Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

High temperature melting of dense molecular hydrogen from machine-learning interatomic potentials trained on quantum Monte Carlo

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read A quantum Monte Carlo-trained machine-learning potential predicts that solid molecular hydrogen survives above 1600 K up to 180 GPa, with no melting maximum before the liquid turns atomic.

desk verdict The qualitative high-melting result is credible and consistent across two ML potentials, but the precise curve and 5 K error bars are not supported given the model energy error is comparable to the latent heat. read the letter →

arxiv 2411.15665 v1 pith:6Q2RFEHM submitted 2024-11-23 physics.chem-ph

Shubhang Goswami , Scott Jensen , Yubo Yang , Markus Holzmann , Carlo Pierleoni , David M. Ceperley This is my paper · ORCID
classification physics.chem-ph
keywords machinelearninginteratomicpotentialquantumMonteCarlohydrogenmeltingcurvehighpressurepathintegralmoleculardynamicsClausius-Clapeyronequationtwo-phasecoexistenceliquid-liquidtransition
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 claims that the melting curve of dense molecular hydrogen is much higher than the widely used density-functional-theory (DFT) calculations suggest. Using a machine-learned interatomic potential, MACE-M18, trained on diffusion Monte Carlo (QMC) energies and forces with deliberately larger weight on total energies, the authors predict that solid molecular hydrogen remains stable above 1600 K at pressures up to 180 GPa, with the melting temperature rising monotonically from about 966 K at 50 GPa to 1777 K at 180 GPa for classical protons, and from 954 K at 50 GPa to about 1635 K near the quantum molecular-atomic crossover at 172 GPa. They find no melting-point maximum before the liquid dissociates into atomic hydrogen at roughly 173 GPa (quantum) or 189 GPa (classical). If correct, DFT melting curves that peak near 800-1150 K at 70-125 GPa underestimate the stability of the solid, and the liquid-liquid transition is largely hidden inside the crystalline phase.

What carries the argument

The load-bearing object is the MACE-M18 model, an equivariant message-passing neural-network interatomic potential trained on 16,290 QMC configurations of 96 protons with a loss weighting $\lambda_E/\lambda_F = 220\,\text{Å}^{-2}$ that strongly favors reproducing total energies over forces. This choice matters because solid-liquid coexistence is controlled by the internal-energy difference between phases, and QMC energies have smaller errors than QMC forces. The melting curve itself comes from combining two-phase (TP) coexistence simulations, which put upper and lower bounds on $T_m$, with the Clausius-Clapeyron relation $dT_m/dP = D(T,P)$ built from homogeneous liquid and solid energy/volume runs; a Markov-chain Monte Carlo over polynomial coefficients enforces both the slope data and the TP bounds. Quantum proton effects are included through the mass-derivative relation $dP/dx = -\Delta k/(x\Delta v)$, evaluated with path-integral molecular dynamics, which shifts the classical curve to the quantum one.

What would settle it

Evaluate the average QMC-minus-MACE energy correction separately for liquid and solid configurations along the predicted melting line, for example at 130 GPa and 1529 K. If the phase-resolved correction differs by more than a few meV per atom, the melting temperature moves by more than the quoted 5 K uncertainty. A direct thermodynamic-integration free-energy difference between the hcp solid and liquid at the same conditions would settle the same question, as would a clean experimental melting measurement between 150 and 170 GPa showing liquid at temperatures below about 1500 K.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the melting line of dense molecular hydrogen, computed with a MACE potential trained on QMC data with strong energy weighting, keeps rising throughout 50-180 GPa instead of turning over into a maximum. In the authors' fit the classical melting temperature is a quadratic function of pressure, $T_m(P)=509+9.935P-0.01606P^2$ (P in GPa), and the quantum hydrogen line is a quartic with coefficients {374, 14.51, -0.0661, 1.67e-4, -1.54e-7} over 50-170 GPa; the liquid volume is about 1.4% larger than the solid along the line and the latent heat grows from roughly 6.6 to 15.9 kJ/mol of H2. The slope of the melting curve decreases with pressure but stays positive, so no melting maximum occurs before a molecular-to-atomic transition in the liquid at about 172-173 GPa (quantum) and 189 GPa (classical), where the melting curve's character changes. The results agree with the authors' earlier DPMD-based finding that the solid survives above 900 K, sharpen that claim to above 1600 K, and disagree with DFT-based melting curves with maxima near 820-1150 K at 70-125 GPa.

Load-bearing premise

The model's residual root-mean-square energy error of 27 meV per atom is assumed to be statistically uncorrelated with phase, so it does not bias the liquid-solid free-energy difference; because the latent heat is only about 34 meV per atom at 50 GPa, even a small phase-correlated part of that error would shift the predicted melting temperature by tens of kelvin.

Editorial extensions

If this is right

  • Solid molecular hydrogen remains stable above 1600 K up to 180 GPa, so the melting line has no maximum in this pressure range.
  • DFT-based melting curves with maxima of 820-1150 K at 70-125 GPa underestimate the solid's stability and should be revisited.
  • The molecular-atomic (liquid-liquid) crossover at about 172-173 GPa for quantum hydrogen occurs at the edge of the solid region, so the melting curve changes character there rather than at a melting maximum.
  • Combining two-phase bounds with Clausius-Clapeyron slopes reduces melting-temperature uncertainty to about 5 K, a precision that makes the melting curve a sharp benchmark for other methods.
  • Proton zero-point motion lowers the melting line by tens of kelvin, and the mass-derivative method simultaneously provides a deuterium melting curve.

Reading between the lines

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

  • If the high melting line is right, some dynamic diamond-anvil-cell melting signatures above 50 GPa that were interpreted as a melting maximum may instead reflect the solid-liquid boundary at higher temperature or sample gradients; re-analysis of those signals with the new curve would be a direct test.
  • The energy-weighted training recipe is transferable: for any material whose phase diagram hinges on solid-liquid coexistence, machine-learned potentials should fit total energies with high weight rather than forces alone, because force-only fits can miss the phase energy difference.
  • The model was trained only up to 200 GPa and on molecular configurations, so the predicted drop of the melting curve just above the crossover is the least certain part; extending QMC training into the dissociated liquid would test whether the melting curve falls steeply or has a maximum near 190 GPa.
  • The same classical-to-quantum mass shift can be rerun for deuterium, giving a testable isotope shift of the melting line that experiments could measure.
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 reports a new machine-learned interatomic potential (MACE-M18) for dense molecular hydrogen, trained on 16,290 diffusion Monte Carlo configurations from the authors' published database, with an increased weight on total energies relative to forces. Using two-phase coexistence simulations, Clausius-Clapeyron integration anchored to those bounds, microcanonical two-phase runs, and a mass-perturbation path-integral method, the authors obtain classical and quantum melting lines for 50–180 GPa. They predict that molecular solid hydrogen remains stable above 1600 K in this pressure range, that there is no melting-point maximum before a molecular-to-atomic crossover in the liquid at roughly 189 GPa (classical) or 172–173 GPa (quantum), and they compare their melting line with experimental estimates and with DFT-based curves. The manuscript emphasizes that the training weighting of total energies is important for phase-equilibrium predictions.

Significance. If correct, the central result has substantial impact: it would strengthen the case that DFT-based melting curves for dense hydrogen are too low by several hundred kelvin and that experimental melting maxima near 70–100 GPa are artifacts. The paper's strengths are the use of QMC-level electronic structure data, the multi-method consistency checks (two-phase, Clausius-Clapeyron, microcanonical, and mass-derivative), the benchmarking against experimental equations of state and CEIMC pair correlations, and the public availability of the database and code. The qualitative conclusion that the solid remains stable above 1600 K is supported by two independent ML potentials trained on the same QMC data; however, the quantitative precision claimed for the melting line (about 5 K) is not supported by the evidence in the manuscript, because the only propagated errors are statistical fitting errors.

major comments (4)
  1. [Section IV, Tables II–III, and Eq. (5)] The quoted "about 5 K" melting-temperature uncertainty is only the MCMC spread of the polynomial fit given the two-phase bounds and Clausius-Clapeyron slopes; it does not propagate the MACE-M18 model's 27 meV/atom RMS energy error (Table I). That error is the same order as the latent heat per atom along the melting line (34 meV/atom at 50 GPa and about 82 meV/atom at 180 GPa from Table II). An RMS error averaged over the full database does not bound the phase-correlated part of the energy error that enters the liquid-solid free-energy difference, and no phase-resolved error estimates are provided. A phase-correlated energy error of only 10–20 meV/atom would shift the Clapeyron slope and, integrated over the 50–180 GPa range, move the melting temperature by the 100–300 K scale that separates this work from DFT-based curves. The authors should either propagate this model uncertainty into the melting line or demonstrate explicitly why it cancels between phases.
  2. [Section V, Fig. 9] Two ML potentials trained on the same QMC database, MACE-M18 and DPMD, differ by 50–160 K in the classical melting temperature and by up to 80 K in the quantum melting temperature. This is direct evidence of model-form uncertainty, and it is an order of magnitude larger than the quoted 5 K statistical error. The microcanonical runs in Section III.C also show run-to-run spreads of up to 40 K at fixed pressure. The paper should either report the melting curve with an uncertainty of at least this size or provide a quantitative argument for why the M18 predictions are more trustworthy than those of DPMD beyond the lower RMS errors in Table I.
  3. [Appendix A] The training data treat energies and forces inconsistently: the total energies receive an RPA finite-size correction, while the forces are linearly extrapolated as 2*DMC - VMC and receive no finite-size correction. Since the two-phase dynamics and the Clausius-Clapeyron volume and energy differences depend on both forces and energies, this correction asymmetry is a possible systematic bias that is not included in the error budget. The authors should quantify the sensitivity of the melting curve to this choice, for example by training a model on force data with no extrapolation or with an alternative correction and recomputing coexistence at one or two pressures.
  4. [Section V and Fig. 6] The molecular-atomic crossover in the liquid is identified at roughly 172–173 GPa (quantum) and 189 GPa (classical), but the text also states that the melting-curve study "does not include atomic liquid configurations." Because the training database is predominantly molecular and the model has no guarantee of extrapolation, the pair-correlation evidence for abrupt dissociation at these pressures should be labeled as extrapolative. This caveat matters for the secondary claim that there is no melting maximum before the crossover, because a shift in the crossover pressure would affect the maximum-free interpretation.
minor comments (5)
  1. [Section IV, Results] The sentence "the internal energy of the solid is higher than that of the liquid" is inconsistent with Table II, in which u_s is more negative than u_l (since L > 0); the text should say that the liquid has the higher internal energy, which is also consistent with the positive Clausius-Clapeyron slope.
  2. [Section VI] "Claudius-Clapeyron" should be "Clausius-Clapeyron."
  3. [Eq. (2)] The typesetting of Eq. (2) is ambiguous; write D = T / [P + (u_l - u_s)/(v_l - v_s)] explicitly.
  4. [Text near Fig. 9] "This differene between the two models" should be "This difference between the two models."
  5. [Table I caption] The caption would be clearer if it stated that the errors are per-atom RMS values after dividing the total-energy errors by N^{1/2}, since this normalization is unusual for readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: QMC-trained potential plus MD/thermodynamic integration yields the melting curve; the main risks are model accuracy, not logical circularity.

full rationale

The claimed derivation chain is: QMC energies and forces (first-principles, not defined by melting) are used to train the MACE-M18 potential; the potential then drives two-phase, Clausius-Clapeyron, microcanonical, and mass-perturbation simulations, whose outputs (melting temperature, volumes, latent heats) are statistically combined. None of these steps takes the melting temperature as a fitted input. The Clausius-Clapeyron relation (Eqs. 2-3) integrates slopes computed from model energy and volume differences, and the MCMC polynomial (Eqs. 4-5) anchors the absolute scale with two-phase bounds; this is a standard self-consistent thermodynamic construction, not a renaming of training labels. The database and prior DPMD melting line are taken from Niu et al. (ref 3), and the paper self-cites that work, but the present MACE-M18 melting curve is a fresh simulation result; the prior DPMD curve is compared, not imposed. External checks (experimental equation of state at 300 K, lower-pressure melting data, CEIMC pair-correlation functions) provide independent support. The most serious caveat, that the 27 meV/atom RMS energy error is comparable to the roughly 34 meV/atom latent heat, is a potential phase-correlated bias in the ML potential; that is an accuracy and robustness threat, not a circularity, because the error is not fitted to the melting curve and the training labels do not depend on the melting temperature. No step reduces an output to an input by construction.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central result rests on the accuracy of the QMC database, the machine-learning model trained on it, and the analytic smoothness assumptions used to stitch together the melting curve from two-phase bounds and Clausius-Clapeyron slopes. No new physical entities are introduced.

free parameters (4)
  • lambda_E / lambda_F ratio = 220 A^-2
    Chosen to trade off energy error versus force error during MACE training (Fig. 1); it changes the trained M18 potential and hence the melting curve.
  • Classical melting curve polynomial coefficients = {509, 9.935, -0.01606} (quadratic in P/GPa)
    Fit to two-phase bounds and Clausius-Clapeyron slopes over 50 to 170 GPa; defines the reported classical Tm(P).
  • Quantum melting curve polynomial coefficients = {374, 14.51, -0.0661, 1.67E-4, -1.54E-7} (quartic in P/GPa)
    Fit to Clausius-Clapeyron slopes and quantum reference points from the mass-derivative method; defines the reported quantum Tm(P).
  • MACE architecture (hidden irreps and cutoff radius) = 128x0e + 128x1o, cutoff 4.0 to 4.4 A
    Architecture complexity and cutoff chosen by monitoring training errors and MD runtime; affects model transferability and the resulting phase behavior.
assumptions (7)
  • domain assumption Born-Oppenheimer approximation: electrons stay in the ground state for fixed proton positions.
    Invoked in Appendix A and used for both QMC training data and MD; the authors estimate the error as about 10 K per atom at low density with negligible effect on the melting line.
  • domain assumption Diffusion Monte Carlo with the Slater-Jastrow trial wavefunction and fixed-node approximation gives accurate energies and forces for dense hydrogen.
    DMC is used to generate the training database; accuracy depends on trial wavefunction quality and the fixed-node error, which are not rigorously bounded.
  • domain assumption The mixed-estimator bias in QMC forces is corrected by a linear temperature-dependent term and the remaining bias is an order of magnitude smaller.
    Appendix A: forces are linearly extrapolated (2DMC-VMC) and a linear mixed-estimator correction is applied; residual bias is asserted to be small.
  • domain assumption The MACE cutoff radius of 4.0 to 4.4 Å captures all relevant interactions for melting free energies.
    The potential is treated as short-ranged; no long-range dispersion tail is included, which could affect the solid-liquid energy difference.
  • domain assumption The melting curve is analytic (no triple point) in 50 to 180 GPa for classical and 50 to 170 GPa for quantum hydrogen, so a polynomial describes Tm(P).
    Section III.B: the polynomial form is assumed because the Clausius-Clapeyron derivatives are nearly linear; the authors exclude pressures near the molecular-atomic crossover.
  • domain assumption The quantum melting curve can be obtained from the classical one using the mass-derivative relation dP/dx = -Delta k / (x Delta v) with smooth behavior of Pm(T,x).
    Section III.D: this relation is used to shift the classical curve using PIMD data at only three temperatures (1142, 1342, and 1500 K) and inverse-mass values 1/3, 1/2, and 1.
  • domain assumption Proton exchange statistics (para-ortho) do not affect the melting line.
    Section III.D: PIMD uses distinguishable-particle path integrals; the authors argue quantum statistics are not important for the melting line of molecular hydrogen at high pressure.

how reviews work

0 comments
Cite this review

Pith. "Pith review of High temperature melting of dense molecular hydrogen from machine-learning interatomic potentials trained on quantum Monte Carlo." pith.science (2026). https://pith.science/paper/6Q2RFEHM

@misc{pith2026241115665,
  author       = {Pith},
  title        = {Pith review of: High temperature melting of dense molecular hydrogen from machine-learning interatomic potentials trained on quantum Monte Carlo},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6Q2RFEHM}},
  note         = {Machine review of arXiv:2411.15665}
}
read the original abstract

We present results and discuss methods for computing the melting temperature of dense molecular hydrogen using a machine learned model trained on quantum Monte Carlo data. In this newly trained model, we emphasize the importance of accurate total energies in the training. We integrate a two phase method for estimating the melting temperature with estimates from the Clausius-Clapeyron relation to provide a more accurate melting curve from the model. We make detailed predictions of the melting temperature, solid and liquid volumes, latent heat and internal energy from 50 GPa to 180 GPa for both classical hydrogen and quantum hydrogen. At pressures of roughly 173 GPa and 1635K, we observe molecular dissociation in the liquid phase. We compare with previous simulations and experimental measurements.

Figures

Figures reproduced from arXiv: 2411.15665 by the authors.

Figure 1
Figure 1. FIG. 1. RMS force errors versus the RMS energy errors as [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. A comparison of the c/a ratio (top panel), equation of state [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. A comparison of the pair correlation function of CEIMC [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Example of the classical MD results showing volume per [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Pair correlation functions for classical hydrogen at 1800 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Melting temperature versus pressure (lower panel), the [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 7
Figure 7. Figure 7: We find that the character of the melting line changes [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison of melting lines derived from two ML models [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    physics.comp-ph 2025-01 conditional novelty 6.0 of 10

    At 1200 K, deep variational free energy calculations find a transition from atomic liquid to molecular crystal in dense hydrogen at about 180 GPa, with kinks in pressure and entropy.

Reference graph

Works this paper leans on

66 extracted references · 38 canonical work pages · cited by 1 Pith paper

  1. [1]

    author author J. M. \ McMahon , author M. A. \ Morales , author C. Pierleoni ,\ and\ author D. M. \ Ceperley ,\ title title The properties of hydrogen and helium under extreme conditions , \ https://doi.org/10.1103/RevModPhys.84.1607 journal journal Rev. Mod. Phys. \ volume 84 ,\ pages 1607--1653 ( year 2012 ) NoStop

    work page doi:10.1103/revmodphys.84.1607

  2. [2]

    Wigner \ and\ author H

    author author E. Wigner \ and\ author H. B. \ Huntington ,\ title title On the Possibility of a Metallic Modification of Hydrogen , \ https://doi.org/10.1063/1.1749590 journal journal The Journal of Chemical Physics \ volume 3 ,\ pages 764--770 ( year 1935 ) ,\ https://arxiv.org/abs/https://pubs.aip.org/aip/jcp/article-pdf/3/12/764/18789026/764\_1\_online...

    doi:10.1063/1.1749590

  3. [3]

    Niu , author Y

    author author H. Niu , author Y. Yang , author S. Jensen , author M. Holzmann , author C. Pierleoni ,\ and\ author D. M. \ Ceperley ,\ title title Stable solid molecular hydrogen above 900 k from a machine-learned potential trained with diffusion quantum monte carlo , \ https://doi.org/10.1103/PhysRevLett.130.076102 journal journal Phys. Rev. Lett. \ volu...

    doi:10.1103/physrevlett.130.076102

  4. [4]

    Rahman ,\ title title Correlations in the motion of atoms in liquid argon , \ https://doi.org/10.1103/PhysRev.136.A405 journal journal Phys

    author author A. Rahman ,\ title title Correlations in the motion of atoms in liquid argon , \ https://doi.org/10.1103/PhysRev.136.A405 journal journal Phys. Rev. \ volume 136 ,\ pages A405--A411 ( year 1964 ) NoStop

    doi:10.1103/physrev.136.a405 1964

  5. [5]

    Parrinello \ and\ author A

    author author M. Parrinello \ and\ author A. Rahman ,\ title title Crystal structure and pair potentials: A molecular-dynamics study , \ https://doi.org/10.1103/PhysRevLett.45.1196 journal journal Phys. Rev. Lett. \ volume 45 ,\ pages 1196--1199 ( year 1980 ) NoStop

    doi:10.1103/physrevlett.45.1196

  6. [6]

    Parrinello \ and\ author A

    author author M. Parrinello \ and\ author A. Rahman ,\ title title Study of an F center in molten KCl , \ https://doi.org/10.1063/1.446740 journal journal The Journal of Chemical Physics \ volume 80 ,\ pages 860--867 ( year 1984 ) ,\ https://arxiv.org/abs/https://pubs.aip.org/aip/jcp/article-pdf/80/2/860/18946078/860\_1\_online.pdf https://pubs.aip.org/ai...

    doi:10.1063/1.446740

  7. [7]

    Car \ and\ author M

    author author R. Car \ and\ author M. Parrinello ,\ title title Unified approach for molecular dynamics and density-functional theory , \ https://doi.org/10.1103/PhysRevLett.55.2471 journal journal Phys. Rev. Lett. \ volume 55 ,\ pages 2471--2474 ( year 1985 ) NoStop

    doi:10.1103/physrevlett.55.2471

  8. [8]

    It assumes that the electrons are in their ground state for a given arrangement of protons and can be integrated out

    note Our calculations, as well as all others that we are aware of, start with the Born-Oppenheimer approximation (BOA). It assumes that the electrons are in their ground state for a given arrangement of protons and can be integrated out. We are left with the problem of finding the equilibrium properties of the protons interacting with a PES. We refer to c...

    work page

Show all 66 references
  1. [9]

    Pierleoni , author K

    author author C. Pierleoni , author K. T. \ Delaney , author M. A. \ Morales , author D. M. \ Ceperley ,\ and\ author M. Holzmann ,\ title title Trial wave functions for high-pressure metallic hydrogen , \ https://doi.org/https://doi.org/10.1016/j.cpc.2008.01.041 journal journ...

    doi

  2. [10]

    author author M. a. \ Morales , author C. Pierleoni , author E. Schwegler ,\ and\ author D. M. \ Ceperley ,\ title title Evidence for a first-order liquid-liquid transition in high-pressure hydrogen from ab initio simulations , \ https://doi.org/10.1073/pnas.1007309107 journal...

    doi

  3. [11]

    Rillo , author M

    author author G. Rillo , author M. A. \ Morales , author D. M. \ Ceperley ,\ and\ author C. Pierleoni ,\ title title Coupled Electron-Ion Monte Carlo simulation of hydrogen molecular crystals , \ http://arxiv.org/abs/1708.07344 journal journal J. Chem Phys. \ volume 148 ,\ pag...

    2018 arXiv

  4. [12]

    Gorelov , author M

    author author V. Gorelov , author M. Holzmann , author D. M. \ Ceperley ,\ and\ author C. Pierleoni ,\ title title Energy gap closure of crystalline molecular hydrogen with pressure , \ https://doi.org/10.1103/PhysRevLett.124.116401 journal journal Phys. Rev. Lett. \ volume 12...

    doi

  5. [13]

    Dion , author H

    author author M. Dion , author H. Rydberg , author E. Schr\"oder , author D. C. \ Langreth ,\ and\ author B. I. \ Lundqvist ,\ title title Van der waals density functional for general geometries , \ https://doi.org/10.1103/PhysRevLett.92.246401 journal journal Phys. Rev. Lett....

    doi

  6. [14]

    note See Supplementary Material section in ref Niu2023 NoStop

  7. [15]

    Zhang , author J

    author author L. Zhang , author J. Han , author H. Wang , author W. A. \ Saidi , author R. Car ,\ and\ author W. E ,\ title title End-to-end symmetry preserving inter-atomic potential energy model for finite and extended systems , \ in\ https://papers.nips.cc/paper/2018/hash/e...

    2018

  8. [16]

    Zhang , author J

    author author L. Zhang , author J. Han , author H. Wang , author R. Car ,\ and\ author W. E ,\ title title Deep Potential Molecular Dynamics: A Scalable Model with the Accuracy of Quantum Mechanics , \ https://doi.org/10.1103/PhysRevLett.120.143001 journal journal Phys. Rev. L...

    doi

  9. [17]

    Batatia , author D

    author author I. Batatia , author D. P. \ Kovacs , author G. N. C. \ Simm , author C. Ortner ,\ and\ author G. Csanyi ,\ title title MACE : Higher order equivariant message passing neural networks for fast and accurate force fields , \ in\ https://openreview.net/forum?id=YPpSn...

    2022

  10. [18]

    Batatia , author S

    author author I. Batatia , author S. Batzner , author D. P. \ Kov \'a cs , author A. Musaelian , author G. N. C. \ Simm , author R. Drautz , author C. Ortner , author B. Kozinsky ,\ and\ author G. Cs \'a nyi ,\ https://doi.org/10.48550/arXiv.2205.06643 title The design space o...

    doi arXiv

  11. [19]

    author author D. M. \ Ceperley , author S. Jensen , author Y. Yang , author H. Niu , author C. Pierleoni ,\ and\ author M. Holzmann ,\ title title Training models using forces computed by stochastic electronic structure methods , \ https://doi.org/10.1088/2516-1075/ad2eb0 jour...

    doi

  12. [20]

    Ji , author B

    author author C. Ji , author B. Li , author W. Liu , author J. S. \ Smith , author A. Majumdar , author W. Luo , author R. Ahuja , author J. Shu , author J. Wang , author S. Sinogeikin , author Y. Meng , author V. B. \ Prakapenka , author E. Greenberg , author R. Xu , author X...

    doi

  13. [21]

    Pierleoni , author M

    author author C. Pierleoni , author M. A. \ Morales , author G. Rillo , author M. Holzmann ,\ and\ author D. M. \ Ceperley ,\ title title Liquid–liquid phase transition in hydrogen by coupled electron–ion Monte Carlo simulations , \ https://doi.org/10.1073/pnas.1603853113 jour...

    doi

  14. [22]

    Zhang \ and\ author E

    author author Y. Zhang \ and\ author E. J. \ Maginn ,\ title title A comparison of methods for melting point calculation using molecular dynamics simulations , \ https://doi.org/10.1063/1.3702587 journal journal The Journal of Chemical Physics \ volume 136 ,\ pages 144116 ( ye...

    doi

  15. [23]

    Wang , author M

    author author X. Wang , author M. Yang , author X. Gai , author Y. Sun , author B. Cao , author J. Chen , author M. Liang , author F. Tian ,\ and\ author L. Li ,\ title title A comprehensive investigation on the accuracy and efficiency of methods for melting temperature calcul...

  16. [24]

    Liberatore , author C

    author author E. Liberatore , author C. Pierleoni ,\ and\ author D. M. \ Ceperley ,\ title title Liquid-solid transition in fully ionized hydrogen at ultra-high pressures , \ https://doi.org/10.1063/1.3586808 journal journal The Journal of Chemical Physics \ volume 134 ,\ page...

    doi

  17. [25]

    author author K. K. \ Ly \ and\ author D. M. \ Ceperley ,\ https://arxiv.org/abs/2409.19484 title Melting of atomic hydrogen and deuterium with path-integral monte carlo , \ ( year 2024 ),\ https://arxiv.org/abs/2409.19484 arXiv:2409.19484 [cond-mat.mtrl-sci] NoStop

    2024 arXiv

  18. [26]

    author author D. A. \ Kofke ,\ title title Direct evaluation of phase coexistence by molecular simulation via integration along the saturation line , \ https://doi.org/10.1063/1.465023 journal journal The Journal of Chemical Physics \ volume 98 ,\ pages 4149--4162 ( year 1993 ...

    doi

  19. [27]

    author author V. V. \ Kechin ,\ title title Melting curve equations at high pressure , \ https://doi.org/10.1103/PhysRevB.65.052102 journal journal Phys. Rev. B \ volume 65 ,\ pages 052102 ( year 2001 ) NoStop

    2001 doi

  20. [28]

    Boninsegni , author C

    author author M. Boninsegni , author C. Pierleoni ,\ and\ author D. M. \ Ceperley ,\ title title Isotopic shift of helium melting pressure: Path integral Monte Carlo study , \ https://doi.org/10.1103/PhysRevLett.72.1854 journal journal Phys. Rev. Lett. \ volume 72 ,\ pages 185...

    doi

  21. [29]

    Filippi \ and\ author D

    author author C. Filippi \ and\ author D. M. \ Ceperley ,\ title title Path-integral monte carlo calculation of the kinetic energy of condensed lithium , \ https://doi.org/10.1103/PhysRevB.57.252 journal journal Phys. Rev. B \ volume 57 ,\ pages 252--257 ( year 1998 ) NoStop

    doi

  22. [30]

    author author D. M. \ Ceperley ,\ title title Path integral monte carlo methods for fermions , \ in\ @noop booktitle Monte Carlo and Molecular Dynamics of Condensed Matter Systems ,\ editor edited by\ editor K. Binder \ and\ editor G. Ciccotti \ ( publisher Editrice Compositor...

    1996

  23. [31]

    Gregoryanz , author A

    author author E. Gregoryanz , author A. F. \ Goncharov , author K. Matsuishi , author H.-k. \ Mao ,\ and\ author R. J. \ Hemley ,\ title title Raman Spectroscopy of Hot Dense Hydrogen , \ https://doi.org/10.1103/PhysRevLett.90.175701 journal journal Phys. Rev. Lett. \ volume 9...

    doi

  24. [32]

    Datchi , author P

    author author F. Datchi , author P. Loubeyre ,\ and\ author R. LeToullec ,\ title title Extended and accurate determination of the melting curves of argon, helium, ice ( h _ 2 O), and hydrogen ( h _ 2 ) , \ https://doi.org/10.1103/PhysRevB.61.6535 journal journal Phys. Rev. B ...

    doi

  25. [33]

    author author I. F. \ Silvera \ and\ author V. V. \ Goldman ,\ title title The isotropic intermolecular potential for H 2 and D 2 in the solid and gas phases , \ https://doi.org/10.1063/1.437103 journal journal J. Chem. Phys. \ volume 69 ,\ pages 4209--4213 ( year 1978 ) NoStop

    doi

  26. [34]

    author author D. A. \ Young \ and\ author M. Ross ,\ title title Theoretical calculation of thermodynamic properties and melting curves for hydrogen and deuterium , \ https://doi.org/10.1063/1.441058 journal journal J. Chem. Phys. \ volume 74 ,\ pages 6950--6955 ( year 1981 ) NoStop

    doi

  27. [35]

    Diatschenko , author C

    author author V. Diatschenko , author C. W. \ Chu , author D. H. \ Liebenberg , author D. A. \ Young , author M. Ross ,\ and\ author R. L. \ Mills ,\ title title Melting curves of molecular hydrogen and molecular deuterium under high pressures between 20 and 373 k , \ https://...

    doi

  28. [36]

    Deemyad \ and\ author I

    author author S. Deemyad \ and\ author I. F. \ Silvera ,\ title title Melting Line of Hydrogen at High Pressures , \ https://doi.org/10.1103/PhysRevLett.100.155701 journal journal Phys. Rev. Lett. \ volume 100 ,\ pages 155701 ( year 2008 ) NoStop

    doi

  29. [37]

    author author M. I. \ Eremets \ and\ author I. A. \ Trojan ,\ title title Evidence of maximum in the melting curve of hydrogen at megabar pressures , \ https://doi.org/10.1134/S0021364009040031 journal journal JETP Lett. \ volume 89 ,\ pages 174--179 ( year 2009 ) NoStop

    doi

  30. [38]

    Subramanian , author A

    author author N. Subramanian , author A. F. \ Goncharov , author V. V. \ Struzhkin , author M. Somayazulu ,\ and\ author R. J. \ Hemley ,\ title title Bonding changes in hot fluid hydrogen at megabar pressures , \ https://doi.org/10.1073/pnas.1102760108 journal journal Proc. N...

    doi

  31. [39]

    author author R. T. \ Howie , author P. Dalladay-Simpson ,\ and\ author E. Gregoryanz ,\ title title Raman spectroscopy of hot hydrogen above 200 GPa , \ https://doi.org/10.1038/nmat4213 journal journal Nat. Mater. \ volume 14 ,\ pages 495--499 ( year 2015 ) NoStop

    doi

  32. [40]

    \ Zha , author H

    author author C.-s. \ Zha , author H. Liu , author J. S. \ Tse ,\ and\ author R. J. \ Hemley ,\ title title Melting and High P - T Transitions of Hydrogen up to 300 GPa , \ https://doi.org/10.1103/PhysRevLett.119.075302 journal journal Phys. Rev. Lett. \ volume 119 ,\ pages 07...

    doi

  33. [41]

    author author A. F. \ Goncharov , author R. T. \ Howie ,\ and\ author E. Gregoryanz ,\ title title Hydrogen at extreme pressures (Review Article) , \ https://doi.org/10.1063/1.4807051 journal journal Low Temperature Physics \ volume 39 ,\ pages 402--408 ( year 2013 ) ,\ https:...

    doi

  34. [42]

    author author E. A. \ Kraut \ and\ author G. C. \ Kennedy ,\ title title New melting law at high pressures , \ https://doi.org/10.1103/PhysRev.151.668 journal journal Phys. Rev. \ volume 151 ,\ pages 668--675 ( year 1966 ) NoStop

    1966 doi

  35. [43]

    author author J. P. \ Perdew , author K. Burke ,\ and\ author M. Ernzerhof ,\ title title Generalized gradient approximation made simple , \ https://doi.org/10.1103/PhysRevLett.77.3865 journal journal Phys. Rev. Lett. \ volume 77 ,\ pages 3865--3868 ( year 1996 ) NoStop

    doi

  36. [44]

    author author S. A. \ Bonev , author E. Schwegler , author T. Ogitsu ,\ and\ author G. Galli ,\ title title A quantum fluid of metallic hydrogen suggested by first-principles calculations , \ https://doi.org/10.1038/nature02968 journal journal Nature \ volume 431 ,\ pages 669-...

    doi

  37. [45]

    Liu , author E

    author author H. Liu , author E. R. \ Hern \' a ndez , author J. Yan ,\ and\ author Y. Ma ,\ title title Anomalous Melting Behavior of Solid Hydrogen at High Pressures , \ https://doi.org/10.1021/jp403885h journal journal J. Phys. Chem. C \ volume 117 ,\ pages 11873--11877 ( y...

    doi

  38. [46]

    Zong , author H

    author author H. Zong , author H. Wiebe ,\ and\ author G. J. \ Ackland ,\ title title Understanding high pressure molecular hydrogen with a hierarchical machine-learned potential , \ https://doi.org/10.1038/s41467-020-18788-9 journal journal Nat. Commun. \ volume 11 ,\ pages 5...

    doi

  39. [47]

    Cheng , author G

    author author B. Cheng , author G. Mazzola , author C. J. \ Pickard ,\ and\ author M. Ceriotti ,\ title title Evidence for supercritical behaviour of high-pressure liquid hydrogen , \ https://doi.org/10.1038/s41586-020-2677-y journal journal Nature \ volume 585 ,\ pages 217--2...

    doi

  40. [48]

    Behler \ and\ author M

    author author J. Behler \ and\ author M. Parrinello ,\ title title Generalized neural-network representation of high-dimensional potential-energy surfaces , \ https://doi.org/10.1103/PhysRevLett.98.146401 journal journal Phys. Rev. Lett. \ volume 98 ,\ pages 1--4 ( year 2007 ) NoStop

    doi

  41. [49]

    author author A. D. \ Becke ,\ title title Density-functional exchange-energy approximation with correct asymptotic behavior , \ https://doi.org/10.1103/PhysRevA.38.3098 journal journal Phys. Rev. A \ volume 38 ,\ pages 3098--3100 ( year 1988 ) NoStop

    doi

  42. [50]

    Gorelov , author C

    author author V. Gorelov , author C. Pierleoni ,\ and\ author D. M. \ Ceperley ,\ title title Benchmarking vdW-DF first-principles predictions against Coupled Electron-Ion Monte Carlo for high-pressure liquid hydrogen , \ https://doi.org/10.1002/ctpp.201800185 journal journal ...

    doi

  43. [51]

    author author R. C. \ Clay , author M. Holzmann , author D. M. \ Ceperley ,\ and\ author M. A. \ Morales ,\ title title Benchmarking density functionals for hydrogen-helium mixtures with quantum Monte Carlo: Energetics, pressures, and forces , \ https://doi.org/10.1103/PhysRev...

    doi

  44. [52]

    author author R. M. \ Martin , author L. Reining ,\ and\ author D. M. \ Ceperley ,\ @noop title Interacting Electrons: Theory and Computational Approaches \ ( publisher Cambridge University Press ,\ year 2016 ) NoStop

    2016

  45. [53]

    Chiesa , author D

    author author S. Chiesa , author D. M. \ Ceperley ,\ and\ author S. Zhang ,\ title title Accurate, Efficient, and Simple Forces Computed with Quantum Monte Carlo Methods , \ https://doi.org/10.1103/PhysRevLett.94.036404 journal journal Phys. Rev. Lett. \ volume 94 ,\ pages 036...

    doi

  46. [54]

    author author M. A. \ Morales , author J. M. \ McMahon , author C. Pierleoni ,\ and\ author D. M. \ Ceperley ,\ title title Nuclear Quantum Effects and Nonlocal Exchange-Correlation Functionals Applied to Liquid Hydrogen at High Pressure , \ https://doi.org/10.1103/PhysRevLett...

    doi

  47. [55]

    author author P. M. \ Celliers , author M. Millot , author S. Brygoo , author R. S. \ McWilliams , author D. E. \ Fratanduono , author J. R. \ Rygg , author A. F. \ Goncharov , author P. Loubeyre , author J. H. \ Eggert , author J. L. \ Peterson , author N. B. \ Meezan , autho...

    doi

  48. [56]

    note The data and code can be found in the following url https://qmc-hamm.hub.yt/index.html NoStop

  49. [57]

    Pierleoni \ and\ author D

    author author C. Pierleoni \ and\ author D. Ceperley ,\ title The coupled electron-ion monte carlo method , \ in\ https://doi.org/10.1007/3-540-35273-2_18 booktitle Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology Volume 1 ,\ editor edited b...

    doi

  50. [58]

    Giannozzi , author S

    author author P. Giannozzi , author S. Baroni , author N. Bonini , author M. Calandra , author R. Car , author C. Cavazzoni , author D. Ceresoli , author G. L. \ Chiarotti , author M. Cococcioni , author I. Dabo , author A. Dal Corso , author S. de Gironcoli , author S. Fabris...

  51. [59]

    Giannozzi , author O

    author author P. Giannozzi , author O. Andreussi , author T. Brumme , author O. Bunau , author M. Buongiorno Nardelli , author M. Calandra , author R. Car , author C. Cavazzoni , author D. Ceresoli , author M. Cococcioni , author N. Colonna , author I. Carnimeo , author A. Dal...

  52. [60]

    author author A. P. \ Thompson , author H. M. \ Aktulga , author R. Berger , author D. S. \ Bolintineanu , author W. M. \ Brown , author P. S. \ Crozier , author P. J. \ in 't Veld , author A. Kohlmeyer , author S. G. \ Moore , author T. D. \ Nguyen , author R. Shan , author M...

  53. [61]

    Ceriotti , author J

    author author M. Ceriotti , author J. More ,\ and\ author D. E. \ Manolopoulos ,\ title title i-pi: A python interface for ab initio path integral molecular dynamics simulations , \ https://doi.org/10.1016/J.CPC.2013.10.027 journal journal Comput. Phys. Commun. \ volume 185 ,\...

    2013 doi

  54. [62]

    a np \" a \

    author author J. Kim , author A. D. \ Baczewski , author T. D. \ Beaudet , author A. Benali , author M. C. \ Bennett , author M. A. \ Berrill , author N. S. \ Blunt , author E. J. L. \ Borda , author M. Casula , author D. M. \ Ceperley , author S. Chiesa , author B. K. \ Clark...

  55. [63]

    a np \" a \

    author author P. R. C. \ Kent , author A. Annaberdiyev , author A. Benali , author M. C. \ Bennett , author E. J. \ Landinez Borda , author P. Doak , author H. Hao , author K. D. \ Jordan , author J. T. \ Krogel , author I. Kyl \" a np \" a \" a , author J. Lee , author Y. Luo...

  56. [64]

    Lin , author F

    author author C. Lin , author F. H. \ Zong ,\ and\ author D. M. \ Ceperley ,\ title title Twist-averaged boundary conditions in continuum quantum monte carlo algorithms , \ https://doi.org/10.1103/PhysRevE.64.016702 journal journal Phys. Rev. E \ volume 64 ,\ pages 016702 ( ye...

    doi

  57. [65]

    Grinberg , author N

    author author I. Grinberg , author N. J. \ Ramer ,\ and\ author A. M. \ Rappe ,\ title title Transferable relativistic dirac-slater pseudopotentials , \ https://doi.org/10.1103/PhysRevB.62.2311 journal journal Phys. Rev. B \ volume 62 ,\ pages 2311--2314 ( year 2000 ) NoStop

    doi

  58. [66]

    Holzmann , author R

    author author M. Holzmann , author R. C. \ Clay , author M. A. \ Morales , author N. M. \ Tubman , author D. M. \ Ceperley ,\ and\ author C. Pierleoni ,\ title title Theory of finite size effects for electronic quantum Monte Carlo calculations of liquids and solids , \ https:/...

    doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.