Dynamic exchange-correlation potentials for the electron gas in dimensionality $D=3\mathrm{}$ and $D=2\mathrm{}$

Recent progress in the formulation of a fully dynamical local approximation to time-dependent density functional theory appeals to the longitudinal and transverse components of the exchange and correlation kernel in the linear current-density response of the homogeneous fluid at long wavelength. Both components are evaluated for the electron gas in dimensionality $D=3\mathrm{}$ and $D=2\mathrm{}$ by an approximate decoupling in the equation of motion for the current density, which accounts for processes of excitation of two electron-hole pairs. Each pair is treated in the random phase approximation, but the role of exchange and correlation is also examined; in addition, final-state exchange processes are included phenomenologically so as to satisfy the exactly known high-frequency behaviors of the kernel. The transverse and longitudinal spectra involve the same decay channels and are similar in shape. A two-plasmon threshold in the spectrum for two-pair excitations in $D=3\mathrm{}$ leads to a sharp minimum in the real part of the exchange and correlation kernel at twice the plasma frequency. In $D=2\mathrm{}$ the same mechanism leads to a broad spectral peak and to a broad minimum in the real part of the kernel, as a consequence of the dispersion law of the plasmon vanishing at long wavelength. The numerical results have been fitted to simple analytic functions.