Two-temperature warm dense hydrogen as a test of quantum protons driven by orbital-free density functional theory electronic forces

Kang Dongdong; Luo Kai; Runge Keith; Trickey S. B.

doi:10.1063/5.0025164

Volume 5 Issue 6

Nov. 2020

Turn off MathJax

Article Contents

Article Navigation > Matter and Radiation at Extremes > 2020 > 5(6): 064403

Kang Dongdong, Luo Kai, Runge Keith, Trickey S. B.. Two-temperature warm dense hydrogen as a test of quantum protons driven by orbital-free density functional theory electronic forces[J]. Matter and Radiation at Extremes, 2020, 5(6): 064403. doi: 10.1063/5.0025164

Citation:

Kang Dongdong, Luo Kai, Runge Keith, Trickey S. B.. Two-temperature warm dense hydrogen as a test of quantum protons driven by orbital-free density functional theory electronic forces[J]. Matter and Radiation at Extremes, 2020, 5(6): 064403. doi: 10.1063/5.0025164

Citation:

Kang Dongdong, Luo Kai, Runge Keith, Trickey S. B.. Two-temperature warm dense hydrogen as a test of quantum protons driven by orbital-free density functional theory electronic forces[J]. Matter and Radiation at Extremes, 2020, 5(6): 064403. doi: 10.1063/5.0025164

PDF( 1643 KB)

Two-temperature warm dense hydrogen as a test of quantum protons driven by orbital-free density functional theory electronic forces

doi: 10.1063/5.0025164

Kang Dongdong^{1
,
,
,},
Luo Kai^2
,,
Runge Keith^3
,,
Trickey S. B.^{4
,
,}

1.
Department of Physics, National University of Defense Technology, Changsha, Hunan 410073, People’s Republic of China
2.
Earth and Planets Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Rd. NW, Washington, DC 20015, USA
3.
Department of Materials Science and Engineering, University of Arizona, Tucson, Arizona 85721, USA
4.
Quantum Theory Project, Department of Physics and Department of Chemistry, University of Florida, Gainesville, Florida 32611, USA

More Information

Corresponding author: a)Author to whom correspondence should be addressed: ddkang@nudt.edu.cn
Received Date: 2020-08-14
Accepted Date: 2020-10-14

Available Online: 2020-11-01

Publish Date: 2020-11-15

Abstract

Abstract

We consider a steady-state (but transient) situation in which a warm dense aggregate is a two-temperature system with equilibrium electrons at temperature T_e, ions at T_i, and T_e ≠ T_i. Such states are achievable by pump–probe experiments. For warm dense hydrogen in such a two-temperature situation, we investigate nuclear quantum effects (NQEs) on structure and thermodynamic properties, thereby delineating the limitations of ordinary ab initio molecular dynamics. We use path integral molecular dynamics (PIMD) simulations driven by orbital-free density functional theory (OFDFT) calculations with state-of-the-art noninteracting free-energy and exchange-correlation functionals for the explicit temperature dependence. We calibrate the OFDFT calculations against conventional (explicit orbitals) Kohn–Sham DFT. We find that when the ratio of the ionic thermal de Broglie wavelength to the mean interionic distance is larger than about 0.30, the ionic radial distribution function is meaningfully affected by the inclusion of NQEs. Moreover, NQEs induce a substantial increase in both the ionic and electronic pressures. This confirms the importance of NQEs for highly accurate equation-of-state data on highly driven hydrogen. For T_e > 20 kK, increasing T_e in the warm dense hydrogen has slight effects on the ionic radial distribution function and equation of state in the range of densities considered. In addition, we confirm that compared with thermostatted ring-polymer molecular dynamics, the primitive PIMD algorithm overestimates electronic pressures, a consequence of the overly localized ionic description from the primitive scheme.

FullText(HTML)

References(83)

References

[1]	S. T. Weir, A. C. Mitchell, and W. J. Nellis, “Metallization of fluid molecular hydrogen at 140 GPa (1.4 Mbar),” Phys. Rev. Lett. 76, 1860 (1996).10.1103/physrevlett.76.1860 doi: 10.1103/physrevlett.76.1860
[2]	P. M. Celliers, G. W. Collins, L. B. Da Silva et al., “Shock-induced transformation of liquid deuterium into a metallic fluid,” Phys. Rev. Lett. 84, 5564 (2000).10.1103/physrevlett.84.5564 doi: 10.1103/physrevlett.84.5564
[3]	H. U. Rahman, F. J. Wessel, P. Ney et al., “Shock waves in a Z-pinch and the formation of high energy density plasma,” Phys. Plasmas 19, 122701 (2012).10.1063/1.4769264 doi: 10.1063/1.4769264
[4]	M. D. Knudson, M. P. Desjarlais, A. Becker et al., “Direct observation of an abrupt insulator-to-metal transition in dense liquid deuterium,” Science 348, 1455 (2015).10.1126/science.aaa7471 doi: 10.1126/science.aaa7471
[5]	M. D. Knudson and M. P. Desjarlais, “High-precision shock wave measurements of deuterium: Evaluation of exchange-correlation functionals at the molecular-to-atomic transition,” Phys. Rev. Lett. 118, 035501 (2017).10.1103/physrevlett.118.035501 doi: 10.1103/physrevlett.118.035501
[6]	P. Loubeyre, S. Brygoo, J. Eggert et al., “Extended data set for the equation of state of warm dense hydrogen isotopes,” Phys. Rev. B 86, 144115 (2012).10.1103/physrevb.86.144115 doi: 10.1103/physrevb.86.144115
[7]	A. L. Kritcher, T. Döppner, D. Swift et al., “Probing matter at Gbar pressures at the NIF,” High Energy Density Phys. 10, 27 (2014).10.1016/j.hedp.2013.11.002 doi: 10.1016/j.hedp.2013.11.002
[8]	R. Nora, W. Theobald, R. Betti et al., “Gigabar spherical shock generation on the OMEGA laser,” Phys. Rev. Lett. 114, 045001 (2015).10.1103/physrevlett.114.045001 doi: 10.1103/physrevlett.114.045001
[9]	P. Sperling, E. J. Gamboa, H. J. Lee et al., “Free-electron x-ray laser measurements of collisional-damped plasmons in isochorically heated warm dense matter,” Phys. Rev. Lett. 115, 115001 (2015).10.1103/physrevlett.115.115001 doi: 10.1103/physrevlett.115.115001
[10]	M. S. Murillo and M. W. C. Dharma-wardana, “Temperature relaxation in hot dense hydrogen,” Phys. Rev. Lett. 100, 205005 (2008).10.1103/physrevlett.100.205005 doi: 10.1103/physrevlett.100.205005
[11]	Q. Ma, J. Dai, D. Kang et al., “Molecular dynamics simulation of electroneion temperature relaxation in dense hydrogen: A scheme of truncated Coulomb potential,” High Energy Density Phys. 13, 34 (2014).10.1016/j.hedp.2014.09.004 doi: 10.1016/j.hedp.2014.09.004
[12]	Q. Ma, J. Dai, D. Kang et al., “Extremely low electron-ion temperature relaxation rates in warm dense hydrogen: Interplay between quantum electrons and coupled ions,” Phys. Rev. Lett. 122, 015001 (2019).10.1103/physrevlett.122.015001 doi: 10.1103/physrevlett.122.015001
[13]	Q. Zeng and J. Dai, “Structural transition dynamics of the formation of warm dense gold: From an atomic scale view,” Sci. China-Phys. Mech. Astron. 63, 263011 (2020).10.1007/s11433-019-1466-2 doi: 10.1007/s11433-019-1466-2
[14]	S. I. Anisimov, B. L. Kapeliovich, and T. L. Perel’man, “Electron emission from metal sufraces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39, 375 (1974).
[15]	J. Hohlfeld, S.-S. Wellershoff, J. Güdde et al., “Electron and lattice dynamics following optical excitation of metals,” Chem. Phys. 251, 237 (2000).10.1016/s0301-0104(99)00330-4 doi: 10.1016/s0301-0104(99)00330-4
[16]	M. S. Murillo, “Using Fermi statistics to create strongly coupled ion plasmas in atom traps,” Phys. Rev. Lett. 87, 115003 (2001).10.1103/physrevlett.87.115003 doi: 10.1103/physrevlett.87.115003
[17]	M. Lyon, S. D. Bergeson, G. Hart et al., “Stronly-coupled plasmas formed from laser-heated solids,” Sci. Rep. 5, 15693 (2015).10.1038/srep15693 doi: 10.1038/srep15693
[18]	J. Clérouin, G. Robert, P. Arnault et al., “Evidence for out-of-equilibrium states in warm dense matter probed by x-ray Thomson scattering,” Phys. Rev. E 91, 011101(R) (2015).10.1103/physreve.91.011101 doi: 10.1103/physreve.91.011101
[19]	R. N. Barnett and U. Landman, “Born-Oppenheimer molecular-dynamics simulations of finite systems: Structure and dynamics of (H2O)2,” Phys. Rev. B 48, 2081 (1993).10.1103/physrevb.48.2081 doi: 10.1103/physrevb.48.2081
[20]	D. Marx and J. Hutter, “Ab initio molecular dynamics: Theory and implementation,” in Modern Methods and Algorithms of Quantum Chemistry, edited by J. Grotendorst (John von Neumann Institute for Computing, Jülich, 2000), p. 301ff.
[21]	J. S. Tse, “Ab initio molecular dynamics with density functional theory,” Annu. Rev. Phys. Chem. 53, 249 (2002).10.1146/annurev.physchem.53.090401.105737 doi: 10.1146/annurev.physchem.53.090401.105737
[22]	D. Marx and J. Hutter, Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods (Cambridge University Press, Cambridge, 2009).
[23]	N. J. Hartley, P. Belancourt, D. A. Chapman et al., “Electron-ion temperature equilibration in warm dense tantalum,” High Energy Density Phys. 14, 1 (2015).10.1016/j.hedp.2014.10.003 doi: 10.1016/j.hedp.2014.10.003
[24]	A. Ng, P. Sterne, S. Hansen et al., “Dc conductivity of two-temperature warm dense gold,” Phys. Rev. E 94, 033213 (2016).10.1103/physreve.94.033213 doi: 10.1103/physreve.94.033213
[25]	T. Ogitsu, A. Fernandez-Pañella, S. Hamel et al., “Ab initio modeling of nonequilibrium electron-ion dynamics of iron in the warm dense matter regime,” Phys. Rev. B 97, 214203 (2018).10.1103/physrevb.97.214203 doi: 10.1103/physrevb.97.214203
[26]	L. Harbour, G. D. Förster, M. W. C. Dharma-wardana et al., “Ion-ion dynamic structure factor, acoustic modes, and equation of state of two-temperature warm dense aluminum,” Phys. Rev. E 97, 043210 (2018).10.1103/physreve.97.043210 doi: 10.1103/physreve.97.043210
[27]	Zh. A. Moldabekov, S. Groth, T. Dornheim et al., “Structural characteristics of strongly coupled ions in a dense quantum plasma,” Phys. Rev. E 98, 023207 (2018).10.1103/physreve.98.023207 doi: 10.1103/physreve.98.023207
[28]	N. Nettelmann, B. Holst, A. Kietzmann et al., “Ab initio equation of state data for hydrogen, helium, and water and the internal structure of Jupiter,” Astrophys. J. 683, 1217–1228 (2008).10.1086/589806 doi: 10.1086/589806
[29]	J. D. Lindl, P. Amendt, R. L. Berger et al., “The physics basis for ignition using indirect-drive targets on the National Ignition Facility,” Phys. Plasmas 11, 339 (2004).10.1063/1.1578638 doi: 10.1063/1.1578638
[30]	D. Kang, Y. Hou, Q. Zeng et al., “Unified first-principles equations of state of deuterium-tritium mixtures in the global inertial confinement fusion region,” Matter Radiat. Extremes 5, 055401 (2020).10.1063/5.0008231 doi: 10.1063/5.0008231
[31]	U. Zastrau, P. Sperling, M. Harmand et al., “Resolving ultrafast heating of dense cryogenic hydrogen,” Phys. Rev. Lett. 112, 105002 (2014).10.1103/physrevlett.112.105002 doi: 10.1103/physrevlett.112.105002
[32]	U. Zastrau, P. Sperling, A. Becker et al., “Equilibration dynamics and conductivity of warm dense hydrogen,” Phys. Rev. E 90, 013104 (2014).10.1103/physreve.90.013104 doi: 10.1103/physreve.90.013104
[33]	U. Zastrau, P. Sperling, C. Fortmann-Grote et al., “Ultrafast electron kinetics in short pulse laser-driven dense hydrogen,” J. Phys. B: At., Mol. Opt. Phys. 48, 224004 (2015).10.1088/0953-4075/48/22/224004 doi: 10.1088/0953-4075/48/22/224004
[34]	N. D. Mermin, “Thermal properties of the inhomogeneous electron gas,” Phys. Rev. 137, A1441 (1965).10.1103/physrev.137.a1441 doi: 10.1103/physrev.137.a1441
[35]	D. M. Ceperley, “Path integrals in the theory of condensed helium,” Rev. Mod. Phys. 67, 279 (1995).10.1103/revmodphys.67.279 doi: 10.1103/revmodphys.67.279
[36]	M. Zaghoo, R. J. Husband, and I. F. Silvera, “Striking isotope effect on the metallization phase lines of liquid hydrogen and deuterium,” Phys. Rev. B 98, 104102 (2018).10.1103/physrevb.98.104102 doi: 10.1103/physrevb.98.104102
[37]	S. Biermann, D. Hohl, and D. Marx, “Quantum effects in solid hydrogen at ultra-high pressure,” Solid State Commun. 108, 337 (1998).10.1016/s0038-1098(98)00388-3 doi: 10.1016/s0038-1098(98)00388-3
[38]	H. Kitamura, S. Tsuneyuki, T. Ogitsu et al., “Quantum distribution of protons in solid molecular hydrogen at megabar pressures,” Nature 404, 259 (2000).10.1038/35005027 doi: 10.1038/35005027
[39]	G. Geneste, M. Torrent, F. Bottin et al., “Strong isotope effect in phase II of dense solid hydrogen and deuterium,” Phys. Rev. Lett. 109, 155303 (2012).10.1103/physrevlett.109.155303 doi: 10.1103/physrevlett.109.155303
[40]	F. J. Bermejo, K. Kinugawa, J. Dawidowski et al., “Beyond classical molecular dynamics: Simulation of quantum-dynamics effects at finite temperatures; the case of condensed molecular hydrogen,” Chem. Phys. 317, 198 (2005).10.1016/j.chemphys.2005.04.010 doi: 10.1016/j.chemphys.2005.04.010
[41]	J. Chen, X.-Z. Li, Q. Zhang et al., “Quantum simulation of low-temperature metallic liquid hydrogen,” Nat. Commun. 4, 2064 (2013).10.1038/ncomms3064 doi: 10.1038/ncomms3064
[42]	H. Y. Geng, R. Hoffmann, and Q. Wu, “Lattice stability and high-pressure melting mechanism of dense hydrogen up to 1.5 TPa,” Phys. Rev. B 92, 104103 (2015).10.1103/physrevb.92.104103 doi: 10.1103/physrevb.92.104103
[43]	M. A. Morales, J. M. McMahon, C. Pierleoni et al., “Nuclear quantum effects and nonlocal exchange-correlation functionals applied to liquid hydrogen at high pressure,” Phys. Rev. Lett. 110, 065702 (2013).10.1103/physrevlett.110.065702 doi: 10.1103/physrevlett.110.065702
[44]	D. Kang, H. Sun, J. Dai et al., “Nuclear quantum dynamics in dense hydrogen,” Sci. Rep. 4, 5484 (2014).10.1038/srep05484 doi: 10.1038/srep05484
[45]	D. Kang and J. Dai, “Dynamic electron-ion collisions and nuclear quantum effects in quantum simulation of warm dense matter,” J. Phys.: Condens. Matter 30, 073002 (2018).10.1088/1361-648x/aa9e29 doi: 10.1088/1361-648x/aa9e29
[46]	S. Azadi, R. Singh, and T. D. Kühne, “Nuclear quantum effects induce metallization of dense solid molecular hydrogen,” J. Comput. Chem. 39, 262 (2018).10.1002/jcc.25104 doi: 10.1002/jcc.25104
[47]	J. Hinz, V. V. Karasiev, S. X. Hu et al., “Fully consistent density functional theory determination of the insulator-metal transition boundary in warm dense hydrogen,” Phys. Rev. Res. 2, 032065 (2020).10.1103/physrevresearch.2.032065 doi: 10.1103/physrevresearch.2.032065
[48]	B. Lu, D. Kang, D. Wang et al., “Towards the same line of liquid-liquid phase transition of dense hydrogen from various theoretical predictions,” Chin. Phys. Lett. 36, 103102 (2019).10.1088/0256-307x/36/10/103102 doi: 10.1088/0256-307x/36/10/103102
[49]	W. Kohn and L. J. Sham, “Self-consistent equations including exchange and correlation effects,” Phys. Rev. 140, A1133 (1965).10.1103/physrev.140.a1133 doi: 10.1103/physrev.140.a1133
[50]	S. Zhang, H. Wang, W. Kang et al., “Extended application of Kohn-Sham first-principles molecular dynamics method with plane wave approximation at high energy: From cold materials to hot dense plasmas,” Phys. Plasmas 23, 042707 (2016).10.1063/1.4947212 doi: 10.1063/1.4947212
[51]	C. Gao, S. Zhang, W. Kang et al., “Validity boundary of orbital-free molecular dynamics method corresponding to thermal ionization of shell structure,” Phys. Rev. B 94, 205115 (2016).10.1103/physrevb.94.205115 doi: 10.1103/physrevb.94.205115
[52]	V. V. Karasiev, T. Sjostrom, and S. B. Trickey, “Generalized-gradient-approximation noninteracting free-energy functionals for orbital-free density functional calculations,” Phys. Rev. B 86, 115101 (2012).10.1103/physrevb.86.115101 doi: 10.1103/physrevb.86.115101
[53]	V. V. Karasiev, D. Chakraborty, O. A. Shukruto et al., “Nonempirical generalized gradient approximation free-energy functional for orbital-free simulations,” Phys. Rev. B 88, 161108(R) (2013).10.1103/physrevb.88.161108 doi: 10.1103/physrevb.88.161108
[54]	K. Luo, V. V. Karasiev, and S. B. Trickey, “Towards accurate orbital-free simulations: A generalized gradient approximation for the noninteracting free energy density functional,” Phys. Rev. B 101, 075116 (2020).10.1103/physrevb.101.075116 doi: 10.1103/physrevb.101.075116
[55]	T. Sjostrom and J. Daligault, “Gradient corrections to the exchange-correlation free energy,” Phys. Rev. B 90, 155109 (2014).10.1103/physrevb.90.155109 doi: 10.1103/physrevb.90.155109
[56]	V. V. Karasiev, L. Calderín, and S. B. Trickey, “Importance of finite-temperature exchange correlation for warm dense matter calculations,” Phys. Rev. E 93, 063207 (2016).10.1103/physreve.93.063207 doi: 10.1103/physreve.93.063207
[57]	V. V. Karasiev, S. X. Hu, M. Zaghoo et al., “Exchange-correlation thermal effects in shocked deuterium: Softening the principal Hugoniot and thermophysical properties,” Phys. Rev. B 99, 214110 (2019).10.1103/physrevb.99.214110 doi: 10.1103/physrevb.99.214110
[58]	K. Ramakrishna, T. Dornheim, and J. Vorberger, “Influence of finite temperature exchange-correlation effects in hydrogen,” Phys. Rev. B 101, 195129 (2020).10.1103/physrevb.101.195129 doi: 10.1103/physrevb.101.195129
[59]	F. J. Bermejo, K. Kinugawa, C. Cabrillo et al., “Quantum effects on liquid dynamics as evidenced by the presence of well-defined collective excitations in liquid para-hydrogen,” Phys. Rev. Lett. 84, 5359 (2000).10.1103/physrevlett.84.5359 doi: 10.1103/physrevlett.84.5359
[60]	D. Chandler and P. G. Wolynes, “Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids,” J. Chem. Phys. 74, 4078 (1981).10.1063/1.441588 doi: 10.1063/1.441588
[61]	D. Marx and M. Parrinello, “Ab initio path integral molecular dynamics: Basic ideas,” J. Chem. Phys. 104, 4077 (1996).10.1063/1.471221 doi: 10.1063/1.471221
[62]	M. Suzuki, “Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations,” Phys. Lett. A 146, 319 (1990).10.1016/0375-9601(90)90962-n doi: 10.1016/0375-9601(90)90962-n
[63]	R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Intergrals (McGraw-Hill, New York, 1965).
[64]	M. Parrinello and A. Rahman, “Study of an F center in molten KCl,” J. Chem. Phys. 80, 860 (1983).10.2172/5968486 doi: 10.2172/5968486
[65]	M. E. Tuckerman, “Path integration via molecular dynamics,” in Quantum Simulations of Complex Many-Body Systems: From Theory to Algorithms, edited by J. Grotendorst, D. Marx, and A. Muramatsu (John von Neumann Institute for Computing, Jülich, 2002), pp. 269–298.
[66]	I. R. Craig and D. E. Manolopoulos, “Quantum statistics and classical mechanics: Real time correlation functions from ring polymer molecular dynamics,” J. Chem. Phys. 121, 3368 (2004).10.1063/1.1777575 doi: 10.1063/1.1777575
[67]	A. Pérez, M. E. Tuckerman, and M. H. Müser, “A comparative study of the centroid and ring-polymer molecular dynamics methods for approximating quantum time correlation functions from path integrals,” J. Chem. Phys. 130, 184105 (2009).10.1063/1.3126950 doi: 10.1063/1.3126950
[68]	M. Rossi, M. Ceriotti, and D. E. Manolopoulos, “How to remove the spurious resonances from ring polymer molecular dynamics,” J. Chem. Phys. 140, 234116 (2014).10.1063/1.4883861 doi: 10.1063/1.4883861
[69]	M. E. Tuckerman, D. Marx, M. L. Klein et al., “Efficient and general algorithms for path integral Car-Parrinello molecular dynamics,” J. Chem. Phys. 104, 5579 (1996).10.1063/1.471771 doi: 10.1063/1.471771
[70]	K. Luo, V. V. Karasiev, and S. B. Trickey, “A simple generalized gradient approximation for the noninteracting kinetic energy density functional,” Phys. Rev. B 98, 041111(R) (2018).10.1103/physrevb.98.041111 doi: 10.1103/physrevb.98.041111
[71]	V. V. Karasiev, T. Sjostrom, J. Dufty et al., “Accurate homogeneous electron gas exchange-correlation free energy for local spin-density calculations,” Phys. Rev. Lett. 112, 076403 (2014).10.1103/physrevlett.112.076403 doi: 10.1103/physrevlett.112.076403
[72]	S. Groth, T. Dornheim, T. Sjostrom et al., “Ab initio exchange-correlation free energy of the uniform electron gas at warm dense matter conditions,” Phys. Rev. Lett. 119, 135001 (2017).10.1103/physrevlett.119.135001 doi: 10.1103/physrevlett.119.135001
[73]	V. V. Karasiev, J. W. Dufty, and S. B. Trickey, “Nonempirical semilocal free-energy density functional for matter under extreme conditions,” Phys. Rev. Lett. 120, 076401 (2018).10.1103/physrevlett.120.076401 doi: 10.1103/physrevlett.120.076401
[74]	C. Pierleoni, M. A. Morales, G. Rillo et al., “Liquid–liquid phase transition in hydrogen by coupled electron–ion Monte Carlo simulations,” Proc. Nat. Acad. Sci. U. S. A. 113, 4953 (2016).10.1073/pnas.1603853113 doi: 10.1073/pnas.1603853113
[75]	V. Kapil, M. Rossi, O. Marsalek et al., “i-PI 2.0: A universal force engine for advanced molecular simulations,” Comput. Phys. Commun. 236, 214 (2019).10.1016/j.cpc.2018.09.020 doi: 10.1016/j.cpc.2018.09.020
[76]	M. Chen, J. Xia, C. Huang et al., “Introducing PROFESS 3.0: An advanced program for orbital-free density functional theory molecular dynamics simulations,” Comput. Phys. Commun. 190, 228 (2015).10.1016/j.cpc.2014.12.021 doi: 10.1016/j.cpc.2014.12.021
[77]	V. V. Karasiev, T. Sjostrom, and S. B. Trickey, “Finite-temperature orbital-free DFT molecular dynamics: Coupling Profess and Quantum Espresso,” Comput. Phys. Commun. 185, 3240 (2014).10.1016/j.cpc.2014.08.023 doi: 10.1016/j.cpc.2014.08.023
[78]	V. V. Karasiev, S. B. Trickey, and J. W. Dufty, “Status of free-energy representations for the homogeneous electron gas,” Phys. Rev. B 99, 195134 (2019).10.1103/physrevb.99.195134 doi: 10.1103/physrevb.99.195134
[79]	M. Ceriotti, M. Parrinello, T. E. Markland et al., “Efficient stochastic thermostatting of path integral molecular dynamics,” J. Chem. Phys. 133, 124104 (2010).10.1063/1.3489925 doi: 10.1063/1.3489925
[80]	P. Giannozzi, O. Andreussi, T. Brumme et al., “Advanced capabilities for materials modelling with Quantum Espresso,” J. Phys.: Condens. Matter 29, 465901 (2017).10.1088/1361-648x/aa8f79 doi: 10.1088/1361-648x/aa8f79
[81]	M. A. Morales, C. Pierleoni, and D. M. Ceperley, “Equation of state of metallic hydrogen from coupled electron-ion Monte Carlo simulations,” Phys. Rev. E 81, 021202 (2010).10.1103/physreve.81.021202 doi: 10.1103/physreve.81.021202
[82]	E. Liberatore, C. Pierleoni, and D. M. Ceperley, “Liquid-solid transition in fully ionized hydrogen at ultra-high pressures,” J. Chem. Phys. 134, 184505 (2011).10.1063/1.3586808 doi: 10.1063/1.3586808
[83]	R. W. Hall and B. J. Berne, “Nonergodicity in path integral molecular dynamics,” J. Chem. Phys. 81, 3641 (1984).10.1063/1.448112 doi: 10.1063/1.448112