Dynamical exchange-correlation potentials for an electron liquid

The imaginary parts of the exchange-correlation kernels f_xc^L,T(ω) 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 Imf_xc^L,T(ω) and with the compressibility and the third moment sum rules, we construct simple interpolation formulas for Imf_xc^L,T(ω) 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 Imf_xc^L(ω) 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 Imf_xc^L(ω) and Imf_xc^T(ω) at small ω.