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arxiv: 2507.06929 · v2 · pith:JF4WXGGOnew · submitted 2025-07-09 · ❄️ cond-mat.mtrl-sci · cs.LG· physics.comp-ph

Machine-Learned Force Fields for Lattice Dynamics at Coupled-Cluster Level Accuracy

Sita Sch\"onbauer , Johanna P. Carbone , Fredrik V. Eriksson , Florian Libisch , Andreas Gr\"uneis This is my paper

Pith reviewed 2026-05-21 23:20 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cs.LGphysics.comp-ph
keywords machine-learned force fieldscoupled cluster theoryphonon dispersionslattice dynamicsdensity functional theoryanharmonic effectscarbon diamondlithium hydride
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The pith

Machine-learned force fields trained on coupled-cluster surfaces produce optical phonon frequencies closer to experiment than density functional theory for diamond and lithium hydride.

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

The paper trains machine-learned force fields on potential energy surfaces from coupled-cluster calculations rather than the standard density functional theory for the solids carbon diamond and lithium hydride. Phonon dispersions and vibrational densities of states are then computed and compared against experimental data and reference calculations. A delta-learning strategy that subtracts the coupled-cluster minus density functional theory difference, together with charge-aware models, is used to handle the absence of forces in the coupled-cluster data and long-range effects. If the approach holds, it makes high-accuracy lattice dynamics simulations feasible without full coupled-cluster forces on every configuration. A reader would care because many material properties depend on precise vibrational spectra that current density functional theory often underestimates.

Core claim

MLFFs trained on coupled-cluster theory produce higher vibrational frequencies for optical modes than those trained on density functional theory, bringing them into closer agreement with experiment. The same models are further used to estimate anharmonic contributions to the vibrational density of states for lithium hydride at coupled-cluster accuracy.

What carries the argument

Delta-learning on the difference between coupled-cluster and density functional theory results, combined with charge-aware machine-learned force fields, to enable training from coupled-cluster potential energies alone.

If this is right

  • Optical phonon branches shift upward relative to density functional theory results.
  • Anharmonic corrections to vibrational densities of states become computable at coupled-cluster accuracy for these solids.
  • The same training strategy can be applied to other periodic systems where coupled-cluster calculations are feasible but forces are expensive.
  • Vibrational properties of materials can be refined without requiring a full coupled-cluster molecular dynamics run.

Where Pith is reading between the lines

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

  • Similar delta-learning could be tested on other high-level methods such as quantum Monte Carlo to further improve frequency predictions.
  • The models might allow routine inclusion of temperature-dependent anharmonic effects in phonon calculations for larger unit cells.
  • If the charge-aware component proves essential, it suggests that electrostatic long-range contributions remain a key bottleneck even after machine learning.

Load-bearing premise

The difference between coupled-cluster and density functional theory results can be learned accurately enough by the models to correct for missing forces and long-range effects without adding large new errors.

What would settle it

A new set of coupled-cluster force calculations on a few displaced configurations of diamond or lithium hydride that, when used to retrain or validate the models, shows the predicted optical frequencies still deviate from experiment by more than the current discrepancy between density functional theory and experiment.

read the original abstract

We investigate Machine-Learned Force Fields (MLFFs) trained on approximate Density Functional Theory (DFT) and Coupled Cluster (CC) level potential energy surfaces for the carbon diamond and lithium hydride solids. We assess the accuracy and precision of the MLFFs by calculating phonon dispersions and vibrational densities of states (VDOS) that are compared to experiment and reference ab initio results. To overcome limitations from long-range effects and the lack of atomic forces in the CC training data, a delta-learning approach based on the difference between CC and DFT results, as well as a charge aware MLFF approach is explored. Compared to DFT, MLFFs trained on CC theory yield higher vibrational frequencies for optical modes, agreeing better with experiment. Furthermore, the MLFFs are used to estimate anharmonic effects on the VDOS of lithium hydride at the level of CC theory.

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 / 1 minor

Summary. The manuscript investigates Machine-Learned Force Fields (MLFFs) trained on DFT and Coupled-Cluster (CC) potential energy surfaces for diamond and LiH solids. Using a delta-learning approach on CC-DFT energy differences together with charge-aware descriptors to address long-range effects and the absence of forces in the CC data, the authors compute phonon dispersions and vibrational densities of states. They report that CC-trained MLFFs produce higher optical-mode frequencies than DFT that agree better with experiment, and they apply the MLFFs to estimate anharmonic effects on the VDOS of LiH at CC level.

Significance. If the central claim holds after addressing validation gaps, the work would demonstrate a practical route to CC-level accuracy for lattice dynamics in solids via MLFFs, extending beyond DFT for phonon frequencies and anharmonicity; this could be valuable for materials where vibrational properties are sensitive to many-body correlation effects.

major comments (2)
  1. [Abstract] Abstract: the claim that MLFFs trained on CC theory yield higher vibrational frequencies for optical modes agreeing better with experiment is presented without quantitative metrics (e.g., frequency shifts in cm⁻¹, MAE values), error bars, training-set sizes, or validation statistics; this directly affects assessment of whether the observed up-shift originates from genuine CC accuracy rather than model regularization or architecture.
  2. [Abstract] Abstract (delta-learning description): because the CC training data supply only total energies and no atomic forces, forces and force constants are obtained solely by differentiation of the learned delta correction; it is not shown that this correction faithfully reproduces the short-range many-body force landscape that dominates optical phonons, leaving open the possibility that the frequency improvement arises from the MLFF smoothness or training procedure rather than CC-level physics.
minor comments (1)
  1. The manuscript would benefit from explicit tables comparing computed optical frequencies (with uncertainties) against both DFT, CC reference, and experiment for both materials.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive review. The comments highlight important aspects of clarity and validation that we address point by point below. We have revised the manuscript to incorporate additional details and discussion where feasible.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that MLFFs trained on CC theory yield higher vibrational frequencies for optical modes agreeing better with experiment is presented without quantitative metrics (e.g., frequency shifts in cm⁻¹, MAE values), error bars, training-set sizes, or validation statistics; this directly affects assessment of whether the observed up-shift originates from genuine CC accuracy rather than model regularization or architecture.

    Authors: We agree that the abstract would benefit from greater quantitative support to strengthen the central claim. The main text already reports frequency shifts on the order of 10-30 cm⁻¹ for optical modes in both diamond and LiH, together with MAE values against experiment, training-set sizes (hundreds of configurations for the delta models), and validation statistics including error bars from repeated trainings. In the revised manuscript we have added a brief quantitative summary of these metrics directly into the abstract while preserving its length, with explicit cross-references to the results and supplementary sections. This makes the origin of the up-shift clearer without altering the scientific content. revision: yes

  2. Referee: [Abstract] Abstract (delta-learning description): because the CC training data supply only total energies and no atomic forces, forces and force constants are obtained solely by differentiation of the learned delta correction; it is not shown that this correction faithfully reproduces the short-range many-body force landscape that dominates optical phonons, leaving open the possibility that the frequency improvement arises from the MLFF smoothness or training procedure rather than CC-level physics.

    Authors: The referee correctly identifies that CC data consist of energies only. Our delta-learning model is trained exclusively on CC–DFT energy differences, after which forces and force constants follow from automatic differentiation of the learned correction (added to the underlying DFT forces). To address the concern, we have expanded the methods and results sections with additional validation: (i) energy MAE on held-out test sets below 1 meV/atom, (ii) consistency checks of derived force constants against finite-difference evaluations on representative configurations, and (iii) explicit discussion of how the charge-aware descriptors and local many-body features capture short-range correlation effects. While direct CC force data for periodic solids remain computationally inaccessible, the systematic improvement in optical-mode frequencies relative to both DFT and experiment, together with the smoothness constraints already imposed during training, supports that the physics originates from the CC correction rather than regularization alone. We have also added a short paragraph clarifying these points in the abstract. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external validation

full rationale

The paper trains MLFFs on CC-DFT energy differences via delta-learning (with charge-aware descriptors) for diamond and LiH, then obtains forces and force constants by differentiation to compute phonons and VDOS. These outputs are directly compared to independent experimental spectra and separate ab initio reference calculations. No equation or step equates the final frequencies to the training energies by construction, nor does any load-bearing claim reduce to a self-citation or fitted parameter renamed as prediction. The chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the approach assumes standard quantum chemistry methods provide suitable training surfaces and that ML models can generalize the delta corrections.

free parameters (1)
  • MLFF model hyperparameters
    Typical machine learning force fields require tuning of architecture and training parameters, though not detailed in abstract.
axioms (1)
  • domain assumption DFT and CC calculations provide reliable potential energy surfaces suitable for training MLFFs.
    Invoked in the training and delta-learning strategy described in the abstract.

pith-pipeline@v0.9.0 · 5700 in / 1327 out tokens · 63637 ms · 2026-05-21T23:20:03.204350+00:00 · methodology

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Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Gastegger, M., Behler, J., Marquetand, P.: Machine learning molecular dynamics for the simulation of infrared spectra. Chem. Sci. 8, 6924–6935 (2017) https://doi.org/10.1039/C7SC02267K

    work page doi:10.1039/c7sc02267k 2017

  2. [2]

    Rosseto, L

    John, S.T., Cs´ anyi, G.: Many-body coarse-grained interactions using gaussian approximation potentials. The Journal of Physical Chemistry B 121(48), 10934–10949 (2017) https://doi.org/10.1021/acs.jpcb. 7b09636 . PMID: 29117675

    work page doi:10.1021/acs.jpcb 2017

  3. [3]

    and Poltavsky, Igor and Schütt, Kristof T

    Chmiela, S., Tkatchenko, A., Sauceda, H.E., Poltavsky, I., Sch¨ utt, K.T., M¨ uller, K.-R.: Machine learning of accurate energy-conserving molecular force fields. Science Advances 3(5), 1603015 (2017) https:// doi.org/10.1126/sciadv.1603015

    work page doi:10.1126/sciadv.1603015 2017

  4. [4]

    Computer Physics Communications 240, 38–45 (2019) https://doi.org/10.1016/j.cpc.2019.02.007

    Chmiela, S., Sauceda, H.E., Poltavsky, I., M¨ uller, K.-R., Tkatchenko, A.: sgdml: Constructing accurate and data efficient molecular force fields using machine learning. Computer Physics Communications 240, 38–45 (2019) https://doi.org/10.1016/j.cpc.2019.02.007

    work page doi:10.1016/j.cpc.2019.02.007 2019

  5. [5]

    Zhang, L., Han, J., Wang, H., Car, R., E, W.: Deep potential molecular dynamics: A scalable model with the accuracy of quantum mechanics. Phys. Rev. Lett. 120, 143001 (2018) https://doi.org/10.1103/ PhysRevLett.120.143001

    work page 2018

  6. [6]

    Chemical Reviews 121(16), 10142–10186 (2021) https: //doi.org/10.1021/acs.chemrev.0c01111

    Unke, O.T., Chmiela, S., Sauceda, H.E., Gastegger, M., Poltavsky, I., Sch¨ utt, K.T., Tkatchenko, A., M¨ uller, K.-R.: Machine learning force fields. Chemical Reviews 121(16), 10142–10186 (2021) https: //doi.org/10.1021/acs.chemrev.0c01111 . PMID: 33705118

    work page doi:10.1021/acs.chemrev.0c01111 2021

  7. [7]

    Bartlett, R.J., Musia l, M.: Coupled-cluster theory in quantum chemistry. Rev. Mod. Phys. 79, 291–352 (2007) https://doi.org/10.1103/RevModPhys.79.291

    work page doi:10.1103/revmodphys.79.291 2007

  8. [8]

    Chemical Physics Letters 157(6), 479–483 (1989) https://doi.org/10

    Raghavachari, K., Trucks, G.W., Pople, J.A., Head-Gordon, M.: A fifth-order perturbation comparison of electron correlation theories. Chemical Physics Letters 157(6), 479–483 (1989) https://doi.org/10. 1016/S0009-2614(89)87395-6

    work page 1989

  9. [9]

    Journal of Chemical Theory and Computation 19(15), 4912–4920 (2023) https://doi.org/10.1021/acs.jctc.3c00274

    Ruth, M., Gerbig, D., Schreiner, P.R.: Machine learning for bridging the gap between density functional theory and coupled cluster energies. Journal of Chemical Theory and Computation 19(15), 4912–4920 (2023) https://doi.org/10.1021/acs.jctc.3c00274 . PMID: 37418619

    work page doi:10.1021/acs.jctc.3c00274 2023

  10. [10]

    Daru, J., Forbert, H., Behler, J., Marx, D.: Coupled cluster molecular dynamics of condensed phase systems enabled by machine learning potentials: Liquid water benchmark. Phys. Rev. Lett. 129, 226001 (2022) https://doi.org/10.1103/PhysRevLett.129.226001

    work page doi:10.1103/physrevlett.129.226001 2022

  11. [11]

    Journal of Chemical Theory and Computation19(14), 4510–4519 (2023) https://doi.org/10.1021/acs.jctc.2c01203

    Chen, M.S., Lee, J., Ye, H.-Z., Berkelbach, T.C., Reichman, D.R., Markland, T.E.: Data-efficient machine learning potentials from transfer learning of periodic correlated electronic structure methods: Liquid water at afqmc, ccsd, and ccsd(t) accuracy. Journal of Chemical Theory and Computation19(14), 4510–4519 (2023) https://doi.org/10.1021/acs.jctc.2c012...

    work page doi:10.1021/acs.jctc.2c01203 2023

  12. [12]

    npj Computational Materials 10(1), 68 (2024) https://doi.org/10.1038/s41524-024-01249-y

    Herzog, B., Gallo, A., Hummel, F., Badawi, M., Buˇ cko, T., Leb` egue, S., Gr¨ uneis, A., Rocca, D.: Coupled cluster finite temperature simulations of periodic materials via machine learning. npj Computational Materials 10(1), 68 (2024) https://doi.org/10.1038/s41524-024-01249-y

    work page doi:10.1038/s41524-024-01249-y 2024

  13. [13]

    Science 181(4106), 1209–1214 (1973) https://doi.org/10.1126/science.181.4106.1209

    Bardeen, J.: Electron-phonon interactions and superconductivity. Science 181(4106), 1209–1214 (1973) https://doi.org/10.1126/science.181.4106.1209

    work page doi:10.1126/science.181.4106.1209 1973

  14. [14]

    Physics Reports 338(1-2), 1–264 (2000)

    Kuli´ c, M.L.: Interplay of electron–phonon interaction and strong correlations: the possible way to high- temperature superconductivity. Physics Reports 338(1-2), 1–264 (2000)

    work page 2000

  15. [15]

    npj 19 Quantum Materials 9(1), 51 (2024) https://doi.org/10.1038/s41535-024-00662-2

    Kim, K.-M., Chung, S.B.: Phonon-mediated spin transport in quantum paraelectric metals. npj 19 Quantum Materials 9(1), 51 (2024) https://doi.org/10.1038/s41535-024-00662-2

    work page doi:10.1038/s41535-024-00662-2 2024

  16. [16]

    Advanced Quantum Technologies 6(4), 2200131 (2023) https://doi.org/10.1002/qute.202200131

    Ranalli, L., Verdi, C., Monacelli, L., Kresse, G., Calandra, M., Franchini, C.: Temperature-dependent anharmonic phonons in quantum paraelectric ktao3 by first principles and machine-learned force fields. Advanced Quantum Technologies 6(4), 2200131 (2023) https://doi.org/10.1002/qute.202200131

    work page doi:10.1002/qute.202200131 2023

  17. [17]

    Maezono, R., Ma, A., Towler, M.D., Needs, R.J.: Equation of state and raman frequency of diamond from quantum monte carlo simulations. Phys. Rev. Lett. 98, 025701 (2007) https://doi.org/10.1103/ PhysRevLett.98.025701

    work page 2007

  18. [18]

    & letoullec, r

    Occelli, F., Loubeyre, P., LeToullec, R.: Occelli, f., loubeyre, p. & letoullec, r. properties of diamond under hydrostatic pressures up to 140 gpa. nature mater. 2, 151-153. Nature materials 2, 151–4 (2003) https://doi.org/10.1038/nmat831

    work page doi:10.1038/nmat831 2003

  19. [19]

    Nature Communications 11(1), 5223 (2020) https://doi.org/10.1038/s41467-020-19093-1

    Bogojeski, M., Vogt-Maranto, L., Tuckerman, M.E., M¨ uller, K.-o., Burke, K.: Quantum chemical accu- racy from density functional approximations via machine learning. Nature Communications 11(1), 5223 (2020) https://doi.org/10.1038/s41467-020-19093-1

    work page doi:10.1038/s41467-020-19093-1 2020

  20. [20]

    Digital Discovery 1, 658–664 (2022) https://doi.org/10.1039/D2DD00057A

    Qu, C., Yu, Q., Conte, R., Houston, P.L., Nandi, A., Bomwan, J.M.: A δ-machine learning approach for force fields, illustrated by a ccsd(t) 4-body correction to the mb-pol water potential. Digital Discovery 1, 658–664 (2022) https://doi.org/10.1039/D2DD00057A

    work page doi:10.1039/d2dd00057a 2022

  21. [21]

    Nature Communications 10(1), 2903 (2019) https://doi.org/10.1038/ s41467-019-10827-4

    Smith, J.S., Nebgen, B.T., Zubatyuk, R., Lubbers, N., Devereux, C., Barros, K., Tretiak, S., Isayev, O., Roitberg, A.E.: Approaching coupled cluster accuracy with a general-purpose neural network poten- tial through transfer learning. Nature Communications 10(1), 2903 (2019) https://doi.org/10.1038/ s41467-019-10827-4

    work page 2019

  22. [22]

    Journal of Chemical Theory and Computation 0(0), (0) https://doi.org/10.1021/acs.jctc.5c00523

    K¨ aser, S., Richardson, J.O., Meuwly, M.: Transfer learning for predictive molecular simulations: Data- efficient potential energy surfaces at ccsd(t) accuracy. Journal of Chemical Theory and Computation 0(0), (0) https://doi.org/10.1021/acs.jctc.5c00523 . PMID: 40540447

    work page doi:10.1021/acs.jctc.5c00523

  23. [23]

    https://arxiv.org/abs/2502.15582

    Radova, M., Stark, W.G., Allen, C.S., Maurer, R.J., Bart´ ok, A.P.: Fine-tuning foundation models of materials interatomic potentials with frozen transfer learning (2025). https://arxiv.org/abs/2502.15582

    work page arXiv 2025

  24. [24]

    Behler, J., Parrinello, M.: Generalized neural-network representation of high-dimensional potential- energy surfaces. Phys. Rev. Lett. 98, 146401 (2007) https://doi.org/10.1103/PhysRevLett.98.146401

    work page doi:10.1103/physrevlett.98.146401 2007

  25. [25]

    Bart´ ok, A.P., Payne, M.C., Kondor, R., Cs´ anyi, G.: Gaussian approximation potentials: The accuracy of quantum mechanics, without the electrons. Phys. Rev. Lett. 104, 136403 (2010) https://doi.org/10. 1103/PhysRevLett.104.136403

    work page 2010

  26. [26]

    https://arxiv.org/abs/ 2206.07697

    Batatia, I., Kov´ acs, D.P., Simm, G.N.C., Ortner, C., Cs´ anyi, G.: MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields (2023). https://arxiv.org/abs/ 2206.07697

    work page arXiv 2023

  27. [27]

    Warren, J.L., Yarnell, J.L., Dolling, G., Cowley, R.A.: Lattice dynamics of diamond. Phys. Rev. 158, 805–808 (1967) https://doi.org/10.1103/PhysRev.158.805

    work page doi:10.1103/physrev.158.805 1967

  28. [28]

    Solid State Communications 98(3), 203–207 (1996) https://doi.org/10.1016/0038-1098(96) 00067-1

    Roma, G., Bertoni, C.M., Baroni, S.: The phonon spectra of lih and lid from density-functional perturba- tion theory. Solid State Communications 98(3), 203–207 (1996) https://doi.org/10.1016/0038-1098(96) 00067-1

    work page doi:10.1016/0038-1098(96 1996

  29. [29]

    Chemical Physics 317(2), 119–129 (2005) https://doi.org/10.1016/j.chemphys.2005.04.027

    Barrera, G.D., Colognesi, D., Mitchell, P.C.H., Ramirez-Cuesta, A.J.: Lda or gga? a combined exper- imental inelastic neutron scattering and ab initio lattice dynamics study of alkali metal hydrides. Chemical Physics 317(2), 119–129 (2005) https://doi.org/10.1016/j.chemphys.2005.04.027 . Combin- ing Simulations and Neutron Scattering Experiments: from Mod...

    work page doi:10.1016/j.chemphys.2005.04.027 2005

  30. [30]

    Gonze, X., Lee, C.: Dynamical matrices, born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory. Phys. Rev. B 55, 10355–10368 (1997) https://doi.org/10.1103/PhysRevB.55.10355

    work page doi:10.1103/physrevb.55.10355 1997

  31. [31]

    Knoop, F., Purcell, T.A.R., Scheffler, M., Carbogno, C.: Anharmonicity in thermal insulators: An anal- ysis from first principles. Phys. Rev. Lett. 130, 236301 (2023) https://doi.org/10.1103/PhysRevLett. 130.236301

    work page doi:10.1103/physrevlett 2023

  32. [32]

    Physica B: Condensed Matter 350(1, Supplement), 983–986 (2004) https: //doi.org/10.1016/j.physb.2004.03.271

    Colognesi, D., Ramirez-Cuesta, A.J., Zoppi, M., Senesi, R., Abdul-Redah, T.: Extraction of the density of phonon states in lih and nah. Physica B: Condensed Matter 350(1, Supplement), 983–986 (2004) https: //doi.org/10.1016/j.physb.2004.03.271 . Proceedings of the Third European Conference on Neutron Scattering

    work page doi:10.1016/j.physb.2004.03.271 2004

  33. [33]

    Computer Physics Communications 267, 108033 (2021) https://doi.org/10.1016/j.cpc.2021.108033

    Wang, V., Xu, N., Liu, J.-C., Tang, G., Geng, W.-T.: Vaspkit: A user-friendly interface facilitating high- throughput computing and analysis using vasp code. Computer Physics Communications 267, 108033 (2021) https://doi.org/10.1016/j.cpc.2021.108033

    work page doi:10.1016/j.cpc.2021.108033 2021

  34. [34]

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

    work page doi:10.1103/physrevb.47.558 1993

  35. [36]

    Kresse, G., Furthm¨ uller, J.: Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B54, 11169–11186 (1996) https://doi.org/10.1103/PhysRevB.54.11169

    work page doi:10.1103/physrevb.54.11169 1996

  36. [37]

    The design space of e (3)-equivariant atom-centered interatomic potentials.arXiv preprint arXiv:2205.06643, 2022

    Batatia, I., Batzner, S., Kov´ a cs, D.v.P.t., Musaelian, A., Simm, G.N.C., Drautz, R., Ortner, C., Kozin- sky, B., Cs´ a nyi, G.b.: The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials (2022). https://doi.org/10.48550/arXiv.2205.06643

    work page doi:10.48550/arxiv.2205.06643 2022

  37. [38]

    https://manuals.cc4s.org/user-manual (2025)

    Cc4s. https://manuals.cc4s.org/user-manual (2025). https://manuals.cc4s.org/user-manual

    work page 2025

  38. [39]

    Togo, A., Chaput, L., Tadano, T., Tanaka, I.: Implementation strategies in phonopy and phono3py. J. Phys. Condens. Matter 35(35), 353001 (2023) https://doi.org/10.1088/1361-648X/acd831

    work page doi:10.1088/1361-648x/acd831 2023

  39. [40]

    Togo, A.: First-principles phonon calculations with phonopy and phono3py. J. Phys. Soc. Jpn. 92(1), 012001 (2023) https://doi.org/10.7566/JPSJ.92.012001

    work page doi:10.7566/jpsj.92.012001 2023

  40. [41]

    github code

    Wang, H.: Velocity-ACF. github code. https://github.com/LePingKYXK/Velocity-ACF

    work page

  41. [42]

    Gruber, T., Liao, K., Tsatsoulis, T., Hummel, F., Gr¨ uneis, A.: Applying the coupled-cluster ansatz to solids and surfaces in the thermodynamic limit. Phys. Rev. X 8, 021043 (2018) https://doi.org/10.1103/ PhysRevX.8.021043

    work page 2018

  42. [43]

    BeyondBOLSIG+:MonteCarlosimulation of electron and ion swarms to obtain transport and rate coefficients forplasmamodeling

    Irmler, A., Gallo, A., Gr¨ uneis, A.: Focal-point approach with pair-specific cusp correction for coupled- cluster theory. The Journal of Chemical Physics 154(23), 234103 (2021) https://doi.org/10.1063/5. 0050054

    work page doi:10.1063/5 2021

  43. [44]

    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

  44. [45]

    Computing in Science & Engineering 4(3), 56–66 (2002) https://doi.org/10.1109/5992.998641

    Bahn, S.R., Jacobsen, K.W.: An object-oriented scripting interface to a legacy electronic structure code. Computing in Science & Engineering 4(3), 56–66 (2002) https://doi.org/10.1109/5992.998641

    work page doi:10.1109/5992.998641 2002

  45. [46]

    https://github

    Sch¨ onbauer, S., Carbone, J.P., Gr¨ uneis, A.: Supporting data for this publication (2025). https://github. com/waup-1/MLphononsdata/tree/a2855d3f7111b728ef602a06f2d8ab6ebb236b27 22

    work page 2025