Mapping from current densities to vector potentials in time-dependent current density functional theory

Abstract

Under reasonable assumptions the time- dependent particle density nsrW , td and the current density jW srW , td of a many-particle system that evolves under the action of external scalar and vector potentials VsrW , td and AW srW , td and is initially in the quantum state ucs0dl can be reproduced in another many-particle system with a different two-particle interaction, subjected to external potentials V8srW , td and AW 8srW , td and starting from an initial state uc8s0dl, which yields the same density and current as ucs0dl. Here we show that given the initial state of this other many-particle system, the potentials V8srW , td and AW 8srW , td, if they exist, are uniquely determined up to gauge transformations that do not alter the initial state. As a special case, we obtain a simpler proof of the Runge-Gross theorem for time-dependent current density functional theory. This theorem provides a formal basis for the application of time- dependent current density functional theory to transport problems.