Density-Functional Theory for Time-Dependent Systems

A density-functional formalism comparable to the Hohenberg-Kohn-Sham theory of the ground state is developed for arbitrary time-dependent systems. It is proven that the single-particle potential $v\left(\overrightarrow{\mathrm{r}}t\right)$ leading to a given $v$-representable density $n\left(\overrightarrow{\mathrm{r}}t\right)$ is uniquely determined so that the corresponding map $v\to n$ is invertible. On the basis of this theorem, three schemes are derived to calculate the density: a set of hydrodynamical equations, a stationary action principle, and an effective single-particle Schrödinger equation.