Thomas-Fermi simulations of dense plasmas without pseudopotentials

The Thomas-Fermi model for warm and hot dense matter is widely used to predict material properties such as the equation of state. However, for practical reasons current implementations use pseudopotentials for the electron-nucleus interaction instead of the bare Coulomb potential. This complicates the calculation and quantities such as free energy cannot be converged with respect to the pseudopotential parameters. We present a method that retains the bare Coulomb potential for the electron-nucleus interaction and does not use pseudopotentials. We demonstrate that accurate free energies are obtained by checking variational consistency. Examples for aluminum and iron plasmas are presented.

Figure 1

An example of a typical pseudopotential using the method of Ref. [2]. The Coulomb potential $-Z/r$ is replaced by a pseudopotential $V_{\mathrm{ps}}\left(r\right)$ below some cutoff radius $r_c$. Calculated properties should be converged with respect to pseudopotential parameters, in this case $r_c$. $a_B$ is the Bohr radius.

Figure 2

An example of a $n_e^{\mathrm{PA}}\left(r\right)$ for solid density aluminum at 1 eV generated by the model of Refs. [13, 14]. The inset shows the same electron density but multiplied by $4\pi r^2.n_e^{\mathrm{PA}}\left(r\right)$ is an accurate approximation to the full three-dimensional electron density near the nuclei, a fact guaranteed by the realization that the effective potential felt by the electrons near the nuclei is dominated by the Coulomb nuclear potential.

Figure 3

A two-dimensional slice of electron densities in a cubic volume for simple cubic aluminum at 1 eV and solid density. The slice is taken through a crystal plane with an atom at the center. (a) Full electron density ${log}_{10}{n}_e\left(\mathit{r}\right)\left[a_B^{-3}\right]$; (b) electron density from superposition approximation ${log}_{10}{n}_e^{\mathrm{super}}\left(\mathit{r}\right)\left[a_B^{-3}\right]$; (c) absolute difference $\mathrm{\Delta }{n}_e\left(\mathit{r}\right)=n_e\left(\mathit{r}\right)-n_e^{\mathrm{super}}\left(\mathit{r}\right)\left[a_B^{-3}\right]$; (d) percentage difference $\mathrm{\Delta }{n}_e\left(\mathit{r}\right)/n_e\left(\mathit{r}\right)$.

Figure 4

A two-dimensional slice of electron densities in a cubic volume for a snapshot of an iron plasma simulation at 10 eV and $22.5{\mathrm{g}/\mathrm{cm}}^3$ with 27 atoms in the simulation cell. (a) Full electron density ${log}_{10}{n}_e\left(\mathit{r}\right)\left[a_B^{-3}\right]$; (b) electron density from the superposition approximation ${log}_{10}{n}_e^{\mathrm{super}}\left(\mathit{r}\right)\left[a_B^{-3}\right]$; (c) absolute difference $\mathrm{\Delta }{n}_e\left(\mathit{r}\right)=n_e\left(\mathit{r}\right)-n_e^{\mathrm{super}}\left(\mathit{r}\right)\left[a_B^{-3}\right]$; (c) percentage difference $\mathrm{\Delta }{n}_e\left(\mathit{r}\right)/n_e\left(\mathit{r}\right)$. Note the considerably reduced range of $\mathrm{\Delta }{n}_e\left(\mathit{r}\right)$ compared to $n_e\left(\mathit{r}\right)$.

Figure 5

Convergence of excess pressure $P_{\mathrm{ex}}^{\mathrm{vir}}$ with respect to the number of FFT points in one dimension $N$. The total number of FFT points is $N^3$. Convergence is shown for solid density aluminum at three temperatures. The horizontal dashed lines indicate $\pm 5\%$ from the converged value.