Local Density Approximation Exchange-correlation Free-energy Functional*

Restricted path integral Monte-Carlo (RPIMC) simulation data for the homogeneous electron gas at finite temperatures [1] are used to fit the exchange-correlation free energy as a function of the density and temperature. Together with a new finite-$T$ spin-polarization interpolation, this provides the local spin density approximation (LSDA) for the exchange-correlation free-energy functional required by finite-$T$ density functional theory. We discuss and compare different methods of fitting to the RPIMC data. The new function reproduces the RPIMC data in the fitting range of Wigner-Seitz radius and temperature, satisfies correct high-density, low- and high-$T$ asymptotic limits and is applicable beyond the range of fitting data.\\[4pt] [1] Phys. Rev. Lett. \textbf{110}, 146405 (2013).