Influence of the noninteracting density response function on the exchange-only kernel in time-dependent density-functional theory

A formally exact expression for the interaction density response function $\chi (\mathbf{r},{\mathbf{r}}^{\prime },\omega )$ exists in terms of (i) its noninteracting counterpart ${\chi }_0(\mathbf{r},{\mathbf{r}}^{\prime },\omega )$ and (ii) an exchange $\left(x\right)$-correlation $\left(c\right)$ kernel $f_{xc}(\mathbf{r},{\mathbf{r}}^{\prime },\omega )$. In the absence of a first-principles theory for the $\omega$ dependence of $f_{xc}$, the adiabatic approximation is most frequently made in this term, to construct a workable time-dependent density-functional theory. In the present study, a proposal is put forward to avoid the adiabatic approximation by working in the exchange-only limit in which $f_{xc}$ is set equal to $f_x(\mathbf{r},{\mathbf{r}}^{\prime },\omega )$. We then refer to a result for the exchange energy given by Pines and Nozieres to motivate the assumption that $f_x=f_x\left[\mathrm{Im}{\chi }_0(\mathbf{r},{\mathbf{r}}^{\prime },\omega )\right]$. The essential proposal here is therefore that the integral equation to be solved for the interacting density response function $\chi$ is in the exchange-only case characterized entirely by the noninteracting response function ${\chi }_0$.