Hydrogen-molecule spectrum by the many-body GW approximation and the 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 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 similar to 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.