Finite-temperature density functional theory (DFT) has become of topical interest, partly due to the increasing ability to create novel states of warm-correlated matter (WCM).Warm-dense matter (WDM), ultra-fast matter (UFM), and high-energy density matter (HEDM) may all be regarded as subclasses of WCM. Strong electron-electron, ion-ion and electron-ion correlation effects and partial degeneracies are found in these systems where the electron temperature T e is comparable to the electron Fermi energy E F . Thus, many electrons are in continuum states which are partially occupied. 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 ≠ 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 DFT dominated by solid-state theory or quantum chemistry. Many basic questions arise in trying to implement DFT for WCM. Hohenberg-Kohn-Mermin theory can be adapted for treating these systems if suitable finite- T exchange-correlation (XC) functionals can be constructed. They are functionals of both the one-body electron density ne and the one-body ion densities ρj . Here, j counts many species of nuclei or charge states. 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 in situ , directly from the pair-distribution functions. This eliminates the need for any XC-functionals. 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.
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