Pith. sign in

REVIEW 4 major objections 5 minor 80 references

Neural Canonical Transformations for Quantum Anharmonic Solids of Lithium

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Neural canonical transformations reveal that quantum anharmonicity lowers lithium's bcc-fcc transition temperature, and that the high-pressure oC88 phase is stabilized by the electronic potential energy surface rather than by nuclear…

desk verdict Solid application of a known method with a plausible but not yet airtight claim that oC88 stability comes from electronic structure, not nuclear quantum effects. read the letter →

arxiv 2412.12451 v3 pith:RGQVYTSG submitted 2024-12-17 cond-mat.mtrl-sci physics.comp-ph

Qi Zhang , Xiaoyang Wang , Rong Shi , Xinguo Ren , Han Wang , Lei Wang This is my paper · ORCID
classification cond-mat.mtrl-sciphysics.comp-ph PACS 64.70.kd63.20.-e71.15.Mb
keywords neuralcanonicaltransformationquantumanharmonicitylithiumphasediagrambcc-fcctransitionoC88structurehybridfunctionalcorrectionnormalizingflowvariationalfreeenergy
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

This paper asks how nuclear quantum motion and anharmonicity affect the phase stability of solid lithium, and whether the puzzling high-pressure oC88 phase can be explained at all. The authors use neural canonical transformations, a variational density-matrix method that combines normalizing-flow phonon wave functions with learned occupation probabilities, to compute free energies of bcc, fcc, cI16, and oC88 lithium. They find that quantum anharmonicity lowers the bcc-fcc transition temperature to 84, 142, and 196 K at 0, 1, and 2 GPa, closer to experiment than classical molecular dynamics. They also find that anharmonic effects actually widen the free-energy gap between oC88 and cI16, and that it is only after applying an HSE06 hybrid-functional energy correction that oC88 becomes stable, predicting a cI16-oC88 transition near 62 GPa at 100 K. The message is that for lithium, the electronic potential energy surface, not nuclear quantum or thermal effects, is what stabilizes oC88.

What carries the argument

The load-bearing object is the neural canonical transformation ansatz for the variational density matrix, $\rho = \sum_n p_n |\Psi_n\rangle\langle\Psi_n|$, with occupation probabilities $p_n$ modeled by a product spectrum ansatz and phonon wave functions $|\Psi_n\rangle = U_\theta |\Phi_n\rangle$ built by a normalizing flow that maps phonon coordinates $q$ to quasi-phonon coordinates $\xi = f_\theta(q)$ and enters the wave function through a Jacobian determinant. This gives orthogonal, non-Gaussian wave functions for excited vibrational states of hundreds-atom supercells, so the anharmonic potential energy surface is treated exactly rather than through a Taylor expansion or a Gaussian variational density matrix. The free energy $F = \mathbb{E}[k_B T \ln p_n + \text{local energy}]$ is minimized jointly over the probability and wave-function parameters, and constant-pressure calculations are closed by computing the stress tensor; the anharmonic phonon frequencies and zero-point energies appearing in the phase-stability comparison are read off from single-phonon excitation energies of the optimized states. The Born-Oppenheimer surface is represented by a machine-learned potential trained on PBE density functional theory, which supplies energies and forces for the large supercells.

What would settle it

Perform full structure relaxations of cI16 and oC88 with the HSE06 hybrid functional at 60, 70, and 80 GPa, and check whether oC88's enthalpy falls below cI16 around 62 GPa; if it does not, the paper's stabilization mechanism fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that quantum anharmonicity and electronic-structure accuracy play opposite roles in lithium's high-pressure phase diagram. Neural canonical transformations show that anharmonicity lowers the zero-point energy of cI16 by about 4.92 meV/atom but oC88 by only about 1.67 meV/atom, so nuclear quantum effects actually worsen oC88's stability relative to cI16. A single-point HSE06 calculation on the NCT-optimized structures reverses the picture: it lowers oC88 relative to cI16 by 6.17 meV/atom compared with PBE, enough to place the cI16-oC88 transition at roughly 62 GPa and 100 K, in line with experiments that put oC88 between about 62 and 70 GPa. In the same framework, the bcc-fcc transition temperature comes out at 84, 142, and 196 K at 0, 1, and 2 GPa, consistently below classical MD values of 144, 185, and 218 K and closer to the experimental low-pressure boundary. The paper also reports that NCT-optimized fractional coordinates of the cI16 Wyckoff position reproduce the experimental values reported for the 16c site.

Load-bearing premise

The high-pressure conclusion assumes that a single HSE06 energy correction, computed once at 70 GPa on structures relaxed with a PBE-trained machine-learned potential, gives the correct relative stability of cI16 and oC88 across the whole 60-80 GPa window without re-relaxing the structures under HSE.

Editorial extensions

If this is right

  • The bcc-fcc phase boundary in lithium is set roughly 40-60 K lower than classical molecular dynamics predicts, so quantum anharmonicity must be included when comparing computed with measured transition temperatures.
  • oC88's stability window between cI16 and oC40 is controlled by the electronic potential energy surface, meaning future crystal-structure searches for lithium should not rely on PBE alone for poor metallic phases.
  • Anharmonicity actually increases the free-energy difference between oC88 and cI16, so nuclear quantum effects are not the missing ingredient that earlier studies invoked.
  • Neural canonical transformations deliver anharmonic phonon spectra and zero-point energies for supercells of several hundred atoms, making such calculations feasible for other quantum solids like hydrogen, helium, and hydrides.
  • The cI16-oC88 transition near 62 GPa and 100 K predicted after the HSE06 correction matches the experimentally observed narrow stability range of oC88.

Reading between the lines

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

  • Because the HSE06 correction in the paper is a single-point calculation at 70 GPa on geometries relaxed with a PBE-trained potential, repeating the correction at 60 and 80 GPa on re-relaxed structures would directly test whether the predicted 62 GPa transition is an artifact of the single-point approximation.
  • The same variational machinery could be applied to other poor metals where PBE over-stabilizes metallic states; hybrid-functional corrections might expose narrow stability windows that classical and PBE-based structure searches miss.
  • The reported bcc-fcc transition temperatures carry an unquantified systematic error from the machine-learned potential (0.2-0.5 meV/atom), so a higher-accuracy potential could move the 84 K boundary by tens of kelvin; this is worth testing with path-integral calculations on a denser DFT dataset.
  • If anharmonicity consistently softens the more symmetric phase (cI16 here) while the electronic correction favors the less symmetric one (oC88), then the two effects oppose each other; in other light-element solids the same competition could produce phase sequences that depend sensitively on both functional accuracy and nuclear mass.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript presents neural canonical transformations (NCT), a variational density-matrix method that combines normalizing-flow phonon wave functions with a product ansatz for phonon occupation probabilities, and applies this method to solid lithium. Using a deep-potential surrogate for the PBE Born-Oppenheimer energy surface, the authors compute anharmonic free energies for bcc/fcc at 0-2 GPa and for cI16/oC88 at high pressure. They report that quantum anharmonicity lowers the bcc-fcc transition temperature relative to classical molecular dynamics (84, 142, and 196 K versus 144, 185, and 218 K at 0, 1, and 2 GPa), that the predicted cI16 fractional coordinates agree with experiment, and that a single-point HSE06 electronic-structure correction makes oC88 more stable than cI16 below roughly 62 GPa at 100 K, leading them to conclude that the experimentally observed oC88 phase is stabilized by electronic-structure effects rather than by nuclear quantum or thermal effects.

Significance. If fully supported, these results would be significant for both methodology and lithium phase diagram physics. The paper introduces an open-source, variational framework that goes beyond Gaussian phonon ansätze, and the low-pressure conclusion that quantum anharmonicity lowers the bcc-fcc transition temperature is a concrete, falsifiable prediction. The high-pressure claim—that the poor-metal oC88 structure is stabilized by the electronic-structure functional—offers a new resolution to a known discrepancy. The strengths of the manuscript include the principled variational formalism, the reproducible open-source code, the use of two independent electronic-structure codes for the HSE correction, and the explicit reporting of statistical errors. However, the high-pressure conclusion rests on untested assumptions about the pressure transferability and structural transferability of a single-point HSE correction, and the bcc-fcc transition temperatures are not error-propagated with respect to the quoted deep-potential systematic errors.

major comments (4)
  1. [Fig. 3(c) and SM §SII.D, Table S2] The central high-pressure conclusion is based on a single-point HSE06 correction evaluated at 70 GPa and applied in Fig. 3(c) as a pressure-independent shift over the 60-90 GPa range. The predicted cI16-oC88 transition occurs near 62 GPa, 8 GPa away from the pressure at which the correction is computed, and no HSE calculation at other pressures or HSE-level volume relaxation is reported. Because the correction (6.17-7.67 meV/atom) is larger than the PBE-level free-energy difference it is correcting (approximately 3.96 meV/atom at 70 GPa), a modest pressure dependence of the HSE shift could move or eliminate the predicted transition. The authors should assess pressure transferability, for example by computing HSE at 62 and 80 GPa or by re-relaxing both structures under HSE.
  2. [SM §SII.D, Table S2; main-text section on high-pressure structural stability] The HSE stabilization is computed as a potential-energy difference at fixed PBE/NCT-optimized structures, while the NCT free-energy contributions (zero-point energy, anharmonic entropy, thermal terms) are taken unchanged from the PBE surface. This assumes that HSE and PBE have essentially the same equilibrium volumes, internal coordinates, and phonon curvatures for both the 432-atom cI16 and 352-atom oC88 cells. The claim that oC88 is stabilized by the potential energy surface 'rather than thermal or quantum nuclear effects' is therefore only as strong as this assumption. A concrete test would be to re-relax both structures with HSE, or at least to recompute the electronic energy along NCT-sampled configurations, and to verify that the relative free energy remains negative when the HSE surface is used.
  3. [SM §SII.B and Fig. 2] The bcc-fcc transition temperatures are extracted from Gibbs free-energy differences of order 0.1-0.6 meV/atom, whereas the quoted systematic error of the low-pressure deep-potential model is 0.2-0.5 meV/atom. Since the manuscript itself notes that an error of 1 meV can shift the transition temperature by more than 100 K, the reported transition temperatures of 84, 142, and 196 K carry unquantified systematic uncertainty. The comparison with classical MD on the same energy surface supports the direction of the quantum-anharmonicity correction, but the precise transition temperatures, and the claimed agreement with experiment (77-88 K at 0 GPa), need either an error analysis with multiple DP models or direct DFT validation at selected thermodynamic points.
  4. [Main text 'High-pressure structural stability' and SM §SII.D] The claim that the poor-metal oC88 phase is stabilized by HSE is not benchmarked against a higher-level electronic-structure method. For a system whose relative stability changes by about 6-8 meV/atom when switching from PBE to HSE, the possibility of overcorrection is a genuine correctness risk. Before asserting the mechanism as established, the authors should validate the HSE energy ordering with an independent higher-level method, such as quantum Monte Carlo or RPA, or at least with a different hybrid-functional parameterization.
minor comments (5)
  1. [Fig. 3(b) and main text] The text states that the anharmonic zero-point energy of oC88 decreases by 1.67 meV/atom, but the harmonic and anharmonic values in the figure (89.43 and 88.10 meV/atom) give a decrease of 1.33 meV/atom; this inconsistency should be corrected.
  2. [Fig. 3(a)] The caption and legend contain typographical errors: 'Neural cannonical transformation' should read 'Neural canonical transformation', and the space-group symbol 'c2mb' in the oC88 panel should be 'C2mb'.
  3. [SM Fig. S4(b)] The caption reads 'Ironic entropy' and should read 'Ionic entropy'.
  4. [Figs. 1 and 2] Several axis labels and legends contain garbled or missing symbols, such as 'opt only' and 'T emperature'; these should be cleaned before publication.
  5. [SM §SII.B] The energy-level cutoff K=20 for the product spectrum ansatz is stated, but no convergence test with respect to K is reported; a brief convergence statement would strengthen the variational free-energy values.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: free energies are variationally computed, transition temperatures come from crossings of independent free-energy curves, and the HSE correction is an external electronic-structure input rather than a fitted target.

full rationale

The derivation chain is self-contained at the level tested. The NCT free energy is obtained by minimizing a variational density-matrix functional (Eqs. 2-4) with no parameter fitted to the experimental transition temperatures; the bcc-fcc transition temperatures are read off crossings of independently computed Gibbs free-energy curves for the two phases on the same deep-potential BOES, and the classical MD comparisons use the same energy surface. The high-pressure oC88 stabilization claim likewise does not reduce to an input: the PBE-based NCT free-energy difference is an independently minimized variational quantity, and the HSE correction is a separate single-point electronic-structure calculation (SM Table S2) that changes the potential-energy difference by an ab initio computed value (about 6.17 meV/atom in ABACUS and 7.67 meV/atom in FHI-aims) rather than a fitted shift. The paper explicitly reports that the DP model's accuracy is cross-checked against direct DFT calculations, so the machine-learned potential is used as a surrogate for PBE DFT and is independently validated. Self-citations to the authors' prior NCT work ([37-39]) and to the deep-potential model ([29,46,47]) occur, but they are not load-bearing in a circular sense: the NCT code is open-sourced and benchmarked against an exact one-dimensional anharmonic solution, and the DP model is checked against explicit DFT results. The most consequential approximation, applying the 70 GPa single-point HSE correction as a constant shift across 60-90 GPa without HSE re-relaxation, is a transferability/accuracy assumption about electronic structure, not a case where the prediction is built into the input. Such an approximation is a correctness risk rather than circularity, and per the review rules it is noted here rather than scored as circular. No step was found in which an equation reduces to its own input, a fitted parameter is renamed as a prediction, or a self-citation is invoked to forbid alternatives.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claims rest on a machine-learned surrogate for PBE DFT and on HSE single-point corrections. No new physical entities are introduced, but the accuracy of the potential and the functional are external inputs that are not independently certified within the paper.

free parameters (2)
  • Deep potential (DP) model weights = Trained parameters from Li-DP-Hyb2/Hyb3 (Ref. 29)
    The Born-Oppenheimer energy surface is replaced by a machine-learned potential fitted to PBE DFT data. The reported systematic errors are 0.2-0.5 meV/atom for bcc/fcc and <1 meV/atom for cI16/oC88, directly affecting the small free energy differences that drive the phase stability conclusions.
  • Energy level cutoff K = 20
    Truncation of the number of phonon occupation levels per mode in the product spectrum ansatz. This is a hand-chosen numerical cutoff; the paper does not report a systematic convergence study with respect to K.
assumptions (5)
  • domain assumption Born-Oppenheimer approximation
    Electronic and nuclear motion are decoupled; standard for lattice dynamics but can fail in metals with strong electron-phonon coupling.
  • domain assumption Variational ansatz expressiveness
    The combination of a normalizing flow for wave functions and a product spectrum for occupations can closely approximate the thermal density matrix of the anharmonic solid. Supported only by a 1D benchmark in SM, not by direct validation on lithium.
  • domain assumption HSE06 hybrid functional provides accurate relative energies
    The oC88 stabilization conclusion relies on HSE single-point corrections being more accurate than PBE for poor metals. No higher-level method (e.g., QMC or RPA) is used to validate.
  • domain assumption Machine-learned potential fidelity
    The DP model faithfully reproduces PBE DFT forces and energies; the paper quotes error ranges but does not show per-structure validation plots against direct DFT for each phase.
  • domain assumption Finite-size supercells represent thermodynamic limit
    Supercells of 250-432 atoms are used; finite-size effects are stated to be consistent with MD but not systematically extrapolated.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Neural Canonical Transformations for Quantum Anharmonic Solids of Lithium." pith.science (2026). https://pith.science/paper/RGQVYTSG

@misc{pith2026241212451,
  author       = {Pith},
  title        = {Pith review of: Neural Canonical Transformations for Quantum Anharmonic Solids of Lithium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RGQVYTSG}},
  note         = {Machine review of arXiv:2412.12451}
}
read the original abstract

Lithium is a typical quantum solid, characterized by cubic structures at ambient pressure. As the pressure increases, it forms more complex structures and undergoes a metal-to-semiconductor transformation, complicating theoretical and experimental analyses. We employ the neural canonical transformation approach, an \textit{ab initio} variational method based on probabilistic generative models, to investigate the quantum anharmonic effects in lithium solids at finite temperatures. This approach combines a normalizing flow for phonon excited-state wave functions with a probabilistic model for the occupation of energy levels, optimized jointly to minimize the free energy. Our results indicate that quantum anharmonicity lowers the \textit{bcc}-\textit{fcc} transition temperature compared to classical molecular dynamics predictions. At high pressures, the predicted fractional coordinates of lithium atoms in the \textit{cI16} structure show good quantitative agreement with experimental observations. Finally, contrary to previous beliefs, we find that the poor metallic \textit{oC88} structure is stabilized by the potential energy surface obtained via high-accuracy electronic structure calculations, rather than thermal or quantum nuclear effects.

Figures

Figures reproduced from arXiv: 2412.12451 by the authors.

Figure 1
Figure 1. FIG. 1: Numerical results for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (a) Gibbs free energy (Eq [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: (a) Fractional coordinates of the Wycko [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

80 extracted references · 67 canonical work pages

  1. [1]

    Cazorla and J

    C. Cazorla and J. Boronat, Rev. Mod. Phys. 89, 035003 (2017)

    work page 2017

  2. [2]

    Monserrat, N

    B. Monserrat, N. D. Drummond, and R. J. Needs, Phys. Rev. B 87, 144302 (2013)

    work page 2013

  3. [3]

    Kapil, E

    V . Kapil, E. Engel, M. Rossi, and M. Ceriotti, Journal of Chemical Theory and Computation 15, 5845 (2019), pMID: 31532997

    work page 2019

  4. [4]

    J. A. Flores-Livas, L. Boeri, A. Sanna, G. Profeta, R. Arita, and M. Eremets, Physics Reports 856, 1 (2020), a perspective on conventional high-temperature superconductors at high pres- sure: Methods and materials

    work page 2020

  5. [5]

    Monacelli, R

    L. Monacelli, R. Bianco, M. Cherubini, M. Calandra, I. Errea, and F. Mauri, Journal of Physics: Condensed Matter33, 363001 (2021)

    work page 2021

  6. [6]

    Azadi, B

    S. Azadi, B. Monserrat, W. M. C. Foulkes, and R. J. Needs, Phys. Rev. Lett. 112, 165501 (2014)

    work page 2014

  7. [7]

    Borinaga, I

    M. Borinaga, I. Errea, M. Calandra, F. Mauri, and A. Bergara, Phys. Rev. B 93, 174308 (2016)

    work page 2016

  8. [8]

    Monacelli, I

    L. Monacelli, I. Errea, M. Calandra, and F. Mauri, Nature Physics 17, 63 (2021)

    work page 2021

Show all 80 references
  1. [9]

    Monacelli, M

    L. Monacelli, M. Casula, K. Nakano, S. Sorella, and F. Mauri, Nature Physics 19, 845 (2023)

    2023

  2. [10]

    Pederiva and G

    F. Pederiva and G. V . Chester, Journal of low temperature physics 113, 741 (1998)

    1998

  3. [11]

    Vitali, M

    E. Vitali, M. Rossi, L. Reatto, and D. E. Galli, Phys. Rev. B 82, 174510 (2010)

    2010

  4. [12]

    Monserrat, N

    B. Monserrat, N. D. Drummond, C. J. Pickard, and R. J. Needs, Phys. Rev. Lett. 112, 055504 (2014)

    2014

  5. [13]

    Errea, M

    I. Errea, M. Calandra, and F. Mauri, Phys. Rev. Lett. 111, 177002 (2013)

    2013

  6. [14]

    Errea, M

    I. Errea, M. Calandra, and F. Mauri, Phys. Rev. B 89, 064302 (2014)

    2014

  7. [15]

    Errea, M

    I. Errea, M. Calandra, C. J. Pickard, J. R. Nelson, R. J. Needs, Y . Li, H. Liu, Y . Zhang, Y . Ma, and F. Mauri, Nature532, 81 (2016)

    2016

  8. [16]

    Errea, F

    I. Errea, F. Belli, L. Monacelli, A. Sanna, T. Koretsune, T. Tadano, R. Bianco, M. Calandra, R. Arita, F. Mauri, et al., Nature 578, 66 (2020)

    2020

  9. [17]

    C. L. Guillaume, E. Gregoryanz, O. Degtyareva, M. I. McMa- hon, M. Hanfland, S. Evans, M. Guthrie, S. V . Sinogeikin, and H. Mao, Nature Physics 7, 211 (2011)

    2011

  10. [18]

    Hutcheon and R

    M. Hutcheon and R. Needs, Phys. Rev. B 99, 014111 (2019)

    2019

  11. [19]

    Jerabek, A

    P. Jerabek, A. Burrows, and P. Schwerdtfeger, Chemical Com- munications 58, 13369 (2022)

    2022

  12. [20]

    D. M. Ri ffe and J. D. Christensen, (2024), arXiv:2406.09527 [cond-mat.mtrl-sci]

    2024 arXiv

  13. [21]

    M. K. Phuthi, Y . Huang, M. Widom, and V . Viswanathan, (2024), arXiv:2406.15491 [cond-mat.mtrl-sci] . 6

    2024 arXiv

  14. [22]

    G. J. Ackland, M. Dunuwille, M. Martinez-Canales, I. Loa, R. Zhang, S. Sinogeikin, W. Cai, and S. Deemyad, Science 356, 1254 (2017)

    2017

  15. [23]

    A. M. J. Schae ffer, W. B. Talmadge, S. R. Temple, and S. Dee- myad, Phys. Rev. Lett. 109, 185702 (2012)

    2012

  16. [24]

    Frost, J

    M. Frost, J. B. Kim, E. E. McBride, J. R. Peterson, J. S. Smith, P. Sun, and S. H. Glenzer, Phys. Rev. Lett.123, 065701 (2019)

    2019

  17. [25]

    Matsuoka and K

    T. Matsuoka and K. Shimizu, Nature 458, 186 (2009)

    2009

  18. [26]

    J. Lv, Y . Wang, L. Zhu, and Y . Ma, Phys. Rev. Lett.106, 015503 (2011)

    2011

  19. [27]

    Marqu ´es, M

    M. Marqu ´es, M. I. McMahon, E. Gregoryanz, M. Hanfland, C. L. Guillaume, C. J. Pickard, G. J. Ackland, and R. J. Nelmes, Phys. Rev. Lett. 106, 095502 (2011)

    2011

  20. [28]

    F. A. Gorelli, S. F. Elatresh, C. L. Guillaume, M. Marqu´es, G. J. Ackland, M. Santoro, S. A. Bonev, and E. Gregoryanz, Phys. Rev. Lett. 108, 055501 (2012)

    2012

  21. [29]

    X. Wang, Z. Wang, P. Gao, C. Zhang, J. Lv, H. Wang, H. Liu, Y . Wang, and Y . Ma, Nature Communications14, 2924 (2023)

    2023

  22. [30]

    G. J. Martyna, A. Hughes, and M. E. Tuckerman, The Journal of Chemical Physics 110, 3275 (1999)

    1999

  23. [31]

    J. A. Barker, The Journal of Chemical Physics 70, 2914 (1979)

    1979

  24. [32]

    J. M. Bowman, The Journal of Chemical Physics 68, 608 (1978)

    1978

  25. [33]

    J. M. Bowman, Accounts of Chemical Research19, 202 (1986)

    1986

  26. [34]

    Gerber and M

    R. Gerber and M. Ratner, Chemical Physics Letters 68, 195 (1979)

    1979

  27. [35]

    Siciliano, L

    A. Siciliano, L. Monacelli, and F. Mauri, Phys. Rev. B 110, 134111 (2024)

    2024

  28. [36]

    Monacelli, A

    L. Monacelli, A. Siciliano, and N. Marzari, (2024), arXiv:2410.08882 [cond-mat.mtrl-sci]

    2024

  29. [37]

    H. Xie, L. Zhang, and L. Wang, Journal of Machine Learning 1, 38 (2022)

    2022

  30. [38]

    H. Xie, L. Zhang, and L. Wang, SciPost Physics 14 (2023), 10.21468/scipostphys.14.6.154

    2023 doi

  31. [39]

    Zhang, R.-S

    Q. Zhang, R.-S. Wang, and L. Wang, The Journal of Chemical Physics 161, 024103 (2024)

    2024

  32. [40]

    L. Dinh, D. Krueger, and Y . Bengio, arXiv preprint arXiv:1410.8516 (2014)

    2014 arXiv

  33. [41]

    L. Dinh, J. Sohl-Dickstein, and S. Bengio, arXiv preprint arXiv:1605.08803 (2016)

    2016 arXiv

  34. [43]

    Papamakarios, arXiv preprint arXiv:1910.13233 (2019)

    G. Papamakarios, arXiv preprint arXiv:1910.13233 (2019)

    2019 arXiv

  35. [44]

    Papamakarios, E

    G. Papamakarios, E. Nalisnick, D. J. Rezende, S. Mohamed, and B. Lakshminarayanan, Journal of Machine Learning Re- search 22, 1 (2021)

    2021

  36. [45]

    Saleh, ´A

    Y . Saleh, ´A. F. Corral, A. Iske, J. K ¨upper, and A. Yachmenev, arXiv preprint arXiv:2308.16468 (2023)

    2023 arXiv

  37. [46]

    H. Wang, L. Zhang, J. Han, and W. E, Computer Physics Com- munications 228, 178 (2018)

    2018

  38. [47]

    J. Zeng, D. Zhang, D. Lu, P. Mo, Z. Li, Y . Chen, M. Rynik, L. Huang, Z. Li, S. Shi, Y . Wang, H. Ye, P. Tuo, J. Yang, Y . Ding, Y . Li, D. Tisi, Q. Zeng, H. Bao, Y . Xia, J. Huang, K. Muraoka, Y . Wang, J. Chang, F. Yuan, S. L. Bore, C. Cai, Y . Lin, B. Wang, J. Xu, J.-X. Zhu...

    2023

  39. [48]

    The code of neural canonical transformations for lithium solids, implemented with the Python package JAX [57], is available at https://github.com/zhangqi94/lithium

  40. [49]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996)

    1996

  41. [50]

    R. M. Martin, Electronic structure: basic theory and practical methods (Cambridge university press, 2020)

    2020

  42. [51]

    Baroni, S

    S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001)

    2001

  43. [52]

    Souvatzis, O

    P. Souvatzis, O. Eriksson, M. I. Katsnelson, and S. P. Rudin, Phys. Rev. Lett. 100, 095901 (2008)

    2008

  44. [53]

    Souvatzis, O

    P. Souvatzis, O. Eriksson, M. Katsnelson, and S. Rudin, Com- putational Materials Science 44, 888 (2009)

    2009

  45. [54]

    See Supplemental Material for details of the neural canonical transformation approach and additional results, which includes Refs. [67–81]

  46. [55]

    Becca and S

    F. Becca and S. Sorella, Quantum Monte Carlo Approaches for Correlated Systems (Cambridge University Press, 2017)

    2017

  47. [56]

    D. P. Kingma and J. Ba, arXiv preprint arXiv:1412.6980 (2014)

    2014 arXiv

  48. [57]

    JAX: composable trans- formations of Python+NumPy programs,

    J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, “JAX: composable trans- formations of Python+NumPy programs,” (2018)

    2018

  49. [58]

    Martyn and B

    J. Martyn and B. Swingle, Phys. Rev. A 100, 032107 (2019)

    2019

  50. [59]

    D. Wu, L. Wang, and P. Zhang, Phys. Rev. Lett. 122, 080602 (2019)

    2019

  51. [60]

    J.-G. Liu, L. Mao, P. Zhang, and L. Wang, Machine Learning: Science and Technology 2, 025011 (2021)

    2021

  52. [61]

    O. H. Nielsen and R. M. Martin, Phys. Rev. B 32, 3780 (1985)

    1985

  53. [62]

    S ¨oderlind, B

    P. S ¨oderlind, B. Grabowski, L. Yang, A. Landa, T. Bj ¨orkman, P. Souvatzis, and O. Eriksson, Phys. Rev. B85, 060301 (2012)

    2012

  54. [63]

    Hanfland, K

    M. Hanfland, K. Syassen, N. Christensen, and D. Novikov, Na- ture 408, 174 (2000)

    2000

  55. [64]

    Y . Ma, M. Eremets, A. R. Oganov, Y . Xie, I. Trojan, S. Medvedev, A. O. Lyakhov, M. Valle, and V . Prakapenka, Nature 458, 182 (2009)

    2009

  56. [65]

    Y . Ma, A. R. Oganov, and C. W. Glass, Phys. Rev. B76, 064101 (2007)

    2007

  57. [66]

    H. E. Sauceda, V . Vassilev-Galindo, S. Chmiela, K.-R. M¨uller, and A. Tkatchenko, Nature Communications 12, 442 (2021)

    2021

  58. [67]

    C. S. Barrett, Phys. Rev. 72, 245 (1947)

    1947

  59. [68]

    Rao and K

    C. Rao and K. Rao, Progress in Solid State Chemistry 4, 131 (1967)

    1967

  60. [69]

    Eger and E

    M. Eger and E. Gross, Annals of Physics 24, 63 (1963)

    1963

  61. [70]

    Eger and E

    F. Eger and E. Gross, Il Nuovo Cimento (1955-1965) 34, 1225 (1964)

    1964

  62. [71]

    B. S. DeWitt, Phys. Rev. 85, 653 (1952)

    1952

  63. [72]

    Hutchinson, Communications in Statistics - Simulation and Computation 18, 1059 (1989)

    M. Hutchinson, Communications in Statistics - Simulation and Computation 18, 1059 (1989)

    1989

  64. [73]

    S. S. Schoenholz and E. D. Cubuk, in Advances in Neural In- formation Processing Systems, V ol. 33 (Curran Associates, Inc., 2020)

    2020

  65. [74]

    Kresse and J

    G. Kresse and J. Furthm ¨uller, Phys. Rev. B 54, 11169 (1996)

    1996

  66. [75]

    More information can be found at https: //abacus.ustc.edu.cn/main.htm

    (), Atomic-orbital Based Ab-initio Computation at USTC (ABACUS) is an open-source package for density functional theory calculations. More information can be found at https: //abacus.ustc.edu.cn/main.htm

  67. [76]

    D. R. Hamann, Phys. Rev. B 88, 085117 (2013)

    2013

  68. [77]

    (), Optimized Norm-Conservinng Vanderbilt PSeudopotential (ONCVPSP) code is available at http: //www.mat-simresearch. com

  69. [78]

    More information can be found at https: //fhi-aims.org

    Fritz Haber Institute ab initio materials simulations (FHI-aims) is a quantum mechanics software package based on numeric atom-centered orbitals. More information can be found at https: //fhi-aims.org

  70. [79]

    V . Blum, R. Gehrke, F. Hanke, P. Havu, V . Havu, X. Ren, K. Reuter, and M. Sche ffler, Computer Physics Communica- tions 180, 2175 (2009)

    2009

  71. [80]

    J. Heyd, G. E. Scuseria, and M. Ernzerhof, The Journal of 7 Chemical Physics 118, 8207 (2003)

    2003

  72. [81]

    ∇µ ln pn · kBT ln pn + E q∼|Ψn(q)|2 h Evib n (q) i!# , ∇θF = 2 E n∼pn

    K. He, X. Zhang, S. Ren, and J. Sun, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2016). 8 Supplemental Material: Neural Canonical Transformations for Quantum Anharmonic Solids of Lithium Contents References 5 SI. Neural Canonical Tr...

    2016

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.