Center of mass and relative motion in time dependent density functional theory

Abstract

It is shown that the exchange-correlation part of the action functional $A_{xc}[ρ(\vec r,t)]$ in time-dependent density functional theory , where $ρ(\vec r,t)$ is the time-dependent density, is invariant under the transformation to an accelerated frame of reference $ρ(\vec r,t) \to ρ' (\vec r,t) = ρ(\vec r + \vec x (t),t)$, where $\vec x (t)$ is an arbitrary function of time. This invariance implies that the exchange-correlation potential in the Kohn-Sham equation transforms in the following manner: $V_{xc}[ρ'; \vec r, t] = V_{xc}[ρ; \vec r + \vec x (t),t]$. Some of the approximate formulas that have been proposed for $V_{xc}$ satisfy this exact transformation property, others do not. Those which transform in the correct manner automatically satisfy the ``harmonic potential theorem", i.e. the separation of the center of mass motion for a system of interacting particles in the presence of a harmonic external potential. A general method to generate functionals which possess the correct symmetry is proposed.