Frequency Dependence of the Exact Exchange-Correlation Kernel of Time-Dependent Density-Functional Theory

We present a scheme for numerically reconstructing the exact and the Kohn-Sham time-dependent density-density response functions, and from their inverse the numerical representation of the exact frequency-dependent exchange-correlation kernel $f_{\mathrm{xc}}$ of time-dependent Kohn-Sham density-functional theory. We investigate the exact $f_{\mathrm{xc}}$ in detail for a system that is known to show strong memory effects. The reconstructed kernel fulfills the appropriate sum rule. Using it in linear response calculations, we show how the exact $f_{\mathrm{xc}}$, due to its frequency dependence, yields the exact excitation energies, including the ones of double excitation character.

Figure 1

${\partial }_z{v}_{\mathrm{xc},0}(z)$ (dotted black) compared to $\int f_{\mathrm{xc}}(z,z^{\prime },\omega ){\partial }_{z^{\prime }}{n}_0(z^{\prime })d{z}^{\prime }$, according to Eq. (6), for $\omega =0$ (real part red, full line, coinciding with dotted curve) and the first and second excitation frequencies $\omega \approx {\omega }_{01}$ and ${\omega }_{02}$ (real part designated by a green and a blue full line, respectively, slightly below the dotted line). Only one of these lines is visible as the lines coincide on the scale of the plot. The corresponding imaginary parts (dashed red, green, blue) are all negligible, as they should be, and therefore just one dashed zero-line can be seen in the figure.

Figure 2

Top panel: Lowest ${\lambda }_m(\omega )$ parts of the eigenvalues of $\mathrm{\Omega }(\omega )$ (green diamonds). Intersections with the solid red parabola ${\omega }^2$ line represent solutions of (7). For comparison both the exact ${\omega }_{0n}^2$ as obtained from the static interacting Schrödinger equation [24] (solid horizontal lines) and the KS ${\omega }_{s,0j}^2$ (short dashed horizontal lines) are shown. Bottom panel: The solutions that are found when neglecting off-diagonal elements of the Casida matrix are plotted as in the top panel. See main text for a discussion of the color coding.