On the relation between the scalar and tensor exchange-correlation kernels of the time-dependent density-functional theory

TL;DR

This paper derives a simple relation between scalar and tensor exchange-correlation kernels in time-dependent density functional theory, facilitating easier approximations and connections between the two formulations.

Contribution

The authors establish a straightforward formula linking scalar and tensor xc kernels under the weak dynamic xc assumption, aiding practical applications.

Findings

Derived a relation between $f_{xc}$ and $ abla imes abla imes \, ext{tensor kernel}

Provides a bridge for approximating tensor kernels using scalar kernels

Simplifies the theoretical framework for time-dependent density functional theory

Abstract

The scalar $f_{xc}$ and tensor $\hat{f}_{xc}$ exchange-correlation (xc) kernels are key ingredients of the time-dependent density functional theory and the time-dependent current density functional theory, respectively. We derive a comparatively simple relation between these two kernels under the assumption that the dynamic xc can be considered "weak". We expect our formula to serve as a convenient bridge between the scalar $f_{xc}$ which directly enters many applications and the tensor $\hat{f}_{xc}$ which, due to its locality in space, is much easier to approximate.