitation energy Ω(p=1) and size of the excitation d (x,p=1) exc computed from BSE@evGW0 and TDDFT (both in TDA; effect of TDA investigated in App. G) employing the PBEh functional with varying amount of exact exchange (exact exchange fraction shown in the legend). (…

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