  An efficient all-electron G$^0$W$^0$ method and a quasiparticle
selfconsistent GW (QSGW) method for molecules are proposed in the molecular
orbital space with the full random phase approximation. The convergence with
basis set is examined. As an application, the ionization energy ($I$) and
electron affinity ($A$) of a series of conjugated molecules (up to 32 atoms)
are calculated and compared to experiment. The QSGW result improves the
G$^0$W$^0$ result and both of them are in significantly better agreement with
experimental data than those from Hartree-Fock (HF) and hybrid density
functional calculations, especially for $A$. The nearly correct energy gap and
suppressed self-interaction error by the HF exchange make our method a good
candidate for investigating electronic and transport properties of molecular
systems.
