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

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