Fast and Universal Kohn-Sham Density Functional Theory Algorithm for Warm Dense Matter to Hot Dense Plasma

Understanding many processes, e.g., fusion experiments, planetary interiors, and dwarf stars, depends strongly on microscopic physics modeling of warm dense matter and hot dense plasma. This complex state of matter consists of a transient mixture of degenerate and nearly free electrons, molecules, and ions. This regime challenges both experiment and analytical modeling, necessitating predictive ab initio atomistic computation, typically based on quantum mechanical Kohn-Sham density functional theory (KS-DFT). However, cubic computational scaling with temperature and system size prohibits the use of DFT through much of the warm dense matter regime. A recently developed stochastic approach to KS-DFT can be used at high temperatures, with the exact same accuracy as the deterministic approach, but the stochastic error can converge slowly and it remains expensive for intermediate temperatures ($<50\mathrm{eV}$). We have developed a universal mixed stochastic-deterministic algorithm for DFT at any temperature. This approach leverages the physics of KS-DFT to seamlessly integrate the best aspects of these different approaches. We demonstrate that this method significantly accelerated self-consistent field calculations for temperatures from 3 to 50 eV, while producing stable molecular dynamics and accurate diffusion coefficients.

Figure 1

(Top) Density of states (DOS) of 64 atoms diamond structure carbon ($3.51\mathrm{g}/\mathrm{cc}$) at 10 eV. See text for explicit form. DDFT (dash-dot line), SDFT (dashed line), MDFT (solid line), and MDFT deterministic component (solid fill under) are shown. (Bottom) The occupied DOS. Individual states are represented as Lorentzians with a 2 eV broadening. Fermi energy, $E_f$, shown as dashed vertical line.

Figure 2

Single disordered carbon snapshot $3.51\mathrm{g}/\mathrm{cc}$. (a) Top: comparison of SCF calculation times on 512 threads. See text for parameters. Middle: comparison of relative error in free energy ($A$). Bottom: comparison of electron chemical potential ($\mu$). 0 shown as an eye guide. (b) Average standard deviation of forces (solid lines, filled markers) and SCF time on 256 threads (dashed lines, open markers) for different $N_{\chi }$ ($x$ axis) and $N_{\psi }$. $N_{\psi }=0$, 128, and 256 are shown as black circles, red diamonds, and blue squares, respectively.

Figure 3

Velocity autocorrelation function from DDFT (solid lines), MDFT (dash-dotted line), and two SDFT (dotted and dashed line): $10\mathrm{g}/\mathrm{cc}$ 10 eV results shown on the right, while $3.51\mathrm{g}/\mathrm{cc}$ 5 eV results are shown on the left. The self-diffusion coefficients ($D$, integral of VACF) are shown in the key with units of $0.001{\mathrm{cm}}^2/\mathrm{s}$ with the estimated error range (see the Supplemental Material [42] for full simulation parameters).