  We check the ab initio GW approximation and Bethe-Salpeter equation (BSE)
many-body methodology against the exact solution benchmark of the hydrogen
molecule H$_2$ ground state and excitation spectrum, and in comparison with the
configuration interaction (CI) and time-dependent Hartree-Fock methods. The
comparison is made on all the states we could unambiguously identify from the
excitonic wave functions' symmetry. At the equilibrium distance $R = 1.4 \,
a_0$, the GW+BSE energy levels are in good agreement with the exact results,
with an accuracy of 0.1~0.2 eV. GW+BSE potential-energy curves are also in good
agreement with the CI and the exact result up to $2.3 \, a_0$. The solution no
longer exists beyond $3.0 \, a_0$ for triplets ($4.3 \, a_0$ for singlets) due
to instability of the ground state. We tried to improve the GW reference ground
state by a renormalized random-phase approximation (r-RPA), but this did not
solve the problem.
