  The exchange-correlation hole and potential of the homogeneous electron gas
have been investigated within the random-phase approximation, employing the
plasmon-pole approximation for the linear density response function. The
angular dependence as well as the time dependence of the exchange-correlation
hole are illustrated for a Wigner-Seitz radius $r_s=4$ (atomic unit). It is
found that there is a substantial cancellation between exchange and correlation
potentials in space and time, analogous to the cancellation of exchange and
correlation self-energies. Analysis of the sum rule explains why it is more
advantageous to use a non-interacting Green function than a renormalized one
when calculating the response function within the random-phase approximation
and consequently the self-energy within the well-established $GW$
approximation. The present study provides a starting point for more accurate
and comprehensive calculations of the exchange-correlation hole and potential
of the electron gas with the aim of constructing a model based on the local
density approximation as in density functional theory.
