GW approach to the calculation of electron self-energies in semiconductors

Various approximation schemes concerning the calculation of the electron self-energy M for a semiconductor in the (bubble) GW scheme of Hedin [Phys. Rev. 139, A796 (1965)] are discussed. It is shown by using a contour-deformation procedure in the complex energy plane that M, as obtained in the first iteration cycle of the GW scheme, is Hermitian for real energies <3Eg/2, where Eg is the unperturbed energy gap, and non-Hermitian for E>3Eg/2. The Taylor expansion for M around the midgap energy value E=0 has a convergence radius of 3Eg/2. Extended use of a (truncated) Taylor series at E>3Eg/2 is not capable of giving the non-Hermitian part of M, while there is also no guarantee that the Hermitian part is correctly obtained in this way.