Evidence for a First-Order Liquid-Liquid Transition in High-Pressure

Evidence for a first-order liquid-liquid transition in SEE COMMENTARY high-pressure hydrogen from ab initio simulations

Miguel A. Moralesa, Carlo Pierleonib,c, Eric Schweglerd, and D. M. Ceperleya,b,e,1

aPhysics Department, University of Illinois at Urbana-Champaign, Urbana, IL 61801; bInstitute of Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL 61801; cConsorzio Nazionale Italiano di Struttura della Materia (CNISM) and Physics Department, University of L’Aquila, Via Vetoio, 67100 L’Aquila Italy; dLawrence Livermore National Laboratory, Livermore, CA 94550; and eNational Center of Supercomputing Applications, Urbana, IL 61801

Contributed by David Ceperley, May 26, 2010 (sent for review February 11, 2010)

Using quantum simulation techniques based on either density accuracy is difficult to assess. In general, accurate temperature functional theory or quantum Monte Carlo, we find clear evidence measurements under these extreme conditions are notoriously dif- of a first-order transition in liquid hydrogen, between a low con- ficult to achieve. Moreover, the existing experimental data is ductivity molecular state and a high conductivity atomic state. sparse and a rapid, yet continuous, change in conductivity with in- Using the temperature dependence of the discontinuity in the creasing pressure cannot be completely ruled out from the data. electronic conductivity, we estimate the critical point of the transi- The majority of the theoretical techniques that have been used tion at temperatures near 2,000 K and pressures near 120 GPa. to study dense liquid hydrogen fall into two groups: methods Furthermore, we have determined the melting curve of molecular based on effective models (the chemical picture) and simulation hydrogen up to pressures of 200 GPa, finding a reentrant melting methods starting from protons and electrons (first-principles line. The melting line crosses the metalization line at 700 K and approach). In general, the effective models exhibit a clear first- order LLT that persists for temperatures well above 10,000 K 220 GPa using density functional energetics and at 550 K and – 290 GPa using quantum Monte Carlo energetics. (16 19). However, the presence of a first-order phase transition in the effective models is now recognized to be spurious (20) and the most recent implementations now specifically include phase transition ∣ quantum Monte Carlo ∣ density functional theory ∣ ad hoc terms to smooth out the transition region to ensure plasma phase transition ∣ melting positive compressibility (21). The vast majority of first-principles simulations of extended ince the pioneering work of Wigner and Huntington (1), on systems are performed using density functional theory (DFT). Sthe metallization of solid molecular hydrogen by pressure, Although realizations of DFTcontain many approximations, cur- there has been a great effort to understand the molecular dissocia- rent implementations provide a reasonable compromise between tion process in high-pressure hydrogen from both experiment and accuracy and computational requirements. At high pressures, theory. In the solid, at low temperatures, metallization has been where experimental results are scarce, more accurate electronic expected to occur in conjunction with a transition to a solid atomic structure methods are needed to assess the quality of DFT pre- state, although a transition to exotic phases such as quantum fluids dictions. In this work, we use the accurate quantum Monte Carlo (2) or metallic molecular phases may also be possible (3–5). (QMC) method to test the predictions of DFT in the dissociation For dense hydrogen in the liquid phase, metallization (prob- and metallization regime of the phase diagram of hydrogen at ably accompanied by molecular dissociation) can occur either high pressures. QMC has been shown to give very accurate results through a continuous process (a crossover) or through a sharp, in extended systems and currently has the capability to reach near first-order transition, often called the plasma phase transition. chemical accuracy in many systems (22). Numerous experiments have been performed in the liquid phase Although there are numerous publications of DFT simulation using dynamic compression techniques to measure both the prin- studies of hydrogen, there is currently no consensus about its pre- cipal Hugoniot in hydrogen and deuterium [using for example: dictions regarding the existence of a first-order transition in the gas guns (6), laser-driven compression (7–9), magnetically driven liquid. Two of the earlier studies using Car–Parrinello molecular flyers (10, 11), and converging explosives (12)] and to measure dynamics (MD), by Scandolo (23) and Bonev et al. (24), showed off-Hugoniot properties at lower temperatures [electrical conduc- evidence of a first-order transition, for (P ¼ 125 GPa, T ¼ tivity measurements using shock reverberation (13) and multiple 1;500 K) in the former and (P ¼ 52 GPa, T ¼ 2;800–3;000 K) – shocks (14), compressibility measurements using explosive-driven in the latter. The use of the Car Parrinello method in these si- generators (15), to mention a few]. The conductivity measure- mulations put in question the validity of their results because ments by Nellis and coworkers (13) produced the first evidence hydrogen metallizes at these conditions, which can potentially of minimum metallic conductivity in fluid hydrogen at a pressure led to problems with this approach. As a result, the majority of the subsequent first-principles simulations of hydrogen have of 140 GPa and temperatures on the order of 3,000 K. Until – recently, there were no experimental indications of a first-order relied on the use of Born Oppenheimer molecular dynamics (BOMD) to describe the onset of metallization. For example, liquid-liquid transition (LLT) in hydrogen. Even though results the BOMD simulations reported by Vorberger et al. (25) and could not rule out the existence of the transition and most studies Holst et al. (26) exhibited a gradual crossover with no evidence

had been performed at fairly high temperatures, there was no PHYSICS of a sharp transition, and a recent BOMD study by Tamblyn and sign of a sharp, first-order behavior. The only direct experimental Bonev (27) indicated that a first-order transition is likely but still evidence of a LLT is from the work of Fortov et al. (15) where an open question. In addition a previous exploration by coupled reverberating shocks produced with high explosives were used to ramp compress hydrogen, presumably reaching temperatures 3 in the range of 3–8 × 10 K. Using highly resolved flash X-ray Author contributions: M.A.M., C.P., E.S., and D.M.C. designed research, performed diagnostics, they were able to measure the compressibility of research, analyzed data, and wrote the paper. the liquid and found a 20% increase in density in the regime where The authors declare no conflict of interest. the conductivity increases sharply by approximately 5 orders of Freely available online through the PNAS open access option. magnitude. It is important to note that in these experiments the See Commentary on page 12743. temperature is estimated using a “confined atom” model whose 1To whom correspondence should be addressed. E-mail: [email protected].

www.pnas.org/cgi/doi/10.1073/pnas.1007309107 PNAS ∣ July 20, 2010 ∣ vol. 107 ∣ no. 29 ∣ 12799–12803 Downloaded by guest on September 27, 2021 3500 Results and Discussion Primary LLT - DFT Hugoniot LLT - QMC Liquid-Liquid Phase Transition. Fig. 2 shows a comparison of the 3000 Scandolo (2003) BOMD melting curve 1,000 K isotherm in hydrogen as a function of density for the dif- Eremets, et al. (2009) 2500 ferent simulation methods used here. A plateau in the pressure Critical point Bonev, et al. (2004) 0 02 ∕ 3 Datchi, et al. (2000) over a narrow range of densities (on the order of . g cm , Liquid H Gregoryanz, et al. (2003) which corresponds to a volume change of approximately 2%) 2000 Deemyad, et al. (2008) Weir, et al. (1996) is clearly seen in both CEIMC and BOMD simulation results, in- Mazin, et al. (1997) dicating the existence of a first-order transition. In this region 1500 ðdP∕dVÞ ¼ 0 Liquid H T , which implies a diverging isothermal compressi- 2 −1

Temperature (K) Coexistence bility, κT ¼ −V ðdV∕dPÞT . We observe similar behavior at T ¼ 1000 point 1;500 K for CEIMC and at T ¼ 1;500 K, 800 K, and 600 K for BOMD. The transition occurs systematically at higher pressures 500 Solid H2 I in CEIMC as compared to BOMD. This is reasonable given that III most DFTexchange-correlation functionals have band closure at 0 II 0 50 100 150 200 250 300 densities that are too low and tend to favor delocalized electronic Pressure (GPa) states over localized ones, presumably due to self-interaction er- rors (42). The estimated latent heat (enthalpy change per atom) Fig. 1. Phase diagram of hydrogen including the melting curve of the associated with the transition at T ¼ 1;000 K is 440 (90) K and molecular solid (phase-I) and the predicted liquid-liquid phase transition. 570 (160) K for the DFT and QMC transition lines respectively. The experimental measurements of the melting line are shown as red crosses – (28), green triangles (29), turquoise X’s (30), and violet stars (31). The melting The corresponding values obtained from the Clausius Clapeyron ΔH ¼ TΔVðdP ∕dTÞ line determined from Car–Parrinello DFT simulations is shown with orange equation LLT are 380 (90) K and 410 (90) K. triangles (32) and from BOMD (present work) with a solid black line. A fit In this case, the derivatives of the transition pressure with tem- of the BOMD melting curve to the Kechin formula (33) is shown with a perature are estimated numerically. There is good agreement dashed black line. The liquid-liquid transition determined from DFT and between the two independent estimations, which suggest that QMC calculations are shown with blue triangles and squares, respectively our results are thermodynamically consistent. The overall loca- (the blue lines are simply a guide to the eye). The dashed blue line is an tion of the transition region found here is in agreement with a extrapolation of the QMC transition line assuming a constant offset of recent BOMD study reported in ref. 27. However, in this case the QMC and DFT transition pressures. The blue circle corresponds to the ðdP∕dVÞ ¼ 0 volume discontinuity found with DFT simulations along the 1,500 K isotherm a T region was not found, most likely due to inade- in Ref. (23). The turquoise diamond is the onset of metallic conductivity in quate sampling across the transition region. reverberating shock wave measurements (13). Finally thin solid lines repre- It is interesting to note that as the transition region is sent the experimental measured transition lines between three crystalline approached, the simulations exhibit quite strong size effects. molecular phases at low temperature (34). To obtain converged results in the BOMD simulations (sampled electron-ion Monte Carlo method (CEIMC) (35) also suggested at the Γ-point), we had to use a system with 432 atoms. Our liquid the occurrence of a continuous molecular dissociation. These simulations with 432 atoms produced radial distribution functions studies have led to the general belief that the results reported that were in excellent agreement with larger system sizes, but in the earlier simulations were influenced by the use of the showed significant differences with 128 and 256 atom simula- Car–Parrinello method and that the transition region as de- tions. A similar observation of strong size effects was made in scribed by DFT is indeed smooth (36). ref. 27, which indicated that the orientational order of the liquid In this work we consider again the onset of molecular dissocia- is particularly sensitive as well. For a much smaller system size of tion in liquid hydrogen at high pressure. We employ first principle 64 atoms, we observed first-order behavior during the metalliza- simulations based on DFT within BOMD and QMC within the tion transition only when a 3 × 3 × 3 grid of k-points was used; CEIMC approach (37, 38) with improved trial wave function simulations at the Γ-point showed no evidence of significant (39, 40). To pinpoint a possible transition we trace isotherms 300 using a much finer grid in density than in previous studies. In 80

addition, we calculate the melting line of the molecular solid 70

using free energy calculations in both liquid and solid phases with 60 250 DFT. Fig. 1 shows the resulting phase diagram. A sharp, first- 50

order phase transition, is observed at both levels of theory. At Pressure (GPa) 40

high temperatures, the transition ends at a critical point around 0.45 0.5 0.55 0.6 0.65 T ≈ 2;000 K above which metallization is a continuous process 200 density (g cm-1) with pressure. The values of the transition pressure depend on the simulation method, with the more accurate predictions Pressure (GPa) coming from QMC simulations. At low temperature the DFT 150 transition line meets the melting curve in a liquid-liquid-solid multiphase coexistence point at approximately 700 K whereas CEIMC BOMD the extrapolation of the QMC transition line and the melting line PIMD 100 at higher pressures should meet at 290 GPa and 550 K. The 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 results presented here resolve the discrepancies among previous density (g cm-1) DFT calculations of hydrogen, reconcile the theoretical understanding of the metallization in hydrogen and the results of Fig. 2. Pressure isotherms of hydrogen at T ¼ 1;000 K from first principles the dynamical compression in the liquid phase (13, 15) and simulation methods. Blue asterisks and red squares are simulation results for the static compression* in the solid phase. classical hydrogen nuclei from CEIMC and BOMD, respectively. Black circles are results from DFT-based PIMD simulations. A plateau in the pressure is clearly present in all cases, indicating the existence of a sharp first-order *In the solid phase the highest pressure reported so far in the diamond anvil cell experi- transition. All simulation methods agree in the nature of the transition, ments is of 320 GPa ; the sample still remains in the insulating state (41). This pressure but disagree on the transition pressure. (Inset) Pressure isotherm from BOMD value is compatible with our estimate of the pressure of metallization extrapolated from simulations at T ¼ 3;000 K, where no evidence of first-order behavior is seen. QMC data at melting. Error bars are much smaller than the symbol sizes, typically <0.1 GPa.

12800 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1007309107 Morales et al. Downloaded by guest on September 27, 2021 discontinuities. In the case of the CEIMC simulations we use an overlap between molecules. Close to the transition, the molecular

integration over the boundary conditions of the simulation cell state is transient and very weakly bound; molecules form and SEE COMMENTARY with 64 (4 × 4 × 4) k-points. Simulations with 54 atoms produced break on small time scales of the order of the collision frequency radial distribution functions that were indistinguishable from (25, 27). Whereas the description of molecular and atomic simulations with 128 atoms and 64 k-points, for both the metallic fractions based on somewhat arbitrary bonding state criteria and insulating sides of the transition. has been used to describe the transition region, and has led to The simulation results presented so far have assumed that the conclude in favor of a continuous dissociation transition (35), protons are classical particles. To test the validity of this assump- we believe that it is more indicative to use the discontinuity of tion and to study its influence on the nature of the transition re- the electrical conductivity: It coincides precisely with the onset gion, we performed path integral molecular dynamics (PIMD) of the ðdP∕dVÞT ¼ 0 plateau. simulations with DFT at T ¼ 1;000 K. The path integrals were Fig. 4 shows the (time averaged) DC conductivities as a func- discretized with a time step of ð8;000 KÞ−1, which corresponds tion of pressure for CEIMC, BOMD, and PIMD simulations. The to 8 beads at T ¼ 1;000 K, enough to obtain converged results. calculated conductivities are in rough agreement with experimen- A pressure isotherm from the PIMD simulations is shown in tal observations, as also shown by previous simulations (26, 43). Fig. 2. The inclusion of zero point motion in hydrogen produces At 2,000 K and above, the conductivity increases smoothly as a a decrease in the transition pressure by approximately 40 GPa, function of pressure, but for T ≤ 1;500 K there is a sharp and but does not change the nature of the transition; a clear plateau, discontinuous increase in the conductivity at the transition. indicating a first-order transition still exists in the pressure The size of the discontinuity increases with decreasing tempera- isotherm. ture in both BOMD and CEIMC simulations. By extrapolating to Fig. 3 shows the proton-proton distribution function and elec- the pressure where the size of the discontinuity in conductivity tronic conductivity at the two ends of the transition for the DFT goes to zero, we can estimate the critical point of the transition. isotherm at T ¼ 1;000 K. The transition has strong effects on Above this temperature, the conductivity and the dissociation as a electronic properties, in particular, those related to electron function of density is continuous. At lower temperatures, a first- localization. The molecular fraction changes slowly across the order transition is encountered as we go across the dissociation transition; in fact, partial dissociation is seen at densities well be- regime with a volume discontinuity and a sharp metallization of low the transition. The molecular state in these conditions is very the liquid. The transition corresponds to a sharp change from different from that found at lower densities where there is no semiconducting to metallic behavior caused by the sudden collapse of the molecular state. Even though strong short range correlations persist after the transition is crossed, the resulting 2 14 atomic liquid is metallic. Our findings are in agreement with a = 0.878 g/cm3 ] 12

-1 recent qualitative analysis of the hydrogen phase diagram by 1.5 10 Tamblyn et al. (44), where the authors propose that the semicon- cm) 8 1 ( ductor-to-metal and dissociation transitions may merge in a cri- 3 6 tical point. Notice that as mentioned before, the location of the 0.5 ) [10 4 transition is different when computed with CEIMC and BOMD. ( 2 0 0 Radial Distribution Function Radial Distribution 0 1 2 3 4 5 0 2 4 6 8 10 12 14 10000 r (a.u.) (eV) 7500 3000K 1.6 14 = 0.897 g/cm3 5000 1.4 ] 12 -1 2500 1.2 10

1 cm) 8 0 0.8 ( 3 6 7500 2000K 0.6 0.4 ) [10 4 5000 ( 0.2 2 ] 2500 0 0 -1 Radial Distribution Function Radial Distribution 0 1 2 3 4 5 0 2 4 6 8 10 12 14 0 1500K r (a.u.) (eV) 7500

1.6 3 5000 = 1.088 g/cm 20 1.4 ]

-1 2500 1.2 15 1 cm) 0 ( 0.8 Conductivity [( cm) 1000K 3 10 7500 0.6

) [10 5000 0.4 5 ( 0.2 2500 0 0 Radial Distribution Function Radial Distribution 0 1 2 3 4 5 0 2 4 6 8 10 12 14 0 r (a.u.) (eV) 7500 800K PHYSICS 5000 Fig. 3. The proton-proton radial distribution function (left column) and the electronic conductivity (right column) at three densities close to the DFT pla- 2500 T ¼ 1;000 teau region for the isotherm at K. The top two rows correspond to 0 the two ends of the plateau whereas the bottom row shows a density deep 0 50 100 150 200 250 300 inside the metallic regime. A sharp increase in DC conductivity from essen- Pressure (GPa) tially zero to approximately 4;000 ðΩ cmÞ−1 (i.e., above the Mott criterium for minimum metallic conductivity) is found within this transition region. Fig. 4. The DC electronic conductivity of hydrogen as a function of pressure The atomic fraction also increases across the transition, but the rate of calculated using the Kubo–Greenwood formula and DFT. The circles, squares, change is much smaller than that of the conductivity. Finally at higher density and diamonds correspond to averages over protonic configurations sampled a Drude-like behavior of the dynamical conductivity is observed. from the BOMD, CEIMC, and PIMD simulations, respectively.

Morales et al. PNAS ∣ July 20, 2010 ∣ vol. 107 ∣ no. 29 ∣ 12801 Downloaded by guest on September 27, 2021 We expect the CEIMC results to be more accurate because they liquid and the solid are in excellent agreement with QMC calcu- avoid the approximate exchange-correlation functional. None- lations in this regime. Given this, we have chosen to extrapolate theless, both methods produce the same qualitative picture of the DFT melting line to higher pressure using Kechin’s equation, b the transition. Tm ¼ 14.025 Kð1 þ P∕aÞ expð−P∕cÞ (a ¼ 0.1129 GPa, b ¼ 0.7155, c ¼ 149 GPa) (32, 33), and extrapolate the QMC-LLT Melting Line. As indicated in Fig. 1, at low temperatures the pre- line to higher pressures assuming that the shift between the dicted LLT is expected to meet the melting line of the molecular QMC- and DFT-LLT lines are systematic; they meet at a pressure solid. In this work, we determine the melting line of the system of 290 GPa and a temperature of 550 K. The uncertainty in this by comparing the free energy of the liquid phase and the free prediction is admittedly large because the two lines have similar energy of the molecular solid phase (phase I, hcp rotationally slopes and such an extrapolation can be unreliable. However, this disordered). For computational reasons, we limit our study to gives a QMC-based estimate of the location of the multiphase the DFT level of theory. It is well known that the semilocal ex- coexistence point‡. This prediction is also compatible with a change-correlation functionals in DFT underestimate the band reported liquid state at T ¼ 364 K and P ¼ 335 GPa, obtained gap in most semiconductors and favor delocalized states. This ex- via an alternative QMC-based ab initio method (47). plains why the LLT is predicted at lower pressure with DFT than In summary, we have used both DFT- and QMC-based simula- with QMC. For pressures below the metallization, the band gap is tions to examine the onset of molecular dissociation in hydrogen finite and the ground state properties should be accurately repro- under high-pressure conditions. Our simulation results clearly in- duced. As soon as the DFT-band gap closes, the nature of the dicate that for temperatures below 2,000 K, there is a range of ground state in this theory changes significantly and the predic- densities where ðdP∕dρÞT ¼ 0 and the electrical conductivity of tions from DFT become inaccurate. This discrepancy will con- the fluid increases in a discontinuous fashion. These observations tinue until the true band gap of the system closes. At higher are consistent with the existence of a first-order phase transition densities, DFT will again produce reliable results. This is consis- in liquid hydrogen that ends in a critical point at approximately tent with our finding here and in previous work (40) and limits 2,000 K. By determining the melting line of hydrogen with the range of pressure for which DFT can be used to predict the highly accurate DFT-based free energy calculations we predict melting line (Fig. 1). that there is a liquid-liquid-solid multiphase coexistence point Using thermodynamic integration with BOMD (45), we per- in the hydrogen phase diagram at a pressure of approximately formed free energy calculations in the solid and liquid phases 290 GPa and a temperature of 550 K that corresponds to the in- to determine the melting line at high pressures. We neglected tersection of the liquid-liquid, metal-insulator, phase transition quantum effects on the nuclei for these calculations. The melting with the melting curve. line of hydrogen should be well represented by our calculations with classical protons for pressures below 200 GPa, because the Methods The CEIMC method was used to perform the QMC simulations. CEIMC is a system remains insulating. Unfortunately, we were unable to first-principles simulation method that uses fixed-node reptation QMC to study the influence of nuclear quantum effects on the melting line solve the electronic problem within the Born–Oppenheimer approximation. due to the large computational demands of the PIMD method. QMC methods for hydrogen do not rely on pseudopotential approximations We used 432 atoms for the calculations in the liquid and 360 and treat electron correlation explicitly by solving the many-electron pro- atoms in the solid, assuming an hcp molecular crystal with no blem directly. At present, it has the potential to be more accurate than orientational order. Variable cell MD simulations at selected DFT while being considerably less expensive than traditional quantum pressures were used to equilibrate the hcp cell as a function of chemistry methods. We refer to previous publications for details about – density in the solid. These were subsequently used in simulations the CEIMC method (37 40). In this work, we used a more accurate wavefunc- tion than in previous studies by using orbitals obtained from DFT with a back- at constant volume. Using coupling constant integration with flow transformation. Both Jastrow and backflow functions contained, in BOMD, we calculated the free energy difference between the addition to the analytic forms given in (48), short range components with system described by DFT and a reference system described by free parameters optimized to minimize the energy of a set of configurations a suitable effective potential. In the solid, we used a reference at any given density. This trial wave function provides a very accurate descrip- Einstein crystal (46) to calculate the free energy of the reference tion of hydrogen, while maintaining high efficiency. solid. In addition, for both phases, we performed BOMD simula- DFT-based first-principles simulations were performed using the Born– tions in a dense grid of points for pressures between 15 and Oppenheimer molecular dynamics method, with the CPMD simulation pack- 200 GPa, and temperatures between 500 and 1000 K. The result- age (49). The BOMD and PIMD simulations were performed with 432 atoms ing set of pressures and energies, together with the results from for the liquid and 360 atoms for the solid in the canonical ensemble, using a massive Nose–Hoover chain thermostat (50). The electronic structure was de- the coupling constant integration, were used to fit a functional of scribed with the Perdew–Burke–Ernzerhof exchange-correlation functional the Gibbs free energy in each phase. The melting line shown in and a Troullier–Martins norm conserving nonlocal pseudopotential with a Fig. 1 was obtained from the fitted functions and is in good agree- core radius of rc ¼ 0.5 a:u. to represent hydrogen. The equations of motion ment with previous DFT simulations that were calculated with a were integrated with a time step of 8 a.u. at the Γ-point with a plane-wave combination of Car–Parrinello MD and the two-phase coexis- cutoff of 90 Ry in the DFT simulations. The path integrals in the PIMD simula- −1 tence approach (32). tions were discretized with a time step of ð8;000 KÞ , which corresponds to 8 The melting line intersects the DFT-LLT line at 700 K and imaginary time slices at T ¼ 1;000 K and is sufficient to obtain converged 220 GPa and should result in a liquid-liquid-solid multiphase results for this system. Electronic conductivities were calculated using DFT and the standard Kubo–Greenwood relation (51). coexistence point†. Unfortunately, the use of DFT to extend the melting curve to higher pressures is likely to be highly inac- ACKNOWLEDGMENTS. This research was supported in part by the National curate as it would involve an insulator-metal transition. However, Nuclear Security Administration under the Stewardship Science Academic the maximum in the melting line and the subsequent change in Alliances program through US Department of Energy (DOE) Grant DE- slope is a robust prediction of DFT because at this point the sys- FG52-06NA26170 and the Lawrence Livermore National Laboratory under tem is far from the LLTand the thermodynamic properties of the Contract DE-AC5207NA27344. M.A.M. acknowledges support of a Stockpile Stewardship Graduate Fellowship; and C.P. thanks the Institute of Condensed

†This point could be either a triple point if the two liquid phases coexist with a single ‡It should be mentioned that the rotationally invariant hcp-molecular phase (phase I) of solid phase, or a quadruple point if the metal-insulator transition extends below freezing. hydrogen has been observed to transform into a symmetry broken phase (phase III) at P ≲ Note that quadruple points are compatible with Gibb’s phase rule in a two component 160 GPa (52–54). Despite a number of investigations (55, 56), the structure of phase III is system. The characterization of this point requires a study of metal-insulator transition still under debate. Moreover experiments have been limited to T ≲ 250 K and it is not in the crystal phase, a work which is in progress. known whether phase III will extend in temperature up to the melting point.

12802 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1007309107 Morales et al. Downloaded by guest on September 27, 2021 Matter Theory at the University of Illinois at Urbana-Champaign for a short from the US DOE INCITE program, the National Center for Supercomputer term visit, and acknowledges financial support from Ministero dell’Università Applications, Lawrence Livermore National Laboratory, and CASPUR (Italy) e della Ricerca, Italy (Grant PRIN2007). Computer time was made available in the framework of Competitive HPC Grants 2009. SEE COMMENTARY

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