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

The scalar fxc and tensor 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." A calculation of the frequency-dependent dielectric function of silicon using this relation in conjunction with Vignale-Kohn f xc demonstrates a potential of our method to account for the dynamic many-body effects within the rigorous scheme of time-dependent density functional theory. Our formula provides a bridge between the scalar fxc, which directly enters many applications, and the tensor f xc which, due to its locality in space, is much easier to approximate.