Recently developed extended Kohn-Sham theory and extended quasiparticle (QP) theory are both fundamental and important, in particular, in treating the initially excited eigenstates from first-principles. However, still the QP theory including $GW\left(\mathrm{\Gamma }\right)$ methods has been applied only to the initial ground state by almost all authors. In this paper, we want to emphasize that these theories allow one to apply $GW\left(\mathrm{\Gamma }\right)$ methods directly to an initially excited eigenstate. Using only basic and elementary arguments without using the Green's function, we explicitly demonstrate that the QP wave functions can be normalized to unity and the self-energy can be hermitized for the initially excited eigenstate. Our paper supports the conventional procedure commonly used in first-principles calculations. These theories hold when relaxation from excited states can be ignored. We propose the normalized $GW$ (NGW) approach and perform self-consistent calculations for initially excited (cationic and photoabsorbed) states of several light atoms. The resulting QP energies show good agreement with the available experimental data for excited states, as well as with the expected relationships among different QP energies, supporting the validity of the method.

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• Electronic structure
• Photoemission
• Electron affinity
• First-principles calculations
• GW method
• Perturbation theory