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Recent progress in the formulation of a fully dynamical local approximationto time-dependent Density Functional Theory appeals to the longitudinal andtransverse components of the exchange and correlation kernel in the linearcurrent-density response of the homogeneous fluid at long wavelength. Bothcomponents are evaluated for the electron gas in dimensionality D=3 and D=2 byan approximate decoupling in the equation of motion for the current density,which accounts for processes of excitation of two electron-hole pairs. Eachpair is treated in the random phase approximation, but the role of exchange andcorrelation is also examined; in addition, final-state exchange processes areincluded phenomenologically so as to satisfy the exactly known high-frequencybehaviours of the kernel. The transverse and longitudinal spectra involve thesame decay channels and are similar in shape. A two-plasmon threshold in thespectrum for two-pair excitations in D=3 leads to a sharp minimum in the realpart of the exchange and correlation kernel at twice the plasma frequency. InD=2 the same mechanism leads to a broad spectral peak and to a broad minimum inthe real part of the kernel, as a consequence of the dispersion law of theplasmon vanishing at long wavelength. The numerical results have been fitted tosimple analytic functions.Comment: 13 pages, 11 figures included. Accepted for publication in Phys. Rev.

physical review. b, condensed matter

Dynamic exchange-correlation potentials for the electron gas in dimensionality<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:mn /></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mn /></mml:math>

Dynamic exchange-correlation potentials for the electron gas in dimensionalityD=3andD=2