Dynamical exchange-correlation potentials for an electron liquid

The imaginary parts of the exchange-correlation kernels $f_{\mathrm{xc}}^{L,T}\left(\omega \right)$ in the longitudinal and transverse current-current response functions of a homogeneous electron liquid are calculated exactly at low frequency, to leading order in the Coulomb interaction. Combining these new results with the previously known high-frequency behaviors of $\mathrm{Im}{f}_{\mathrm{xc}}^{L,T}\left(\omega \right)$ and with the compressibility and the third moment sum rules, we construct simple interpolation formulas for $\mathrm{Im}{f}_{\mathrm{xc}}^{L,T}\left(\omega \right)$ in three and two spatial dimensions. A feature of our interpolation formulas is that they explicitly take into account the two-plasmon component of the excitation spectrum: our longitudinal spectrum $\mathrm{Im}{f}_{\mathrm{xc}}^L\left(\omega \right)$ is thus intermediate between the Gross-Kohn interpolation, which ignores the two-plasmon contribution, and a recent approximate calculation by Nifosì, Conti, and Tosi, which probably overestimates it. Numerical results for both the real and imaginary parts of the exchange-correlation kernels at typical electron densities are presented, and compared with those obtained from previous approximations. We also find an exact relation between $\mathrm{Im}{f}_{\mathrm{xc}}^L\left(\omega \right)$ and $\mathrm{Im}{f}_{\mathrm{xc}}^T\left(\omega \right)$ at small $\omega .$