GW approximation with LSDA+U method and applications to NiO, MnO, and V$_2$O$_3$

A GW approximation (GWA) method named U+GWA is proposed, where we can start GWA with more localized wave functions obtained by the local spin-density approximation (LSDA)+U method. Then GWA and U+GWA are applied to MnO, NiO, and V$_2$O$_3$ in antiferromagnetic phase. The band gaps and energy spectra show an excellent agreement with the experimentally observed results and are discussed in details. The calculated width of d-bands of V$_2$O$_3$ is much narrower than that of the observed one which may be a mixture of t$_{2g}^2$ multiplet and single electron t$_{2g}$ level. GWA or U+GWA does not work also in the paramagnetic phase of V$_2$O$_3$ and the reason for thois is clarified. The method of the unique choice of on-site Coulomb interaction is discussed in details. The criterion for whether we should adopt GWA or U+GWA is discussed and is assessed with the help of the off-diagonal elements of the self-energy.