  Atomic hydrogen provides a unique test case for computational electronic
structure methods, since its electronic excitation energies are known
analytically. With only one electron, hydrogen contains no electronic
correlation and is therefore particularly susceptible to spurious
self-interaction errors introduced by certain computational methods. In this
paper we focus on many-body perturbation-theory (MBPT) in Hedin's GW
approximation. While the Hartree-Fock and the exact MBPT self-energy are free
of self-interaction, the correlation part of the GW self-energy does not have
this property. Here we use atomic hydrogen as a benchmark system for GW and
show that the self-interaction part of the GW self-energy, while non-zero, is
small. The effect of calculating the GW self-energy from exact wavefunctions
and eigenvalues, as distinct from those from the local-density approximation,
is also illuminating.
