Satellite Excitations and Final State Interactions in Atomic Photoionization

: Satellite excitations and ﬁnal state conﬁguration interactions appear due to the many-electron correlations and result in a photoelectron spectrum complex ﬁnal state structure instead of single lines corresponding to one-hole states. In the present work, both processes are considered in a framework of the many-body perturbation theory, and two techniques, namely the spectral function and CI (conﬁguration interaction) methods are considered. It is shown that for the calculation of satellite lineshapes and low-energy Auger decay, the spectral function method is more appropriate, but in the case of strong ﬁnal state interactions, the methods of solution of Dyson equation or secular matrix are superior. The results obtained for satellites and low energy Auger decay in the Ne 1s, Ne 2p photoelectron spectra, the Co 3s, and the Th 5p photoelectron spectra are in agreement with the experimental data.


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][2][3][4][5], and strong deviation of multiplet structure of the 3s spectra of 3d metals from that obtained in Hartree-Fock approximation [6][7][8][9][10][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][13][14][15][16][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][22][23][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][29][30] and extended to valence and core Auger transitions [31][32][33][34][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.

Theoretical Approach
The relaxation of atomic orbitals caused by ionization of the k-shell is described by the excitations of electron-hole pairs under the action of suddenly switched potential of the k-hole [27]. The corresponding Feynman diagram is shown in Figure 1a, where a wavy line stands for the interaction, determined by the Coulomb integral, right (left) directed arrows denote particles (holes), which are added to the initial atomic configuration. Coulomb integral, corresponding to a wavy line may be written as: where λ and µ are multipole values of direct and exchange parts, and angular weight factors α λ γ and β µ γ depend on the coupling scheme γ. The case k = i corresponds to monopole shakeup satellites, and the case k = i corresponds to Auger decay or to final state configuration interaction. The Coulomb integrals were calculated using the atomic wave functions P v (r), obtained by the Hartree-Fock method [46,47] using the standard formula: To calculate the angular part of the Feynman diagram, one must plot an angular momentum graph [48], in which the interaction vertex is topologically equivalent to the Feynman diagram and the free particle and hole lines are connected in correspondence to the coupling scheme [35]. Figure 1b shows the angular momentum graph for the satellite excitation in the case of the LS-coupling scheme. The spin momentum graph is obtained by removing an interaction line [48]. Additional arrows are also added in the case of coupling of holes [49]. When reducing the angular momentum graph of Figure 1b together with the spin momentum graph and adding factors for Coulomb interaction vertex [48] one obtains: where f = 1 if electrons l i and l j are non-equivalent, f = √ 2 if electrons l i and l j are equivalent, and [a] denotes 2a + 1.
For the exchange graph, one must interchange lines of iand j-holes at the Coulomb interaction line, what results in additional phase factors, and one obtains the weight factor before exchange integral: where f = 1, if electrons l i and l j are non-equivalent, and f = 0, if electrons l i and l j are equivalent. Formulas (3) and (4) differ from the formulas for Auger decay [50], obtained by coupling of initial hole's and final electron's angular momentums and spins into L ij and S ij , by inessential common phase factor only. The k-vacancy spectrum, which includes the main line, discrete shake-up, continuum shake-off excitations, and Auger decay is represented by a spectral function: where E is the energy parameter of k-hole, which runs over all relevant values and Σ k (E)− is the self-energy of k-vacancy.
In the second order of perturbation theory, the self-energy is represented by the Feynman diagrams of Figure 2a-c, and is defined by its real and imaginary parts as follows: It is assumed, that the sums in Equations (6) and (7) include integration over continuum energies ε s also.
The integral of the spectral function equals to the unity and the intensity of all spectral distribution is proportional the photoionization cross-section σ k (ω) calculated in Hartree-Fock approximation [27].
The positions E ν of discrete satellites are the solutions of the Dyson equation: The intensities of the main line and shake-up satellite relative to all spectral distribution are proportional to the spectroscopic factors: If the discrete line is in the continuum of another transition, its spectroscopic factor may be calculated as an integral of the spectral function. Spectroscopic factors of all main lines in photoelectron spectra calculated by spectral function [18] and "overlapping" [19] methods are approximately the same and all are close to the value 0.8, and in general agree with the experiment [20]. The second-order diagrams and Formula (7) represent the broadening of the satellite due to the direct transitions from one-hole states to the shake-off continuum k −1 → k −1 l −1 q . If there exist at the same energy a discrete shake-up satellite k −1 j −1 s it is broadened by the decay into underlying continuum k −1 j −1 s → k −1 l −1 q . This process is represented by the Feynman diagram in Figure 2f and the contribution to the imaginary part of the self-energy is written as: Figure 2d,e represent the interference between two ways of excitation of shake-off continuum, namely direct k −1 → k −1 l −1 q , and via shake-up resonance The contribution of these diagrams to the spectral function is written as: It is seen from Formulas (5), (7), (10) and (11), that contributions of decay channels to the spectral function are additive (in the numerator of Equation (5)) and the spectral functions for decay channels can be separated. Thus one can obtain the spectral function for low-energy Auger decay [31].

Noble Gases
Calculation of spectral functions in the simplified forth order approach (see Formulas (5)-(7), (10) and (11), and Figure 2) made it possible to reproduce strongly asymmetrical satellite lineshapes and to estimate intensities and lineshapes of low energy Auger decay. Figure 3 shows theoretical lineshape of the satellite 2s −1 2p −1 ( 1 P)3s( 2 P), which is broadened by the decay into three continua 2p −2 ( 3 P)εp( 2 P), 2p −2 ( 1 D)εp( 2 P), and 2p −2 ( 1 S)εp( 2 P). The spectral densities of the satellite transferred to these three low energy Auger transitions are also shown. According to our calculations, the most intense is the transition to 2p −2 1 D εp continuum, the intensity of transition to 2p −2 1 S εp is significantly smaller, and a transitions to 2p −2 3 P εp continuum is almost completely depressed. This prediction was confirmed by Kaneyashi et al. [36], who obtained the relative intensities 1 D: 1 S: 3 P = 1500:400:300.  Figure 4 shows the satellite state 1s −1 2s −1 3 S 3s 2 S , which is asymmetrically broadened due to decay into 1s −1 2p −1 3 P εp 2 S and 1s −1 2p −1 1 P εp 2 S continua. Theoretical curve reproduces experimental strongly asymmetrical lineshape having a Fano profile [12]. Theoretical lineshapes of low-energy Auger decay of this state are shown in Figure 5. It is seen in Figure 5, that the intensity of decay into the singlet channel 1s −1 2p −1 1 P εp 2 S is larger than the intensity of decay into the triplet channel 1s −1 2p −1 3 P εp 2 S . This result is in qualitative agreement with the experimental data of Hikosaka et al. [37].

Atoms in Chemical Compounds
The photoionization of 3s-levels of 3d elements with the configuration 3d N 2S+1 L due to the interaction between 3s-hole and unfilled 3d-shell results in a line split in two components. These components correspond to two states: a low spin state with total spin S t = S − 1/2 and a high spin state with total spin S t = S + 1/2. It can be shown that this splitting is proportional to the spin S of the initial 3d N state. However, for Co 3+ (3d 6 ( 5 D)) using atomic Hartree-Fock wavefunctions [46,47], we obtained exchange integral 3s3s R 2 3d3d = 0.492 a.u., resulting in a splitting 13.4 eV between the states 4 D and 6 D, whereas experimental splitting is just 4.7 eV. Furthermore, the ratio of the spectral line intensities is not equal to a statistical ratio (2S + 2):2S = 6:4 [42]. In some works, this problem was solved by multiplication of the exchange integral by some scaling factor [10,11]. However, this formal decrease of the exchange integral is due to the interaction of the initial one-hole state 3s −1 3d N with more complex two-hole-one-particle states 3p −2 3d N+1 [7][8][9]. This interaction is represented by the Feynman diagram shown in Figure 2a. Since holes i and j in the final state are equivalent, the exchange part is absent. The corresponding angular momentum graphs are shown in Figure 6. In this graph, all lines correspond to holes and a black square denotes fractional parentage coefficients for the 3d 4l+2−N hole configurations. Using graphical methods for calculating angular momentum graphs [48], we obtain the following formula for the weight factor before the Coulomb integral: where l 1 , l 2 , and l 3 correspond to s-, p-, and d-electrons, respectively, G LS L 1 S 1 is the fractional parentage coefficient for the hole configurations of the 3d shell; N = 4l + 2 − N is the number of holes in the 3d shell in the initial state. In our case, we obtained coefficients α 1 for the interaction of the term 3s −1 3d 6 ( 5 D)( 4 D) of initial state with excited states 3p −2 ( 1 D)3d 7 ( 4 F)( 4 D) and 3p −2 ( 1 D)3d 7 ( 4 P)( 4 D) 0.683 and 0.446, respectively. Formula (12) makes it possible to draw some qualitative conclusions about the configuration interaction under consideration. In the first line of the first 6j-symbol l 1 = 0 and l 3 = 2. Thus, according to the triangle rule for the first line of this 6j-symbol there is only one possible value for L 2 , i.e., L 2 = 2. Therefore, only the term 1 D among three possible terms 3 P, 1 D, and 1 S, of the 3p −2 shell is involved in the configuration interaction. This circumstance makes it possible not only to simplify the calculations but also to qualitatively estimate the effect of the configuration interaction on the splitting of the 3s-line in the photoelectron spectrum. Consider atoms with the shell more than half-filled N ≥ 2l + 1 in the state of the ground term 3d N ( 2S+1 L). According to the table of fractional parentage coefficients [51], the addition of one electron to the ground term results in terms of the configuration 3d N+1 with the spin S 1 = S − 1/2. As shown above, the interaction of the state 3s −1 3d N with the states 3p −2 3d N+1 is possible only if the 3p −2 shell has the term 1 D. For this reason and because the relation S 1 = S − 1/2 is fulfilled for ground terms when N ≥ 2l + 1 the high-spin state S t = S + 1/2 of 3s −1 3d N configuration does not interact with the configuration 3p −2 3d N +1 . Therefore, the configuration interaction at N ≥ 2l + 1 affects only the position of the low-spin state, resulting in the reduction of the splitting between low-spin and high-spin states.
In the case under consideration (the ground term 5 D of the 3d 6 shell), the interaction between the excited states involving two terms 4 P and 4 F of the configuration 3d 7 is also possible. The Feynman diagrams of such an interaction are shown in Figure 2d,e. The corresponding formula for the self-energy part is written as: (13) where sums run over all distinct terms γ and δ of the configuration under consideration. Since only term 1 D of 3p −2 configuration is possible, the sum runs over terms of 3d N+1 configuration with S 1 = S t .
The experimental 3s-photoelectron spectrum of Co 3+ ion in paramagnetic BiCoO 3 [42] is shown in Figure 7. Theoretical splitting of the 3s-line with account for configuration interaction (CI) 5.1 eV is in good agreement with experimental value 4.7 eV and with theoretical result 5.4 eV [7], obtained with account for a larger number of configurations. In the Hartree-Fock approximation (HF), the value of splitting equals 13.4 eV. The spectroscopic factor of the low-spin state calculated by Formula (9) equals 0.75. Taking into account the statistical ratio of the high-spin to low-spin component 1.5, we obtain theoretical ratio 2, which is equal to the experimental value. It was also pointed out that interaction with exited configuration 3p-4f is important [9]. However, the value of leading Coulomb integral in this case 3s3p R 1 3p4 f = −0.0088 a.u. is significantly less then 3s3p R 1 3p3d = 0.650 a.u., used in the present work for the main channel of CI. That is why the main features of the spectrum can be reproduced by our method, which is equivalent to solving the secular matrix of dimension 3. Figure 7. Experimental 3s-photoelectron spectrum of Co 3+ ion in paramagnetic BiCoO 3 [42]. Theoretical line splitting in HF approximation (HF) and with account for configuration interaction (CI) are also shown.
In photoelectron spectra of the Th 5p [44] and U 5p [45] of ThO 2 and UO 2 a complex structure is observed instead of two components of the spin doublet 5p 1/2 and 5p 3/2 . In XPS spectrum of ThO 2 , shown in Figure 8 a strong satellite peak with a binding energy of about 20 eV larger than the energy of the 5p 3/2 peak appears. The experimental spinorbit splitting 55.5 eV is significantly larger than the result of our Dirac-Fock calculation of 47.9 eV and the line 5p 1/2 is asymmetrically broadened. These many-electron effects appear due to the interaction of the initial hole states 5p 3/2(1/2) with the more complex two-hole-one-particle states 5d −2 ( 2S+1 L J )5 f (ε f ). In this case, we used an intermediate coupling scheme and the corresponding momentum graph is shown in Figure 9. Reduction of this momentum graph results in the following formula for the weight factor:  Where l 1 , l 2 , and l 3 correspond to 5p-, 5d-holes, and 5f -electron, respectively, j 1 is the total momentum of the 5p −1 hole, L, S, and J correspond to the terms of configuration 5d −2 , the summation is over all half-integer κ in the specified interval, and λ takes values 1 and 3. Formula (14) is a generalization of the non-relativistic formula for Auger decay [50] in the case of intermediate coupling. Note that the first of the 6j-symbols in Formula (14) is also present in the non-relativistic formula [43]. It follows from the triangular condition for elements of the first row of this 6j-symbol, that interaction of the initial vacancy is possible only with three terms 1 D J , 3 F J , and 1 G J of the shell 5d −2 .
Formula (14) was used to calculate the interaction of the initial hole 5p −1 with the excited states 5d −2 5f in the secular matrix. In addition, the interaction between two-holeone-particle states was taken into account. The spectrum of the Th 5p 3/2 electrons was calculated by solving the secular matrix. The eigenvalues of the secular matrix correspond to the positions of the spectral lines, and the squares of the elements of the eigenvector, corresponding to the energy of the main line give the intensities of all lines. In the case of ionization of the 5p 1/2 subshell the levels of the excited 5d −2 2S+1 L 5 f configuration are far from the single-hole state, whereas Auger decay into the 5d −2 2S+1 L εp, ε f continuous states is possible, which leads to the asymmetrical broadening of the Th 5p 1/2 line. To calculate the spectrum of the 5p 1/2 electrons, we used the spectral function method (see Formula (5)). The theoretical spectrum with overall energy shift −9.6 eV representing solid-state effects [44] is shown in Figure 10. The calculated Th 5p spectrum consists of three groups of lines which can be attributed to the 5p 1/2 and 5p 3/2 lines and satellites, i.e., the 5d −2 5 f states. Many-electron effects reduce the binding energies of the 5p 3/2 electrons, and as a result, spin-orbit splitting reaches the value 55 eV, which corresponds to the experimental value 56 eV. It is seen in Figure 10 that the 5p 1/2 line is asymmetrically broadened with raised low binding energy side. The spectroscopic factor of the 5p 1/2 line, estimated as the integral of the spectral function Equation (5) in an interval of 10 eV (which corresponds to the interval where the intensity of the line was experimentally determined) with background subtraction equals 0.69. Thus the theoretical ratio of the intensities I(5p 3/2 ):I(5p 1/2 ), when the satellites are included in 5p 3/2 line, equals 2.9:1. Experiential value of this ratio depends on the background subtraction. When Shirley's method was used, this ratio equals 5:1, but in linear background subtraction, we obtained the ratio 3.6:1. Figure 10. Theoretical Th 5p photoelectron spectrum. The satellite structures of the 5p 3/2 line were calculated by the CI method. The spectral function of 5p 1/2 line was calculated in the second order of perturbation theory. The lineshape 5p 1/2 is slightly asymmetrical, with a raised low-binding energy side.

Conclusions
In the present work, it was shown that for different relations between interaction parameters in the final state, appearing after photoionization, different approaches of MBPT can be used. A spectral function approach with account for the imaginary part of the self-energy in simplified fourth-order make it possible to calculate asymmetrical lineshapes of satellites in photoelectron spectra and of low-energy Auger spectra. Dyson's equation in the third-order makes it possible to calculate the line positions in the case of weak interaction between excited states. In particular, in the case of 3s-spectra of 3d-elements accounting for a very limited number of Feynman diagrams made it possible to reach agreement with the experiment and to draw a qualitative conclusion on many-electron effects for 3d-shell more than half-filled. However, in the case of strong CI in 5p-spectra, the method of secular matrix solution is more appropriate. This series of research was inspired by Miron Ya. Amusia, who was the first person to proposed to use unified MBPT technique for CI and satellite excitations.
Author Contributions: V.G.Y., software and theoretical calculations; Y.A.T., methodology and experimental investigations. All authors have read and agreed to the published version of the manuscript.

Funding:
The work was supported by IGIC RAS state assignment and RRC "KI" state assignment.

Institutional Review Board Statement:
The study was conducted in accordance with the Declaration of Helsinki, and approved by the Institutional Review Board of IGIC RAS in agreement with RRC "KI" 04.05.2022.