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

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

doi:10.1063/5.0025164

Matter and Radiation at Extremes, Volume. 5, Issue 6, 064403(2020)

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

Dongdong Kang¹, Kai Luo², Keith Runge³, and S. B. Trickey⁴

Author Affiliations

¹Department of Physics, National University of Defense Technology, Changsha, Hunan 410073, People’s Republic of China

²Earth and Planets Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Rd. NW, Washington, DC 20015, USA

³Department of Materials Science and Engineering, University of Arizona, Tucson, Arizona 85721, USA

⁴Quantum Theory Project, Department of Physics and Department of Chemistry, University of Florida, Gainesville, Florida 32611, USA

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Figures & Tables(13)

Fig. 1. The state points of the density and ion temperature used in the present simulations. The parameter α takes values of 0.167, 0.373, and 0.681, corresponding to the three curves from top to bottom, respectively. The densities of these state points are 1 g/cm³, 2.5 g/cm³, and 5 g/cm³, respectively.

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Fig. 2. Comparisons of the electronic pressure calculated from OFMD simulations with the different noninteracting free-energy functionals (namely, TTF,⁵² VT84F,⁵³ and LKTF^54,70) vs KSMD simulations, all with the same LPP. T_e = 20 kK and ρ = 1 g/cm³.

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Fig. 3. Comparisons of the radial distribution function from OFMD and PI-OFMD for two-temperature hydrogen at different ion temperatures and densities. The parameter α is 0.167, 0.373, and 0.681 from the top to bottom panels. The electron temperature is 20 kK.

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Fig. 4. Comparisons of RDFs of two-temperature hydrogen obtained from KSMD (LPP and PAW) and OFMD calculations at the state points (5 g/cm³, 879 K) and (1 g/cm³, 300 K), corresponding to dimensionless size α = 0.681, for T_e = 20 kK.

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Fig. 5. Ratio of the first maximum in the RDF for the quantum and classical cases as a function of α.

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Fig. 6. Comparisons of the RDFs of two-temperature hydrogen at two state points (5 g/cm³, 2929 K) and (1 g/cm³, 1000 K), corresponding to dimensionless size α = 0.373, for T_e = 20 kK, 50 kK, and 100 kK. Both OFMD and PI-OFMD results are shown.

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Fig. 7. As in Fig. 6 for the state points (5 g/cm³, 879 K) and (1 g/cm³, 300 K), α = 0.681.

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Fig. 8. Distribution of the radius of gyration of the two-temperature hydrogen at T_e = 20 kK and (from top to bottom) densities 1 g/cm³, 2.5 g/cm³, and 5 g/cm³.

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Fig. 9. Quantum nuclear corrections to the free energy per hydrogen atom at the state points of ion temperature and density used. Results at T_e = 20 kK, 50 kK, and 100 kK are presented for comparison.

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Table 1. Comparison of conventional KSMD and OFMD electronic pressures (GPa) at equilibrium, T_e = T_i = 2 kK. The fractional error for OFMD with respect to the KSMD pressure is shown in parentheses. The KSDT XC free-energy functional was used in all cases.
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Table 1. Comparison of conventional KSMD and OFMD electronic pressures (GPa) at equilibrium, T_e = T_i = 2 kK. The fractional error for OFMD with respect to the KSMD pressure is shown in parentheses. The KSDT XC free-energy functional was used in all cases.
r_s P (KSMD) P (LKTF) P (VT84F) P (TTF)
1.05 1558.7 1453.0(0.068) 1401.7(0.101) 1636.1(−0.049)
1.10 1146.9 1061.0(0.075) 1012.2(0.117) 1224.3(−0.067)
1.25 470.9 416.8(0.115) 377.3(0.199) 536.4(−0.139)

Table 2. NQE corrections to the total free energy and pressure of hydrogen from CEIMC and PI-OFMD calculations at an ion temperature T_i = 2000 K. In PI-OFMD, T_e = 2000 K, while in CEIMC, the NQE corrections are obtained based on the zero-temperature potential-energy surface.⁸¹ The total energy includes the ionic kinetic energy and electronic free energy, F_tot = ϵ_kin + ϵ_pot, as defined by Eqs. (5) and (6), respectively. Corrections are defined as ΔF = (F_tot − F_{tot,classical})/N and ΔP = P − P_classical, respectively. The ratio of the pressure corrections to the pressure obtained classically, ΔP/P, is presented. Statistical errors are reported in parentheses as the uncertainty in the last digit.
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Table 2. NQE corrections to the total free energy and pressure of hydrogen from CEIMC and PI-OFMD calculations at an ion temperature T_i = 2000 K. In PI-OFMD, T_e = 2000 K, while in CEIMC, the NQE corrections are obtained based on the zero-temperature potential-energy surface.⁸¹ The total energy includes the ionic kinetic energy and electronic free energy, F_tot = ϵ_kin + ϵ_pot, as defined by Eqs. (5) and (6), respectively. Corrections are defined as ΔF = (F_tot − F_{tot,classical})/N and ΔP = P − P_classical, respectively. The ratio of the pressure corrections to the pressure obtained classically, ΔP/P, is presented. Statistical errors are reported in parentheses as the uncertainty in the last digit.
r_s α ΔF (mhartree/atom) ΔP (GPa) ΔP/P (%)
CEIMC^a 1.05 0.350 4.0(7) 7(3) 0.4
1.10 0.334 3.8(3) 9(1) 0.7
1.25 0.294 2.8(5) 5(1) 1.0
PI-OFMD 1.05 0.350 7.0(7) 27.7(4) 1.9
1.10 0.334 6.3(7) 23.6(3) 2.2
1.25 0.294 4.1(6) 11.1(2) 2.5

Table 3. Electronic, ionic, and total pressures of two-temperature hydrogen for various densities and electronic and ionic temperatures (T_e, T_i) from OFMD and PI-OFMD calculations. P_e, P_i, and P are the electronic pressure, ionic pressure, and total pressure from PI-OFMD calculations respectively. $P_{e}^{classical}$ , $P_{i}^{classical}$ , and P^classical are their OFMD counterparts with classical ions. The noninteracting free-energy functional LKTF is used. The dependence on the dimensionless size parameter α is also shown.

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Table 3. Electronic, ionic, and total pressures of two-temperature hydrogen for various densities and electronic and ionic temperatures (T_e, T_i) from OFMD and PI-OFMD calculations. P_e, P_i, and P are the electronic pressure, ionic pressure, and total pressure from PI-OFMD calculations respectively. $P_{e}^{classical}$ , $P_{i}^{classical}$ , and P^classical are their OFMD counterparts with classical ions. The noninteracting free-energy functional LKTF is used. The dependence on the dimensionless size parameter α is also shown.

T_e (kK)	α	ρ (g/cm³)	T_i (K)	$P_{e}^{classical}$ (GPa)	P_e (GPa)	ΔP_e/P_e (%)	$P_{i}^{classical}$ (GPa)	P_i (GPa)	ΔP_i/P_i (%)	P^classical (GPa)	P (GPa)	ΔP/P (%)
20	0.167	1.0	5 000	198.99	199.15	0.08	40.68	40.43	−0.60	239.67	239.58	−0.04
20	0.167	2.5	9 204	1815.79	1819.70	0.21	194.40	200.18	2.97	2010.20	2019.88	0.48
20	0.167	5.0	14 610	7389.67	7408.13	0.25	589.51	626.58	6.29	7979.17	8034.71	0.70
20	0.373	1.0	1 000	174.56	178.68	2.36	8.31	12.15	46.21	182.86	190.83	4.36
20	0.373	2.5	1 845	1725.10	1740.63	0.90	38.77	56.40	45.47	1763.87	1797.03	1.88
20	0.373	5.0	2 929	7151.90	7196.04	0.62	117.03	174.99	49.53	7268.92	7371.03	1.40
20	0.681	1.0	300	164.42	172.17	4.71	2.46	9.08	269.10	166.88	181.25	8.61
20	0.681	2.5	553	1689.21	1717.70	1.69	11.46	42.49	270.77	1700.67	1760.19	3.50
20	0.681	5.0	879	7060.90	7135.82	1.06	36.30	129.85	257.71	7097.19	7265.67	2.37
50	0.167	1.0	5 000	258.40	259.23	0.32	41.51	42.52	2.43	299.91	301.75	0.61
50	0.167	2.5	9 204	1921.09	1924.74	0.19	184.84	188.11	1.77	2105.93	2112.84	0.33
50	0.167	5.0	14 610	7557.53	7572.51	0.20	598.92	618.34	3.24	8156.45	8190.84	0.42
50	0.373	1.0	1 000	235.36	239.65	1.82	8.06	12.33	52.98	243.42	251.98	3.52
50	0.373	2.5	1 845	1836.57	1851.04	0.79	38.59	55.28	43.25	1875.16	1906.33	1.66
50	0.373	5.0	2 929	7322.82	7362.95	0.55	119.67	172.68	44.30	7442.49	7535.63	1.25
50	0.681	1.0	300	226.57	233.68	3.14	2.42	9.26	282.64	229.00	242.94	6.09
50	0.681	2.5	553	1801.53	1829.30	1.54	11.22	42.39	277.81	1812.76	1871.69	3.25
50	0.681	5.0	879	7230.73	7304.81	1.02	36.56	130.00	255.58	7267.29	7434.81	2.31
100	0.167	1.0	5 000	466.02	466.47	0.10	41.08	41.89	1.97	507.10	508.36	0.25
100	0.167	2.5	9 204	2248.03	2248.48	0.02	189.55	187.94	−0.85	2437.58	2436.41	−0.05
100	0.167	5.0	14 610	8027.68	8041.09	0.17	593.45	618.83	4.28	8621.13	8659.92	0.45
100	0.373	1.0	1 000	448.47	451.61	0.70	8.41	12.66	50.54	456.88	464.27	1.62
100	0.373	2.5	1 845	2166.42	2181.17	0.68	37.46	55.76	48.85	2203.88	2236.93	1.50
100	0.373	5.0	2 929	7797.50	7839.59	0.54	118.42	175.35	48.07	7915.92	8014.94	1.25
100	0.681	1.0	300	441.51	447.02	1.25	2.41	9.76	304.98	443.92	456.78	2.90
100	0.681	2.5	553	2135.69	2160.99	1.18	11.20	42.68	281.07	2146.89	2203.67	2.64
100	0.681	5.0	879	7708.87	7781.02	0.94	36.15	130.24	260.28	7745.02	7911.26	2.15

Table 4. Comparisons of the electronic pressure P_e obtained from the primitive PIMD algorithm⁶¹ and TRPMD⁶⁸ for hydrogen at ρ = 1 g/cm³. The results of simulations with classical protons are also presented by setting the bead number $P = 1$ . Here, the noninteracting free-energy functional VT84F⁵³ and the finite-temperature XC functional KSDT⁷¹ are used. The standard deviation is shown in parentheses.

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Table 4. Comparisons of the electronic pressure P_e obtained from the primitive PIMD algorithm⁶¹ and TRPMD⁶⁸ for hydrogen at ρ = 1 g/cm³. The results of simulations with classical protons are also presented by setting the bead number $P = 1$ . Here, the noninteracting free-energy functional VT84F⁵³ and the finite-temperature XC functional KSDT⁷¹ are used. The standard deviation is shown in parentheses.

	T_e (kK)	T_i (K)	$P_{e}^{classical}$ (GPa)	P_e (GPa)	ΔP/P (%)
Primitive	20	300	136.8(1.0)	166.5(3.1)	21.7
	20	1000	146.4(1.3)	167.0(2.0)	14.1
	20	5000	169.5(1.0)	184.0(3.2)	8.6
TRPMD	20	300	137.6(0.2)	144.6(0.3)	5.1
	20	1000	146.3(0.5)	150.4(0.7)	2.8
	20	5000	168.2(2.0)	169.2(2.1)	0.6

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

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Paper Information

Category: Fundamental Physics At Extreme Light

Received: Aug. 14, 2020

Accepted: Oct. 14, 2020

Published Online: Nov. 24, 2020

The Author Email:

DOI:10.1063/5.0025164

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Table 1. Comparison of conventional KSMD and OFMD electronic pressures (GPa) at equilibrium, Te = Ti = 2 kK. The fractional error for OFMD with respect to the KSMD pressure is shown in parentheses. The KSDT XC free-energy functional was used in all cases.

Table 1. Comparison of conventional KSMD and OFMD electronic pressures (GPa) at equilibrium, Te = Ti = 2 kK. The fractional error for OFMD with respect to the KSMD pressure is shown in parentheses. The KSDT XC free-energy functional was used in all cases.

Table 1. Comparison of conventional KSMD and OFMD electronic pressures (GPa) at equilibrium, T_e = T_i = 2 kK. The fractional error for OFMD with respect to the KSMD pressure is shown in parentheses. The KSDT XC free-energy functional was used in all cases.

Table 1. Comparison of conventional KSMD and OFMD electronic pressures (GPa) at equilibrium, T_e = T_i = 2 kK. The fractional error for OFMD with respect to the KSMD pressure is shown in parentheses. The KSDT XC free-energy functional was used in all cases.