Dynamical exchange-correlation potentials for an electron liquid

The imaginary parts of the exchange-correlation kernels f(xc)(L,T)(omega) 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 Im f(xc)(L,T)(omega) and with the compressibility and the third moment sum rules, we construct simple interpolation formulas for Im f(xc)(L,T)(omega) 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 Im f(xc)(L)(omega) is thus intermediate between the Gross-Kohn interpolation, which ignores the two-plasmon contribution, and a recent approximate calculation by Nifosi, 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 Im f(xc)(L)(omega) and Im f(xc)(T)(omega) at small omega.