  The combination of the many-body Green's function $GW$ approximation and the
Bethe-Salpeter equation (BSE) formalism has shown to be a promising alternative
to time-dependent density-functional theory (TD-DFT) for computing vertical
transition energies and oscillator strengths in molecular systems. The BSE
formalism can also be employed to compute ground-state correlation energies
thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT).
Here, we study the topology of the ground-state potential energy surfaces (PES)
of several diatomic molecules near their equilibrium bond length. Thanks to
comparisons with state-of-art computational approaches (CC3), we show that
ACFDT@BSE is surprisingly accurate, and can even compete with lower-order
coupled cluster methods (CC2 and CCSD) in terms of total energies and
equilibrium bond distances for the considered systems. However, we sometimes
observe unphysical irregularities on the ground-state PES in relation with
difficulties in the identification of a few $GW$ quasiparticle energies.
