  We report unphysical irregularities and discontinuities in some key
experimentally-measurable quantities computed within the GW approximation of
many-body perturbation theory applied to molecular systems. In particular, we
show that the solution obtained with partially self-consistent GW schemes
depends on the algorithm one uses to solve self-consistently the quasi-particle
(QP) equation. The main observation of the present study is that each branch of
the self-energy is associated with a distinct QP solution, and that each switch
between solutions implies a significant discontinuity in the quasiparticle
energy as a function of the internuclear distance. Moreover, we clearly observe
"ripple" effects, i.e., a discontinuity in one of the QP energies induces
(smaller) discontinuities in the other QP energies. Going from one branch to
another implies a transfer of weight between two solutions of the QP equation.
The case of occupied, virtual and frontier orbitals are separately discussed on
distinct diatomics. In particular, we show that multisolution behavior in
frontier orbitals is more likely if the HOMO-LUMO gap is small.
