Density functional theory (DFT) for electrons at finite temperature is increasingly important in condensed matter and chemistry. Exact conditions that have proven crucial in constraining and constructing accurate approximations for ground-state DFT are generalized to finite temperature, including the adiabatic connection formula. We discuss consequences for functional construction.Because of the small mass ratio between electrons and nuclei, standard electronic structure calculations treat the former as being in their ground state, but routinely account for the finite temperature of the latter, as in ab initio molecular dynamics [1]. But as electronic structure methods are applied in ever more esoteric areas, the need to account for the finite temperature of electrons increases. Phenomena where such effects play a role include rapid heating of solids via strong laser fields [2], dynamo effects in giant planets [3], magnetic [4,5] and superconducting phase transitions [6,7], shock waves [8,9], warm dense matter [10], and hot plasmas [11][12][13].Within density functional theory, the natural framework for treating such effects was created by Mermin [14]. Application of that work to the Kohn-Sham (KS) scheme at finite temperature also yields a natural approximation: treat KS electrons at finite temperature but use groundstate exchange-correlation (XC) functionals. This works well in recent calculations [8,10], where inclusion of such effects is crucial for accurate prediction. This assumes that finite-temperature effects on exchange-correlation are negligible relative to the KS contributions, which may not always be true.The uniform electron gas at finite temperature (also called the one-component plasma) has been well-studied, and has in the past provided the natural starting point for DFT studies of such finite-temperature XC effects, as input into the local density approximation (LDA) at finite T [15]. However, the LDA is too inaccurate for most modern applications of DFT, and almost all recent calculations use a generalized gradient approximation or hybrid with exchange [16]. The errors of LDA would typically be enormous relative to the temperature corrections we seek, especially for correlation, and so could lead to quite misleading results. Accurate calculation of finite temperature contributions requires accurate approximate functionals. Magnetic phase transitions bear an additional difficulty: The low-lying excitations are collective, i.e., magnons whose description requires non-collinear version of spin-DFT. Hence, a finite-temperature version of spin-DFT involving only spin-up and spin-down densities and thus only spin-flip excitations, is bound to fail in predicting, e.g., the critical temperature [4].The most fundamental steps toward both understanding a functional and creating accurate approximations are deriving its inequalities from the variational definition of the functional. These yield both the signs of energy contributions and, via uniform scaling of the spatial coordinates, basic equalities and...

Bounds on the exchange-correlation energy of many-electron systems are derived and tested. By using universal scaling properties of the electron-electron interaction, we obtain the exponent of the bounds in three, two, one, and quasione dimensions. From the properties of the electron gas in the dilute regime, the tightest estimate to date is given for the numerical prefactor of the bound, which is crucial in practical applications. Numerical tests on various low-dimensional systems are in line with the bounds obtained and give evidence of an interesting dimensional crossover between two and one dimensions.