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A parameter-free statistical model for two-dimensional carbon nanostructures

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper argues that the ground-state electron density of carbon nanostructures can be obtained by minimizing a parameter-free bonding free energy that combines the octet rule with bond-entropy maximization.

desk verdict A clever maximum-entropy generalization of Clar's rule that correlates well with DFT on benzenoid PAHs, but the parameter-free premise (U=0) likely fails for antiaromatic systems like cyclobutadiene. read the letter →

arxiv 2412.13588 v1 pith:KLZRYWOB submitted 2024-12-18 cond-mat.mes-hall cond-mat.mtrl-sci

Chang-Chun He , Shao-Gang Xu , Jiarui Zeng , Weijie Huang , Yao Yao , Yu-Jun Zhao , Hu Xu This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-sci
keywords bondingfreeenergyoctetrulecarbonnanostructurespolycyclicaromatichydrocarbonsgraphynetight-bindingmodelbondoccupancynumbersmaximumentropy
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 proposes that for carbon nanostructures, the electron density is set less by bond-specific energetics and more by how evenly electrons can be spread among bonds while every carbon atom satisfies the octet rule. It encodes this idea in a grand canonical ensemble over all Kekulé-like electron allocations and defines a bonding free energy $F_b = \frac{N_{\mathrm{ele}} k_B T_0}{\log N_{\mathrm{ele}}} \sum_i p_i \log p_i$. Minimizing $F_b$ under octet constraints yields occupancy numbers for every C-C bond with no fitted parameters. The authors show these occupancies correlate with DFT bond energies and bond lengths, identify stable magic PAH clusters, and can be used as hopping integrals in a tight-binding model to reproduce energy gaps and band structures. If correct, the model offers a fast, interpretable route to structural and electronic properties that does not require training data.

What carries the argument

The machinery is the grand canonical partition function over bond-occupancy configurations, Eq. (1), whose multinomial structure gives a closed free energy in terms of the probabilities $p_i = n_i / N_{\mathrm{ele}}$. The named central object is the bonding free energy $F_b$, an entropy functional equipped with an equivalent temperature $k_B T_0 / \log N_{\mathrm{ele}}$ chosen to restore extensivity. The octet rule enters as local constraints, one per carbon atom requiring eight surrounding electrons and one per hydrogen requiring two, and the entropy maximum under these constraints fixes the occupancy numbers. Those occupancies then serve as bond-strength descriptors and as hopping integrals in a tight-binding model.

What would settle it

Compute the BFE ordering for a set of PAH isomers and compare it with CCSD(T) or experiment: if any isomer with a less uniform occupancy-number distribution is nevertheless more stable, the entropy-maximization claim fails. Concretely, the model predicts that among C18H12 isomers triphenylene has the most uniform occupancies and the lowest $F_b$; a counterexample would be an isomer with larger occupancy variance but lower energy, or a measured carbon nanobelt whose most stable isomer does not have the most uniform ON distribution.

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Extended reading notes

Core claim

The central claim is that the most stable electron distribution in a carbon nanostructure is the one that maximizes bonding entropy subject only to the octet rule and the total electron count. Writing the grand canonical partition function over all ways to distribute $N_{\mathrm{ele}}$ electrons among bonds gives $F = N_{\mathrm{ele}} k_B T \sum_i p_i \log p_i$; redefining the temperature as $k_B T = k_B T_0 / \log N_{\mathrm{ele}}$ makes this free energy extensive, and the minimum of the resulting $F_b$ selects the occupancy numbers $n_i = 2 p_i$ for C-C bonds. The paper reports that these occupancies reproduce molecular symmetry, correlate linearly with DFT-computed bond energies and bond lengths, rank PAH isomers in agreement with CCSD(T) results, and identify magic clusters. When the occupancies are used as hopping integrals in a tight-binding Hamiltonian, they predict HOMO-LUMO gaps, molecular orbital shapes, and band structures near the Fermi level in agreement with DFT.

Load-bearing premise

The model assumes all C-C bonds have identical internal energy, so a bond's importance comes only from the octet constraints and the entropy of distributing electrons; if bond-specific electronic energies (single versus double or aromatic bonds) matter for stability, the entropy-only ranking can fail.

Editorial extensions

If this is right

  • Relative stabilities of PAH isomers and graphyne allotropes can be ranked from electron counting alone, without DFT or machine-learning parameters.
  • Bond occupancy numbers from the model can be used directly as tight-binding hopping integrals, giving molecular orbital shapes, energy gaps, and band structures that agree with DFT.
  • The mean occupancy per six-membered ring acts as a local aromaticity measure consistent with standard aromaticity descriptors and Clar's rule, including cases where Clar's rule breaks down.
  • Because the model is parameter-free in its core, it can be applied to any sp2 or sp-sp2 carbon framework once the octet constraints are written down, including periodic systems like graphynes.

Reading between the lines

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

  • The same entropy-maximization logic probably transfers to other covalent networks only where bond-specific electronic energy differences are small; for systems where single versus double bond energetics dominate, the $U = 0$ assumption is likely the first thing to break.
  • The model's success on carbon suggests a testable general principle: in valence-constrained covalent networks, the ground-state electron distribution may be closer to a maximum-entropy state than to a single energy-minimized Lewis structure, which could be probed on boron or silicon clusters.
  • A sharper test would be to use the ON-derived hopping integrals without the fitted linear rescaling $\gamma$ and $E_0^g$; if one universal rescaling sufficed across all PAHs, the claim that occupancy numbers encode hopping strengths would be considerably stronger.
  • The equivalence temperature $k_B T_0 / \log N_{\mathrm{ele}}$ is introduced to enforce extensivity; deriving it from a microscopic model of delocalized electrons would turn a useful scheme into a first-principles statistical mechanics.
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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

4 major / 4 minor

Summary. This manuscript introduces a "bonding free energy" (BFE) model in which electrons in a carbon nanostructure are distributed among C-C bonds by minimizing F_b = N_ele k_B T_0 / log(N_ele) * sum_i p_i log(p_i), subject to the octet rule and total electron number. The minimization yields bond occupancy numbers that are compared with DFT charge densities and bond lengths. The authors show correlations between BFE and DFT relative energies for PAHs, cycloarenes, nanobelts, pentagon-containing clusters, and graphynes, and use the ONs as hopping integrals in a tight-binding Hamiltonian to predict gaps, energy levels, orbitals, and band structures. The central claim is that the model is parameter-free and determines electron density and stability without external parameters.

Significance. If the model's claims were fully established, it would provide a simple, computationally cheap descriptor for charge distribution and stability of carbon nanostructures, and a physically motivated way to construct tight-binding hopping integrals. The paper contains many concrete correlations with DFT and experimental trends, and the ON/bond-length relationship is a useful empirical descriptor. The model is also falsifiable and easy to test on new structures, which is a strength. However, the significance is currently limited by the fact that the central derivation rests on an unvalidated U=0 assumption and by the overstatement of the parameter-free nature of the electronic-structure predictions.

major comments (4)
  1. [Eq. (1)] The partition function written in Eq. (1) is not correct as it stands: the sum over n_1,...,n_Nbond of exp(-sum_i (n_i alpha_i + U)) does not equal (sum_i e^{-alpha_i})^{Nele} unless the multinomial degeneracy factor Nele!/(prod_i n_i!) is inserted into the sum. The text states that multinomial coefficients are included, but the displayed first equality omits them; this is a mathematical error in the central derivation, and the subsequent probability p_i = e^{-alpha_i} / sum_j e^{-alpha_j} follows only after the correction. The derivation should be rewritten with the degeneracy factor made explicit.
  2. [Section II.A] The load-bearing assumption U=0 eliminates all bond-specific energetics (single, double, aromatic), so the model is a pure maximum-entropy assignment under octet constraints. This assumption is not derived and is not generally valid. For cyclobutadiene, the octet constraints n_i+n_{i+1}=6 at each carbon and sum_i n_i=12 admit both a uniform solution (all n_i=3) and alternating solutions (n_i=4,2,4,2); the entropy term uniquely selects the uniform solution, predicting a symmetric D4h structure with equal bond orders, whereas the ground state is Jahn-Teller-distorted to a rectangular D2h geometry with alternating bond lengths. The present test set does not include antiaromatic or strongly bond-alternating systems, so the claimed generality is unsupported.
  3. [Section II.C] The statement in the abstract and introduction that the model predicts electronic structure "without relying on external parameters" is contradicted by the procedure in Section II.C: the tight-binding gaps for PAHs are rescaled to DFT gaps using two fitted parameters, gamma = 2.355 and E0_g = 0.01 eV, and the graphyne band structures require a proportionality coefficient fine-tuned to match the DFT band gap. Only the occupancy-number part of the model is parameter-free; the electronic-structure predictions are not. The claims should be reworded and all fitted constants should be explicitly reported as fit parameters.
  4. [Eq. (4)] The substitution k_B T = k_B T_0 / log(N_ele) is introduced ad hoc to enforce extensivity, with no derivation from statistical mechanics. Because this substitution only rescales F_b, it does not affect the location of the minimum, so the entire predictive content rests on the octet constraints and the entropy functional. This should be stated explicitly and justified, or the "parameter-free" claim should be qualified accordingly.
minor comments (4)
  1. [Section II.A] The sentence "the proposed BFE model can easily calculate the resonance weights with external parameters" appears to be a typo and should presumably read "without external parameters".
  2. [Section II.C] The reference to "FIG. 3(a)" for the tight-binding energy-gap correlation should be "FIG. 5(a)", since the linear gap relation is presented in Figure 5.
  3. [Supplementary Material] The derivation of Eq. (4), the extensivity argument, and computational details are repeatedly deferred to the Supplementary Material, which is not included in the submitted manuscript; the main text should at least sketch the derivation so that the reader can verify the central claim.
  4. [General] There are several typographical errors, including "exhbiting" in the Introduction and "prediciton" in the Supplementary Material section; these should be corrected.

Circularity Check

2 steps flagged · score 4.0 of 10

BFE/ON predictions are self-contained, but the electronic-structure claims are partly calibrated: PAH gap rescaling and graphyne hopping coefficient are fitted to the same DFT targets they are said to predict.

  1. fitted input called prediction [Section II C, after Eq. (6) and FIG. 5(a)]
    "The predicted energy gap by the TB model shows a linear correlation with the DFT-calculated energy gap in the HSE06 level, as depicted in FIG. 3(a). This relationship is expressed as EDFT g = γ × ETB g + E0 g , with parameters γ = 2.355 and E0 g = 0.01 eV. Notably, over 70% of the energy gaps predicted by the TB model deviate less than 0.1 eV from those of the DFT calculations as shown in the inset of FIG. 5(a), using only two external parameters, namely γ and E0 g ."

    The quantity called the predicted energy gap is obtained by applying a linear transformation (γ = 2.355, E0 = 0.01 eV) to the TB gaps. These two constants are fit to the very DFT/HSE06 gaps against which the agreement is then reported; no independent training/testing split is described. The 'over 70% within 0.1 eV' statement is therefore an in-sample goodness-of-fit statistic, not an independent prediction. This also contradicts the abstract's claim that predictions are made 'without relying on external parameters,' since the electronic-structure predictions explicitly use two fitted external parameters.

  2. fitted input called prediction [Section II C, graphyne paragraph (FIG. 6)]
    "The TB-based band structures (dashed line) as given in Fig 6(c), which show considerable agreement with DFT results (solid line) near the Fermi level. A single proportionality coefficient is fine-tuned to match the band gap determined by DFT."

    For graphynes, the hopping-integral scale is adjusted so that the TB band gap reproduces the DFT band gap of the same structure. The subsequent claim that the TB band structure shows considerable agreement with DFT near the Fermi level therefore includes, by construction, the fitted band-gap value. The band dispersion away from the gap may retain independent content, but the headline gap agreement is enforced by the fitting step rather than independently predicted.

full rationale

The central BFE derivation leading to Eqs. (1)-(4) is not circular: the occupancy-number distribution is obtained by minimizing a maximum-entropy functional under octet constraints, with no fitted parameters entering that minimization. The model is openly built on the U = 0 ansatz and the maximum-entropy principle; those are strong physical assumptions (and the cyclobutadiene-style symmetric-ON failure is a genuine robustness concern), but they are not cases where a prediction reduces to an input by construction. The genuine circularity is confined to the electronic-structure sections. In Sec. II C, the PAH energy-gap 'prediction' is a linear rescaling of TB gaps with two constants fitted to the same DFT/HSE06 gap data, so the reported agreement is an in-sample calibration. In the graphyne paragraph, a single proportionality coefficient is explicitly fine-tuned to match the DFT band gap and then the band structure is said to agree with DFT. These are fitted inputs renamed as predictions. Because the central stability and electron-density claims retain independent content and are tested against external DFT/CCSD(T) data, the overall circularity score is moderate rather than high.

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

The BFE model introduces no new particles or physical entities; it reinterprets bond occupancies and entropy. The free parameters are the TB calibration constants and the arbitrary temperature scale, which qualify as free parameters because they are either fitted or chosen by hand.

free parameters (4)
  • gamma (TB gap scaling) = 2.355
    Linear scaling between TB energy gaps and DFT HSE06 gaps (Fig. 5a), fitted to the PAH dataset.
  • E0_g (TB gap offset) = 0.01 eV
    Offset in the same linear scaling, fitted to DFT gaps.
  • Graphyne hopping proportionality coefficient = not specified
    A single proportionality coefficient is fine-tuned to match the DFT band gap for gamma-graphyne (Sec. II.C).
  • T0 (equivalent standard temperature) = not specified
    Arbitrary energy scale in Eq. (4); cancels in relative comparisons but is an ad hoc constant.
assumptions (4)
  • domain assumption All C-C bonds have equal internal energy U = 0.
    Invoked in Eq. (1) to make the partition function purely combinatorial; this neglects bond-specific energetics and is the core simplification of the model.
  • domain assumption The octet rule (each carbon has 8 valence electrons, hydrogen 2) is a valid constraint.
    Used to constrain the electron distributions in Sec. II.A; it is an empirical rule for carbon, though the authors cite their prior work on generalized octet rules for other elements.
  • ad hoc to paper The equivalent temperature obeys kBT = kBT0/log(Nele).
    Introduced in Eq. (4) to make the bonding free energy extensive; no physical justification is given in the main text, with details deferred to the SM.
  • ad hoc to paper Hopping integral is proportional to the occupancy number.
    Assumed in Eq. (6) to construct the tight-binding Hamiltonian; this mapping is the key step connecting BFE to electronic structure.

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Cite this review

Pith. "Pith review of A parameter-free statistical model for two-dimensional carbon nanostructures." pith.science (2026). https://pith.science/paper/KLZRYWOB

@misc{pith2026241213588,
  author       = {Pith},
  title        = {Pith review of: A parameter-free statistical model for two-dimensional carbon nanostructures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KLZRYWOB}},
  note         = {Machine review of arXiv:2412.13588}
}
read the original abstract

Energy degeneracy in physical systems may be induced by symmetries of the Hamiltonian, and the resonance of degeneracy states in carbon nanostructures can effectively enhance the stability of the system. Combining the octet rule, we introduce a parameter-free statistical model to determine the physical properties by lifting the energy degeneracy in carbon nanostructures. This model offers a direct path to accurately ascertain electron density distributions in quantum systems, akin to how charge density is used in density functional theory to deduce system properties. Our methodology diverges from traditional quantum mechanics, focusing instead on this unique statistical model by minimizing bonding free energy to determine the fundamental properties of materials. Applied to carbon nanoclusters and graphynes, our model not only precisely predicts bonding energies and electron density without relying on external parameters, but also enhances the prediction of electronic structures through bond occupancy numbers, which act as effective hopping integrals. This innovation offers insights into the structural properties and quantum behavior of electrons across various dimensions.

Figures

Figures reproduced from arXiv: 2412.13588 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Top part is two Kekul´e structures (denoted by [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Total electron allocation model for anthracene [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) The linear relationship between the relative energy predicted by the DFT and [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The relationship between relative energy calculated [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) The linear relationship between the energy gaps predicted by the TB method and those calculated by DFT, with the [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) The linear correlation between BFE and bonding [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Entropy-driven electron density and effective model Hamiltonian for boron systems

    cond-mat.mtrl-sci 2024-12 reject novelty 4.0 of 10

    Minimizing a 'bonding free energy', which equals the entropy of the electron distribution over 2c-2e and 3c-2e bonds under octet constraints, is proposed as a structure-prediction criterion for boron systems.

Reference graph

Works this paper leans on

55 extracted references · 41 canonical work pages · cited by 1 Pith paper

  1. [1]

    author author B. T. \ Matthias , author T. H. \ Geballe ,\ and\ author V. B. \ Compton ,\ title title Superconductivity ,\ https://doi.org/10.1103/RevModPhys.35.1 journal journal Rev. Mod. Phys. \ volume 35 ,\ pages 1 ( year 1963 ) NoStop

    doi:10.1103/revmodphys.35.1

  2. [2]

    author author K. von Klitzing ,\ title title Quantum hall effect: Discovery and application ,\ https://doi.org/https://doi.org/10.1146/annurev-conmatphys-031016-025148 journal journal Annual Review of Condensed Matter Physics \ volume 8 ,\ pages 13 ( year 2017 ) NoStop

    work page doi:10.1146/annurev-conmatphys-031016-025148

  3. [3]

    Tsu \ and\ author L

    author author R. Tsu \ and\ author L. Esaki ,\ title title Tunneling in a finite superlattice ,\ https://doi.org/10.1063/1.1654509 journal journal Appl. Phys. Lett. \ volume 22 ,\ pages 562 ( year 1973 ) NoStop

    work page doi:10.1063/1.1654509 1973

  4. [4]

    R\"ontgen , author M

    author author M. R\"ontgen , author M. Pyzh , author C. V. \ Morfonios , author N. E. \ Palaiodimopoulos , author F. K. \ Diakonos ,\ and\ author P. Schmelcher ,\ title title Latent symmetry induced degeneracies ,\ https://doi.org/10.1103/PhysRevLett.126.180601 journal journal Phys. Rev. Lett. \ volume 126 ,\ pages 180601 ( year 2021 ) NoStop

    doi:10.1103/physrevlett.126.180601

  5. [5]

    author author H. A. \ Jahn , author E. Teller ,\ and\ author F. G. \ Donnan ,\ title title Stability of polyatomic molecules in degenerate electronic states - i—orbital degeneracy ,\ https://doi.org/10.1098/rspa.1937.0142 journal journal Proc. R. Soc. London A: Math. Phys. Sci. \ volume 161 ,\ pages 220 ( year 1937 ) NoStop

    work page arXiv 1937

  6. [6]

    Muller ,\ title title Glossary of terms used in physical organic chemistry (iupac recommendations 1994) ,\ https://doi.org/doi:10.1351/pac199466051077 journal journal Pure Appl

    author author P. Muller ,\ title title Glossary of terms used in physical organic chemistry (iupac recommendations 1994) ,\ https://doi.org/doi:10.1351/pac199466051077 journal journal Pure Appl. Chem. \ volume 66 ,\ pages 1077 ( year 1994 ) NoStop

    work page doi:10.1351/pac199466051077 1994

  7. [7]

    Narita , author X.-Y

    author author A. Narita , author X.-Y. \ Wang , author X. Feng ,\ and\ author K. Müllen ,\ title title New advances in nanographene chemistry ,\ https://doi.org/10.1039/C5CS00183H journal journal Chem. Soc. Rev. \ volume 44 ,\ pages 6616 ( year 2015 ) NoStop

    doi:10.1039/c5cs00183h

  8. [8]

    \ Wang , author A

    author author X.-Y. \ Wang , author A. Narita ,\ and\ author K. M \"u llen ,\ title title Precision synthesis versus bulk-scale fabrication of graphenes ,\ https://doi.org/10.1038/s41570-017-0100 journal journal Nat. Rev. Chem. \ volume 2 ,\ pages 0100 ( year 2017 ) NoStop

    work page doi:10.1038/s41570-017-0100

Show all 55 references
  1. [9]

    Liu , author P

    author author Y. Liu , author P. Kilby , author T. J. \ Frankcombe ,\ and\ author T. W. \ Schmidt ,\ title title The electronic structure of benzene from a tiling of the correlated 126-dimensional wavefunction ,\ https://doi.org/10.1038/s41467-020-15039-9 journal journal Nat. ...

    doi

  2. [10]

    author author E. Clar ,\ title The aromatic sextet ,\ in\ https://doi.org/10.1007/978-94-009-7197-4_4 booktitle Mobile Source Emissions Including Policyclic Organic Species ,\ editor edited by\ editor D. Rondia , editor M. Cooke ,\ and\ editor R. K. \ Haroz \ ( publisher Sprin...

    doi

  3. [11]

    \ Son , author M

    author author Y.-W. \ Son , author M. L. \ Cohen ,\ and\ author S. G. \ Louie ,\ title title Energy gaps in graphene nanoribbons ,\ https://doi.org/10.1103/PhysRevLett.97.216803 journal journal Phys. Rev. Lett. \ volume 97 ,\ pages 216803 ( year 2006 ) NoStop

    doi

  4. [12]

    Cao , author F

    author author T. Cao , author F. Zhao ,\ and\ author S. G. \ Louie ,\ title title Topological phases in graphene nanoribbons: Junction states, spin centers, and quantum spin chains ,\ https://doi.org/10.1103/PhysRevLett.119.076401 journal journal Phys. Rev. Lett. \ volume 119 ...

    doi

  5. [13]

    Randić ,\ title title Aromaticity of polycyclic conjugated hydrocarbons ,\ https://doi.org/10.1021/cr9903656 journal journal Chem

    author author M. Randić ,\ title title Aromaticity of polycyclic conjugated hydrocarbons ,\ https://doi.org/10.1021/cr9903656 journal journal Chem. Rev. \ volume 103 ,\ pages 3449 ( year 2003 ) NoStop

    2003 doi

  6. [14]

    Wassmann , author A

    author author T. Wassmann , author A. P. \ Seitsonen , author A. M. \ Saitta , author M. Lazzeri ,\ and\ author F. Mauri ,\ title title Clar’s theory, -electron distribution, and geometry of graphene nanoribbons ,\ https://doi.org/10.1021/ja909234y journal journal J. Am. Chem....

    doi

  7. [15]

    Randic \ and\ author A

    author author M. Randic \ and\ author A. T. \ Balaban ,\ title title Partitioning of -electrons in rings for clar structures of benzenoid hydrocarbons ,\ https://doi.org/10.1021/ci050196s journal journal J. Chem. Inf. Model. \ volume 46 ,\ pages 57 ( year 2006 ) NoStop

    doi

  8. [16]

    author author Y. Ruiz-Morales ,\ title title The agreement between clar structures and nucleus-independent chemical shift values in pericondensed benzenoid polycyclic aromatic hydrocarbons: an application of the y-rule ,\ https://doi.org/10.1021/jp040179q journal journal J. Ph...

    doi

  9. [17]

    Nishina , author M

    author author N. Nishina , author M. Makino ,\ and\ author J.-i. \ Aihara ,\ title title Aromatic character of irregular-shaped nanographenes ,\ https://doi.org/10.1021/acs.jpca.6b00972 journal journal J. Phys. Chem. A \ volume 120 ,\ pages 2431 ( year 2016 ) NoStop

    doi

  10. [18]

    Strutynski , author A

    author author K. Strutynski , author A. Mateo-Alonso ,\ and\ author M. Melle-Franco ,\ title title Clar rules the electronic properties of 2d -conjugated frameworks: Mind the gap ,\ https://doi.org/https://doi.org/10.1002/chem.201905087 journal journal Chem. Eur. J. \ volume 2...

    doi

  11. [19]

    Wang ,\ title title Quantitative resonance theory based on the clar sextet model ,\ https://doi.org/10.1021/acs.jpca.1c08661 journal journal J

    author author Y. Wang ,\ title title Quantitative resonance theory based on the clar sextet model ,\ https://doi.org/10.1021/acs.jpca.1c08661 journal journal J. Phys. Chem. A \ volume 126 ,\ pages 164 ( year 2022 a ) NoStop

    2022 doi

  12. [20]

    author author Y. Wang ,\ title title Extension and quantification of the fries rule and its connection to aromaticity: Large-scale validation by wave-function-based resonance analysis ,\ https://doi.org/10.1021/acs.jcim.1c00735 journal journal J. Chem. Inf. Model. \ volume 62 ...

    doi

  13. [21]

    Gutman , author S

    author author I. Gutman , author S. Radenkovic , author M. Antic ,\ and\ author J. DJurdjevic ,\ title title A test of clar aromatic sextet theory ,\ @noop journal journal J. Serb. Chem. Soc. \ volume 78 ,\ pages 1539 ( year 2013 ) NoStop

    2013

  14. [22]

    Cyvin \ and\ author I

    author author S. Cyvin \ and\ author I. Gutman ,\ title title Kekul e structures and their symmetry properties ,\ https://doi.org/https://doi.org/10.1016/0898-1221(86)90430-X journal journal Comput. Math. Appl. \ volume 12 ,\ pages 859 ( year 1986 ) NoStop

    doi

  15. [23]

    Garc\' a-Fern\'andez , author J

    author author P. Garc\' a-Fern\'andez , author J. C. \ Wojde , author J. \'I\ niguez ,\ and\ author J. Junquera ,\ title title Second-principles method for materials simulations including electron and lattice degrees of freedom ,\ https://doi.org/10.1103/PhysRevB.93.195137 jou...

    doi

  16. [24]

    author author W. M. C. \ Foulkes \ and\ author R. Haydock ,\ title title Tight-binding models and density-functional theory ,\ https://doi.org/10.1103/PhysRevB.39.12520 journal journal Phys. Rev. B \ volume 39 ,\ pages 12520 ( year 1989 ) NoStop

    doi

  17. [25]

    \ Cao , author Y.-J

    author author Z.-P. \ Cao , author Y.-J. \ Zhao , author J.-H. \ Liao ,\ and\ author X.-B. \ Yang ,\ title title Gap maximum of graphene nanoflakes: a first-principles study combined with the monte carlo tree search method ,\ https://doi.org/10.1039/C7RA06891C journal journal ...

    doi

  18. [26]

    Chandrasekaran , author D

    author author A. Chandrasekaran , author D. Kamal , author R. Batra , author C. Kim , author L. Chen ,\ and\ author R. Ramprasad ,\ title title Solving the electronic structure problem with machine learning ,\ https://doi.org/10.1038/s41524-019-0162-7 journal journal Npj Compu...

    doi

  19. [27]

    author author P. B. \ J rgensen \ and\ author A. Bhowmik ,\ title title Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solids ,\ https://doi.org/10.1038/s41524-022-00863-y journal journal Npj Comput. Mater. \ volume 8 ,\ pages...

    doi

  20. [28]

    author author B. G. \ del Rio , author B. Phan ,\ and\ author R. Ramprasad ,\ title title A deep learning framework to emulate density functional theory ,\ https://doi.org/10.1038/s41524-023-01115-3 journal journal Npj Comput. Mater. \ volume 9 ,\ pages 158 ( year 2023 ) NoStop

    doi

  21. [29]

    author author E. D. \ Glendening \ and\ author F. Weinhold ,\ title title Natural resonance theory: I. general formalism ,\ https://doi.org/https://doi.org/10.1002/(SICI)1096-987X(19980430)19:6<593::AID-JCC3>3.0.CO;2-M journal journal Journal of Computational Chemistry \ volum...

    doi

  22. [30]

    author author M. H. \ Lee ,\ title title Ergodic theory, infinite products, and long time behavior in hermitian models ,\ https://doi.org/10.1103/PhysRevLett.87.250601 journal journal Phys. Rev. Lett. \ volume 87 ,\ pages 250601 ( year 2001 ) NoStop

    doi

  23. [31]

    Aghion , author D

    author author E. Aghion , author D. A. \ Kessler ,\ and\ author E. Barkai ,\ title title From non-normalizable boltzmann-gibbs statistics to infinite-ergodic theory ,\ https://doi.org/10.1103/PhysRevLett.122.010601 journal journal Phys. Rev. Lett. \ volume 122 ,\ pages 010601 ...

    doi

  24. [32]

    author author I. Langmuir ,\ title title The octet theory of valence and its applications with special reference to organic nitrogen compounds ,\ https://doi.org/10.1021/ja01447a011 journal journal J. Am. Chem. Soc. \ volume 42 ,\ pages 274 ( year 1920 ) NoStop

    doi

  25. [33]

    Xu , author C

    author author S. Xu , author C. He , author Y. Zhao , author X. Yang ,\ and\ author H. Xu ,\ title title Generalized octet rule with fractional occupancies for boron ,\ https://doi.org/10.1021/jacs.3c10370 journal journal J. Am. Chem. Soc. \ volume 145 ,\ pages 25003 ( year 20...

    doi

  26. [34]

    Solà ,\ title title Forty years of C lar's aromatic -sextet rule ,\ @noop journal journal Front

    author author M. Solà ,\ title title Forty years of C lar's aromatic -sextet rule ,\ @noop journal journal Front. Chem. \ volume 1 ,\ pages 22 ( year 2013 ) NoStop

    2013

  27. [35]

    author author A. Karton ,\ title title How reliable is DFT in predicting relative energies of polycyclic aromatic hydrocarbon isomers? comparison of functionals from different rungs of jacob's ladder ,\ https://doi.org/https://doi.org/10.1002/jcc.24669 journal journal J. Compu...

    doi

  28. [36]

    author author E. D. \ Glendening , author C. R. \ Landis ,\ and\ author F. Weinhold ,\ title title Resonance theory reboot ,\ https://doi.org/10.1021/jacs.8b12336 journal journal J. Am. Chem. Soc. \ volume 141 ,\ pages 4156 ( year 2019 ) NoStop

    doi

  29. [37]

    Kresse \ and\ author J

    author author G. Kresse \ and\ author J. Furthm\"uller ,\ title title Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set ,\ https://doi.org/10.1103/PhysRevB.54.11169 journal journal Phys. Rev. B \ volume 54 ,\ pages 11169 ( year 19...

    doi

  30. [38]

    Kresse \ and\ author D

    author author G. Kresse \ and\ author D. Joubert ,\ title title From ultrasoft pseudopotentials to the projector augmented-wave method ,\ https://doi.org/10.1103/PhysRevB.59.1758 journal journal Phys. Rev. B \ volume 59 ,\ pages 1758 ( year 1999 ) NoStop

    doi

  31. [39]

    Paier , author M

    author author J. Paier , author M. Marsman , author K. Hummer , author G. Kresse , author I. C. \ Gerber ,\ and\ author J. G. \ Ángyán ,\ title title Screened hybrid density functionals applied to solids ,\ https://doi.org/10.1063/1.2187006 journal journal J. Chem. Phys. \ vol...

    doi

  32. [40]

    Heyd , author G

    author author J. Heyd , author G. E. \ Scuseria ,\ and\ author M. Ernzerhof ,\ title title Hybrid functionals based on a screened Coulomb potential ,\ https://doi.org/10.1063/1.1564060 journal journal J. Chem. Phys. \ volume 118 ,\ pages 8207 ( year 2003 ) NoStop

    doi

  33. [41]

    Miyoshi , author S

    author author H. Miyoshi , author S. Nobusue , author A. Shimizu ,\ and\ author Y. Tobe ,\ title title Non-alternant non-benzenoid kekulenes: the birth of a new kekulene family ,\ https://doi.org/10.1039/C5CS00185D journal journal Chem. Soc. Rev. \ volume 44 ,\ pages 6560 ( ye...

    doi

  34. [42]

    author author J. C. \ Buttrick \ and\ author B. T. \ King ,\ title title Kekulenes , cycloarenes , and heterocycloarenes: addressing electronic structure and aromaticity through experiments and calculations ,\ https://doi.org/10.1039/C6CS00174B journal journal Chem. Soc. Rev. ...

    doi

  35. [43]

    Povie , author Y

    author author G. Povie , author Y. Segawa , author T. Nishihara , author Y. Miyauchi ,\ and\ author K. Itami ,\ title title Synthesis of a carbon nanobelt ,\ https://doi.org/10.1126/science.aam8158 journal journal Science \ volume 356 ,\ pages 172 ( year 2017 ) NoStop

    doi

  36. [44]

    author author C. C. \ Moore ,\ title title Ergodic theorem, ergodic theory, and statistical mechanics ,\ https://doi.org/10.1073/pnas.1421798112 journal journal Proc. Natl. Acad. Sci. U.S.A. \ volume 112 ,\ pages 1907 ( year 2015 ) NoStop

    1907 doi

  37. [45]

    author author J. S. \ Rowlinson ,\ title title Probability, information and entropy ,\ https://doi.org/10.1038/2251196a0 journal journal Nature \ volume 225 ,\ pages 1196 ( year 1970 ) NoStop

    1970 doi

  38. [46]

    Chen , author P

    author author S. Chen , author P. Slattum , author C. Wang ,\ and\ author L. Zang ,\ title title Self-assembly of perylene imide molecules into 1d nanostructures: Methods, morphologies, and applications ,\ https://doi.org/10.1021/acs.chemrev.5b00312 journal journal Chem. Rev. ...

    doi

  39. [47]

    Jiang \ and\ author Z

    author author W. Jiang \ and\ author Z. Wang ,\ title title Molecular carbon imides ,\ https://doi.org/10.1021/jacs.2c04642 journal journal J. Am. Chem. Soc. \ volume 144 ,\ pages 14976 ( year 2022 ) NoStop

    2022 doi

  40. [48]

    Ma , author Y

    author author L. Ma , author Y. Han , author Q. Shi ,\ and\ author H. Huang ,\ title title The design , synthesis and application of rubicene based polycyclic aromatic hydrocarbons (pahs) ,\ https://doi.org/10.1039/D3TC02876C journal journal J. Mater. Chem. C \ volume 11 ,\ pa...

    doi

  41. [49]

    Ruiz-Morales ,\ title title Homo-lumo gap as an index of molecular size and structure for polycyclic aromatic hydrocarbons (pahs) and asphaltenes: A theoretical study

    author author Y. Ruiz-Morales ,\ title title Homo-lumo gap as an index of molecular size and structure for polycyclic aromatic hydrocarbons (pahs) and asphaltenes: A theoretical study. i ,\ @noop journal journal J. Phys. Chem. A \ volume 106 ,\ pages 11283 ( year 2002 ) NoStop

    2002

  42. [50]

    author author Y. Ruiz-Morales ,\ title title Application of the y-rule and theoretical study to understand the topological and electronic structures of polycyclic aromatic hydrocarbons from atomic force microscopy images of soot, coal asphaltenes, and petroleum asphaltenes ,\ ...

    doi

  43. [51]

    author author S. J. \ Cyvin , author J. Brunvoll ,\ and\ author B. N. \ Cyvin ,\ title title Enumeration and classification of coronoid hydrocarbons. 10. double coronoids ,\ https://doi.org/10.1021/ci00067a002 journal journal J. Chem. Inf. Comput. \ volume 30 ,\ pages 210 ( ye...

    doi

  44. [52]

    Malko , author C

    author author D. Malko , author C. Neiss , author F. Vi\ nes ,\ and\ author A. G\"orling ,\ title title Competition for graphene: Graphynes with direction-dependent D irac cones ,\ https://doi.org/10.1103/PhysRevLett.108.086804 journal journal Phys. Rev. Lett. \ volume 108 ,\ ...

    doi

  45. [53]

    author author B. G. \ Kim \ and\ author H. J. \ Choi ,\ title title Graphyne: Hexagonal network of carbon with versatile D irac cones ,\ https://doi.org/10.1103/PhysRevB.86.115435 journal journal Phys. Rev. B \ volume 86 ,\ pages 115435 ( year 2012 ) NoStop

    doi

  46. [54]

    \ Xu , author X.-T

    author author S.-G. \ Xu , author X.-T. \ Li , author Z.-J. \ Chen , author C.-C. \ He , author C. He , author X.-B. \ Yang ,\ and\ author H. Xu ,\ title title Toward hidden materials with directional bonds ,\ https://doi.org/10.1103/PhysRevMaterials.7.084202 journal journal P...

    doi

  47. [55]

    Hu , author C

    author author Y. Hu , author C. Wu , author Q. Pan , author Y. Jin , author R. Lyu , author V. Martinez , author S. Huang , author J. Wu , author L. J. \ Wayment , author N. A. \ Clark , author M. B. \ Raschke , author Y. Zhao ,\ and\ author W. Zhang ,\ title title Synthesis o...

    doi

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