Foundations of time-dependent density functional theory: Some facts together with illustrative examples

The electron density n(r, t), which is the central tool of time-dependent density functional theory, is presently considered to be derivable from a one-body time-dependent potential V (r, t), via one-electron wave functions satisfying a timedependent Schrödinger equation. This is here related via a generalized equation of motion to a Dirac density matrix now involving t. Linear response theory is then surveyed, with a special emphasis on the question of causality with respect to the density dependence of the potential. Extraction of V (r, t) for solvable models is also proposed.