  We examine fractional charge and spin errors in self-consistent Green's
function theory within a second-order approximation (GF2). For GF2 it is known
that the summation of diagrams resulting from the self-consistent solution of
the Dyson equation removes the divergences pathological to second-order
Moller-Plesset theory (MP2) for strong correlations. In the language often used
in density functional theory contexts, this means GF2 has a greatly reduced
fractional spin error relative to MP2. The natural question then is what
effect, if any, does the Dyson summation have on the fractional charge error in
GF2? To this end we generalize our previous implementation of GF2 to open-shell
systems and analyze its fractional spin and charge errors. We find that like
MP2, GF2 possesses only a very small fractional charge error, and consequently
little many electron self-interaction error. This shows that GF2 improves on
the critical failings of MP2, but without altering the positive features that
make it desirable. Furthermore, we find that GF2 has both less fractional
charge and fractional spin errors than typical hybrid density functionals as
well as random phase approximation with exchange.
