The quasiparticle (QP) energies, which are minus of the energies required by removing or produced by adding one electron from/to the system, corresponding to the photoemission or inverse photoemission (PE/IPE) spectra, are determined together with the QP wave functions, which are not orthonormal and even not linearly independent but somewhat similar to the normal spin orbitals in the theory of the configuration interaction, by self-consistently solving the QP equation coupled with the equation for the self-energy. The electron density, kinetic and all interaction energies can be calculated using the QP wave functions. We prove in a simple way that the PE/IPE spectroscopy and therefore this QP theory can be applied to an arbitrary initial excited eigenstate. In this proof, we show that the energy-dependence of the self-energy is not an essential difficulty, and the QP picture holds exactly if there is no relaxation mechanism in the system. The validity of the present theory for some initial excited eigenstates is tested using the one-shot GW approximation for several atoms and molecules. * ohno@ynu.ac.jp

The one-shot GW method beginning with the local density approximation (LDA) enables one to calculate photoemission and inverse photoemission spectra. In order to calculate photoabsorption spectra, one had to additionally solve the Bethe-Salpeter equation (BSE) for the two-particle (electron-hole) Green's function, which doubly induces the evaluation errors. It has been recently reported that the GW + BSE method significantly underestimates the experimental photoabsorption energies (PAEs) of small molecules. In order to avoid these problems, we propose to apply the GW(Γ) method not to the neutral ground state but to the cationic state to calculate PAEs without solving the BSE, which allows the rigorous one-to-one correspondence between the photoabsorption peak and the "extended" quasiparticle level. We applied the self-consistent LGWΓ method including the vertex correction to our method, and found that this method gives the PAEs of B, Na 3 , and Li 3 within the 0.1 eV accuracy.