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Current issues in finite-$T$ density-functional theory and Warm-Correlated Matter

Finite-temperature DFT has become of topical interest, partly due to the increasing ability to create novel states of warm-correlated matter (WCM). Subclasses of WCM are Warm-dense matter (WDM), ultra-fast matter (UFM), and high-energy density matter (HEDM), containing electyrons (e) and ions (i). Strong e-e, i-i and e-i correlation effects and partial degeneracies are found in these systems where the electron temperature $T_e $ is comparable to the electron Fermi energy. The ion subsystem may be solid, liquid or plasma, with many states of ionization with ionic charge $Z_j$. Quasi-equilibria with the ion temperature $T_i\ne T_e$ are common. The ion subsystem in WCM can no longer be treated as a passive "external potential", as is customary in $T=0$ density functional theory (DFT) dominated by solid-state theory or quantum chemistry. Hohenberg-Kohn-Mermin theory can be used for WCMs if finite-$T$ exchange-correlation (XC) functionals are available. They are functionals of both the one-body electron density $n_e$ and the one-body ion densities $ρ_j$. A method of approximately but accurately mapping the quantum electrons to a classical Coulomb gas enables one to treat electron-ion systems entirely classically at any temperature and arbitrary spin polarization, using exchange-correlation effects calculated {\it in situ}, directly from the pair-distribution functions. This eliminates the need for any XC-functionals, or the use of a Born-Oppenheimer approximation. This classical map has been used to calculate the equation of state of WDM systems, and construct a finite-$T$ XC functional that is found to be in close agreement with recent quantum path-integral simulation data. In this review current developments and concerns in finite-$T$ DFT, especially in the context of non-relativistic warm-dense matter and ultra-fast matter will be presented.