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 $A_{xc}[\rho (\vec{r},t)]$ in time-dependent density functional theory , where $\rho (\vec{r},t)$ is the time-dependent density, is invariant under the transformation to an accelerated frame of reference $\rho (\vec{r},t)\to {\rho }^{\prime }(\vec{r},t)=\rho (\vec{r}+\vec{x}(t),t)$, 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: $V_{xc}[{\rho }^{\prime };\vec{r},t]=V_{xc}[\rho ;\vec{r}+\vec{x}(t),t]$. 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.

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