  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.
