Spin-polarized electron liquid at arbitrary temperatures: Exchange-correlation energies, electron-distribution functions, and the static response functions

We use a recently introduced classical mapping of the Coulomb interactions in a quantum electron liquid [Phys. Rev. Lett. 84, 959 (2000)] to present a unified treatment of the thermodynamic properties and the static response of the finite-temperature electron liquid, valid for arbitrary coupling and spin-polarization. The method is based on using a “quantum temperature” $T_q$ such that the distribution functions of the classical electron liquid at $T_q$ leads to the same correlation energy as the quantum electron liquid at $T=0.\mathrm{}$ The functional form of $T_q{(r}_s)$ is presented. The electron-electron pair-distribution functions (PDF’s) calculated using $T_q$ are in good quantitative agreement with available $\mathrm{}(T=0\mathrm{})\mathrm{}$ quantum Monte Carlo results. The method provides a means of treating strong-coupling regimes of $n,T,$ and $\zeta$ currently unexplored by quantum Monte Carlo or Feenberg-functional methods. The exchange-correlation free energies, distribution functions $g_{11}\mathrm{}\left(r\right),$ $g_{12}\mathrm{}\left(r\right),$ $g_{22}\mathrm{}\left(r\right)\mathrm{}$ and the local-field corrections to the static response functions as a function of density n, temperature T, and spin polarization $\zeta$ are presented and compared with any available finite-$T$ results. The exchange-correlation free energy $f_{\mathrm{xc}}(n,T,\zeta ),$ is given in a parametrized form. It satisfies the expected analytic behavior in various limits of temperature, density, and spin polarization, and can be used for calculating other properties like the equation of state, the exchange-correlation potentials, compressibility, etc. The static local-field correction provides a static response function which is consistent with the PDF’s and the relevant sum rules. Finally, we use the finite-$T$ xc-potentials to examine the Kohn-Sham bound- and continuum states at an ${\mathrm{Al}}^{13+}$ nucleus immersed in a hot electron gas to show the significance of the xc-potentials.