Pith. sign in

REVIEW 4 major objections 5 minor 57 references

Deep variational free energy prediction of dense hydrogen solid at 1200K

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read By minimizing a variational free energy built from a flow-model proton distribution and a neural-network electronic wavefunction, this paper finds that dense hydrogen at 1200 K orders into a molecular crystal near 180 GPa, with…

desk verdict A credible variational free-energy calculation finds a molecular solid at 1200 K, but the first-order transition claim outruns the evidence because no liquid branch or finite-size check is reported. read the letter →

arxiv 2501.09590 v2 pith:GILFYDG6 submitted 2025-01-16 physics.comp-ph

Xinyang Dong , Hao Xie , Yixiao Chen , Wenshuo Liang , Linfeng Zhang , Lei Wang , Han Wang This is my paper · ORCID
classification physics.comp-ph
keywords densehydrogenvariationalfreeenergynormalizingflowneural-networkwavefunctionmolecularcrystalatomic-liquidtomolecular-solidtransitionhigh-pressureequationofstate
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 tries to settle what dense hydrogen actually does at 1200 K and pressures around 180 GPa, a regime where DFT-based simulations predict a liquid-liquid transition but quantum Monte Carlo-based machine-learned potentials have recently suggested solid behavior. The authors' approach is to minimize the free energy directly: a normalizing flow models the proton Boltzmann distribution, a neural network models the electronic wavefunction, and both are trained jointly to lower the variational free energy. The optimized solution shows a transition from atomic liquid to molecular crystal at about 180 GPa, signaled by discontinuities in pressure and thermal entropy and by ordering of hydrogen-molecule centers of mass. If this is right, the melting line of hydrogen is higher than previously thought, and many-body electronic accuracy qualitatively changes the predicted phase.

What carries the argument

The load-bearing object is the variational free-energy functional $$F = \mathbb{E}_{X\sim p(X)}\left[k_B T \ln p(X) + E(X)\right],$$ where $p(X)$ is a proton Boltzmann distribution parameterized by a normalizing flow (an invertible neural transformation that turns a simple base distribution into a complex one, with a machine-learned force field supplying the base distribution and an equivariant backflow transformation enforcing translational, periodic, and permutational symmetries), and $E(X)$ is the electronic ground-state energy for fixed protons, obtained from an ensemble variational Monte Carlo wavefunction with twisted boundary conditions that reduce finite-size effects. Jointly minimizing $F$ over the flow parameters and the wavefunction parameters yields both an approximate free energy and the optimized proton ensemble, so a phase change appears as a qualitative change in the distribution the flow converges to.

What would settle it

Run the same variational free energy minimization in larger simulation cells (54, 64, or 128 atoms) and also from a liquid-constrained flow branch; if the crystalline order disappears or the liquid branch has lower free energy at $r_s \approx 1.42$, the claimed transition is a finite-size artifact or a metastable local minimum.

Watch

Extended reading notes

Core claim

The central claim is that at $T = 1200$ K the system undergoes a discontinuous transition from an atomic liquid into a molecular solid as the density parameter crosses $r_s \approx 1.42$, corresponding to a pressure near 180 GPa. The variational free energy calculation finds this by direct minimization rather than by sampling a predetermined potential energy surface: the proton flow model starts from a structureless liquid base and converges to a distribution whose radial distribution function shows a molecular peak near 1.4 Bohr and additional structure at 2 and 3 Bohr; snapshots show the centers of mass of hydrogen molecules arranged in an ordered lattice; and simulated X-ray diffraction spectra display sharp peaks. The transition is marked by a slope change in pressure and a sharp drop in thermal entropy, and the electronic wavefunction achieves roughly 1 milli-Hartree per atom accuracy against quantum Monte Carlo reference data.

Load-bearing premise

The calculation assumes that variational optimization reaches the true global minimum of the free energy and that the 32-atom cubic cell does not artificially stabilize the molecular crystal.

Editorial extensions

If this is right

  • At 1200 K and about 180 GPa, hydrogen's equilibrium state is a molecular crystal whose molecular centers of mass are ordered, not a molecular liquid.
  • The transition from atomic liquid to molecular solid is discontinuous in this calculation, with jumps in pressure and thermal entropy at $r_s \approx 1.42$.
  • The same framework gives direct estimates of free energy and entropy, so equation-of-state data near the transition can be compared quantitatively with other methods and with experiments.
  • Accurate many-body electronic energies beyond DFT are sufficient to change the predicted phase qualitatively, since the liquid base distribution is transformed into a solid only after variational refinement.
  • The detailed crystal structure is not identified from the cubic 32-atom cell; only center-of-mass ordering, not lattice type, is claimed.

Reading between the lines

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

  • If the transition is real, the regime around 180 GPa and 1200 K conventionally attributed to a liquid-liquid transition may actually be a liquid-solid boundary, and experimental Raman anomalies in that region could be melting signatures rather than liquid-liquid signals.
  • The variational method's mode-seeking tendency means the calculation could have converged to a metastable crystalline basin; comparing free energies of the crystal and a deliberately liquid-constrained solution would settle this.
  • Systematic scans of temperature at fixed pressure would turn the single observed transition into a predicted melting line that could be checked by shock or static compression experiments.
  • Because the flow's base distribution is a DFT-level force field and the optimized solution is a crystal, the result implies that many-body electronic correlation, not just the proton sampling method, is what stabilizes the solid.
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 paper reports a deep variational free energy calculation for a 32-atom hydrogen system at 1200 K. The proton Boltzmann distribution is represented by a normalizing flow with a DeePMD base distribution, and the electronic ground-state energy is computed with an ensemble variational Monte Carlo neural-network wavefunction; the two networks are jointly optimized to minimize the free energy in Eq. (2). The wavefunction is benchmarked against existing QMC data and systematically improves the reference energies. At rs ≈ 1.42 (about 180 GPa) the calculation finds a transition to an ordered molecular solid, with reported discontinuities in pressure and entropy. The authors discuss the implications for the high-pressure hydrogen phase diagram and acknowledge limitations due to the 32-atom cubic cell and the mode-seeking character of the variational optimization.

Significance. If the central claim is robust, the paper is significant: it would place a QMC-level electronic structure calculation inside a variational free energy optimization and predict a high-temperature molecular solid in a regime where DFT-based simulations typically predict a liquid-liquid transition. The methodological strengths are concrete: the wavefunction is benchmarked against an external QMC reference (Table II and Fig. 5), the flow-based proton distribution is checked for self-consistency (Appendix D), and the numerical data and inputs are made available in a GitHub repository. The main significance, however, is conditional on establishing that the optimized solid branch is the global free-energy minimum rather than a finite-size-stabilized metastable basin; the current manuscript does not yet provide that evidence.

major comments (4)
  1. [§III.B, Fig. 2; Secs. I and IV; Appendix D] The central claim of a first-order transition to a molecular solid is not supported by a free-energy comparison between distinct branches. The paper itself states in Sec. I that the flow has "mode seeking behavior... can lead to convergence at local minima in the free energy landscape," and Sec. IV concedes that the 32-atom cubic cell "may bias both the existence and structure" of the solid. Appendix D checks only that the trained flow distribution matches the Boltzmann weight of the same E(X) used in the variational objective; a metastable basin would satisfy that self-consistency check equally. No run initialized from a liquid branch at the same thermodynamic state is reported, and no comparison of variational free energies of a liquid solution and the molecular-solid solution is made. Please add such branch comparisons, or otherwise demonstrate that the optimized solid solution is the global minimum, before claiming a phase transition.
  2. [§III.B, Fig. 2] The reported discontinuities in pressure and entropy are not quantified. Fig. 2 shows no error bars for either quantity, and the discontinuity is inferred from adjacent single-branch rs points without hysteresis loops, coexistence calculations, or a Maxwell construction. A first-order transition at fixed temperature would normally show a pressure plateau in the coexistence region rather than a change in slope at a single density. Provide error estimates, finer rs sampling around rs = 1.42, and either hysteresis or free-energy crossings to support the first-order assignment; also specify how the pressure is computed from the variational solution.
  3. [§III.B, Fig. 4; Sec. IV] The molecular-crystal identification rests on a single 32-atom cubic cell. The Bragg peaks in the simulated XRD and the ordering after a quench are consistent with a crystalline state, but a finite cell of this size can artificially stabilize commensurate crystalline order, as the authors themselves acknowledge in Sec. IV. No system-size dependence (for example, 54, 64, or 128 atoms) or cell-shape relaxation is reported. At minimum, the authors should show that the crystalline basin survives larger cells and independent cell shapes, or temper the physical claim accordingly.
  4. [§II.A, Eq. (3); §III.B, Fig. 2(b)] The entropy calculation is insufficiently documented. Eq. (3) shows that the partition function Zbase enters the normalization, but the precise expression used to compute the physical entropy, the annealed importance sampling parameters, and the statistical uncertainty of the AIS estimate are not given. Since the entropy drop at rs = 1.42 is a central piece of evidence, this should be reported. It would also help to clarify why the entropy is computed from a single-branch run rather than as a difference between coexisting phases.
minor comments (5)
  1. [Fig. 4 caption] The caption labels panel (a) as the RDF and panel (b) as the snapshot, while the text refers to panel (a) as the snapshot and panel (b) as the RDF; please reconcile the labeling.
  2. [Appendix A, Eq. (A10)] In the intermediate feature g^{(l)}_I, the term (1/N)∑_I h^{(l)}_I appears twice; if this duplication is intentional, a comment would help, and if not, one occurrence should be removed.
  3. [§II.B] The twist parameter w in Eq. (5) appears as a network input, but the text does not state whether twist averaging is performed; if only one twist is used, this is a finite-size bias worth stating explicitly.
  4. [Abstract] The abstract says the transition is "marked by discontinuities in both the pressure and thermal entropy," but the text reports a "change in slope" in the pressure curve; please align the wording with what is actually computed.
  5. [Throughout] There are several typographical and formatting issues (for example, "di fficulty", "Z¨ urich", "flow ans¨atz"); a careful proofread is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the solid-phase claim emerges from variational free-energy optimization and is not an input of the model.

full rationale

The central derivation is a variational minimization of Eq. (2), with the proton distribution p(X) parameterized by a normalizing flow (Eq. 3) and the internal energy E(X) obtained from a neural-network wavefunction (Eq. 4). Neither p(X) nor E(X) is constructed from the target phase: the DeePMD base distribution at the transition density is liquid-like (Fig. 3), and the wavefunction is benchmarked against external QMC data at 64-atom reference states (Table II). The transition at rs=1.42 emerges from the optimized equation of state rather than from any parameter fitted to that transition, and the pressure and entropy curves are computed from the variational solution, not imposed. The paper's self-citation of Ref. 29 supplies the computational framework but does not justify the physical result; the phase claim rests on the independent variational calculation and on external QMC benchmarks. The acknowledged mode-seeking behavior and the possibility of local free-energy minima (Section I) are soundness and global-optimization limitations, not circular reductions, because a local minimum is still a nontrivial output of the model. Similarly, the finite-size caveat in Section IV concerns whether the 32-atom cubic cell artificially stabilizes the solid, which is a correctness risk rather than a definitional equivalence. Appendix D checks only the self-consistency of the optimized flow with the same energy model used for training, i.e. the stationary-point condition of Eq. (2); the paper does not use this check as the primary evidence for the phase transition, which comes from the RDF and XRD analysis. No step in the derivation chain reduces, by construction or by fitted parameter, to the claimed prediction. The absence of a liquid-branch free-energy comparison is a robustness gap but does not make the derivation circular.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced. The free energy is minimized variationally, so network parameters are optimized rather than fitted to the transition pressure. The key unsupported inputs are the classical-proton and Born-Oppenheimer assumptions, the expressive sufficiency of the ansatze, and the representativeness of the 32-atom cell.

free parameters (1)
  • Neural network architecture hyperparameters (message-passing layers L, feature dimensions d_p1 and d_p2, determinant… = not reported
    These control the expressiveness of the flow and wavefunction ansatze and hence the tightness of the variational free energy bound. The paper provides algorithms but not all numerical values, so the quality of the variational solution depends on choices that are not fully disclosed.
assumptions (4)
  • domain assumption Protons are treated as classical point particles
    Invoked in Section II.A. Nuclear quantum effects at 1200 K are neglected; the conclusion notes they could be added in future work.
  • domain assumption Electrons are assumed to stay in the instantaneous ground state for each proton configuration (Born-Oppenheimer approximation)
    Standard in QMC simulations of hydrogen; stated in Section II.A.
  • ad hoc to paper The neural network ansatze (flow model and wavefunction) are sufficiently expressive to represent both liquid and solid states
    The variational free energy is only an upper bound; if the ansatze lack flexibility, the optimization could miss the true state. The paper gives convergence checks but no expressivity guarantees.
  • domain assumption The 32-atom periodic cell with twist-averaged boundary conditions represents the thermodynamic limit for the solid-liquid phase behavior
    Finite-size effects are acknowledged in Section IV, but no scaling or alternative cell shape is tested.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Deep variational free energy prediction of dense hydrogen solid at 1200K." pith.science (2026). https://pith.science/paper/GILFYDG6

@misc{pith2026250109590,
  author       = {Pith},
  title        = {Pith review of: Deep variational free energy prediction of dense hydrogen solid at 1200K},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GILFYDG6}},
  note         = {Machine review of arXiv:2501.09590}
}
read the original abstract

We perform deep variational free energy calculations to investigate the dense hydrogen system at 1200 K and high pressures. In this computational framework, neural networks are used to model the free energy through the proton Boltzmann distribution and the electron wavefunction. By directly minimizing the free energy, our results reveal the emergence of a crystalline order associated with the center of mass of hydrogen molecules at approximately 180 GPa. This transition from atomic liquid to a molecular solid is marked by discontinuities in both the pressure and thermal entropy. Additionally, we discuss the broader implications and limitations of these findings in the context of recent studies of dense hydrogen under similar conditions.

Figures

Figures reproduced from arXiv: 2501.09590 by the authors.

Figure 1
Figure 1. FIG. 1. The flowchart of pretraining and finetuning of the hydrogen [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The (a) pressure and (b) entropy per particle versus density [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The proton-proton radial distribution function of the DeePMD base and the deep variational free energy prediction computed at [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) The radial distribution function of protons computed at [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of ensemble training curves with and without [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison between the proton distribution sampled from [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

57 extracted references · 51 canonical work pages

  1. [1]

    f(Ξ) = f(Ξ + s), (A1) f(..., ξα,

    periodic features The first step in constructing the neural networks for flow and the electron wavefunction is to introduce a periodic func- tion f that transforms the coordinates of protons and electrons into a form that remains invariant under both spatial transla- tions of all particles by any vector s∈ R3 and the periodic transformation of any individ...

    work page

  2. [2]

    dense layer

    Flow network Algorithm 1 Flow ans¨atz Require: Proton coordinates xI. 1: xI J← xI− xJ 2: f ms I J ← features(xI J) ▷ Eq. (A9) 3: h(0) I ← linear 1 N P J f ms I J h(0) I J← linear f ms I J 4: for i = 0 to L− 1 do 5: g(l) I ← concat h(l) I , 1 N P I h(l) I , 1 N P J h(l) I J ˆh(l) J 6: h(l+1) I ← 1√ 2 h(l) I + 1√ 2 dense layerg(l) I 7: h(l+1) I J ← 1√ 2 h(l...

    work page

  3. [3]

    1: ϕδ i, hα← orbitals(ξα, tα, w, rs) ▷ Algorithm 3 2: eδ i← envelope(hI, riI, rs) ▷ Eq

    Electron wavefunction Algorithm 2 Wavefunction ans¨atz Require: Coordinates ξα, one-hot tα, kpoints km, twist w, scale rs. 1: ϕδ i, hα← orbitals(ξα, tα, w, rs) ▷ Algorithm 3 2: eδ i← envelope(hI, riI, rs) ▷ Eq. (A12) and (A13) 3: ϕδ i← eδ i⊙ ϕδ i 4: Dδ i j← P µ Wµϕδ i,µϕδ j,µ ▷ i∈↑, j∈↓ 5: λδ m← MLP(w, rs) 6: ˆri← ri + linear(hi) 7: Eδ i j← 1√nkV Pnk m=1λ...

    work page

  4. [4]

    Mao and R

    H.-k. Mao and R. J. Hemley, Ultrahigh-pressure transitions in solid hydrogen, Rev. Mod. Phys. 66, 671 (1994)

    work page 1994

  5. [5]

    H.-K. Mao, A. Jephcoat, R. Hemley, L. Finger, C. Zha, R. Hazen, and D. Cox, Synchrotron x-ray di ffraction measure- ments of single-crystal hydrogen to 26.5 gigapascals, Science 239, 1131 (1988)

    work page 1988

  6. [6]

    Loubeyre, R

    P. Loubeyre, R. LeToullec, D. Hausermann, M. Hanfland, R. Hemley, H. Mao, and L. Finger, X-ray diffraction and equa- tion of state of hydrogen at megabar pressures, Nature383, 702 (1996)

    work page 1996

  7. [7]

    Akahama, M

    Y . Akahama, M. Nishimura, H. Kawamura, N. Hirao, Y . Ohishi, and K. Takemura, Evidence from x-ray di ffraction of orienta- tional ordering in phase iii of solid hydrogen at pressures up to 183 gpa, Phys. Rev. B 82, 060101 (2010)

    work page 2010

  8. [8]

    C. Ji, B. Li, W. Liu, J. S. Smith, A. Majumdar, W. Luo, R. Ahuja, J. Shu, J. Wang, S. Sinogeikin, et al. , Ultrahigh- pressure isostructural electronic transitions in hydrogen, Nature 573, 558 (2019)

    work page 2019

Show all 57 references
  1. [9]

    G. J. Ackland and J. S. Loveday, Structures of solid hydrogen at 300 k, Phys. Rev. B 101, 094104 (2020)

    2020

  2. [10]

    Li, High-pressure structures of solid hydrogen: Insights from ab initio molecular dynamics simulations, The Journal of Chemical Physics 160 (2024)

    C. Li, High-pressure structures of solid hydrogen: Insights from ab initio molecular dynamics simulations, The Journal of Chemical Physics 160 (2024)

    2024

  3. [11]

    Wigner and H

    E. Wigner and H. ´a. Huntington, On the possibility of a metallic modification of hydrogen, The Journal of Chemical Physics 3, 764 (1935)

    1935

  4. [12]

    Landau and Y

    L. Landau and Y . B. Zeldovich, On the relation between the liq- uid and the gaseous states of metals, Acta Physicochim. USSR 18, 194 (1943)

    1943

  5. [13]

    S. T. Weir, A. C. Mitchell, and W. J. Nellis, Metallization of fluid molecular hydrogen at 140 gpa (1.4 mbar), Phys. Rev. Lett. 76, 1860 (1996)

    1996

  6. [14]

    V . E. Fortov, R. I. Ilkaev, V . A. Arinin, V . V . Burtzev, V . A. Golubev, I. L. Iosilevskiy, V . V . Khrustalev, A. L. Mikhailov, M. A. Mochalov, V . Y . Ternovoi, and M. V . Zhernokletov, Phase transition in a strongly nonideal deuterium plasma generated by quasi-isentropi...

    2007

  7. [15]

    M. D. Knudson, M. P. Desjarlais, A. Becker, R. W. Lemke, K. Cochrane, M. E. Savage, D. E. Bliss, T. Mattsson, and R. Redmer, Direct observation of an abrupt insulator-to-metal transition in dense liquid deuterium, Science 348, 1455 (2015)

    2015

  8. [16]

    Zaghoo and I

    M. Zaghoo and I. F. Silvera, Conductivity and dissociation in liquid metallic hydrogen and implications for planetary inte- riors, Proceedings of the National Academy of Sciences 114, 11873 (2017)

    2017

  9. [17]

    P. M. Celliers, M. Millot, S. Brygoo, R. S. McWilliams, D. E. Fratanduono, J. R. Rygg, A. F. Goncharov, P. Loubeyre, J. H. Eggert, J. L. Peterson, et al., Insulator-metal transition in dense fluid deuterium, Science 361, 677 (2018)

    2018

  10. [18]

    Lorenzen, B

    W. Lorenzen, B. Holst, and R. Redmer, First-order liquid-liquid phase transition in dense hydrogen, Phys. Rev. B 82, 195107 (2010)

    2010

  11. [19]

    J. Hinz, V . V . Karasiev, S. X. Hu, M. Zaghoo, D. Mej ´ıa- Rodr´ıguez, S. B. Trickey, and L. Calder´ın, Fully consistent den- sity functional theory determination of the insulator-metal tran- sition boundary in warm dense hydrogen, Phys. Rev. Res. 2, 032065 (2020)

    2020

  12. [20]

    van de Bund, H

    S. van de Bund, H. Wiebe, and G. J. Ackland, Isotope quantum effects in the metallization transition in liquid hydrogen, Phys. Rev. Lett. 126, 225701 (2021)

    2021

  13. [22]

    M. A. Morales, C. Pierleoni, E. Schwegler, and D. M. Ceper- ley, Evidence for a first-order liquid-liquid transition in high- pressure hydrogen from ab initio simulations, Proceedings of the National Academy of Sciences 107, 12799 (2010)

    2010

  14. [23]

    Pierleoni, M

    C. Pierleoni, M. A. Morales, G. Rillo, M. Holzmann, and D. M. Ceperley, Liquid–liquid phase transition in hydrogen by cou- pled electron–ion monte carlo simulations, Proceedings of the National Academy of Sciences 113, 4953 (2016)

    2016

  15. [24]

    W. R. Magro, D. M. Ceperley, C. Pierleoni, and B. Bernu, Molecular dissociation in hot, dense hydrogen, Phys. Rev. Lett. 76, 1240 (1996)

    1996

  16. [25]

    Mazzola, S

    G. Mazzola, S. Yunoki, and S. Sorella, Unexpectedly high pres- sure for molecular dissociation in liquid hydrogen by electronic simulation, Nature Communications 5, 3487 (2014)

    2014

  17. [26]

    Cheng, G

    B. Cheng, G. Mazzola, C. J. Pickard, and M. Ceriotti, Evidence for supercritical behaviour of high-pressure liquid hydrogen, Nature 585, 217 (2020)

    2020

  18. [27]

    V . V . Karasiev, J. Hinz, S. Hu, and S. Trickey, On the liquid– liquid phase transition of dense hydrogen, Nature 600, E12 (2021)

    2021

  19. [28]

    Tirelli, G

    A. Tirelli, G. Tenti, K. Nakano, and S. Sorella, High-pressure hydrogen by machine learning and quantum monte carlo, Phys. Rev. B 106, L041105 (2022)

    2022

  20. [29]

    Istas, S

    M. Istas, S. Jensen, Y . Yang, M. Holzmann, C. Pierleoni, and 10 D. M. Ceperley, The liquid-liquid phase transition of hydrogen and its critical point: Analysis from ab initio simulation and a machine-learned potential (2024), arXiv:2412.14953 [cond- mat.stat-mech]

    2024 arXiv

  21. [30]

    H. Niu, Y . Yang, S. Jensen, M. Holzmann, C. Pierleoni, and D. M. Ceperley, Stable solid molecular hydrogen above 900 k from a machine-learned potential trained with di ffusion quan- tum monte carlo, Physical review letters 130, 076102 (2023)

    2023

  22. [31]

    Goswami, S

    S. Goswami, S. Jensen, Y . Yang, M. Holzmann, C. Pier- leoni, and D. M. Ceperley, High temperature melting of dense molecular hydrogen from machine-learning interatomic potentials trained on quantum monte carlo, arXiv preprint arXiv:2411.15665 (2024)

    2024 arXiv

  23. [32]

    Xie, Z.-H

    H. Xie, Z.-H. Li, H. Wang, L. Zhang, and L. Wang, Deep varia- tional free energy approach to dense hydrogen, Phys. Rev. Lett. 131, 126501 (2023)

    2023

  24. [33]

    Pierleoni, D

    C. Pierleoni, D. M. Ceperley, and M. Holzmann, Coupled electron-ion monte carlo calculations of dense metallic hydro- gen, Phys. Rev. Lett. 93, 146402 (2004)

    2004

  25. [34]

    Attaccalite and S

    C. Attaccalite and S. Sorella, Stable liquid hydrogen at high pressure by a novel ab initio molecular-dynamics calculation, Phys. Rev. Lett. 100, 114501 (2008)

    2008

  26. [35]

    Azadi and W

    S. Azadi and W. M. C. Foulkes, Fate of density functional the- ory in the study of high-pressure solid hydrogen, Phys. Rev. B 88, 014115 (2013)

    2013

  27. [36]

    I. A. Kruglov, A. V . Yanilkin, Y . Propad, A. B. Mazitov, P. Rachitskii, and A. R. Oganov, Crystal structure prediction at finite temperatures, npj Computational Materials 9, 197 (2023)

    2023

  28. [37]

    Goodfellow, Y

    I. Goodfellow, Y . Bengio, and A. Courville, Deep Learning (MIT Press, 2016) http://www.deeplearningbook.org

    2016

  29. [38]

    Li and L

    S.-H. Li and L. Wang, Neural network renormalization group, Phys. Rev. Lett. 121, 260601 (2018)

    2018

  30. [39]

    D. Wu, L. Wang, and P. Zhang, Solving statistical mechanics using variational autoregressive networks, Phys. Rev. Lett.122, 080602 (2019)

    2019

  31. [40]

    Zhang, J

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

    2018

  32. [41]

    Gao and S

    N. Gao and S. G ¨unnemann, Ab-initio potential energy surfaces by pairing GNNs with neural wave functions, in International Conference on Learning Representations (2022)

    2022

  33. [42]

    Scherbela, R

    M. Scherbela, R. Reisenhofer, L. Gerard, P. Marquetand, and P. Grohs, Solving the electronic schr ¨odinger equation for mul- tiple nuclear geometries with weight-sharing deep neural net- works, Nature Computational Science 2, 331 (2022)

    2022

  34. [43]

    D. Pfau, J. S. Spencer, A. G. D. G. Matthews, and W. M. C. Foulkes, Ab initio solution of the many-electron schr ¨odinger equation with deep neural networks, Phys. Rev. Res. 2, 033429 (2020)

    2020

  35. [44]

    X. Li, Z. Li, and J. Chen, Ab initio calculation of real solids via neural network ansatz, Nature Communications 13, 7895 (2022)

    2022

  36. [45]

    C. Lin, F. Zong, and D. M. Ceperley, Twist-averaged bound- ary conditions in continuum quantum monte carlo algorithms, Physical Review E 64, 016702 (2001)

    2001

  37. [46]

    Mazzola, R

    G. Mazzola, R. Helled, and S. Sorella, Phase diagram of hydro- gen and a hydrogen-helium mixture at planetary conditions by quantum monte carlo simulations, Phys. Rev. Lett.120, 025701 (2018)

    2018

  38. [47]

    R. M. Neal, Annealed importance sampling, Statistics and com- puting 11, 125 (2001)

    2001

  39. [48]

    H. Xie, L. Zhang, and L. Wang, m* of two-dimensional electron gas: A neural canonical transformation study, SciPost Phys.14, 154 (2023)

    2023

  40. [49]

    Zhang, R.-S

    Q. Zhang, R.-S. Wang, and L. Wang, Neural canonical trans- formations for vibrational spectra of molecules, The Journal of Chemical Physics 161, 024103 (2024)

    2024

  41. [50]

    Github repository, https: //github.com/fermiflow/ SolidHydrogen

  42. [51]

    Casula and S

    M. Casula and S. Sorella, Geminal wave functions with jastrow correlation: A first application to atoms, The Journal of Chem- ical Physics 119, 6500 (2003)

    2003

  43. [52]

    Jumper, R

    J. Jumper, R. Evans, A. Pritzel, T. Green, M. Figurnov, O. Ronneberger, K. Tunyasuvunakool, R. Bates, A. ˇZ´ıdek, A. Potapenko, et al., Highly accurate protein structure predic- tion with alphafold, Nature , 1 (2021)

    2021

  44. [53]

    Zhang, H

    Y . Zhang, H. Wang, W. Chen, J. Zeng, L. Zhang, H. Wang, and E. Weinan, Dp-gen: A concurrent learning platform for the gen- eration of reliable deep learning based potential energy models, Computer Physics Communications 253, 107206 (2020)

    2020

  45. [54]

    M. Chen, G. Guo, and L. He, Systematically improvable op- timized atomic basis sets for ab initio calculations, Journal of Physics: Condensed Matter 22, 445501 (2010)

    2010

  46. [55]

    P. Li, X. Liu, M. Chen, P. Lin, X. Ren, L. Lin, C. Yang, and L. He, Large-scale ab initio simulations based on systematically improvable atomic basis, Computational Materials Science112, 503 (2016)

    2016

  47. [56]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Physical Review Letters 77, 3865 (1996)

    1996

  48. [57]

    Zhang, J

    L. Zhang, J. Han, H. Wang, W. Saidi, and R. Car, End-to-end symmetry preserving inter-atomic potential energy model for finite and extended systems, Advances in Neural Information Processing Systems 31 (2018)

    2018

  49. [58]

    Wirnsberger, G

    P. Wirnsberger, G. Papamakarios, B. Ibarz, S. Racani `ere, A. J. Ballard, A. Pritzel, and C. Blundell, Normalizing flows for atomic solids, Machine Learning: Science and Technology 3, 025009 (2022)

    2022

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.