Self-energy calculation of the hydrogen atom: Importance of the unbound states

We present the calculation of the self-energy of the isolated hydrogen atom within the GW approximation starting from the noninteracting Green's function constructed from the exact wave functions of the hydrogen atom. The error in the electron removal energy of the 1s state is found to be about 0.02 eV, which is much smaller than what one would expect. This small error is explained by the cancellation of the self-screening errors between different l contributions of the self-energy. The unbound continuum states are found to be crucial to get the correct self-energy.