Nonempirical Semilocal Free-Energy Density Functional for Matter under Extreme Conditions

Realizing the potential for predictive density functional calculations of matter under extreme conditions depends crucially upon having an exchange-correlation ($XC$) free-energy functional accurate over a wide range of state conditions. Unlike the ground-state case, no such functional exists. We remedy that with systematic construction of a generalized gradient approximation $XC$ free-energy functional based on rigorous constraints, including the free-energy gradient expansion. The new functional provides the correct temperature dependence in the slowly varying regime and the correct zero-$T$, high-$T$, and homogeneous electron gas limits. Its accuracy in the warm dense matter regime is attested by excellent agreement of the calculated deuterium equation of state with reference path integral Monte Carlo results at intermediate and elevated $T$. Pressure shifts for hot electrons in compressed static fcc Al and for low-density Al demonstrate the combined magnitude of thermal and gradient effects handled well by this functional over a wide $T$ range.

Figure 1

(Left) Behavior of ${\tilde{A}}_x$ and $s_{2x}/s^2\equiv {\tilde{B}}_x/{\tilde{A}}_x$ as functions of $t$. (Right) Comparison of ${\tilde{B}}_c(r_s,t)$ reference data (symbols) and analytical fit (curves) for selected $r_s$ values.

Figure 2

Ratios of deuterium electronic pressure versus $T$ for the free-energy GGA (KDT16) and ground-state PBE $XC$ functionals, to PIMC reference results. AIMD PAW simulations, $\mathrm{\Gamma }$ point only, for 64 atoms (4500 steps, $T\le 125\mathrm{k}\mathrm{K}$) and 32 atoms (4500 steps, $T\ge 125\mathrm{kK}$); time step 25–50 asec.

Figure 3

(Left) Electronic pressure differences $\mathrm{\Delta }P(T)$ for the new KDT16 GGA, SD14 mixed LDA-GGA, corrKSDT LDA, and ground-state PBE $XC$ functionals, all referenced to PZ ground-state LDA values; fcc Al, $3.0\mathrm{g}/{\mathrm{cm}}^3$. (Right) Relative differences.

Figure 4

Relative difference in Al total pressure along six isotherms (10, 15, 20, 30, 40, 60 kK) for KDT16 and PBE $XC$ functionals plotted as $(P^{\mathrm{PBE}}-P^{\mathrm{KDT}16})/P^{\mathrm{KDT}16}\times 100$.