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.

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.

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.

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.

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

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.

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

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.

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.

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 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.

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. Peclassical, Piclassical, 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.

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.