Hydrogen molecule spectrum by many-body GW and Bethe-Salpeter equation

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$_2$ excitation spectrum and in comparison with configuration interaction (CI) and time-dependent Hartree-Fock. The comparison is made on all the states we could unambiguously identify from the excitonic wavefunctions' 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 does not exist anymore 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.