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We present ab initio calculations of the photoemission spectra of bulk sodiumusing different flavors of the cumulant expansion approximation for the Green'sfunction. In particular, we study the dispersion and intensity of the plasmonsatellites. We show that the satellite spectrum is much more sensitive to manydetails than the quasi-particle (QP) spectrum, which suggests that theexperimental investigation of satellites could yield additional informationbeyond the usual studies of the band structure. In particular, a comparison tothe homogeneous electron gas (HEG) shows that the satellites are influenced bythe crystal environment, although the crystal potential in sodium is weak.Moreover, the temperature dependence of the lattice constant is reflected inthe position of the satellites. Details of the screening also play an importantrole; in particular, the contribution of transitions from 2s and 2p semi-corelevels influence the satellites, but not the QP. Moreover, inclusion ofcontributions to the screening beyond the RPA has an effect on the satellites.Finally, we elucidate the importance of the coupling of electrons and holes bycomparing the results of the time-ordered (TOC) and the retarded (RC) cumulantexpansion approximations. Again, we find small but noticeable differences.Since all the small effects add up, our most advanced calculation yields asatellite position which is improved with respect to previous calculations byalmost one eV. This stresses the fact that the calculation of satellites ismuch more delicate than the calculation of a QP band structure.

Cumulant Green's function calculations of plasmon satellites in bulk sodium: influence of screening and the crystal environment