# Ab initio Exchange-Correlation Free Energy of the Uniform Electron Gas at Warm Dense Matter Conditions

## TL;DR

This paper provides the first comprehensive ab initio quantum Monte Carlo results for the exchange-correlation free energy of the warm dense electron gas across various densities, temperatures, and spin polarizations, achieving unprecedented accuracy.

## Contribution

It overcomes previous limitations by extending QMC results to lower temperatures and different spin polarizations, enabling a complete ab initio free energy functional.

## Key findings

- First complete ab initio exchange-correlation free energy functional.
- Achieved accuracy of approximately 0.3%.
- Quantified errors of previous approximate functionals.

## Abstract

In a recent Letter [T.~Dornheim \textit{et al.}, Phys. Rev. Lett. \textbf{117}, 156403 (2016)], we presented the first \textit{ab initio} quantum Monte-Carlo (QMC) results of the warm dense electron gas in the thermodynamic limit. However, a complete parametrization of the exchange-correlation free energy with respect to density, temperature, and spin polarization remained out of reach due to the absence of (i) accurate QMC results below $\theta=k_\text{B}T/E_\text{F}=0.5$ and (ii) of QMC results for spin polarizations different from the paramagnetic case. Here we overcome both remaining limitations. By closing the gap to the ground state and by performing extensive QMC simulations for different spin polarizations, we are able to obtain the first complete \textit{ab initio} exchange-correlation free energy functional; the accuracy achieved is an unprecedented $\sim 0.3\%$. This also allows us to quantify the accuracy and systematic errors of various previous approximate functionals.