**2. 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.

**Figure 1.** Graphs for the satellite's excitation. (**a**) Feynman diagram for the satellite excitation. Arrows directed from left to right (right to left) denote particles (holes), wavy line denotes Coulomb interaction. (**b**) angular momentum graph, corresponding to the Feynman diagram (**a**). Angular momentum graph for the spin part in which interaction line is removed should be included.

Coulomb integral, corresponding to a wavy line may be written as:

$$dL\_{\gamma}(kj\text{is}) = \sum\_{\lambda,\mu} a\_{\gamma}^{\lambda} \left( kj \left| \mathbb{R}^{\lambda} \right| \text{is} \right) + \beta\_{\gamma}^{\mu} (ki \left| \mathbb{R}^{\mu} \right| js), \tag{1}$$

where *λ* and *μ* are multipole values of direct and exchange parts, and angular weight factors *αλγ* and *βμγ* 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 *Pv*(*r*), obtained by the Hartree-Fock method [46,47] using the standard formula:

$$\left(\nu\_1 \nu\_3 \middle| R^{\lambda} \middle| \nu\_2 \nu\_4\right) = \int\_0^\infty P\_{\mathbb{V}\_1}(r) P\_{\mathbb{V}\_2}(r) dr \int\_0^\infty \frac{r^{\lambda}\_<}{r\_>^{\lambda+1}} P\_{\mathbb{V}\_3}(r') P\_{\mathbb{V}\_4}(r') dr' \tag{2}$$

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:

$$\boldsymbol{a}^{\lambda} = \boldsymbol{f}(-1)^{S\_{\bar{\boldsymbol{i}}\bar{\boldsymbol{j}}} + 1 + \lambda} \frac{\left[\boldsymbol{L}\_{\bar{\boldsymbol{i}}\bar{\boldsymbol{j}}} \boldsymbol{S}\_{\bar{\boldsymbol{i}}\bar{\boldsymbol{j}}}\right]^{1/2}}{(2)^{1/2} [\boldsymbol{l}\_{\bar{\boldsymbol{k}}}]^{1/2}} \left\{ \begin{matrix} \lambda & \boldsymbol{l}\_{\bar{\boldsymbol{k}}} & \boldsymbol{l}\_{\bar{\boldsymbol{i}}} \\ \boldsymbol{L}\_{\bar{\boldsymbol{i}}\bar{\boldsymbol{j}}} & \boldsymbol{l}\_{\bar{\boldsymbol{j}}} & \boldsymbol{l}\_{\bar{\boldsymbol{k}}} \end{matrix} \right\} \left\langle \boldsymbol{l}\_{\bar{\boldsymbol{i}}} \middle| \middle| \boldsymbol{C}^{\lambda} \middle| \middle| \boldsymbol{l}\_{\bar{\boldsymbol{k}}} \right\rangle \left\langle \boldsymbol{l}\_{\bar{\boldsymbol{j}}} \middle| \middle| \boldsymbol{C}^{\lambda} \middle| \middle| \right| \mathbf{f}\_{\bar{\boldsymbol{k}}} \right\rangle,\tag{3}$$

where *f* = 1 if electrons *li* and *lj* are non-equivalent, *f* = √2 if electrons *li* and *lj* are equivalent, and [*a*] denotes 2*a* + 1.

For the exchange graph, one must interchange lines of *i*- and *j*-holes at the Coulomb interaction line, what results in additional phase factors, and one obtains the weight factor before exchange integral:

$$\beta^{\mu} = -f(-1)^{L\_{\vec{i}\vec{j}} + l\_{\vec{i}} + l\_{\vec{j}} + \mu} \frac{\left[L\_{\vec{i}\vec{j}}S\_{\vec{i}\vec{j}}\right]^{1/2}}{(2)^{1/2} [l\_{\vec{k}}]^{1/2}} \begin{Bmatrix} \mu & l\_{\vec{k}} & l\_{\vec{j}} \\ L\_{\vec{i}\vec{j}} & l\_{\vec{i}} & l\_{\vec{s}} \end{Bmatrix} \langle l\_{\vec{j}} || \mathbf{C}^{\mu} || l\_{\vec{k}} \rangle \langle l\_{\vec{i}} || \mathbf{C}^{\mu} || l\_{\vec{s}} \rangle \tag{4}$$

where *f* = 1, if electrons *li* and *lj* are non-equivalent, and *f* = 0, if electrons *li* and *lj* 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 *Lij* and *Sij*, 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:

$$A\_k(E) = \frac{1}{\pi} \frac{\text{Im}\Sigma\_k(E)}{\left(E - \varepsilon\_k - \text{Re}\Sigma\_k(E)\right)^2 + \text{Im}\Sigma\_k(E)^2},\tag{5}$$

where *E* is the energy parameter of *k*-hole, which runs over all relevant values and <sup>Σ</sup>*k*(*E*)<sup>−</sup> 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:

$$\operatorname{Re}\Sigma\_k^{(2)}(E) = \sum\_{i,j,s} \frac{\langle kj|\mathcal{U}|is\rangle^2}{E - \varepsilon\_i - \varepsilon\_j + \varepsilon\_s} \tag{6}$$

$$\mathrm{Im}\Sigma\_k^{(2)}(E) = \pi \sum\_{i,j,s} \langle kj|\mathcal{U}|is\rangle^2 \delta\left(E - \varepsilon\_i - \varepsilon\_j + \varepsilon\_s\right) \tag{7}$$

It is assumed, that the sums in Equations (6) and (7) include integration over continuum energies *εs* also.

The integral of the spectral function equals to the unity and the intensity of all spectral distribution is proportional the photoionization cross-section *<sup>σ</sup>k*(*ω*) calculated in Hartree–Fock approximation [27].

The positions *Eν* of discrete satellites are the solutions of the Dyson equation:

$$E\_{\vee} = \varepsilon\_{k} + \text{Re}\Sigma\_{k}(E\_{\vee}) \tag{8}$$

The intensities of the main line and shake-up satellite relative to all spectral distribution are proportional to the spectroscopic factors:

$$f\_{\nu} = \frac{1}{1 - \left. \frac{\partial \text{Re}\Sigma\_k(E)}{\partial E} \right|\_{E = E\_{\nu}}} \tag{9}$$

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].

**Figure 2.** Feynman diagrams for the spectral function in a simplified forth-order approach. (**<sup>a</sup>**–**<sup>c</sup>**) Direct and exchange second-order diagrams, (**d**,**<sup>e</sup>**) direct and exchange third-order diagrams for the decay of satellite into continuum, (**f**) fourth-order diagram (exchange parts are not shown).

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*−<sup>1</sup> → *<sup>k</sup>*−1*l*−<sup>1</sup>*q* . If there exist at the same energy a discrete shake-up satellite *k*−<sup>1</sup> *j*−1*s* it is broadened by the decay into underlying continuum *k*−<sup>1</sup> *j*−1*s* → *<sup>k</sup>*−1*l*−<sup>1</sup>*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:

$$\mathrm{Im}\Sigma\_{k}^{(4)}(E) = \frac{\pi \langle kj | \mathrm{l}I | \mathrm{ks} \rangle^{2}}{\left(E - \varepsilon\_{k} - \varepsilon\_{j} + \varepsilon\_{\mathrm{s}}\right)^{2} \sum\_{l,q} \langle j \mathrm{s} | \mathrm{l}I | lq \rangle^{2} \delta(E - \varepsilon\_{k} - \varepsilon\_{l} + \varepsilon\_{q})} \tag{10}$$

Figure 2d,e represent the interference between two ways of excitation of shake-off continuum, namely direct *k*−<sup>1</sup> → *<sup>k</sup>*−1*l*−<sup>1</sup>*q* , and via shake-up resonance *k*−<sup>1</sup> → *k*−<sup>1</sup> *j*−1*s* → *<sup>k</sup>*−1*l*−<sup>1</sup>*q* . The contribution of these diagrams to the spectral function is written as:

$$\mathrm{Im}\Sigma\_{k}^{(3)}(E) = \frac{2\pi \langle kl| \mathcal{U} | \mathrm{ks} \rangle}{E - \varepsilon\_{k} - \varepsilon\_{j} + \varepsilon\_{s}} \sum\_{l,q} \langle kl| \mathcal{U} | kq \rangle \langle js| \mathcal{U} | lq \rangle \delta(E - \varepsilon\_{k} - \varepsilon\_{l} + \varepsilon\_{q}) \tag{11}$$

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].
