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

TL;DR

This paper demonstrates the invariance of the exchange-correlation action in time-dependent density functional theory under accelerated frame transformations, ensuring correct center of mass motion separation and proposing a method to generate symmetry-preserving functionals.

Contribution

It establishes the invariance property of the exchange-correlation functional under accelerated frames and introduces a method to construct functionals with this symmetry.

Findings

Some approximate $V_{xc}$ satisfy the invariance, others do not.

Correctly transforming $V_{xc}$ ensures the harmonic potential theorem holds.

A general method to generate symmetry-preserving functionals is proposed.

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 $\rho (\vec r,t) \to \rho ' (\vec r,t) = \rho (\vec r + \vec x (t),t)$, 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: $V_{xc}[\rho '; \vec r, t] = V_{xc}[\rho; \vec r + \vec x (t),t]$. 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.