Electrical and thermal conductivity of liquid sodium from first-principles calculations

We report on the electrical and thermal conductivity of liquid sodium at 400 K, calculated using density functional theory with the local density approximation (LDA) and the Kubo-Greenwood formula. We extensively tested system-size errors and k-point sampling, using simulation cells containing up to 2000 atoms. We find that convergence of the results with respect to the size of the system is slow, and at least 1024-atom systems are required to obtain conductivities converged to within a few percent. $\Gamma$-point sampling does not seem to be accurate enough, even for the very largest 2000-atom system. We performed calculations at three densities, including the experimental density ${\rho }_{\mathrm{expt}}=921$ kg m${}^{-3}$, the LDA density ${\rho }_{\mathrm{LDA}}=1046$ kg m${}^{-3}$, and a higher density $\rho =1094$ kg m${}^{-3}$. At the experimental density, the electrical conductivity is underestimated by $\sim 35\%$, at the LDA density it is overestimated by $\sim 18\%$, and at the largest density it is higher than the experimental one by $\sim 50\%$. At the experimental density, we also used the Perdew-Burke-Ernzerhof functional, and found that the conductivity is overestimated by only $\sim 6\%$.

Figure 1

Radial distribution function $g\left(r\right)$ (left panel) and structure factor $S\left(k\right)$ (right panel) of liquid Na at 400 K at the experimental density ${\rho }_{\mathrm{expt}}=921$ kg m${}^{-3}$, computed using simulation cells including 54, 128, 432, and 1024 atoms. Data from simulations with 1458 and 2000 atoms are indistinguishable from those with 1024 atoms and are not shown. All simulations have been performed with the $\Gamma$ point only.

Figure 2

Radial distribution function $g\left(r\right)$ (left panel) and structure factor $S\left(k\right)$ (right panel) of liquid Na at 400 K, computed using 1024-atom simulation cells at the experimental density ${\rho }_{\mathrm{expt}}=921$ kg m${}^{-3}$ (black solid curve) and at the LDA density ${\rho }_{\mathrm{LDA}}=1046$ kg m${}^{-3}$ (dashed violet curve). Experimental data (blue dots) are from Ref. 28.

Figure 3

Optical conductivity $\sigma \left(\omega \right)$ at the experimental density, calculated with the 54-atom system and using various sets of k points.

Figure 4

Optical conductivity $\sigma \left(\omega \right)$ at the experimental density calculated with the $\Gamma$ point only for various system sizes.

Figure 5

Optical conductivity $\sigma \left(\omega \right)$ at the experimental density calculated with the 54-, 128-, 432-, 1024-, 1458-, and 2000-atom systems. Results shown are fully converged with respect to k-point sampling.

Figure 6

Electrical conductivity ${\sigma }_0$ (top panel) obtained from a fit of $\sigma \left(\omega \right)$ to a Drude function for the 54-, 128-, 432-, 1024-, 1458-, and 2000-atom systems at the experimental density ${\rho }_{\mathrm{expt}}=921$ kg m${}^{-3}$. Bottom panel shows the value of $S$ [see Eq. (3)]. Results are shown for various sets of k points.

Figure 7

Electrical conductivity ${\sigma }_0$ (top panel) obtained from a fit of $\sigma \left(\omega \right)$ to a Drude function for the 54-, 128-, 432-, 1024-, 1458-, and 2000-atom systems at the density $\rho =1094$ kg m${}^{-3}$. Bottom panel shows the value of $S$ [see Eq. (3)]. Results are shown for various sets of k points.

Figure 8

Density of represented states N$\left(\epsilon \right)$ (vertical black lines) and derivative of the Fermi distribution $dF/d\epsilon$ divided by its value at the minimum (red curve) for the 128-atom (left panel) and the 1024-atom (right panel) systems. Apparent thicker lines correspond to nearly degenerate eigenvalues.