Excited Electron Dynamics Modeling of Warm Dense Matter

We present a model (the electron force field, or eFF) based on a simplified solution to the time-dependent Schrödinger equation that with a single approximate potential between nuclei and electrons correctly describes many phases relevant for warm dense hydrogen. Over a temperature range of 0 to 100 000 K and densities up to $1\mathrm{g}/{\mathrm{cm}}^3$, we find excellent agreement with experimental, path integral Monte Carlo, and linear mixing equations of state, as well as single-shock Hugoniot curves from shock compression experiments. In principle eFF should be applicable to other warm dense systems as well.

Figure 1

Equation of state for solid deuterium at 300 K and varying density. We show here that the pressures from eFF dynamics are in good agreement with diamond anvil experiments. Here constant number/volume/energy (NVE) dynamics were performed in a cubic periodic box with 64 H atoms. Electrostatics were computed using Ewald summation, and $E_{\mathrm{Pauli}}$ was computed using the minimum image convention. Thermodynamic parameters were averaged over 1 ps following a 200 fs equilibration period, with the time step set to min (0.02, $1/\sqrt{T}$) fs. Masses were set to $m_{\mathrm{nuc}}=m_{\mathrm{elec}}=1.008\mathrm{amu}$; reducing the electron mass to $m_{\mathrm{nuc}}=0.5\mathrm{amu}$ and increasing the nuclear mass by the same amount had a negligible effect on the equations of state.

Figure 2

Proton-proton pair distribution function for liquid ${\mathrm{D}}_2$ for $r_s=2\mathrm{bohr}$ and varying temperature. We show here that eFF predicts a gradual dissociation of molecules into separated atoms as the temperature increases from $T=10000\mathrm{K}$ to 20 000 K. The inset shows the ratio of the amplitude at the first minimum ($g_{\min }$) to the amplitude at the first maximum ($g_{\max }$), indicating that the midpoint is at 16 500 K. The dynamics parameters were the same as in Fig. 1.

Figure 3

Equation of state for liquid deuterium at $r_s=2\mathrm{bohr}$ with varying temperature. We show here that eFF is applicable and consistent with the best theory over the entire temperature range shown. The eFF results (black round points) are in the range with the best QM (PIMC, green solid and dashed lines) above 10 000 K (where PIMC is accurate), and eFF agrees well with the Saumon-Chabrier EOS [20] below 10 000 K (where it is expected to be accurate). The PIMC results shown here are: (a) calculations using free-particle nodes (dashed line [22]), and (b) calculations using antisymmetrized local Gaussian nodes (solid line, open squares [29] and closed squares [23]). Above $T>50000\mathrm{K}$ the eFF uses a harmonic restraint to limit the electron size ($E=\frac{1}{2}{k}_s{s}^2$ for $s>L_{\mathrm{box}}/2$); however, the equation of state is insensitive to the particular value of $k_s$, as long as it keeps the electrons bound. The eFF pressures from $k_s=1$ and from $10\mathrm{hartrees}/{\mathrm{bohr}}^2$ are within 1% of each other over $T=6000$ to 600 000 K.

Figure 4

Shock Hugoniot curve for liquid ${\mathrm{D}}_2$. We show here that eFF agrees well with most experiments: gas gun (red dots), $Z$ machine (green dots), and convergence geometry (orange). The validity of the Nova laser data (blue dots) has been questioned [27]. The PIMC results agree with eFF up to a compression of 4.2, but leads to a lower limiting compression than eFF. To compute the Hugoniot curve, we perform NVE simulations of ${\mathrm{D}}_2$, interpolating to temperatures such that the internal energy, volume, and pressure satisfy $U-U_0+\frac{1}{2}(V-V_0)(P+P_0)=0$. As a starting point, we compute a box of liquid hydrogen with $r_s=3.16\mathrm{bohr}$ (${\rho }_0=0.171\mathrm{g}/{\mathrm{cm}}^3$), $T=19.6\mathrm{K}$; we find $U_0=-0.477043\mathrm{hartrees}/\mathrm{\text{atom}}$ and $P=0$. We note that the eFF Hugoniot curve connects to an eFF low temperature starting point, while the PIMC Hugoniot curve connects to a $U_0$ from a separate calculation [30].