Equilibrium properties of warm dense deuterium calculated by the wave packet molecular dynamics and density functional theory method

A joint simulation method based on the wave packet molecular dynamics and density functional theory (WPMD-DFT) is applied to study warm dense deuterium (nonideal deuterium plasmas). This method was developed recently as an extension of the wave packet molecular dynamics (WPMD) in which the equations of motion are solved simultaneously for classical ions and semiclassical electrons represented as Gaussian wave packets. Compared to the classical molecular dynamics and WPMD simulations, the method of WPMD-DFT provides a more accurate representation of quantum effects such as electron-ion coupling and electron degeneracy. It allows studying nonadiabatic dynamics of electrons and ions in equilibrium and nonequilibrium states while being more accurate and efficient at high densities than WPMD and classical molecular dynamics. In the paper, we discuss particular features of the method such as special boundary conditions and the procedure of isentrope calculation as well as the results obtained by WPMD-DFT for the shock-compressed deuterium. The compression isentrope and principal Hugoniot curves obtained by WPMD-DFT are compared with available experimental data and other simulation approaches to validate the method. It opens up a possibility of further application of the method to study nonequilibrium states and relaxation processes.

Figure 1

Binding energy of the ${\mathrm{H}}_2$ molecule depending on the bond length as obtained by simulations using the original WPMD algorithm (upper dashed red line), the same with corrected energy of isolated two-atom state (upper solid red line), the eFF method with correction (lower dashed green line), and WPMD-DFT with correction (lower solid blue line). The exact value is given by the black dotted line.

Figure 2

The simulation box with a harmonic confining potential (dark red curves). The mean electron density for uniformly distributed electrons is shown by the blue dashed line, and the green dotted line is the sample of an actual electron profile obtained from simulations.

Figure 3

The numbered solid lines show electron density profiles for different values of the wall parameter $k$ (these values are given in the legend). The mean electron density of $\rho =11.9\mathrm{g}/{\mathrm{cm}}^3$ ($n_{\mathrm{e}}=3.55\times {10}^{24}{\mathrm{cm}}^{-3}$) is shown by the dashed line, the temperature is $T=38240$ K. The solid vertical lines are located at the box boundaries $-L_x/2$ and $L_x/2$, the dotted-dashed lines outline the half-box region of $-L_x/4<x<L_x/4$.

Figure 4

The calculated values of pressure for different wall parameters $k$ versus the number electron density calculated using there different ways: from the total number of electrons $n_{\mathrm{e}}=N_{\mathrm{e}}/V$ (circles), by integration of the actual electron profile over the full simulation box (13) (triangles), and by integration over the half-box (13) (asterisks). The deuterium plasma parameters are $\rho =11.9\mathrm{g}/{\mathrm{cm}}^3$ and $T=38240$ K. The solid line represents the corresponding isotherm. The insertion shows the dependence of the mean electron density calculated using Eqs. (13) and (14) on the wall parameter $k$.

Figure 5

Isotherms of deuterium obtained by WPMD-DFT: (a) internal energy, (b) pressure.

Figure 6

Compression isentropes of deuterium in $P\text{−}\rho$ plane obtained by two series of shock-compression experiments [43, 44] (rhombus) and [46, 65, 66, 67] (circles) as well as calculation results obtained by different methods: WPMC-DFT with the Fermi-Zel'dovich approach (brown dotted-dashed line), WPMD-DFT with the compression-relaxation algorithm started from different points (upper green and lower red solid lines), QMD with the Fermi-Zel'dovich approach (dashed black line), WPMC-DFT with density and temperature input from QMD (squares), ideal Fermi gas for electrons (upper blue dotted line), and ideal Boltzmann gas for deutrons (lower pink dotted line).

Figure 7

Total energy of plasma as a function of time for the compression-relaxation algorithm of the isentrope calculation. Colors show the stages of compression, vertical red lines; relaxation, blue (dark gray) sections; and equilibrium states, green (light gray) sections.

Figure 8

Pressure as a function of the electron number density for different compression rates.

Figure 9

Hugoniot curve of shock-compressed deuterium: experimental data from Sandia for the plate impact with $\alpha$ quartz [36] (“Expt. Qtz,” cyan rhombus) and aluminum [35] (“Expt. Al,” green rhombus) standards, experimental data from LNLL [37] (squares), PIMC simulations [77] (asterisks), QMC simulations [76] (triangles), AWPMD simulations [24] (crosses), and the present WPMD-DFT simulations (circles).

Figure 10

Ion-ion pair correlation function for different densities and temperatures. The plasma parameters correspond to (a) selected points from the Hugoniout curve (see Fig. 9 and Table 2); (b) the parameters of QMC simulations [76]. In (b) the solid lines are the present WPMD-DFT simulation, dashed lines are the results from [76].

Figure 11

Visualization of the simulation box for different plasma densities and temperatures. The plasma states correspond to Fig. 10 and Table 2. Red spheres represent positions of ions, blue spheres are the electrons with the spin projection $\frac{1}{2}$, and green spheres are the electrons with the spin projection $-\frac{1}{2}$. The size of electron spheres corresponds to wave packet widths.