**1. Introduction**

The interaction of a single vacancy with more complex states with two vacancies and one excited electron results in a breakdown of the one-electron picture in the 4p photoelectron spectra of lanthanides and the 5p spectra of actinides [1–5], and strong deviation of multiplet structure of the 3s spectra of 3d metals from that obtained in Hartree–Fock approximation [6–11]. The theory of these effects is essential in the investigations of materials by photoelectron spectroscopy. Satellite lines in the photoelectron spectra of noble gases have been the subject of several experimental works and their study provides detailed information on the dynamics of many electron correlations [12–17]. It was obtained theoretically [18,19] and confirmed experimentally [20] that monopole shake-up and shake-off satellites take about 20% of the intensity of the main line. Thus, the account for satellite intensities is required in using theoretical photoionization cross-sections in the elemental analysis by photoelectron spectroscopy. The first satellite calculations were made in the "overlapping" approximation [21–24], in which the satellite intensity was proportional to the square of overlap integral between ground state Hartree–Fock wave functions and relaxed final state wave functions. In addition, a combination of configuration interaction and "overlapping" methods was used for satellite calculation [25,26].

It was an idea of Miron Ya. Amusia to consider the hole potential as a perturbation potential for the ground state wave functions and to use the spectral function of the initial hole (see, e.g., [1]) to calculate the whole spectrum, i.e., main line, shake-up satellites, and shake off continuum [27]. This technique was applied for the shake-up satellites in photoelectron spectra of noble gases [28–30] and extended to valence and core Auger transitions [31–35]. Some predictions of these theoretical results for the valence Auger transitions were confirmed experimentally [36,37]. The creation of a new of HAXPES

(hard x-ray photoelectron spectroscopy) experimental techniques [38,39] caused further development of the theory of many-electron effects in photoionization [40,41]. Similar theoretical approaches were used for the understanding of atomic many-electron effects in photoelectron spectra of atoms in chemical compounds, namely the 3s-spectra of Co [42], the 4p-spectra of Ba [43], the 5p-spectra of Th [44], and U [45].

Thus many-electron effects change the one-electron picture of photoionization [1] and knowledge of the nature of many-electron effects is required for the correct interpretation of XPS data on the compounds under investigation. Furthermore, in the case of noble gases, a comparison of theoretical and experimental results is required to understand photoionization and related phenomena [35]. In the present paper, the many-electron approaches to core relaxation and multiplet splitting are developed and examples of their applications are considered.
