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 rs=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 noninteracting 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.
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Financial support from the Knut and Alice Wallenberg (KAW) Foundation (Grant No. 2017.0061) and the Swedish Research Council (Vetenskapsrådet, VR, Grant No. 2021_-04498) is gratefully acknowledged. We thank Rex Godby forvaluable discussions.