Exchange and correlation in finite-temperature TDDFT

We discuss the finite-temperature generalization of time-dependent density functional theory (TDDFT). The theory is directly analogous to that at temperature T = 0. For example, the finite- T TDDFT exchange-correlation kernel f xc ( T , n ) in the local density approximation can again be expressed as a density derivative of the exchange correlation potential f xc ( T , n ) = [ ∂v xc ( T , n )∕ ∂n ] δ ( r − r ′ ), where n = N ∕ V is the electron number density. An approximation for the kernel f xc ( T , n ) is obtained from the finite- T generalization of the retarded cumulant expansion applied to the homogeneous electron gas. Results for f xc and the loss function are presented for a wide range of temperatures and densities including the warm dense matter regime, where T ≈ T F , the electron degeneracy temperature. The theory also permits a physical interpretation of the exchange and correlation contributions to the theory.