**3. 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 <sup>2</sup>*s*<sup>−</sup>12*p*<sup>−</sup><sup>1</sup>(<sup>1</sup>*P*)<sup>3</sup>*s*(<sup>2</sup>*P*), which is broadened by the decay into three continua <sup>2</sup>*p*<sup>−</sup><sup>2</sup>(<sup>3</sup>*P*)*<sup>ε</sup>p*(<sup>2</sup>*P*), <sup>2</sup>*p*<sup>−</sup><sup>2</sup>(<sup>1</sup>*D*)*<sup>ε</sup>p*(<sup>2</sup>*P*), and <sup>2</sup>*p*<sup>−</sup><sup>2</sup>(<sup>1</sup>*S*)*<sup>ε</sup>p*(<sup>2</sup>*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 <sup>2</sup>*p*<sup>−</sup><sup>2</sup> <sup>1</sup>*D εp* continuum, the intensity of transition to <sup>2</sup>*p*<sup>−</sup><sup>2</sup> <sup>1</sup>*S εp* is significantly smaller, and a transitions to <sup>2</sup>*p*<sup>−</sup><sup>2</sup> <sup>3</sup>*P εp* continuum is almost completely depressed. This prediction was confirmed by Kaneyashi et al. [36], who obtained the relative intensities <sup>1</sup>*D*: <sup>1</sup>*S*: <sup>3</sup>*P* = 1500:400:300.

**Figure 3.** Theoretical lineshape of the satellite <sup>2</sup>*s*<sup>−</sup>12*p*<sup>−</sup><sup>1</sup>(<sup>1</sup>*P*)<sup>3</sup>*s*(<sup>2</sup>*P*) which is broadened by the decay into three continua <sup>2</sup>*p*<sup>−</sup><sup>2</sup>(<sup>3</sup>*P*)*<sup>ε</sup>p*(<sup>2</sup>*P*), <sup>2</sup>*p*<sup>−</sup><sup>2</sup>(<sup>1</sup>*D*)*εp*(<sup>2</sup>*P*) and <sup>2</sup>*p*<sup>−</sup><sup>2</sup>(<sup>1</sup>*S*)*εp*(<sup>2</sup>*P*) (at the satellite energy scale).

Figure 4 shows the satellite state 1*s*<sup>−</sup>12*s*<sup>−</sup><sup>1</sup> <sup>3</sup>*S* 3*s* <sup>2</sup>*S* , which is asymmetrically broadened due to decay into <sup>1</sup>*s*<sup>−</sup>12*p*<sup>−</sup><sup>1</sup> <sup>3</sup>*P εp* <sup>2</sup>*S* and <sup>1</sup>*s*<sup>−</sup>12*p*<sup>−</sup><sup>1</sup> <sup>1</sup>*P εp* <sup>2</sup>*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 <sup>1</sup>*s*<sup>−</sup>12*p*<sup>−</sup><sup>1</sup> <sup>1</sup>*P εp* <sup>2</sup>*S* is larger than the intensity of decay into the triplet channel <sup>1</sup>*s*<sup>−</sup>12*p*<sup>−</sup><sup>1</sup> <sup>3</sup>*P εp* <sup>2</sup>*S* . This result is in qualitative agreemen<sup>t</sup> with the experimental data of Hikosaka et al. [37].

**Figure 4.** Theoretical lineshape of satellite <sup>1</sup>*s*<sup>−</sup>12*s*<sup>−</sup><sup>1</sup> <sup>3</sup>*S*<sup>3</sup>*s* <sup>2</sup>*S* of Ne photoionization (solid line), diamonds experiment [12].

**Figure 5.** Theoretical lineshape of valence Auger decay of satellite state <sup>1</sup>*s*<sup>−</sup>12*s*<sup>−</sup><sup>1</sup> <sup>3</sup>*S*<sup>3</sup>*s* <sup>2</sup>*S* into two continua <sup>1</sup>*s*<sup>−</sup>12*p*<sup>−</sup><sup>1</sup> <sup>3</sup>*Pεp* <sup>2</sup>*S* and <sup>1</sup>*s*<sup>−</sup>12*p*<sup>−</sup><sup>1</sup> <sup>1</sup>*P<sup>ε</sup><sup>p</sup>* <sup>2</sup>*S*.

#### **4. Atoms in Chemical Compounds**

The photoionization of 3*s*-levels of 3*d* elements with the configuration 3*d<sup>N</sup>* <sup>2</sup>*S*+1*L* due to the interaction between 3*s*-hole and unfilled 3*d*-shell results in a line split in two components. These components correspond to two states: a low spin state with total spin *St* = *S* − 1/2 and a high spin state with total spin *St* = *S* + 1/2. It can be shown that this splitting is proportional to the spin *S* of the initial 3*d<sup>N</sup>* state. However, for Co3+(3*d*6(5*D*)) using atomic Hartree–Fock wavefunctions [46,47], we obtained exchange integral -<sup>3</sup>*s*3*sR*<sup>2</sup>3*d*3*<sup>d</sup>* = 0.492 a.u., resulting in a splitting 13.4 eV between the states <sup>4</sup>*D* and <sup>6</sup>*D*, whereas experimental splitting is just 4.7 eV. Furthermore, the ratio of the spectral line intensities is not equal to a statistical ratio (2*S* + 2):2*S* = 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 3*s*<sup>−</sup>13*d<sup>N</sup>* with more complex two-hole-one-particle states <sup>3</sup>*p*<sup>−</sup>23*dN*+<sup>1</sup> [7–9]. 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 3*d*4*l*+2−*<sup>N</sup>* hole configurations. Using graphical methods for calculating angular momentum graphs [48], we obtain the following formula for the weight factor before the Coulomb integral:

$$\begin{split} \boldsymbol{u}^{\lambda} &= (-1)^{L\_{l} + L\_{1} + L\_{2} + l\_{1} + l\_{3}} \overline{\mathbf{G}}\_{\mathcal{L}\_{1} \mathbf{S}\_{1}}^{\rm{L} \rm{L}} \left\{ \begin{array}{cccc} \boldsymbol{L}\_{2} & \boldsymbol{l}\_{1} & \boldsymbol{l}\_{3} \\ \boldsymbol{\lambda} & \boldsymbol{l}\_{2} & \boldsymbol{l}\_{2} \end{array} \right\} \left\{ \begin{array}{cccc} \boldsymbol{L}\_{l} & \boldsymbol{L} & \boldsymbol{l}\_{1} \\ \boldsymbol{l}\_{3} & \boldsymbol{l}\_{2} & \boldsymbol{l}\_{1} \end{array} \right\} \left\{ \begin{array}{cccc} \boldsymbol{S}\_{l} & \boldsymbol{S} & \boldsymbol{1}/2 \\ \boldsymbol{l}\_{3} & \boldsymbol{l}\_{2} & \boldsymbol{l}\_{1} \end{array} \right\} \times \\ & \left( \begin{array}{cccc} \boldsymbol{l}\_{1} & \boldsymbol{\lambda} & \boldsymbol{l}\_{2} \\ \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{0} \end{array} \right) \left( \begin{array}{cccc} \boldsymbol{l}\_{3} & \boldsymbol{\lambda} & \boldsymbol{l}\_{2} \\ \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{0} \end{array} \right) (2\overline{\mathbf{N}})^{1/2} [\boldsymbol{l}\_{1} \boldsymbol{l}\_{3}]^{1/2} [\boldsymbol{l}\_{2}] [\boldsymbol{S}\_{2} \boldsymbol{S} L\_{2} L\_{2}]^{1/2} \end{split} \right\} \tag{12}$$

where *l*1, *l*2, and *l*3 correspond to *s*-, *p*-, and *d*-electrons, respectively, *GLS L*1*S*1 is the fractional parentage coefficient for the hole configurations of the 3*d* shell; *N* = 4*l* + 2 − *N* is the number of holes in the 3*d* shell in the initial state. In our case, we obtained coefficients *α*1 for the interaction of the term <sup>3</sup>*s*<sup>−</sup>13*d*<sup>6</sup>(<sup>5</sup>*D*)(<sup>4</sup>*D*) of initial state with excited states <sup>3</sup>*p*<sup>−</sup><sup>2</sup>(<sup>1</sup>*D*)3*d*<sup>7</sup>(<sup>4</sup>*F*)(<sup>4</sup>*D*) and <sup>3</sup>*p*<sup>−</sup><sup>2</sup>(<sup>1</sup>*D*)3*d*<sup>7</sup>(<sup>4</sup>*P*)(<sup>4</sup>*D*) 0.683 and 0.446, respectively.

**Figure 6.** Angular momentum graphs for the calculation of angular weight factors for the interaction between initial state in photoionization 3*s*<sup>−</sup>13*d<sup>N</sup>* and a the state <sup>3</sup>*p*<sup>−</sup>23*dN*+1. (**a**) Orbital part, (**b**) spin part.

Formula (12) makes it possible to draw some qualitative conclusions about the configuration interaction under consideration. In the first line of the first 6*j*-symbol *l*1 = 0 and *l*3 = 2. Thus, according to the triangle rule for the first line of this 6*j*-symbol there is only one possible value for *L*2, i.e., *L*2 = 2. Therefore, only the term <sup>1</sup>*D* among three possible terms <sup>3</sup>*P*, <sup>1</sup>*D*, and <sup>1</sup>*S*, of the *3p*−<sup>2</sup> 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 3*s*-line in the photoelectron spectrum. Consider atoms with the shell more than half-filled *N* ≥ 2*l* + 1 in the state of the ground term <sup>3</sup>*d<sup>N</sup>*(<sup>2</sup>*S*+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 3*dN*+1 with the spin *S*1 = *S* − 1/2. As shown above, the interaction of the state 3*s*<sup>−</sup>13*d<sup>N</sup>* with the states <sup>3</sup>*p*<sup>−</sup>23*dN*+1 is possible only if the <sup>3</sup>*p*<sup>−</sup><sup>2</sup> shell has the term <sup>1</sup>*D*. For this reason and because the relation *S*1 = *S* − 1/2 is fulfilled for ground terms when *N* ≥ 2*l* + 1 the high-spin state *St* = *S* + 1/2 of 3*s*<sup>−</sup>13*d<sup>N</sup>* configuration does not interact with the configuration <sup>3</sup>*p*<sup>−</sup>23*dN*+1. Therefore, the configuration interaction at *N* ≥ 2*l* + 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 <sup>5</sup>*D* of the 3*d*<sup>6</sup> shell), the interaction between the excited states involving two terms <sup>4</sup>*P* and <sup>4</sup>*F* of the configuration 3*d*<sup>7</sup> 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:

$$\mathrm{Re}\Sigma^{(3)}(E) = \sum\_{\gamma} \frac{\langle k \rangle |\mathcal{U}| \mathrm{js} \rangle\_{\gamma}^{2}}{\mathrm{E} + \left( -\varepsilon\_{j} - \varepsilon\_{j} + \varepsilon\_{\mathrm{s}} \right)\_{\gamma}} + \sum\_{\gamma \neq \delta} \frac{2 \langle k \rangle |\mathcal{U}| \mathrm{js} \rangle\_{\gamma} \langle k \rangle |\mathcal{U}| \mathrm{js} \rangle\_{\delta} \langle \mathrm{s} | \mathrm{U} | \mathrm{js} \rangle\_{\gamma \delta}}{\left[ \mathrm{E} + \left( -\varepsilon\_{j} - \varepsilon\_{j} + \varepsilon\_{\mathrm{s}} \right)\_{\gamma} \right] \left[ \mathrm{E} + \left( -\varepsilon\_{j} - \varepsilon\_{j} + \varepsilon\_{\mathrm{s}} \right)\_{\delta} \right]} \tag{13}$$

where sums run over all distinct terms γ and δ of the configuration under consideration. Since only term <sup>1</sup>*D* of <sup>3</sup>*p*<sup>−</sup><sup>2</sup> configuration is possible, the sum runs over terms of 3*dN*+1 configuration with *S*1 = *S*t.

The experimental 3*s*-photoelectron spectrum of Co3+ ion in paramagnetic BiCoO3 [42] is shown in Figure 7. Theoretical splitting of the 3s-line with account for configuration interaction (CI) 5.1 eV is in good agreemen<sup>t</sup> 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 - 3*s*3*p R*1 3*p*<sup>4</sup> *f* = −0.0088 a.u. is significantly less then - 3*s*3*p R*1 3*p*3*<sup>d</sup>* = 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 Co3+ ion in paramagnetic BiCoO3 [42]. Theoretical line splitting in HF approximation (HF) and with account for configuration interaction (CI) are also shown.

In photoelectron spectra of the Th 5*p* [44] and U 5*p* [45] of ThO2 and UO2 a complex structure is observed instead of two components of the spin doublet 5*p*1/2 and 5*p*3/2. In XPS spectrum of ThO2, shown in Figure 8 a strong satellite peak with a binding energy of about 20 eV larger than the energy of the 5p3/2 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 5*p*1/2 is asymmetrically broadened. These many-electron effects appear due to the interaction of the initial hole states <sup>5</sup>*p*3/2(1/2) with the more complex two-hole-one-particle states <sup>5</sup>*d*−<sup>2</sup>(<sup>2</sup>*S*+<sup>1</sup>*LJ*)<sup>5</sup> *f*(*εf*). 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:

$$\begin{aligned} \mathbf{a}^{\lambda}(L, S, I) &= \sqrt{2} \begin{pmatrix} l\_1 & \lambda & l\_2 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l\_2 & \lambda & l\_3 \\ 0 & 0 & 0 \end{pmatrix} [l\_1 l\_2 l\_3]^{1/2} [L S f\_3]^{1/2} [j\_1]^{-1/2} \begin{Bmatrix} l\_1 & L & l\_3 \\ l\_2 & \lambda & l\_2 \end{Bmatrix} \\ &\times \left\{ \begin{array}{cccc} j\_1 & \kappa & l\_3 \\ 1/2 & j\_3 & \delta \end{array} \right\} \sum\_{\mathbf{x}=L-1/2}^{L+1/2} (-1) \end{aligned} \tag{14}$$

**Figure 8.** Experimental Th5*p* photoelectron spectrum of ThO2 [44].

**Figure 9.** Momentum graph for the calculation of final state configuration interaction in intermediate coupling scheme.

Where *l*1, *l*2, and *l*3 correspond to 5*p*-, 5*d*-holes, and 5*f*-electron, respectively, *j*1 is the total momentum of the <sup>5</sup>*p*<sup>−</sup><sup>1</sup> hole, *L*, *S*, and *J* correspond to the terms of configuration <sup>5</sup>*d*−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 6*j*-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 6*j*-symbol, that interaction of the initial vacancy is possible only with three terms <sup>1</sup>*DJ*, <sup>3</sup>*FJ*, and <sup>1</sup>*GJ* of the shell 5*d*−2.

Formula (14) was used to calculate the interaction of the initial hole <sup>5</sup>*p*<sup>−</sup><sup>1</sup> with the excited states <sup>5</sup>*d*−25*f* in the secular matrix. In addition, the interaction between two-holeone-particle states was taken into account. The spectrum of the Th 5*p*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 5*p*1/2 subshell the levels of the excited <sup>5</sup>*d*−<sup>2</sup> <sup>2</sup>*S*+1*L*<sup>5</sup> *f* configuration are

far from the single-hole state, whereas Auger decay into the 5*d*−<sup>2</sup> <sup>2</sup>*S*+1*L <sup>ε</sup>p*, *εf* continuous states is possible, which leads to the asymmetrical broadening of the Th 5*p*1*/*2 line. To calculate the spectrum of the 5*p*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 5*p* spectrum consists of three groups of lines which can be attributed to the 5*p*1/2 and 5*p*3/2 lines and satellites, i.e., the 5*d*−25 *f* states. Many-electron effects reduce the binding energies of the 5*p*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 5*p*1/2 line is asymmetrically broadened with raised low binding energy side. The spectroscopic factor of the 5*p*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 *<sup>I</sup>*(5*p*3/2):*I*(5*p*1/2), when the satellites are included in 5p3/2 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 5p3/2 line were calculated by the CI method. The spectral function of 5p1/2 line was calculated in the second order of perturbation theory. The lineshape 5p1/2 is slightly asymmetrical, with a raised low-binding energy side.
