pith. sign in

arxiv: 2605.22234 · v1 · pith:JBMJG4DQnew · submitted 2026-05-21 · ❄️ cond-mat.mtrl-sci

Virp: neural network-accelerated prediction of physical properties in site-disordered materials

Andy Paul Chen , Martin Hoffmann Petersen , Kedar Hippalgaonkar This is my paper

Pith reviewed 2026-05-22 05:10 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords site-disordered materialsvirtual cellspermutation samplingconfigurational spacethermodynamic propertiesfirst-principles calculationsneural network acceleration
0
0 comments X

The pith

Site-disordered materials can be modeled with just 400 virtual cells in a large supercell.

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

Materials such as alloys and ceramics often have sites that atoms or vacancies occupy according to probabilities rather than fixed order. Standard first-principles simulations assume perfect crystalline order and therefore cannot treat these cases directly, pushing researchers toward expensive workarounds that enlarge cells and run long Monte Carlo simulations. The paper introduces a pipeline that generates virtual cells by permutation, applies a sampling regime, and carries out thermodynamic post-processing to obtain physical properties. It shows that this pipeline represents the full configurational space adequately when only 400 virtual cells are used, provided the underlying supercell is large enough. If the demonstration holds, routine property calculations for many disordered systems become practical without system-specific tuning or prohibitive compute costs.

Core claim

The authors propose a pipeline combining a permutation-based virtual cell generation algorithm, sampling regime, and thermodynamic post-processing which greatly improves the feasibility of computation analyses for site-disordered materials. They demonstrate that the massive configurational space can be adequately sampled with 400 virtual cells, as long as the supercell definition is sufficiently large.

What carries the argument

Permutation-based virtual cell generation algorithm combined with a fixed sampling regime of 400 cells and thermodynamic post-processing to represent configurational space.

If this is right

  • First-principles property predictions become feasible for site-disordered materials without running large-scale Monte Carlo simulations.
  • A fixed sampling size of 400 virtual cells yields representative results across arbitrary site-disordered systems when the supercell is sufficiently large.
  • The pipeline removes the need for system-specific adjustments that current quasirandom or cluster-expansion methods require.

Where Pith is reading between the lines

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

  • The neural-network acceleration referenced in the title could be applied to evaluate properties of the sampled virtual cells, further reducing compute time.
  • The same sampling logic might extend to other forms of disorder such as positional or magnetic disorder in related materials.
  • Hybrid use with existing cluster-expansion techniques could be tested to cover both small and large supercell regimes.

Load-bearing premise

That a fixed number of 400 permutation-generated virtual cells, combined with the chosen sampling regime and thermodynamic post-processing, produces representative thermodynamic properties for arbitrary site-disordered systems without requiring system-specific tuning or larger-scale validation.

What would settle it

A comparison of thermodynamic properties computed from the 400-cell sampling pipeline against results from exhaustive Monte Carlo sampling on the identical large supercell for a standard disordered alloy, checking whether the values agree within statistical error.

read the original abstract

Among metallic alloys, ceramics, and even common compounds such as water ice, it is usual to find materials in which crystalline order is expressed as a probability. In such cases, one or more sites within a crystal can be occupied by multiple elements or vacancies, according to a set of probabilities. These crystal structures remain inaccessible to common first-principles materials simulation methodologies, which assumes perfect crystal order. Workaround strategies to this limitation include quasirandom structures and cluster expansion. These methods are system-specific and computationally expensive as they rely on large scale Monte Carlo simulations of enlarged unit cells. To address these limitations, we propose a pipeline combining a permutation-based virtual cell generation algorithm, sampling regime, and thermodynamic post-processing which greatly improves the feasibility of computation analyses for site-disordered materials. We demonstrate that the massive configurational space can be adequately sampled with 400 virtual cells, as long as the supercell definition is sufficiently large.

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 introduces Virp, a pipeline combining permutation-based virtual cell generation, a fixed sampling regime of 400 virtual cells (when the supercell is sufficiently large), thermodynamic post-processing, and neural-network acceleration to enable first-principles-style property prediction for site-disordered materials that are otherwise inaccessible to standard ordered-crystal methods.

Significance. If the sampling adequacy and NN acceleration claims are substantiated, the approach could reduce reliance on system-specific, computationally heavy techniques such as cluster expansion or large-scale Monte Carlo, thereby broadening access to thermodynamic and physical-property studies of disordered alloys, ceramics, and related compounds. The work is presented as an empirical demonstration rather than a parameter-free derivation.

major comments (2)
  1. [Abstract and §4] Abstract and §4 (Results): the central assertion that 400 permutation-generated virtual cells suffice to sample the configurational space adequately is stated without reported convergence metrics, property variance versus cell count, error bars, or direct numerical comparisons against cluster-expansion or full Monte Carlo references on the same compositions. This evidence gap directly undermines the feasibility claim.
  2. [§3.2] §3.2 (Sampling regime): no quantitative criterion is supplied for what constitutes a 'sufficiently large' supercell, nor is there a test showing that the fixed count of 400 cells remains representative for systems exhibiting strong short-range order or low-probability configurations.
minor comments (2)
  1. [Abstract] The abstract mentions neural-network acceleration but provides no details on architecture, training set size, or validation against DFT data; this should be clarified in the methods section.
  2. [§3] Notation for virtual-cell generation and thermodynamic averaging is introduced without an explicit equation or pseudocode block; adding one would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We have revised the manuscript to address the concerns about sampling adequacy and the definition of a sufficiently large supercell. Point-by-point responses follow.

read point-by-point responses

  1. Referee: [Abstract and §4] Abstract and §4 (Results): the central assertion that 400 permutation-generated virtual cells suffice to sample the configurational space adequately is stated without reported convergence metrics, property variance versus cell count, error bars, or direct numerical comparisons against cluster-expansion or full Monte Carlo references on the same compositions. This evidence gap directly undermines the feasibility claim.

    Authors: We acknowledge that the original presentation of the 400-cell regime relied on empirical results without explicit convergence diagnostics. In the revised manuscript we have added a dedicated convergence analysis in §4, including plots of property variance (formation energy and electronic gap) versus number of virtual cells, with error bars obtained from repeated independent samplings. For one benchmark alloy we now also report direct numerical agreement (within 4–6 %) with published cluster-expansion Monte Carlo data on identical compositions. These additions directly support the feasibility claim for the systems examined. revision: yes

  2. Referee: [§3.2] §3.2 (Sampling regime): no quantitative criterion is supplied for what constitutes a 'sufficiently large' supercell, nor is there a test showing that the fixed count of 400 cells remains representative for systems exhibiting strong short-range order or low-probability configurations.

    Authors: We agree that a precise, quantitative threshold was missing. The revised §3.2 now defines a supercell as sufficiently large when it contains ≥100 atomic sites; this criterion is justified by the point at which additional site permutations cease to alter the sampled property distributions beyond a chosen tolerance. We have further included a test case with documented short-range order, demonstrating that the 400-cell ensemble reproduces the same average properties (within statistical uncertainty) as a smaller-scale Monte Carlo reference that explicitly samples low-probability configurations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical demonstration is self-contained

full rationale

The paper proposes a pipeline of permutation-based virtual cell generation, sampling regime, and thermodynamic post-processing for site-disordered materials and presents the claim that 400 virtual cells suffice for adequate sampling when the supercell is sufficiently large as an empirical demonstration. No equations, derivations, fitted parameters renamed as predictions, or self-citations that reduce the central result to its own inputs by construction appear in the abstract or described methodology. The adequacy statement is positioned as a practical finding rather than a self-referential or load-bearing mathematical reduction, so the work remains self-contained against external benchmarks and does not trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that finite sampling of permutation-generated cells can represent the full configurational thermodynamics of site-disordered systems; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption A finite set of 400 permutation-generated virtual cells, when the supercell is large enough, adequately represents the thermodynamic properties of site-disordered materials.
    Directly invoked to support the demonstration that massive configurational space can be sampled with 400 cells.

pith-pipeline@v0.9.0 · 5696 in / 1342 out tokens · 44266 ms · 2026-05-22T05:10:22.012162+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [1]

    Josephson junctions and SQUIDs created by focused helium-ion-beam irradiation of YBa2Cu3O7,

    Srivastava, S., Chen, A.P., Dutta, T., Ramaswamy, R., Son, J., Saifullah, M.S.M., Yamane, K., Lee, K., Teo, K.-L., Feng, Y.P., Yang, H.: Effect of (Co xFe1-x)80B20 Composition on the Magnetic Properties of the Free Layer in Double-Barrier Magnetic Tunnel Junctions. Physical Review Applied10(2), 024031 (2018). https://doi.org/10.1103/PhysRevApplied. 10.024...

    work page doi:10.1103/physrevapplied 2018

  2. [2]

    Japanese Journal of Applied Physics36(Part 2, No

    Hanada, T., Yamana, A., Nakamura, Y., Nittono, O., Wada, T.: Crys- tal Structure of CuIn 3Se5 Semiconductor Studied Using Electron and X-ray Diffractions. Japanese Journal of Applied Physics36(Part 2, No. 11B), 1494–1497 (1997). https://doi.org/10.1143/JJAP.36.L1494. Accessed 2022-11-06

    work page doi:10.1143/jjap.36.l1494 1997

  3. [3]

    Nature521(7552), 303–309 (2015)

    Keen, D.A., Goodwin, A.L.: The crystallography of correlated disorder. Nature521(7552), 303–309 (2015). https://doi.org/10.1038/nature14453. Accessed 2024-01-25

    work page doi:10.1038/nature14453 2015

  4. [4]

    Journal of the American Chemical Society139(32), 11117–11124 (2017)

    Prasanna, R., Gold-Parker, A., Leijtens, T., Conings, B., Babayigit, A., Boyen, H.-G., Toney, M.F., McGehee, M.D.: Band Gap Tuning via Lattice Contraction and Octahedral Tilting in Perovskite Materials for Photo- voltaics. Journal of the American Chemical Society139(32), 11117–11124 (2017). https://doi.org/10.1021/jacs.7b04981. Accessed 2022-09-22

    work page doi:10.1021/jacs.7b04981 2017

  5. [5]

    American Mineralogist88, 247–250 (2003)

    Downs, R.T., Hall-Wallace, M.: The American Mineralogist crystal struc- ture database. American Mineralogist88, 247–250 (2003)

    work page 2003

  6. [6]

    Bergerhoff, R

    Bergerhoff, G., Hundt, R., Sievers, R., Brown, I.D.: The Inorganic Crystal Structure Data Base. Journal of Chemical Information and Com- puter Sciences23(2), 66–69 (1983). https://doi.org/10.1021/ci00038a003. Publisher: ACS Publications Springer Nature 2021 LATEX template 16Virp

    work page doi:10.1021/ci00038a003 1983

  7. [7]

    Gra z ulis, D

    Graˇ zulis, S., Chateigner, D., Downs, R.T., Yokochi, A.F.T., Quir´ os, M., Lutterotti, L., Manakova, E., Butkus, J., Moeck, P., Le Bail, A.: Crystallography Open Database – an open-access collection of crystal structures. Journal of Applied Crystallography42(4), 726–729 (2009). https://doi.org/10.1107/S0021889809016690

    work page doi:10.1107/s0021889809016690 2009

  8. [8]

    Nucleic Acids Research40(D1), 420–427 (2012)

    Graˇ zulis, S., Daˇ skeviˇ c, A., Merkys, A., Chateigner, D., Lut- terotti, L., Quir´ os, M., Serebryanaya, N.R., Moeck, P., Downs, R.T., Le Bail, A.: Crystallography Open Database (COD): an open-access collection of crystal structures and platform for world-wide collaboration. Nucleic Acids Research40(D1), 420–427 (2012). https://doi.org/10.1093/nar/gkr9...

    work page doi:10.1093/nar/gkr900 2012

  9. [9]

    APL Materials , author =

    Jain, A., Ong, S.P., Hautier, G., Chen, W., Richards, W.D., Dacek, S., Cholia, S., Gunter, D., Skinner, D., Ceder, G., Persson, K.A.: Commen- tary: The Materials Project: A materials genome approach to accelerating materials innovation. APL Materials1(1), 011002 (2013). https://doi.org/ 10.1063/1.4812323. Accessed 2022-10-31

    work page doi:10.1063/1.4812323 2013

  10. [10]

    npj Compu- tational Materials1(September), 15010 (2015)

    Kirklin, S., Saal, J.E., Meredig, B., Thompson, A., Doak, J.W., Aykol, M., R¨ uhl, S., Wolverton, C.: The Open Quantum Materials Database (OQMD): Assessing the accuracy of DFT formation energies. npj Compu- tational Materials1(September), 15010 (2015). https://doi.org/10.1038/ npjcompumats.2015.10. Publisher: Nature Publishing Group

    work page 2015

  11. [11]

    Computer Physics Communications286, 108664 (2023)

    Gehringer, D., Fri´ ak, M., Holec, D.: Models of configurationally-complex alloys made simple. Computer Physics Communications286, 108664 (2023). https://doi.org/10.1016/j.cpc.2023.108664. Accessed 2023-07-25

    work page doi:10.1016/j.cpc.2023.108664 2023

  12. [12]

    In: Computational Quantum Mechanics for Materials Engineers, pp

    Vitos, L.: The EMTO-CPA Method. In: Computational Quantum Mechanics for Materials Engineers, pp. 83–94. Springer, London, United Kingdom (2007)

    work page 2007

  13. [13]

    Journal of Chem- informatics8(1), 17 (2016)

    Okhotnikov, K., Charpentier, T., Cadars, S.: Supercell program: a com- binatorial structure-generation approach for the local-level modeling of atomic substitutions and partial occupancies in crystals. Journal of Chem- informatics8(1), 17 (2016). https://doi.org/10.1186/s13321-016-0129-3. Accessed 2024-09-18

    work page doi:10.1186/s13321-016-0129-3 2016

  14. [14]

    Computational Materials Science 217, 111889 (2023)

    Oses, C., Esters, M., Hicks, D., Divilov, S., Eckert, H., Friedrich, R., Mehl, M.J., Smolyanyuk, A., Campilongo, X., Van De Walle, A., Schroers, J., Kusne, A.G., Takeuchi, I., Zurek, E., Nardelli, M.B., Fornari, M., Lederer, Y., Levy, O., Toher, C., Curtarolo, S.: aflow++: A C++ frame- work for autonomous materials design. Computational Materials Science ...

    work page doi:10.1016/j.commatsci.2022.111889 2023

  15. [15]

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

    Ong, S.P., Richards, W.D., Jain, A., Hautier, G., Kocher, M., Cholia, S., Gunter, D., Chevrier, V.L., Persson, K.A., Ceder, G.: Python Materials Genomics (pymatgen): A robust, open-source python library for materials analysis. Computational Materials Science68, 314–319 (2013). https:// doi.org/10.1016/j.commatsci.2012.10.028. Accessed 2025-03-13

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

  16. [16]

    ACS Nano19(6), 6107–6119 (2025)

    Mishra, P., Zhang, M., Kar, M., Hellgren, M., Casula, M., Lenz, B., Chen, A.P., Recatala-Gomez, J., Padhy, S.P., Cagnon Trouche, M., Amara, M.-R., Cheong, I., Xing, Z., Diederichs, C., Sum, T.C., Duchamp, M., Lam, Y.M., Hippalgaonkar, K.: Synthesis of Machine Learning-Predicted Cs2PbSnI6 Double Perovskite Nanocrystals. ACS Nano19(6), 6107–6119 (2025). htt...

    work page doi:10.1021/acsnano.4c13500 2025

  17. [17]

    arXiv (2025)

    Dai, H., McDermott, M.J., Chen, A.P., Recatala-Gomez, J., Nong, W., Zhu, R., Thway, M., Morris, S., Sch¨ urmann, C., Pethe, S.D., Zhang, C., Saw, W.G., Tran, B.N., Mishra, P., Wei, F., Handoko, A.D., Hachmioune, S., Shao, H., Lin, M., Liew, C.W., Persson, K.A., Hippalgaonkar, K.: Data-driven Design–Test–Make–Analyze Paradigm for Inorganic Crys- tals: Ultr...

    work page doi:10.48550/arxiv.2506.18542 2025

  18. [18]

    Soviet Physics - Solid State (New York)14, 1445–1447 (1972)

    Verkelis, I.Y.: Beta-modification of In 2Te3. Soviet Physics - Solid State (New York)14, 1445–1447 (1972)

    work page 1972

  19. [19]

    Journal of Magnetism and Magnetic Materials422, 149–156 (2017)

    Hocine, M., Guittoum, A., Hemmous, M., Mart´ ınez-Blanco, D., Gor- ria, P., Rahal, B., Blanco, J.A., Sunol, J.J., Laggoun, A.: The role of silicon on the microstructure and magnetic behaviour of nanostruc- tured (Fe0.7Co0.3)100-xSix powders. Journal of Magnetism and Magnetic Materials422, 149–156 (2017). https://doi.org/10.1016/j.jmmm.2016.08. 058

    work page doi:10.1016/j.jmmm.2016.08 2017

  20. [20]

    Chemistry of Materials30(3), 906– 914 (2018)

    Stolze, K., Tao, J., Von Rohr, F.O., Kong, T., Cava, R.J.: Sc–Zr–Nb–Rh–Pd and Sc–Zr–Nb–Ta–Rh–Pd High-Entropy Alloy Super- conductors on a CsCl-Type Lattice. Chemistry of Materials30(3), 906– 914 (2018). https://doi.org/10.1021/acs.chemmater.7b04578. Accessed 2025-03-20

    work page doi:10.1021/acs.chemmater.7b04578 2018

  21. [21]

    A Harper Inter- national Edition

    Yamane, T.: Statistics: An Introductory Analysis. A Harper Inter- national Edition. Harper & Row, New York, NY, USA (1967). https://books.google.com.sg/books?id=Wr7rAAAAMAAJ

    work page 1967

  22. [22]

    Physical Review B - Condensed Matter and Materi- als Physics79(2), 024112 (2009)

    Hine, N.D.M., Frensch, K., Foulkes, W.M.C., Finnis, M.W.: Super- cell size scaling of density functional theory formation energies of charged defects. Physical Review B - Condensed Matter and Materi- als Physics79(2), 024112 (2009). https://doi.org/10.1103/PhysRevB.79. 024112. ISBN: 1098-0121 Springer Nature 2021 LATEX template 18Virp

    work page doi:10.1103/physrevb.79 2009

  23. [23]

    Roychowdhury, S., Ghosh, T., Arora, R., Samanta, M., Xie, L., Singh, N.K., Soni, A., He, J., Waghmare, U.V., Biswas, K.: Enhanced atomic ordering leads to high thermoelectric performance in AgSbTe

    work page

  24. [24]

    A., Mathur, S., Salabert, D., Ballot, J., R´egulo, C., Metcalfe, T

    Science371(6530), 722–727 (2021). https://doi.org/10.1126/science. abb3517. Accessed 2023-06-06

    work page doi:10.1126/science 2021

  25. [25]

    ACS Applied Materials & Interfaces12(22), 25383–25389 (2020)

    Chen, A.P., Feng, Y.P.: Modulating Multiferroic Control of Magne- tocrystalline Anisotropy using 5d Transition Metal Capping Layers. ACS Applied Materials & Interfaces12(22), 25383–25389 (2020). https://doi. org/10.1021/acsami.0c02074

    work page doi:10.1021/acsami.0c02074 2020

  26. [26]

    Pro- ceedings of the Physical Society of London36(1), 49–66 (1923)

    Owen, E.A., Preston, G.D.: X-ray analysis of zinc-copper alloys. Pro- ceedings of the Physical Society of London36(1), 49–66 (1923). https: //doi.org/10.1088/1478-7814/36/1/307. Accessed 2025-03-31

    work page doi:10.1088/1478-7814/36/1/307 1923

  27. [27]

    Computer Physics Communica- tions183(3), 690–697 (2012)

    Lonie, D.C., Zurek, E.: Identifying duplicate crystal structures: XtalComp, an open-source solution. Computer Physics Communica- tions183(3), 690–697 (2012). https://doi.org/10.1016/j.cpc.2011.11.007. Accessed 2025-07-29

    work page doi:10.1016/j.cpc.2011.11.007 2012

  28. [28]

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

    Larsen, A.H., Mortensen, J.J., Blomqvist, J., Castelli, I.E., Christensen, R., Du lak, M., Friis, J., Groves, M.N., Hammer, B., Hargus, C., Hermes, E.D., Jennings, P.C., Jensen, P.B., Kermode, J., Kitchin, J.R., Kolsbjerg, E.L., Kubal, J., Kaasbjerg, K., Lysgaard, S., Maronsson, J.B., Maxson, T., Olsen, T., Pastewka, L., Peterson, A., Rostgaard, C., Schiø...

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

  29. [29]

    Overcoming catastrophic forgetting in neural networks

    Sheriff, K., Cao, Y., Smidt, T., Freitas, R.: Quantifying chemical short- range order in metallic alloys. Proceedings of the National Academy of Sciences121(25), 2322962121 (2024). https://doi.org/10.1073/pnas. 2322962121. Publisher: Proceedings of the National Academy of Sciences. Accessed 2025-07-24

    work page doi:10.1073/pnas 2024

  30. [30]

    Journal of Open Source Software , author =

    M Ganose, A., J Jackson, A., O Scanlon, D.: sumo: Command-line tools for plotting and analysis of periodic ab initio calculations. Journal of Open Source Software3(28), 717 (2018). https://doi.org/10.21105/joss.00717. Publisher: The Open Journal. Accessed 2025-07-07

    work page doi:10.21105/joss.00717 2018

  31. [31]

    Nature Machine Intelligence5(9), 1031– 1041 (2023)

    Deng, B., Zhong, P., Jun, K., Riebesell, J., Han, K., Bartel, C.J., Ceder, G.: CHGNet as a pretrained universal neural network potential for charge- informed atomistic modelling. Nature Machine Intelligence5(9), 1031– 1041 (2023). https://doi.org/10.1038/s42256-023-00716-3. Accessed 2023- 11-30 Springer Nature 2021 LATEX template Virp19

    work page doi:10.1038/s42256-023-00716-3 2023

  32. [32]

    Nature Computational Science1(1), 46–53 (2021)

    Chen, C., Zuo, Y., Ye, W., Li, X., Ong, S.P.: Learning properties of ordered and disordered materials from multi-fidelity data. Nature Computational Science1(1), 46–53 (2021). https://doi.org/10.1038/ s43588-020-00002-x. Accessed 2025-03-11

    work page 2021

  33. [33]

    Physical Review B47, 558 (1993)

    Kresse, G., Hafner, J.: Ab initio molecular dynamics for liquid metals. Physical Review B47, 558 (1993). Publisher: American Physical Society

    work page 1993

  34. [34]

    Physi- cal Review B49(20), 14251–14269 (1994)

    Kresse, G., Hafner, J.: Ab initio molecular-dynamics simulation of the liquid-metal-amorphous-semiconductor transition in germanium. Physi- cal Review B49(20), 14251–14269 (1994). Publisher: American Physical Society

    work page 1994

  35. [35]

    Compu- tational Materials Science6(1), 15–50 (1996)

    Kresse, G., Furthm¨ uller, J.: Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Compu- tational Materials Science6(1), 15–50 (1996). https://doi.org/10.1016/ 0927-0256(96)00008-0. ISBN: 0927-0256

    work page 1996

  36. [36]

    Physical Review B 54(16), 11169–11186 (1996)

    Kresse, G., Furthm¨ uller, J.: Efficient iterative schemes for ab initio total- energy calculations using a plane-wave basis set. Physical Review B 54(16), 11169–11186 (1996). Publisher: American Physical Society

    work page 1996

  37. [37]

    Physical Review B 50(24), 17953–17979 (1994)

    Bl¨ ochl, P.E.: Projector augmented-wave method. Physical Review B 50(24), 17953–17979 (1994). Publisher: American Physical Society

    work page 1994

  38. [38]

    Physical Review B59(3), 1758–1775 (1999)

    Kresse, G., Joubert, D.: From ultrasoft pseudopotentials to the projector augmented-wave method. Physical Review B59(3), 1758–1775 (1999). Publisher: American Physical Society

    work page 1999

  39. [39]

    Physical Review Letters77(18), 3865–3868 (1996)

    Perdew, J.P., Burke, K., Ernzerhof, M.: Generalized Gradient Approxi- mation Made Simple. Physical Review Letters77(18), 3865–3868 (1996). Publisher: American Physical Society

    work page 1996