图1 不同PIMC模拟覆盖到的均匀电子气体参数范围(横轴为描述密度的无量纲参数$ r_{\mathrm{s}}=a / a_0$，a为自由电子气体中电子的平均间距，a0为玻尔半径，纵轴为描述温度的无量纲参数Θ=T/TF，TF为费米温度。红色点线表示PB-PIMC方法覆盖的温度范围的下界。)

Fig.1 The ranges of applicability for different PIMC method in the study of UEG (The density parameter $ r_{\mathrm{s}}=a / a_0$ with a the average distance of electrons and a0 the Bohr radius, and the temperature parameter $\varTheta=T / T_{\mathrm{F}} $ with TF the Fermi temperature.)

图2 RPIMC模拟得到的氘的冲击雨贡纽曲线[47](图中不同标记的数据点表示采用了不同组合的虚时步长、粒子数以及节点形式的计算结果。)

Fig.2 Hugoniot curve of deuterium calculated with RPIMC method[47] (Data points in different colors and marks represent the results calculated with different time steps τ, particle number N and nodal types.)

图3 等密度下温稠密氘的压强随温度的变化(a) ρ = 10.00 g ·cm-3；(b) ρ = 1 596.49 g ·cm-3 (红色方块表示DPIMC模拟结果[46]，黑色圆圈则表示RPIMC的模拟结果[47]。)

Fig.3 Temperature dependence of deuterium's pressure at constant density of (a) ρ = 10.00 g ·cm-3 and (b) ρ = 1 596.49 g ·cm-3 (The black circles stand for the RPIMC results[47], while the red squares represent the DPIMC results[46].)

图4 DPIMC模拟得到的氘的冲击雨贡纽曲线[86](红色虚线)与RPIMC的模拟结果[47](黑色实线)对比

Fig.4 Hugoniot curve of deuterium calculated with DPIMC method[86], as represented by the red dashed curve. The results of RPIMC simulation[47] are also presented with the black curve, for the comparison purpose

图5 (a) 单一硅原子的内能随温度的变化[61] (模拟采用的周期性边界盒子边长为5.0 Bohr。图中彩色的数据点表示采用各种节点形式的计算结果，作为参照的密度泛函理论(DFT)计算结果由黑色十字表示。); (b) 单一硅原子的压强随温度的变化(图中不同标记的数据点的含义与图(a)一致。)

Fig.5 (a) Temperature dependence of internal energy of a single silicon atom[61] (Periodic cell of 5.0 Bohr is employed in simulation. The data points in different marks represent the results calculated with different nodes. The density functional theory (DFT) results are also shown as the reference, as represented by the black crosses.); (b) Temperature dependence of pressure of a single silicon atom

图6 中Z材料的冲击雨贡纽曲线(a)硼的雨贡纽曲线(黑色实线为RPIMC+DFT-MD模拟给出的结果[51]，蓝色圆圈为实验结果[51]。)；(b)钠的雨贡纽曲线(黑色实线为RPIMC+DFT-MD模拟给出的结果[57]，蓝色方块为文献[57]中引用的实验结果。)；(c)铝的雨贡纽曲线(黑色实线为RPIMC+DFT-MD模拟给出的结果[60]，红色虚线为ext-FPMD方法给出的结果[23]，蓝色菱形为文献[60]中引用的实验结果。)；(d)硅的雨贡纽曲线(黑色实线为RPIMC+DFT-MD模拟给出的结果[62]，红色虚线为OFMD+DFT-MD方法给出的结果[62]，蓝色六边形为文献[62]中引用的实验结果。)

Fig.6 Hugoniot curves of middle Z materials. (a) Hugoniot curve of Boron (The black curve and blue circles represent the results of RPIMC+DFT-MD[51] and experiments[51].); (b) Hugoniot curve of Sodium (The black curve and blue squares represent the results of RPIMC+DFT-MD[57] and the experiments cited within Ref. [57].); (c) Hugoniot curve of Aluminum (The black curve, red dashed curve and blue squares represent the results of RPIMC+DFT-MD[60], ext-FPMD[23] and the experiments cited within Ref. [60].); (d) Hugoniot curve of Silicon (The black curve, red dashed curve and blue squares represent the results of RPIMC+DFT-MD[62], OFMD+DFT-MD[62] and the experiments cited within Ref. [62].)

图7 碳氢材料的冲击雨贡纽曲线(黑色实线为RPIMC+ DFT-MD模拟给出的结果[66]，红色圆圈为ext-FPMD方法给出的结果[24]，绿色方块为DFT-MD模拟给出的结果[88]，蓝色菱形为OFMD模拟给出的结果[88]。)

Fig.7 Hugoniot curve of Hydrocarbon (CH) (The black curve, red circles, green squares, and blue diamonds represent the results of RPIMC+DFT-MD[66], ext-FPMD[24], DFT-MD[88] and OFMD[88], respectively.)

图8 (a) 氮化硼的雨贡纽曲线[89](红色实线表示直接进行PIMC模拟的结果，其中氮化硼初始密度为2.26 g ·cm-3，蓝色点划线表示在同密度下采用线性混合近似得到的结果。)；(b)硅酸镁的雨贡纽曲线[89](红色实线表示直接进行PIMC模拟的结果，其中硅酸镁初始密度为3.208g ·cm-3，绿色点划线表示在同密度下采用线性混合近似得到的结果。)

Fig.8 (a) Hugoniot curve of boron nitride (BN)[89] (The red curve represents the direct result of BN through RPIMC simulations with an initial density of 2.26g ·cm-3. The blue dash dot line represents the result with linear mixed approximation of elemental B and N.); (b) Hugoniot curve of magnesium silicate (MgSiO3)[89] (The red curve represents the RPIMC result of MgSiO3 with an initial density of 3.208 g ·cm-3, while the green dash dot curve represents the result with linear mixed approximation of elemental Mg, Si, and O.)