Influence of nuclear quantum effects on metastable states of warm dense hydrogen

Abstract

Previously, it has been effectively shown that nuclear quantum effects (NQE) considerably influence the prediction of the warm dense hydrogen phase diagram. Here, we study the influence of NQE on the parameters of the conducting state formation in warm dense hydrogen using path integral molecular dynamics (PIMD) within the framework of the finite temperature density functional theory approach (FTDFT). Both molecular and conducting metastable branches are obtained along the 700 and 1000 K isotherms in the PBE functional, as well as 500 K in the BLYP functional, which are expected for the first order phase transition [Landau and Lifshitz. Statistical Physics. Vol. 5. Elsevier, 2013]. Proton-proton pair correlation functions (PCF) and conductivity are calculated to diagnose metastable states. The results are compared with isotherms obtained for “classical” protons within the framework of quantum molecular dynamics (QMD). The latent heat of the transition to the conducting state is estimated based on the calculation of pairwise entropy. Strong overlapping of metastable and stable branches of the isotherm points is found in warm dense hydrogen.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-48839-y.

Introduction

The nature of the formation of a conducting state in fluid hydrogen is still a controversial issue^1–7. Although most theoretical studies have suggested that this is a first-order phase transition^2,8–22, experimental data do not provide a clear answer. Some measurement results have demonstrated features of a first-order phase transition^23–29, while others only have shown a sharp increase in conductivity and reflectivity^30–32. The conducting state of fluid hydrogen was achieved by dynamic compression at a pressure of 140 GPa and a temperature around 2600 K in experiments with lifetimes³⁰. Direct experimental observation of a first-order phase transition in shock-compressed fluid hydrogen was claimed in works^23,28, where an almost density discontinuity was obtained, and the conductivity increased sharply by approximately 5 orders of magnitude along the isentrope. While the initial experiment²³ showed the flattening of the isentrope’s slope based on the equation of state calculated within the chemical models of plasma, the updated measurement results from the work of have revealed the existence of the plateau on the isentrope. However, the initio simulation of thermodynamic properties of warm dense hydrogen/deuterium^6,7 under conditions considered in the experiment²³ has not revealed such a density discontinuity. The experiments^24–26, which employ diamond anvil cells (DAC) for the static compression of fluid hydrogen, also focused on observing the first-order phase transition that separates two distinct coexisting phases of hydrogen with different densities and conductivities.

Properties of warm dense hydrogen are investigated by means of the QMD and quantum simulation techniques³ based on density functional theory (DFT) with different exchange–correlation functionals, such as PBE³³, vdW-DF1³⁴, vdW-DF2³⁵, BLYP^36,37, HSE³⁸, SCAN³⁹ and also PIMD based on the DFT: DFT with PBE in the works^{10,16,17,19,20}, HSE in the work of Bergermann et al.⁴⁰, SCAN and SCAN-L with long-range dispersion rVV10 part²¹, PIMD with PBE by Morales et al.¹¹ and with vdW-DF2 by Morales et al.¹³, PIMD and BLYP²². Simulation of conducting fluid hydrogen is also carried out within the framework of various Monte Carlo methods, such as Quantum Monte Carlo¹⁵, Path Integral Monte Carlo (PIMC)^41–43 and Coupled Electron–Ion Monte Carlo (CEIMC)^14,18,44, in which evidence of a weak first-order thermodynamic transition is observed, although without the formation of stable aggregates such as ⁴⁵. Among first-principles modeling methods, quantum Monte-Carlo methods have better balance accuracy and computational resources when studying the phase transition.

in fluid hydrogen. This approach does not have the disadvantage of the DFT associated with the strong dependence of the results on the type of exchange–correlation functional, in particular, the work by Gorelov et al.⁴⁶ suggests that the agreement with experiment of optical properties in DFT calculations may be due to a fortunate (essentially coincidental) error cancellation. In all these studies, a first-order phase transition has been diagnosed by a plateau on the isotherm and a sharp increase in electrical conductivity, which associates these results with experimental observations from the Refs.^{23,27,28,30,31}. It has also been noted in the Refs.^10,11,13 that hydrogen transitions from a molecular to an atomic state.

The analytical thermodynamic models for the equation of state for hydrogen in the region of the fluid–fluid phase transition has been suggested in Ref.⁴⁷ using reaction energy with six parameters and fitting it to existing experimental data and the DFT calculation results. Similar approach is also used in Ref.¹ to determine accurate values of the critical point for the results of the molecular dynamic modeling. More detailed phenomenological approach employing description of microscopic processes of chemical bonding and repulsion in dense system with possibility of the formation of dimer species has been presented in recent work⁴⁸.

Since the density jump on the isotherm obtained in most theoretical works within the QMD approach does not exceed , metastable states are an important argument in favor of a first-order phase transition. The existence of metastable states accompanying the plasma phase transition (PPT) has been predicted in the papers^49,50, where the transition has been studied within a simple chemical model of plasma. The first-order PPT suggested in Ref.⁵¹ results from the competition among the long-range effective Coulomb attraction, the short-range strong quantum repulsion, and temperature. In addition to the difference in specific volumes, the PPT is related to a sharp change in the degrees of ionization, which leads to a difference in conductivity as well. The isotherms at the PPT region of parameters differ from the classical van der Waals profile by a relatively small density jump and a sloped shape with a significant overlap in density of two branches of metastable states. A modern chemical model provides qualitatively the same result⁵².

The initio studies mentioned above either do not consider metastable states or have found some indications of the metastability, but only in the regions close to the plateau without any attempts to reach the spinodal points. The method for calculating metastable states with the QMD approach has been suggested in Ref.⁵³, where the metastable branches for molecular fluid hydrogen have been obtained along the 700 and 1000 K isotherms. The profile of the isotherms, according to Ref.⁵³, consists of a strong overlapping of the equilibrium and molecular metastable branches, which indicates the similarity of the fluid–fluid phase transition in warm dense hydrogen with the PPT.

It should be noted that the hydrogen atom is a “light” particle with a de Broglie thermal wavelength close to 0.66 at a temperature of 700 K⁵⁴, which is comparable to the atomic separation in the hydrogen molecule of . Therefore, classical treatment of nuclei could be inapplicable for fluid hydrogen at relatively low temperatures, and it is necessary to consider the proton as a quantum particle. The PIMD approach allows for taking into account the influence of NQE associated with the zero-point energy (ZPE) in the hydrogen molecule on the thermodynamic and structural properties of fluid hydrogen as it has been shown in Refs.^11,13,21,22.

This work aims to assess only the influence of NQE on the region of existence of metastable states and the parameters of the phase transition in fluid hydrogen. It should be noted that the results obtained in this work are qualitative rather than quantitative, and we do not attempt to find the accurate location of the phase boundary on the phase diagram. The results include the calculated equation of state with metastable branches and pair correlation functions (PCFs), phase diagram, latent heat, and electrical conductivity. The main modeling methods and parameters are presented in a section below, where one could also find a brief description of the PIMD approach, parameters of the DFT calculations, the method of metastable states exploration, basic relations for the pairwise entropy and electrical conductivity.

Results

Isotherms and metastable states

We verify our approach by carrying out simulations at the same conditions as in Ref.¹³. Morales et al. calculate the isotherm at 1000 K using VASP with the vdW-DF2 functional applying PIMD type of dynamics for protons and provide the pair correlation function for hydrogen atoms. For the vdW-DF2 functional, hydrogen is still in the molecular fluid state at the density . They show that the treatment of NQE makes the molecular peak less pronounced than in the classical case. The same result for the PCF is obtained in this work using PIMD + VASP with the vdW-DF2 functional. The results are shown in Fig. 1(a): black squares are taken from Morales et al.¹³, and the red curve corresponds to the current work.

Fig. 1.

The pair correlation function for hydrogen fluid at from Morales et al.¹³ (black circles), obtained using i-PI + VASP with vdW-DF2, and our results, obtained using pimd + VASP with vdW-DF2 (red curve), are shown on the top left part of Figure. PCFs for the stable (black), metastable molecular (olive) and conducting (dark blue) branches at , calculated using i-PI + VASP with PBE, are presented on the bottom left part. The following two figures represent the isotherms at 1000 K (left) and 700 K (right) calculated with the NQE using PIMD + VASP with PBE (stable states—filled circles, metastable states—open circles) and from Norman et al.⁵³ calculated for classical nuclei using VASP with PBE (stable states—filled squares, metastable states—open squares). The last figure shows the isotherm at 500 K obtained with the NQE using PIMD + VASP with BLYP.

After verifying the approach, two isotherms at temperatures 700 and 1000 K are obtained using PIMD + VASP with the PBE functional. The results are shown in Fig. 1c,d compared with⁵³. The treatment of NQE reduces the phase transition pressure by approximately 50 GPa , which is in agreement with the results obtained in the works^11,13 Both molecular and conducting metastable branches are obtained here. The conducting non-molecular metastable branch does not appear in our previous simulations with classical nuclei in the paper⁵³ and other papers.

It should be noted that our isotherms obtained within PIMD + PBE approach with strong overlap of stable and both metastable branches are qualitatively similar to results of the QMD simulation with classical protons from Ref.⁵³ as well as to the isotherms provided by the chemical model of plasma⁵². Such profile of the isotherm initially predicted in Ref.⁴⁹ is one of the key indications to the PPT mechanism of formation of conducting fluid.

The structure of hydrogen along the stable and metastable branches is confirmed by the shapes of PCFs, as shown in Fig. 1(b). As one can see from Fig. 1(c) for the specific volume of (which corresponds to the density of ) at , it is possible to examine PCFs for stable and metastable states (both molecular and non-molecular conducting). The stable and molecular metastable PCFs have a distinct first peak at a distance of , which coincides with the proton separation in the hydrogen molecule and confirms its molecular structure. While the first peak of the metastable conducting state is considerably shifted to higher distances, this is a direct indication of a sharp decrease in stable hydrogen molecules in the conducting state.

Notably, we observe the collapse of metastable branches and reach spinodal points, corresponding to the condition . The location of these points does not depend on the number of particles, as shown in the papers^55–57. Thus, the total size of the metastable region in pressure values, including the molecular metastable and non-molecular conducting metastable regions, can be estimated. The values of and are counted up and down from the phase equilibrium pressure, respectively, and are restricted by spinodal points at or the last reachable metastable point for at . The value of at 700 K is 37 GPa , including and . At the higher temperature of 1000K , the total size and molecular metastable region decrease to and , while the non-molecular metastable region increases to . Therefore, at , the molecular and conducting metastable regions become almost symmetrical relative to the plateau on the isotherm.

Comparison of the sizes with our previous results from Ref.⁵³ obtained for classical nuclei at the same temperature reveals that the inclusion of the NQE leads to a decrease in the molecular metastable region from 47 to 30 GPa at and from 32 to 24 GPa at . However, if we compare our isotherms for the PIMD at and for the QMD at , we notice that these isotherms almost coincide in both the position of the plateau and the value of . Therefore, the influence of the NQE is related to a temperature shift of the phase equilibrium curve by the value of toward lower temperatures rather than a pressure decrease, as suggested in Refs.^11,13.

According to the data presented in works^22,58 the BLYP functional provides results that are closer to the Quantum Monte Carlo in comparison with PBE. In the present work we used the BLYP functional in order to perform the simulations with.

PIMD and compute the 500 K isotherm, which is shown in Fig. 1(e). The isotherm demonstrates the plateau at the pressure 315 GPa and discontinuity in the specific volume, which can be considered as a clear evidence of the first-order phase transition. The phase transition for the BLYP functional is considerably weaker than for PBE, and, therefore, it is more difficult to diagnose it. Despite a small jump in the specific volume the isotherm obtained shows wide range of pressures from 280 up to 350 GPa , which correspond to the metastable region. Metastable regions on the isotherm have been obtained following exactly the same procedure used for PBE. However, here we couldn’t reach the spinodal points where 0.

For each state point (fixed specific volume and temperature) we analyzed equilibrated trajectories. The position of each H atom in a frame was taken as its centroid: for PIMD this is the bead-average of the ring polymer, ; distances were computed with periodic boundary conditions. Association was quantified by a single-linkage cluster analysis: at a given linking length , atoms and are connected if , which partitions all atoms into disjoint clusters of size . From each trajectory we accumulated the time-averaged cluster-size fractions . The cutoff was chosen as , the first minimum of the equilibrium radial distribution function computed from the same trajectories, which separates intramolecular pairs from the first coordination shell. The quantity plotted in Fig. 2 is : the time-averaged fraction of atoms belonging to size- 2 clusters at .

Fig. 2.

The fraction of molecules vs. specific volume for two exchangecorrelation functionals at 700 K for PBE and at 500 K for BLYP. The stable points are shown with filled points, while metastable extensions of both branches are shown with open circles.

In the molecular fluid this geometric criterion gives . Upon decreasing specific volume and crossing into the conducting regime, drops rapidly yet remains nonzero, reflecting transient dimers that persist in the metallic fluid. We did not impose a minimum lifetime for HH bonds; adding a dynamical persistence criterion a posteriori would be expected to reduce further in the conducting phase.

Phase diagram

The phase diagram of warm dense hydrogen/deuterium is shown in Fig. 3 for the region of pressures and temperatures where the formation of conducting fluid hydrogen is diagnosed both theoretically and experimentally. The difference between hydrogen and deuterium is essentially associated with NQE because in classical description of nuclei the phase coexistence P–T lines should coincide⁵⁹.The black solid line—the experimental melting curve of hydrogen⁶⁰. The results obtained in this work are shown via red stars. The blue shaded area is the region of metastable states obtained. The pink shaded area corresponds to the region of metastable molecular fluid hydrogen obtained in Ref.⁵³ for the stable phase equilibrium curve calculated in Ref.¹⁰ (orange line) using quantum molecular dynamics method with the PBE exchange–correlation functional. Theoretical data are depicted by dots connected by lines. Squares correspond to the results of the PIMD approach with the CEIMC method for the exchange–correlation energy from Ref.¹⁴. The half-filled black squares represent the results obtained.

Fig. 3.

Phase diagram of warm dense hydrogen. Results from this work (red stars, blue metastable region) are compared with the experimental data for hydrogen and deuterium and ab initio calculations for hydrogen from the literature. The pink area shows a metastable region from Ref.⁵³.

by Geng et al.¹⁹ for classical protons within the QMD + PBE approach. Diamonds are the simulation results from²² for quantum protons within the PIMD approach with the BLYP exchange–correlation functional. Experimental data are shown by points. The orange filled circle corresponds to the minimum metallic conductivity measured in Ref.³⁰. Octagons correspond to the plateau on the isentrope observed in the experiment of Fortov et al.²³. The half-filled green points are results obtained at the NIF²⁹ corresponding to the beginning of absorption (squares) and a sharp increase of reflectivity (circles). The downward red filled triangles correspond to the experimental data from²⁷ obtained at the Sandia Z machine. The purple filled pentagons represent data measured in the experiment with the DAC²⁶.

All data related to the phase boundary between insulating molecular and conducting non-molecular hydrogen (both experimental and theoretical) presented in Fig. 3 are above the melting curve (solid black line), which corresponds to the experiments⁶⁰ approximated by the empirical Kechin’s equation⁶¹. A first-order fluid–fluid phase transition in hydrogen is observed below the critical temperature .

Our results for the phase equilibrium curve calculated within the PIMD approach are depicted by stars, which correspond to the values of pressure on the plateau on the isotherm shown in the previous subsection Isotherms and metastable states. The critical temperature estimated in this work, , is less than the value obtained in Ref.¹⁰ within the QMD method for the same PBE exchange–correlation functional.

The blue shaded area covers the region of metastable states: molecular metastable hydrogen is to the right of the phase boundary. Conducting non-molecular metastable states are also obtained in this work, which differs from⁵³, where only molecular metastable states were obtained within the QMD approach without considering NQEs.

The fluid–fluid phase boundaries obtained within initio methods shown in Fig. 3 (QMD + PBE)^10,19 (PIMD + BLYP)²², (PIMD + CEIMC)¹⁴ including this work (PIMD + PBE) are characterized by a discontinuity in volume and electrical conductivity. While these approaches give qualitatively similar results, the location of the phase equilibrium curve strongly depends on the method of approximating the exchange–correlation energy, which can shift the phase boundary by a pressure value as high as 100 GPa .

Heat of transition and pairwise entropy

The latent heat of transition can be estimated from the Clausius-Clapeyron ( ) relation:

1

where is the change in specific volume. Thus, is calculated using the values from Fig. 3 (red stars). Morales et al.¹¹ obtained the heat of phase transition using QMD with the PBE functional and quantum Monte Carlo (QMC). The value of at 1000 K lies in the region of atom. Tian et al.²⁰ estimated the heat of transition at 1000 K using the same method. The value varies within atom, depending on the size of the unit cell. In this work, we obtain a value of equal to 0.031 eV . This result is closer to Ref.²⁰, since convergence with respect to calculation parameters¹⁰ is achieved. Additionally, the latent heat can be obtained directly from the enthalpy change: , considering the density jump at the phase transition pressure. This method yields a value of 0.034 eV , which matches the result of the C–C estimation.

Equation (5) is widely used for estimating the excess entropy change and building correlations with dynamical properties (e.g.,^62–64). We are interested in applying this equation to obtain the pairwise entropy at 700 K and 1000 K at different densities. The PCFs and the integrals (5) for both 700 K and 1000 K are provided in the Supporting Information. The resulting values of the dependence are presented at the bottom of Fig. 4, combined with the pressure dependencies (on top). The red and blue areas indicate the molecular and conducting fluids coexistence regions.

Fig. 4.

The pressure (top) and pairwise entropy (bottom) dependencies on density for two isotherms at 700 K (blue) and 1000 K (red), obtained in the PBE functional. The areas show the molecular and conducting fluids coexistence regions.

The pairwise entropy monotonically grows as the density increases. The rapid pair entropy growth can be explained by the ionization and dissociation of molecules. At the transition pressure, the pairwise entropy exhibits a significant change. This change can be used to calculate the pairwise contribution to the heat of transition, . The values of are equal to 0.020 eV /atom at 700 K and 0.022 eV /atom at 1000 K . Thus, we conclude that the latent heat consists of pairwise contribution . Higher-order contributions , etc., could arise after the phase transition, since Geng et al.¹⁹ have pointed to the formation of clusters.

By applying the pairwise entropy calculation method in this work, we directly connect the structural changes during the phase transition and the latent heat. The thermodynamic methods used previously do not provide such a relation.

Electrical conductivity

The electrical conductivity is also a marker of the phase transition from the molecular to the conducting fluid. It increases with increasing pressure, and the transition value of has been observed for this process.

Electrical conductivities have been previously calculated across the isotherms using both QMD and PIMD^{10,11,13,19,53}. In 2010, Morales et al.¹¹ compared QMD, the CEIMC method, and the PIMD approaches in predicting across the 1000 isotherm. All the methods represented a significant increase in at different phase transition pressures. Tian et al.²⁰ used the PBE functional with classical nuclei and obtained a sharp increase in conductivity at and 1000 K. Recently, van de Bund and colleagues²² calculated the conductivity for hydrogen and deuterium from snapshots taken from the PIMD trajectories. They showed that the conductivity steeply increases by two orders of magnitude at the transition pressure, from (molecular phase) to (conducting phase). However, they did not report the electrical conductivities at metastable branches.

Previously, in Ref.⁵³, we predicted conductivity values for the molecular phase along the metastable branches for classical nuclei. The values increased with pressure, and a difference between equilibrium and metastable states of several orders of magnitude was observed.

Figure 5 shows the results on the electrical conductivity at , obtained in this work. The calculations of electrical conductivity were carried out for the PIMD trajectories at stable states and the RPMD trajectories at metastable states. The values of were averaged over different beads and ionic configurations for each bead. We also observed a significant growth of along the stable part of the isotherm, which corresponds to the transition to the conducting phase. However, the increase in along the metastable branches (both molecular and non-molecular) was more gradual.

Fig. 5.

The electrical conductivity dependence on pressure at 700 K (PBE) and 500 K (BLYP) for the results with quantum nuclear effects (open and filled circles), as well as the results from the work⁵³ (open and filled squares).

The jump in conductivity was much more pronounced for the results obtained at within the QMD approach in Ref.⁵³ compared to the PIMD data at the same temperature, as shown in Fig. 5. However, the QMD dependence of on pressure at was much closer to the PIMD conductivities at . This is an additional argument, along with the phase equilibrium pressure and the size of the molecular metastable region, which points to a temperature shift by the value due to the inclusion of the NQE.

Comparison of the conductivities obtained within the PIMD and the QMD for the metastable molecular hydrogen shows that the inclusion of the NQE considerably increases the value of by two orders of magnitude compared to the system with classical protons. Therefore, the metastable molecular state also becomes conducting, with values approaching those of the non-molecular stable states as pressure increases.

The calculations with the BLYP functional also reflect the significant growth by the orders of magnitude of the electrical conductivity at the transition pressure. These results for the 500 K isotherm are shown with green filled circles in Fig. 5. Also, we are able to find both metastable branches—molecular and conducting, shown with open circles. The observation of the electrical conductivity jumps in both exchange–correlation functionals—PBE and BLYP—strengthens the idea about the existence of phase transition in warm dense hydrogen in this area of phase diagram.

Discussion

The path integral molecular dynamics and quantum simulation techniques based on density functional theory are applied to study warm dense hydrogen in the range of temperatures and densities where the phase transition is observed experimentally. We calculate the equation of state, including metastable states, proton-proton pair correlation functions, the static fraction of atoms combined in hydrogen molecules, pairwise entropy and latent heat, as well as the electrical conductivity. The calculations are performed for two exchange–correlation functionals, PBE and BLYP.

Based on the analysis of the obtained results, a method based on the PIMD approach with DFT for diagnosing metastable states is developed. Molecular and non-molecular conducting metastable states are demonstrated along isotherms at 700 K and 1000 K for the PBE functional and along the isotherm at 500 K for BLYP. At 1000 K , the metastable branches reach the spinodal points, and therefore the size of the metastable region is estimated to be 24 GPa . At 700 K , the size of the metastable region is 37 GPa , but the spinodal point is reached only for the conducting state. For the BLYP functional, the spinodal points are not reached within the studied range, and the size of the metastable region can be roughly estimated to be at least 80 GPa .

The heat of transition is calculated using three techniques: enthalpy change, Clausius-Clapeyron relation, and pairwise entropy. The last method of latent heat calculation is used for the first time. The dependence exhibits a rapid increase in the region of density discontinuity on the isotherm. This effect is directly related to the structural changes at the transition due to the decomposition of molecules. The quantity provides the major contribution ( atom ) to , while the total averaged over the Clausius-Clapeyron relation and enthalpy change is equal to atom . The inclusion of nuclear quantum effects does not affect the value of the latent heat.

The electrical conductivity increases sharply at the stable part of the isotherm at 700 K , indicating the phase transition from the molecular to the conducting state with values exceeding . This result is consistent with the available experimental data³⁰. Along the metastable branches, the conductivity grows more gradually. The dependence obtained in this work within the PIMD approach at 700 K lies close to the QMD results reported at .

The isotherms obtained in this work exhibit a substantial overlap between the stable and metastable branches, which is qualitatively similar to the results obtained for classical protons within QMD in previous studies⁵³. This similarity suggests the plasma nature of the phase transition in fluid hydrogen. Quantitatively, by comparing the present results, including the equation of state, the size of the molecular metastable region, and the pressure dependence of electrical conductivity obtained.

within the PIMD framework, with the corresponding QMD data for the PBE functional, it is found that the treatment of nuclear quantum effects is equivalent to an effective temperature shift of approximately 300 K .

Methods

Path integral molecular dynamics

The NQE are treated using the PIMD methods^66,67. This family of methods is based on the isomorphism between the approximate quantum partition function of particles and the classical partition function of particles with the following Hamiltonian:

2

where is the number of beads (Trotter slices) in the quantum polymer, and are a fictitious momentum and mass, respectively, , and is the interaction potential between beads with the same .

Each polymer ring represents a single atom, which becomes distributed via this ring . The interatomic potential is realized such that each bead of a polymer ring acts only with the corresponding bead of another polymer ring. The interaction between beads inside a polymer is presented via a harmonic potential that acts only between nearest neighbors.

Calculations of the electronic structure and forces acting on beads are carried out within the framework of finite temperature density functional theory (FTDFT)⁶⁸ in the Vienna Ab-initio Simulation Package (VASP) v. 5.3.5^69–72, recently applied for various materials and structures (e.g.^73–76 ). The forces are calculated by the Hellmann–Feynman theorem. The exchange-correlation functional chosen is the Perdew-Burke-Ernzerhof (PBE) functional³³. The plane wave basis set with an energy cutoff of 1200 eV is used for solving the Kohn–Sham equations. The computational cell contains 512 hydrogen atoms. The k -mesh consists of the Baldereschi mean value point⁷⁷. The validity of the PBE functional and the sufficiency of a single k-point for the 512-atom supercell are checked in Ref.¹⁰ for the pressure calculation.

It should be noted that application of the non-local exchange correlation functions, such as vdw-DF1 or vdw-DF2, instead of local PBE functional allows taking into account the dispersion interaction, which increases the pressure and density range of the phase equilibrium^27,78. The approach based on PIMD and DFT with non-local exchange correlation functionals provides better agreement with the data measured in the experiments with dynamical compression in Sandia Z-machine²⁷ and National Ignition Facility (NIF)²⁹ in comparison with the PBE. However, even such advanced mentioned earlier as well as methods based on SCAN²¹ and HSE⁴⁰ exchange–correlation functionals do not provide complete description of the experimental data with dynamic compression of hydrogen/deuterium. In particular none of such approaches provide the slope of the phase equilibrium curve suggested in the experiments. On the other hand, the data measured with static compression in the DAC^24–26 are in much better agreement with the results of the DFT with different types of the exchange–correlation functionals, including PBE. Since PBE provides the lowest computation cost in comparison with the mentioned functionals, in the current work we have obtained the majority of results with using of this functional. Along with PBE we have done some calculations also with BLYP functional which has been used previously in the work²² for the theoretical study of the phase transition in fluid hydrogen.

The calculation of the equilibrium part of the equation of state is performed within the PIMD (8 beads) in the canonical NVT ensemble using massive Nosé-Hoover thermostat chains^79–81 with 4 chains, the integration time step is 0.5 fs. The pressure values are averaged along the MD trajectory within each bead slice and finally are also averaged over beads. The MD trajectory consists of between 5000 and 10,000 time steps, which correspond to from 2.5 to 5 ps.

Both the PIMD and the RPMD calculations could be performed in the ⁸² and PIMD⁸³ software packages. These packages are widely used for treating NQE. For example, i-PI has been applied for determining the insulator–metal transition boundary in warm dense hydrogen²¹, studying solid molecular hydrogen using a machine-learned potential⁸⁴, and investigating the stability of ⁸⁵. The PIMD package has been used for studying quantum and temperature effects on the crystal structure of ⁸⁶ and describing hydrogen diffusivity in metals⁸⁷.

The PIMD package has two options for interacting with VASP: a client-host (as i-PI) or as a binary built with VASP inside. The latter option makes it very user-friendly and outperforms i-PI in terms of computational efficiency. We carry out verification for hydrogen fluid at and , available from the work of David Ceperley and colleagues¹³.

Metastable states exploration

The approach for calculating metastable states within the QMD has been initially proposed in Ref.⁵³, which employs the ideas from classical MD or Monte-Carlo simulations^88–91. Depending on the type of metastable states (molecular or conducting), the first metastable point on the isotherm can be obtained using the configuration of the stable molecular/conducting hydrogen.

with a changed supercell volume: increasing for the conducting state and decreasing for the molecular one. After relaxation, the final configuration of the new metastable trajectory is used as the initial one for another trajectory.

Figure 6 shows the evolution of pressure in the two-phase point in the NVT ensemble at and . Fluid hydrogen transits between molecular (red) and conducting (blue) states. The inset demonstrates PCFs averaged over the colored regions. The height of molecular peak varies depending on the phase. We reproduced the metastable branches of the isotherms by consequently increasing/decreasing the density by and relaxing every new configuration.

Fig. 6.

The pressure fluctuations at and during the simulation in the NVT ensemble within the PIMD + DFT(PBE) approach. The dashed line shows the average value, while the colored regions correspond to the metastable molecular (red) and conducting (blue) states. The inset demonstrates PCFs averaged over blue and red regions.

Approaching extreme spinodal points on the isotherms is a rather common procedure in modeling of different systems, where the inter-particle forces are calculated as derivatives of the potential energy by distances. However, it is not the case in our simulations, where the forces are calculated by the Hellmann–Feynman theorem at every step of the numerical integration of Newton’s equations. This procedure introduces additional perturbations that could limit the possibilities to model short-living metastable states.

Once the metastable states are obtained from the NVT run, they appear to be in the partial equilibrium during the NVE run with the corresponding lifetimes^89,90,92. Even the small perturbations from the thermostat could destroy the metastable states, as it is mentioned previously in various works and books, authored by Landau and Lifshitz⁹³, Skripov⁸⁸, Haile⁹⁴, Tester and Modell⁹⁵ and Depenedetti⁹⁶. The lifetimes of the metastable states seem to be small, so the accurate procedure of obtaining them should be applied. Because of that we run the simulations of metastable branches without the thermostat.

Another factor which decreases the lifetime of metastable states is the size of the system, since the increase of the volume increases the probability of the appearance of high-density phase nucleation centres. In the work⁵³ it has been shown that the computationally reachable size of the metastable region for the molecular branch of the isotherm at decreases with the increase of the simulation cell. For both considered systems the condition of zero derivative of pressure with respect to the specific volume was not fulfilled. The result shows that in order to reach the spinodal points in some cases the system size has to be decreased.

In the work by Lorenzen et al.¹⁰, the authors discovered the hysteresis at isotherm. However, these points are only the very beginning of the metastable branches. The end of metastable branches should meet the criteria . The points by Lorenzen et al. do not reach this limit. In this work, we are able to observe the limit at 1000 K for both molecular and conducting branches (discussed in the part Isotherms and metastable states). Also the indication to the existence of metastable states are obtained in works^14,21,40, however, only the initial points close to the plateau have been considered as well as in Ref.¹⁰.

Due to the reasons discussed above, the thermostat has to be turned off for calculating the metastable states. However, in the NVE part, we have to carefully monitor the change in the average temperature. In our calculations, the drift of average.

temperature does not exceed 15 K . We use Ring Polymer MD (RPMD)⁹⁷ with 8 beads, which treats physical masses as fictitious . This choice leads to the physical dynamics of all beads. The correlation functions in RPMD should be averaged over all beads. The original formulation of RPMD can be found in the Refs.^97,98.

Pairwise entropy

For a system in thermodynamic equilibrium at (number) density and temperature T, the entropy can be represented as the sum of the entropy of an ideal gas at the same density and temperature, , and the excess part. The ideal-gas part is, in effect, the contribution from uncorrelated particle motion, and the rest can be expanded in terms of two-particle, three-particle, etc. correlations^99,100:

3

The excess entropy, , is thus:

4

The two-particle contribution, , is in many cases the dominant contribution to the excess entropy^100,101. The three particle entropy typically constitutes of for atomic liquids and hard-sphere systems in a gas or liquid state and rises up to of the pair entropy when the system approaches the freezing point^102–104. For a homonuclear system, the pairwise entropy can be directly computed from the pair correlation function (PCF) even for polyatomic molecules¹⁰⁸ :

5

Electrical conductivity

The real part of the frequency-dependent dynamic conductivity for a given ionic configuration is defined by the Kubo-Greenwood formula^109,110

6

where is the system volume, is the electron mass, is the Planck constant. The summation is carried out over all electron states . The contributions of the sum terms with (intraband transitions) are taken into account as well as the contributions of terms with (interband transitions). Summation over index multiplied by is an averaging over three spatial coordinates. The summation is also carried out over all -points in the Brillouin zone, taking into account the weight of -point. The summation over -points provides non-zero contribution for the intraband part ( ) according to Refs.^111,112. The factor 2 before the weights takes into account the spin-degeneracy of the system. is the FermiDirac distribution function, which defines the occupation of state , at temperature T. is the energy level corresponding to the wave function .

The finite volume of the considered system leads to a discrete spectrum of the eigenvalues . Therefore, an approximation for the δ-function included in formula (6) has to be used. To solve this problem, the Gaussian function with a finite width is applied:

7

The value of is taken to be as small as possible without the appearance of local oscillations of arising due to the discrete spectrum of eigenvalues^111,113,114. In this work, the width was chosen to be 0.1 eV.

The static conductivity at ionic configuration is determined as the limit:

8

The KG4VASP package¹¹² is used for calculating the static conductivity based on formulas (6), (8). The conductivity values are computed also with using the PBE functional, as it has been applied in Ref.¹⁰. A Monkhorst–Pack¹¹⁵ -mesh.

is used, which allows for reaching the convergence of the results. The results are averaged over a set of ionic configurations and over all beads.

For each bead, the conductivity values are averaged over the 8 ionic configurations separated by 500 timesteps intervals at the very end of the PIMD trajectory. Thus, each value is averaged over 64 configurations. For each configuration, we minimize the conductivity as a function of the effective width of the delta function by the Fibonacci method. Thus, the resulting Gaussian smearing value varies in ranges from 0.1 to 1.2 eV depending on the ionic configuration.

Author contributions

I.S. and G.N. conducted research of the phase diagram and metastable states, did the electric conductivity calculations. V.L. and N.K. carried out the study of local structure, pair entropy behaviour, fractions of H₂$ molecules. All the authors contributed equally to the visualisation of results and writing of the manuscript. All authors reviewed the manuscript.

Funding

The study is supported by the Ministry of Science and Higher Education of the Russian Federation: phase diagram investigation and conceptualization within the State Assignment No. 075–00270-26–00 (G.N.); the study of local structure, pair entropy calculations, fractions of molecules within the Agreement No. 075–03-2026–305 16.01.2026 (V.L. and N.K.). N.K. acknowledges the framework of the Basic Research Program at HSE University for the optimization of ab initio calculations in VASP on NVIDIA Tesla V100, as well as the installation of the pimd package. I.S. is supported by the European Union—NextGenerationEU under the Italian Ministry of University and Research (MUR) projects PRIN2022 PNRRP2022MC742PNRR, CUP E53D23018440001 (methods, metastable states investigation, electric conductivity calculations).

Data availability

The input simulation scripts and the raw data that support the study could be reasonably requested from V.L. or N.K.

Declarations

Competing interests

The authors declare no competing interests.