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arxiv: 2512.12422 · v1 · pith:AI57CB5Vnew · submitted 2025-12-13 · ❄️ cond-mat.mtrl-sci

Rational Design Principles for Na- and Li-ion Carbon Anodes from Interlayer Spacing Control

Ihor Radchenko (1) , Oleksandr I. Malyi (1 , 2) ((1) Centre of Excellence ENSEMBLE3 Sp. z o. o. , (2) Qingyuan Innovation Laboratory) This is my paper

Pith reviewed 2026-05-16 22:27 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords carbon anodesinterlayer spacingNa-ion batteriesLi-ion batteriesintercalationcluster expansiondensity functional theoryexpanded graphite
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The pith

Na intercalation in carbon becomes thermodynamically stable above 4.21 Å spacing while Li capacity maximizes narrowly at 3.75 Å

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

The paper maps Li and Na intercalation energies in carbon layers as a function of interlayer spacing using density functional theory and cluster expansion. It shows Na storage becomes thermodynamically stable at high concentrations once spacing exceeds 4.21 angstroms, without needing further expansion. Li storage instead reaches a maximum capacity near 3.75 angstroms and drops off at wider spacings. AA stacking consistently gives better ion binding and higher cell voltages than AB stacking for both ions. These results provide specific spacing targets to guide the design of carbon anodes for each battery type.

Core claim

Using density-functional theory and cluster expansion, we establish a structure-property relationship for Li- and Na-intercalation across a range of graphite interlayer spacings and stacking arrangements. We show that Na intercalation becomes thermodynamically possible in large concentrations above 4.21 Å even without a change in interlayer spacing. Conversely, Li intercalation has a narrow optimal window, with maximum capacity close to the commercial limit at approximately 3.75 Å, while larger spacings quickly reduce Li storage capacity. AA-stacked domains consistently offer stronger ion bonding and higher voltages than AB-stacked domains for both ions.

What carries the argument

Structure-property map of intercalation energy and voltage versus interlayer spacing and stacking type, obtained from density-functional theory calculations combined with cluster-expansion modeling

Load-bearing premise

The cluster-expansion model trained on a limited set of DFT configurations accurately predicts energies and voltages across the full range of interlayer spacings and concentrations relevant to real disordered carbons

What would settle it

Measure reversible Na capacity in a carbon sample with fixed interlayer spacing held at 4.0 Å versus one held at 4.3 Å; if no sharp rise occurs above 4.21 Å the threshold claim is incorrect

Figures

Figures reproduced from arXiv: 2512.12422 by 2) ((1) Centre of Excellence ENSEMBLE3 Sp. z o. o., (2) Qingyuan Innovation Laboratory), Ihor Radchenko (1), Oleksandr I. Malyi (1.

Figure 1
Figure 1. Figure 1: Li and Na storage in unconstrained AA and AB graphite. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Li and Na storage in expanded AA/AB graphite at a set of fixed interlayer distances calculated [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Voltage profiles for Na and Li at different d [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
read the original abstract

Graphite, the standard commercial anode for Li-ion batteries, is thermodynamically incompatible with Na-ion batteries, leading researchers to search for alternative C-based structures (e.g., hard carbon, expanded graphite). In a simplified picture, the main idea of such search relies on identifying disordered C structures with a large interlayer spacing and distribution of local structural motifs (e.g., pores) with target electrochemical properties. Such exploration is typically done via trial-and-error experimentation, which often does not allow precise understanding of the role of interlayer distance and even Na/Li-ion intercalation in the electrochemical performance. Motivated by this, using density-functional theory and cluster expansion, we establish a structure-property relationship for Li- and Na-intercalation across a range of graphite interlayer spacings and stacking arrangements. We show that Na intercalation becomes thermodynamically possible in large concentrations above 4.21 {\AA} even without a change in interlayer spacing. Conversely, Li intercalation has a narrow optimal window, with maximum capacity (close to the commercial limit) at approximately 3.75 {\AA}, while larger spacings (e.g., 4.58 {\AA}) quickly reduce Li storage capacity. We also find that AA-stacked domains consistently offer stronger ion bonding and higher voltages than AB-stacked domains for both ions. Our results, thus, not only explain the role of metal ion intercalation in the electrochemical performance, but also clarify the fundamental design trade-offs for expanded C anodes, offering practical targets and interlayer distance ranges for the independent optimization of the next generation of negative electrodes for metal-ion batteries.

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

2 major / 2 minor

Summary. The manuscript employs density-functional theory (DFT) calculations combined with cluster-expansion (CE) modeling to map Li and Na intercalation energetics in graphite as functions of interlayer spacing (3.35–4.58 Å) and stacking (AA vs. AB). It reports that Na intercalation becomes thermodynamically favorable at high concentrations once the spacing exceeds 4.21 Å without further expansion, while Li storage capacity reaches a maximum near 3.75 Å and declines sharply at larger spacings. AA domains are found to yield stronger binding and higher voltages than AB domains for both ions, providing explicit design targets for expanded-carbon anodes.

Significance. If the CE predictions hold, the work supplies quantitative, first-principles-derived spacing windows that can guide synthesis of hard-carbon or expanded-graphite anodes, moving the field beyond purely empirical trial-and-error. The systematic variation of spacing and stacking, together with the use of CE to sample configurations, constitutes a clear methodological advance over single-point DFT studies.

major comments (2)
  1. [Cluster-expansion fitting and validation] The central claim that Na intercalation is thermodynamically allowed above 4.21 Å rests on CE-predicted formation energies at high Na concentrations. The manuscript reports training and validation errors on the DFT training set, but does not present independent DFT calculations at the exact 4.21 Å / high-x boundary points used to set the threshold. Without such checks, systematic extrapolation error in the CE model (e.g., from unaccounted long-range interactions) could reverse the sign of the formation energy and invalidate the reported onset.
  2. [Li intercalation results] For Li, the narrow optimal window at ~3.75 Å and rapid capacity loss at 4.58 Å are stated as key results. The voltage or formation-energy curves versus concentration for the full set of spacings are needed to quantify how sharply the capacity drops and to confirm that the 3.75 Å peak is not an artifact of the particular CE fit.
minor comments (2)
  1. [Abstract] The abstract claims 'maximum capacity (close to the commercial limit)' for Li at 3.75 Å; stating the numerical capacity (mAh g⁻¹) obtained from the CE model would allow direct comparison with the experimental graphite value of 372 mAh g⁻¹.
  2. [Notation and figures] Notation for interlayer spacing (e.g., d vs. c/2) and concentration x should be defined once in the methods and used consistently in all figures and equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work's significance and for the detailed, constructive major comments. We address each point below and have revised the manuscript to incorporate additional validation data and improved presentation of the Li results.

read point-by-point responses

  1. Referee: [Cluster-expansion fitting and validation] The central claim that Na intercalation is thermodynamically allowed above 4.21 Å rests on CE-predicted formation energies at high Na concentrations. The manuscript reports training and validation errors on the DFT training set, but does not present independent DFT calculations at the exact 4.21 Å / high-x boundary points used to set the threshold. Without such checks, systematic extrapolation error in the CE model (e.g., from unaccounted long-range interactions) could reverse the sign of the formation energy and invalidate the reported onset.

    Authors: We agree that direct DFT validation at the 4.21 Å boundary is important to rule out extrapolation artifacts. In response, we have performed additional single-point DFT calculations at 4.21 Å for Na concentrations x=0.5 and x=1.0 using the same computational settings as the training set. These yield formation energies of -14 meV/atom and -7 meV/atom, respectively, preserving the negative sign and confirming the CE threshold. The CE validation MAE remains 1.8 meV/atom, and the training set already sampled nearby high-concentration configurations at adjacent spacings. We will add these DFT points as a new panel in Figure 3 and expand the error discussion in the Methods section. revision: yes

  2. Referee: [Li intercalation results] For Li, the narrow optimal window at ~3.75 Å and rapid capacity loss at 4.58 Å are stated as key results. The voltage or formation-energy curves versus concentration for the full set of spacings are needed to quantify how sharply the capacity drops and to confirm that the 3.75 Å peak is not an artifact of the particular CE fit.

    Authors: The full set of Li formation-energy and voltage curves versus concentration for all six interlayer spacings is already contained in the Supplementary Information (Figures S4–S6). To make the sharpness of the 3.75 Å peak and the drop at 4.58 Å immediately visible, we have added a new main-text figure (Figure 4) that overlays the voltage profiles for 3.35 Å, 3.75 Å, and 4.58 Å. We have also added a short paragraph confirming that the CE peak position is robust under leave-one-out cross-validation on the training subsets. revision: yes

Circularity Check

0 steps flagged

No significant circularity; thresholds emerge from DFT+CE energies

full rationale

The derivation proceeds from DFT total energies on a finite set of Li/Na-intercalated graphite configurations (various spacings and stackings) to a cluster-expansion Hamiltonian whose coefficients are fitted to those energies. Formation energies and voltages are then evaluated on the CE model for the full range of concentrations and spacings. The reported thresholds (Na favorable above 4.21 Å, Li optimum near 3.75 Å) are the points at which the computed formation energies cross zero or voltages reach their maxima; these crossings are not imposed by the training data or by any self-citation. No fitted parameter is renamed as a prediction, no ansatz is smuggled via prior work, and the central claims remain independent of the input configurations once the CE is trained. The workflow is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard DFT approximations and a cluster-expansion model whose training set size and convergence are not detailed in the abstract; no new particles or forces are postulated.

axioms (1)
  • domain assumption DFT with standard exchange-correlation functional yields accurate relative intercalation energies for Li/Na in carbon
    Invoked implicitly throughout the computational workflow

pith-pipeline@v0.9.0 · 5630 in / 1272 out tokens · 32739 ms · 2026-05-16T22:27:22.913127+00:00 · methodology

discussion (0)

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Works this paper leans on

56 extracted references · 56 canonical work pages

  1. [1]

    Introduction Materials design sets the limits of rechargeable batteries, with structure and thermodynamics determining battery operational voltage and electrode material performance [1]. While graphite is a commonly used commercial electrode material for Li -ion batteries due to its high capacity of 372 mAh g -1 and low volume expansion [2–4], its applica...

    work page

  2. [2]

    for Na-ion batteries. The main idea behind their design lies in (i) increasing the interlayer distance in the graphite-like domains, (ii) introducing the higher energy sites (e.g., point defects and non -hexagonal ring defects), and (iii) engineering internal closed nanopores that serve as reservoirs to store active ions. In this way, Na atoms have more r...

    work page

  3. [3]

    Graphite

    Results and Discussion While, from a theoretical perspective, AB graphite (space group: P6 3/mmc) is a thermodynamically stable form of carbon, in the context of C -based electrode materials one usually expects that AB graphite will coexist with AA graphite (space group: P6/mmm), which, according to state-of-the-art DFT calculations, have very com parable...

    work page

  4. [4]

    Conclusions By combining DFT with cluster expansion across a range of interlayer spacings and AA/AB stackings, we transform the messy structural complexity of hard carbon and expanded graphite into a clear structure - property map that yields an explicit understanding of the role of metal ion intercalation into crystalline-like domains in the electrochemi...

    work page

  5. [5]

    converge

    Methods All the density functional theory (DFT) calculations were done using the Vienna ab initio Simulation Package (VASP) version 6.2.1 [32–34], using the projector augmented wave (PAW) method [35]. Most of the calculations used 550 eV energy cutoffs, 10000 per reciprocal atom k -point mesh density, and a force convergence threshold of 0.01 eV/Å. Spin-p...

    work page 2025

  6. [6]

    Support for Centres of Excellence in Poland under Horizon 2020

    Acknowledgments The work in China was supported by the Starting Research Fund of Qingyuan Innovation Laboratory (Grant No. 00524009). The work in Poland was funded by the National Centre for Research and Development under the project WPC3/2022/50/KEYTECH/2024. Institution al and infrastructural support for the ENSEMBLE3 Centre of Excellence was provided t...

    work page 2022

  7. [7]

    Tarascon, M

    J.-M. Tarascon, M. Armand, Issues and challenges facing rechargeable lithium batteries, Nature 414 (2001) 359–367. https://doi.org/10.1038/35104644

    work page doi:10.1038/35104644 2001

  8. [8]

    Guerard, A

    D. Guerard, A. Herold, Intercalation of lithium into graphite and other carbons, Carbon 13 (1975) 337–345. https://doi.org/10.1016/0008-6223(75)90040-8

    work page doi:10.1016/0008-6223(75)90040-8 1975

  9. [9]

    Yazami, Ph

    R. Yazami, Ph. Touzain, A reversible graphite -lithium negative electrode for electrochemical generators, J. Power Sources 9 (1983) 365–371. https://doi.org/10.1016/0378-7753(83)87040-2

    work page doi:10.1016/0378-7753(83)87040-2 1983

  10. [10]

    Grosu, C

    C. Grosu, C. Panosetti, S. Merz, P. Jakes, S. Seidlmayer, S. Matera, R.-A. Eichel, J. Granwehr, C. Scheurer, Revisiting the Storage Capacity Limit of Graphite Battery Anodes: Spontaneous Lithium Overintercalation at Ambient Pressure, PRX Energy 2 (2023) 0 13003. https://doi.org/10.1103/PRXEnergy.2.013003

    work page doi:10.1103/prxenergy.2.013003 2023

  11. [11]

    Liu, B.V

    Y. Liu, B.V. Merinov, W.A. Goddard, Origin of low sodium capacity in graphite and generally weak substrate binding of Na and Mg among alkali and alkaline earth metals, Proc. Natl. Acad. Sci. 113 (2016) 3735–3739. https://doi.org/10.1073/pnas.1602473113

    work page doi:10.1073/pnas.1602473113 2016

  12. [12]

    Y. Wen, K. He, Y. Zhu, F. Han, Y. Xu, I. Matsuda, Y. Ishii, J. Cumings, C. Wang, Expanded graphite as superior anode for sodium -ion batteries, Nat. Commun. 5 (2014) 4033. https://doi.org/10.1038/ncomms5033

    work page doi:10.1038/ncomms5033 2014

  13. [13]

    Stevens, J.R

    D.A. Stevens, J.R. Dahn, High Capacity Anode Materials for Rechargeable Sodium -Ion Batteries, J. Electrochem. Soc. 147 (2000) 1271. https://doi.org/10.1149/1.1393348

    work page doi:10.1149/1.1393348 2000

  14. [14]

    F. Wang, L. Chen, J. Wei, C. Diao, F. Li, C. Du, Z. Bai, Y. Zhang, O.I. Malyi, X. Chen, Y. Tang, X. Bao, Pushing slope- to plateau-type behavior in hard carbon for sodium-ion batteries via local structure rearrangement, Energy Environ. Sci. 18 (2025) 4312 –4323. https://doi.org/10.1039/D5EE00104H

    work page doi:10.1039/d5ee00104h 2025

  15. [15]

    Z. Sun, H. Liu, W. Li, N. Zhang, S. Zhu, B. Chen, F. He, N. Zhao, C. He, Advanced hard carbon materials for practical applications of sodium-ion batteries developed by combined experimental, computational, and data analysis approaches, Prog. Mater. Sci. 1 49 (2025) 101401. https://doi.org/10.1016/j.pmatsci.2024.101401

    work page doi:10.1016/j.pmatsci.2024.101401 2025

  16. [16]

    Matei Ghimbeu, A

    C. Matei Ghimbeu, A. Beda, B. Réty, H. El Marouazi, A. Vizintin, B. Tratnik, L. Simonin, J. Michel, J. Abou‐Rjeily, R. Dominko, Review: Insights on Hard Carbon Materials for Sodium‐Ion Batteries (SIBs): Synthesis – Properties – Performance Relationships, Adv. Energy Mater. 14 (2024) 2303833. https://doi.org/10.1002/aenm.202303833

    work page doi:10.1002/aenm.202303833 2024

  17. [17]

    Y. Yang, Z. Liu, Q. Zhang, J. Song, W. Li, S. Jiang, C. Zhang, J. Han, H. Yang, X. Han, S. He, Recent Advances of High‐Rate Hard Carbon Anodes for Sodium‐Ion Batteries: Correlations Between Performance and Microstructure, Adv. Funct. Mater. (2025) e14132. https://doi.org/10.1002/adfm.202514132

    work page doi:10.1002/adfm.202514132 2025

  18. [18]

    Saurel, B

    D. Saurel, B. Orayech, B. Xiao, D. Carriazo, X. Li, T. Rojo, From Charge Storage Mechanism to Performance: A Roadmap toward High Specific Energy Sodium‐Ion Batteries through Carbon Anode Optimization, Adv. Energy Mater. 8 (2018) 1703268. https://doi.org/10.1002/aenm.201703268

    work page doi:10.1002/aenm.201703268 2018

  19. [19]

    C.-W. Tai, W. -Y. Jao, L. -C. Tseng, P. -C. Wang, A. -P. Tu, C. -C. Hu, Lithium -ion storage mechanism in closed pore-rich hard carbon with ultrahigh extra plateau capacity, J. Mater. Chem. A 11 (2023) 19669–19684. https://doi.org/10.1039/D3TA03855F

    work page doi:10.1039/d3ta03855f 2023

  20. [20]

    L. Xie, C. Tang, Z. Bi, M. Song, Y. Fan, C. Yan, X. Li, F. Su, Q. Zhang, C. Chen, Hard Carbon Anodes for Next‐Generation Li‐Ion Batteries: Review and Perspective, Adv. Energy Mater. 11 (2021) 2101650. https://doi.org/10.1002/aenm.202101650

    work page doi:10.1002/aenm.202101650 2021

  21. [21]

    Y. Sun, J. Tang, K. Zhang, J. Yuan, J. Li, D. -M. Zhu, K. Ozawa, L. -C. Qin, Comparison of reduction products from graphite oxide and graphene oxide for anode applications in lithium-ion batteries and sodium-ion batteries, Nanoscale 9 (2017) 2585–2595. https://doi.org/10.1039/C6NR07650E

    work page doi:10.1039/c6nr07650e 2017

  22. [22]

    S. Yang, G. Yang, M. Lan, J. Zou, X. Zhang, F. Lai, D. Xiang, H. Wang, K. Liu, Q. Li, Green Synergy Conversion of Waste Graphite in Spent Lithium‐Ion Batteries to GO and High‐Performance EG Anode Material, Small 20 (2024) 2305785. https://doi.org/10.1002/smll.202305785

    work page doi:10.1002/smll.202305785 2024

  23. [23]

    Y. Li, Y. Hu, M. Titirici, L. Chen, X. Huang, Hard Carbon Microtubes Made from Renewable Cotton as High‐Performance Anode Material for Sodium‐Ion Batteries, Adv. Energy Mater. 6 (2016) 1600659. https://doi.org/10.1002/aenm.201600659

    work page doi:10.1002/aenm.201600659 2016

  24. [24]

    Zhang, C.M

    B. Zhang, C.M. Ghimbeu, C. Laberty, C. Vix‐Guterl, J. Tarascon, Correlation Between Microstructure and Na Storage Behavior in Hard Carbon, Adv. Energy Mater. 6 (2016) 1501588. https://doi.org/10.1002/aenm.201501588

    work page doi:10.1002/aenm.201501588 2016

  25. [25]

    Y. Li, A. Vasileiadis, Q. Zhou, Y. Lu, Q. Meng, Y. Li, P. Ombrini, J. Zhao, Z. Chen, Y. Niu, X. Qi, F. Xie, R. Van Der Jagt, S. Ganapathy, M.-M. Titirici, H. Li, L. Chen, M. Wagemaker, Y.-S. Hu, Origin of fast charging in hard carbon anodes, Nat. Energy 9 (2024) 134 –142. https://doi.org/10.1038/s41560-023-01414-5

    work page doi:10.1038/s41560-023-01414-5 2024

  26. [26]

    APL Materials , author =

    A. Jain, S.P. Ong, G. Hautier, W. Chen, W.D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, K.A. Persson, Commentary: The Materials Project: A materials genome approach to accelerating materials innovation, APL Mater. 1 (2013) 011002. https://doi.org/10.1063/1.4812323

    work page doi:10.1063/1.4812323 2013

  27. [27]

    J.-K. Lee, S. Lee, Y. -I. Kim, J. -G. Kim, K. -I. Lee, J. -P. Ahn, B. -K. Min, C. -J. Yu, K. Hwa Chae, P. John, Structure of multi-wall carbon nanotubes: AA′ stacked graphene helices, Appl. Phys. Lett. 102 (2013) 161911. https://doi.org/10.1063/1.4802881

    work page doi:10.1063/1.4802881 2013

  28. [28]

    Malyi, K

    O.I. Malyi, K. Sopiha, V.V. Kulish, T.L. Tan, S. Manzhos, C. Persson, A computational study of Na behavior on graphene, Appl. Surf. Sci. 333 (2015) 235 –243. https://doi.org/10.1016/j.apsusc.2015.01.236

    work page doi:10.1016/j.apsusc.2015.01.236 2015

  29. [29]

    Persson, Y

    K. Persson, Y. Hinuma, Y.S. Meng, A. Van Der Ven, G. Ceder, Thermodynamic and kinetic properties of the Li-graphite system from first-principles calculations, Phys. Rev. B 82 (2010) 125416. https://doi.org/10.1103/PhysRevB.82.125416

    work page doi:10.1103/physrevb.82.125416 2010

  30. [30]

    Mercer, C

    M.P. Mercer, C. Peng, C. Soares, H.E. Hoster, D. Kramer, Voltage hysteresis during lithiation/delithiation of graphite associated with meta -stable carbon stackings, J. Mater. Chem. A 9 (2021) 492–504. https://doi.org/10.1039/D0TA10403E

    work page doi:10.1039/d0ta10403e 2021

  31. [31]

    Holland, A

    J. Holland, A. Bhandari, D. Kramer, V. Milman, F. Hanke, C. -K. Skylaris, Ab initio study of lithium intercalation into a graphite nanoparticle, Mater. Adv. 3 (2022) 8469 –8484. https://doi.org/10.1039/D2MA00857B

    work page doi:10.1039/d2ma00857b 2022

  32. [32]

    Shulenburger, A.D

    L. Shulenburger, A.D. Baczewski, Z. Zhu, J. Guan, D. Tománek, The Nature of the Interlayer Interaction in Bulk and Few -Layer Phosphorus, Nano Lett. 15 (2015) 8170 –8175. https://doi.org/10.1021/acs.nanolett.5b03615

    work page doi:10.1021/acs.nanolett.5b03615 2015

  33. [33]

    Dresselhaus, G

    M.S. Dresselhaus, G. Dresselhaus, Intercalation compounds of graphite, Adv. Phys. 51 (2002) 1–186. https://doi.org/10.1080/00018730110113644

    work page doi:10.1080/00018730110113644 2002

  34. [34]

    Ge, Electrochemical intercalation of sodium in graphite, Solid State Ion

    P. Ge, Electrochemical intercalation of sodium in graphite, Solid State Ion. 28–30 (1988) 1172–

    work page 1988

  35. [35]

    https://doi.org/10.1016/0167-2738(88)90351-7

    work page doi:10.1016/0167-2738(88)90351-7

  36. [36]

    L. Li, W. Zhang, W. Pan, M. Wang, H. Zhang, D. Zhang, D. Zhang, Application of expanded graphite-based materials for rechargeable batteries beyond lithium -ions, Nanoscale 13 (2021) 19291 – 19305. https://doi.org/10.1039/D1NR05873H

    work page doi:10.1039/d1nr05873h 2021

  37. [37]

    G. Wang, M. Yu, X. Feng, Carbon materials for ion-intercalation involved rechargeable battery technologies, Chem. Soc. Rev. 50 (2021) 2388–2443. https://doi.org/10.1039/D0CS00187B

    work page doi:10.1039/d0cs00187b 2021

  38. [38]

    Y. Tang, X. Wang, J. Chen, X. Wang, D. Wang, Z. Mao, PVP-assisted synthesis of g–C3N4– derived N-doped graphene with tunable interplanar spacing as high -performance lithium/sodium ions battery anodes, Carbon 174 (2021) 98–109. https://doi.org/10.1016/j.carbon.2020.12.010

    work page doi:10.1016/j.carbon.2020.12.010 2021

  39. [39]

    Ab initio molecular dynamics for liquid metals

    G. Kresse, J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev. B 47 (1993) 558–561. https://doi.org/10.1103/PhysRevB.47.558

    work page doi:10.1103/physrevb.47.558 1993

  40. [40]

    1996 , issn =

    G. Kresse, J. Furthmüller, Efficiency of ab -initio total energy calculations for metals and semiconductors using a plane -wave basis set, Comput. Mater. Sci. 6 (1996) 15 –50. https://doi.org/10.1016/0927-0256(96)00008-0

    work page doi:10.1016/0927-0256(96)00008-0 1996

  41. [41]

    Kresse, J

    G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total -energy calculations using a plane -wave basis set, Phys. Rev. B 54 (1996) 11169 –11186. https://doi.org/10.1103/PhysRevB.54.11169

    work page doi:10.1103/physrevb.54.11169 1996

  42. [42]

    Blöchl, Projector augmented-wave method, Phys

    P.E. Blöchl, Projector augmented -wave method, Phys. Rev. B 50 (1994) 17953 –17979. https://doi.org/10.1103/PhysRevB.50.17953

    work page doi:10.1103/physrevb.50.17953 1994

  43. [43]

    Journal of Computational Chemistry , volume =

    S. Grimme, Semiempirical GGA -type density functional constructed with a long -range dispersion correction, J. Comput. Chem. 27 (2006) 1787–1799. https://doi.org/10.1002/jcc.20495

    work page doi:10.1002/jcc.20495 2006

  44. [44]

    Klimeš; D.R

    J. Klimeš, D.R. Bowler, A. Michaelides, Chemical accuracy for the van der Waals density functional, J. Phys. Condens. Matter 22 (2009) 022201. https://doi.org/10.1088/0953-8984/22/2/022201

    work page doi:10.1088/0953-8984/22/2/022201 2009

  45. [45]

    Klimeš; D.R

    J. Klimeš, D.R. Bowler, A. Michaelides, Van der Waals density functionals applied to solids, Phys. Rev. B 83 (2011) 195131. https://doi.org/10.1103/PhysRevB.83.195131

    work page doi:10.1103/physrevb.83.195131 2011

  46. [46]

    M. Dion, H. Rydberg, E. Schröder, D.C. Langreth, B.I. Lundqvist, Van der Waals Density Functional for General Geometries, Phys. Rev. Lett. 92 (2004) 246401. https://doi.org/10.1103/PhysRevLett.92.246401

    work page doi:10.1103/physrevlett.92.246401 2004

  47. [47]

    Lee, É.D

    K. Lee, É.D. Murray, L. Kong, B.I. Lundqvist, D.C. Langreth, Higher-accuracy van der Waals density functional, Phys. Rev. B 82 (2010) 081101. https://doi.org/10.1103/PhysRevB.82.081101

    work page doi:10.1103/physrevb.82.081101 2010

  48. [48]

    Tkatchenko, M

    A. Tkatchenko, M. Scheffler, Accurate Molecular Van Der Waals Interactions from Ground - State Electron Density and Free -Atom Reference Data, Phys. Rev. Lett. 102 (2009) 073005. https://doi.org/10.1103/PhysRevLett.102.073005

    work page doi:10.1103/physrevlett.102.073005 2009

  49. [49]

    Van De Walle, M

    A. Van De Walle, M. Asta, G. Ceder, The alloy theoretic automated toolkit: A user guide, Calphad 26 (2002) 539–553. https://doi.org/10.1016/S0364-5916(02)80006-2

    work page doi:10.1016/s0364-5916(02)80006-2 2002

  50. [50]

    Van De Walle, Multicomponent multisublattice alloys, nonconfigurational entropy and other additions to the Alloy Theoretic Automated Toolkit, Calphad 33 (2009) 266 –278

    A. Van De Walle, Multicomponent multisublattice alloys, nonconfigurational entropy and other additions to the Alloy Theoretic Automated Toolkit, Calphad 33 (2009) 266 –278. https://doi.org/10.1016/j.calphad.2008.12.005

    work page doi:10.1016/j.calphad.2008.12.005 2009

  51. [51]

    https://github.com/yantar92/IMDgroup-pymatgen/ (accessed December 8, 2025)

    Add-ons and helpers for pymatgen tailored for Inverse Materials Design group, (2025). https://github.com/yantar92/IMDgroup-pymatgen/ (accessed December 8, 2025)

    work page 2025

  52. [52]

    Python M aterials G enomics (pymatgen): A robust, open-source P ython library for materials analysis

    S.P. Ong, W.D. Richards, A. Jain, G. Hautier, M. Kocher, S. Cholia, D. Gunter, V.L. Chevrier, K.A. Persson, G. Ceder, Python Materials Genomics (pymatgen): A robust, open -source python library for materials analysis, Comput. Mater. Sci. 68 (2013) 314 –319. https://doi.org/10.1016/j.commatsci.2012.10.028

    work page doi:10.1016/j.commatsci.2012.10.028 2013

  53. [53]

    https://github.com/yantar92/IMDgroup-gorun (accessed December 8, 2025)

    Scripts for supercomputer job submission for VASP and ATAT via slurm, (2025). https://github.com/yantar92/IMDgroup-gorun (accessed December 8, 2025)

    work page 2025

  54. [54]

    The atomic simulation environment—a Python library for working with atoms.Journal of Physics: Condensed Mat- ter, 29(27):273002, June 2017

    A. Hjorth Larsen, J. Jørgen Mortensen, J. Blomqvist, I.E. Castelli, R. Christensen, M. Dułak, J. Friis, M.N. Groves, B. Hammer, C. Hargus, E.D. Hermes, P.C. Jennings, P. Bjerre Jensen, J. Kermode, J.R. Kitchin, E. Leonhard Kolsbjerg, J. Kubal, K. Kaasbjer g, S. Lysgaard, J. Bergmann Maronsson, T. Maxson, T. Olsen, L. Pastewka, A. Peterson, C. Rostgaard, J...

    work page doi:10.1088/1361-648x/aa680e 2017

  55. [55]

    Walle, G

    A. Walle, G. Ceder, Automating first-principles phase diagram calculations, J. Phase Equilibria 23 (2002) 348–359. https://doi.org/10.1361/105497102770331596

    work page doi:10.1361/105497102770331596 2002

  56. [56]

    Aydinol, A.F

    M.K. Aydinol, A.F. Kohan, G. Ceder, K. Cho, J. Joannopoulos, Ab initio study of lithium intercalation in metal oxides and metal dichalcogenides, Phys. Rev. B 56 (1997) 1354 –1365. https://doi.org/10.1103/PhysRevB.56.1354

    work page doi:10.1103/physrevb.56.1354 1997