  In the context of inhomogeneous one-dimensional finite systems, recent
numerical advances [Phys. Rev. B 103, 125155 (2021)] allow us to compute the
exact coupling-constant dependent exchange-correlation kernel
$f^\lambda_\text{xc}(x,x',\omega)$ within linear response time-dependent
density functional theory. This permits an improved understanding of
ground-state total energies derived from the adiabatic-connection
fluctuation-dissipation theorem (ACFDT). We consider both `one-shot' and
`self-consistent' ACFDT calculations, and demonstrate that chemical accuracy is
reliably preserved when the frequency dependence in the exact functional
$f_\text{xc}[n](\omega=0)$ is neglected. This performance is understood on the
grounds that the exact $f_\text{xc}[n]$ varies slowly over the most relevant
$\omega$ range (but not in general), and hence the spatial structure in
$f_\text{xc}[n](\omega=0)$ is able to largely remedy the principal issue in the
present context: self-interaction (examined from the perspective of the
exchange-correlation hole). Moreover, we find that the implicit orbitals
contained within a self-consistent ACFDT calculation utilizing the adiabatic
exact kernel $f_\text{xc}[n](\omega=0)$ are remarkably similar to the exact
Kohn-Sham orbitals, thus further establishing that the majority of the physics
required to capture the ground-state total energy resides in the spatial
dependence of $f_\text{xc}[n]$ at $\omega = 0$.
