Exchange-correlation potentials in the adiabatic connection fluctuation-dissipation framework

We provide the expression of the exchange-correlation potential in the adiabatic connection fluctuation-dissipation (ACFD) framework, for arbitrary time-dependent (TD) kernels. We investigate the asymptotic behavior of the ACFD potential in three relevant approximations: the random-phase approximation, the exact-exchange kernel in two-electron systems, and the adiabatic local-density approximation. We show that these potentials have the expected $-{1/r+Q/r}^3-\alpha {/(2r}^4)$ tail (in closed-shell systems with spherical symmetry), where Q and $\alpha$ depend on the TD kernel and reflect the physics included in each approximation. We also discuss approximate ACFD potentials that are much simpler to compute than the exact ones while being likely of reasonable accuracy.