Abstract

In this paper, we reported the energy positions, quantum defect and effective nuclear charge of the $4s^{2}4p^2\left({}^3{P}_2\right)nd$ , $4s^{2}4p^2\left({}^1{D}_2\right)ns$ , $4s^{2}4p^2\left({}^2{D}_2\right)ns\left({}^2{D}_2\right)$ , $4s^{2}4p^2\left({}^1{D}_2\right)nd\left({}^2{P}_{1/2}^o\right)$ and $4s^{2}4p^2\left({}^2{D}_2\right)ns\left({}^{2}S\right)$ Rydberg series of Se⁺ ions and $4s^{2}4p^2\left({}^2{P}_{1/2}^o\right)nd\left({}^1{P}_1^o\right)$ Rydberg series of Rb⁺. Calculations are performed using the semi-empirical Modified Atomic Orbital theory (MAOT) from n = 55. Very good agreements are seen between the present calculations and the available theoretical and experimental data. The present predicted data up to n = 55 may be of great importance to the atomic physics community, particularly in understanding the chemical evolution of Selenium in the Universe. Furthermore, the accurate data presented in this work may serve as a valuable guideline for future experimental investigations and alternative theoretical studies.

Keywords

Rydberg Series, Resonance Energy, Quantum Defect, Photoionization, Effective Charge and Modified Atomic Orbital Theory

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1. Introduction

The study of photoionization for atoms and ions is a fundamental process that allows to understand the chemical evolution of the universe elements. Thus, photoionization is very important in many fields of astrophysics, especially stars, astrophysical plasmas, fusion with inertial confinement. Almost half of the neutron capture elements (atomic number Z > 30 atomic number) in the universe, are created by slow nucleosynthesis [1]-[5]. Both theoretical models and experimental data have explored the photoionization of Se⁺, Se²⁺, Se³⁺, Se⁴⁺, and Se⁵⁺ ions to shed light on selenium’s chemical development in space. Synchrotron Radiation (SR) provided the experimental means to determine absolute cross-sections for single photoionization. These experiments produced precise resonance energies and quantum defects related to various Rydberg series identified in the spectra of Se⁺, Se³⁺ and Se⁵⁺ [6] and Se²⁺ [7]. From the theoretical point of view, studies have been carried out to compare with the measurements of simple photoionization section made by previous experiments. Indeed, the Breit-Pauli and Dirac-Coulomb matrix approximations was used by McLaughlin et al. [8] to calculate the first theoretical resonance energies and quantum defects for the Rydberg dominant series of the Se⁺ ion. Sakho et al. [9] completed predictions on the resonance energies that belong to numerous Rydberg series of the Se²⁺ and Se⁵⁺ ions using Screening Constant by Unit Nuclear Charge (SCUNC) method. In the same way, Sakho [10] calculated high resonance energies belonging to many Rydberg series of Se³⁺ and Rb⁺ using the SCUNC formalism. In the same way, Sakho calculated high resonance energies belonging to many Rydberg series of Se⁺ using the SCUNC formalism [11]. The data derived from Rubidium photoionization serves as a critical resource for simulating astrophysical processes, specifically the nucleosynthesis of heavy elements [12]. In this work, we intend to provide precise data on the photoionization of Se⁺ and Rb⁺ ions that may be useful guideline for the physical atomic community. In addition, we aim to demonstrate the possibilities of using the semi-empirical procedure of MAOT [13]-[16] to reproduce experimental data

with high precision .Thus, we calculated the resonance energies and quantum defect for the $4s^{2}4p^2\left({}^3{P}_2\right)nd$ , $4s^{2}4p^2\left({}^1{D}_2\right)ns$ , $4s^{2}4p^2\left({}^2{D}_2\right)ns\left({}^2{D}_2\right)$ , $4s^{2}4p^2\left({}^1{D}_2\right)nd\left({}^2{P}_{1/2}^o\right)$ and $4s^{2}4p^2\left({}^2{D}_2\right)ns\left({}^{2}S\right)$ Rydberg series originating from the $4s^{2}4p^3\left({}^2{P}_{\frac{1}{2},\frac{3}{2}}^o\right)$ and $4s^{2}4p^3\left({}^2{D}_{\frac{3}{2},\frac{5}{2}}^o\right)$ states of Se⁺ and $4s^{2}4p^2\left({}^2{P}_{1/2}^o\right)nd\left({}^1{P}_1^o\right)$ Rydberg series of Rb⁺ converging to the ($3d^{10}4s^{2}4p^{5}2P_{\circ 1/2}$ ) serie limits in Rb²⁺. Section 2 presents the theoretical procedure adopted in this work with a brief description of the MAOT formalism [13]-[16] and the analytical expressions used in the calculations. In Section 3, we present and discuss our current findings, comparing them with existing data from the literature (experimental and theoretical data). In Section 4, we present a conclusion and summary of this study.

2. Theory

2.1. Brief Description of the MOAT Formalism

In the framework of the Modified Atomic Orbital Theory (MAOT), total energy of a $\left(\nu \ell \right)$ -given orbital is expressed in the form [13]-[15]

$E=-\frac{{\left[Z-\sigma \left(\ell \right)\right]}^2}{{\nu }^2}$ (1)

For an atomic system of several electrons M, the total energy is given by (in Rydberg):

$E=-{\displaystyle \sum _{i=1}^M\frac{{\left[Z-{\sigma }_i\left(\ell \right)\right]}^2}{{\nu }_i^2}}$ (2)

With respect to the usual spectroscopic notation ${\left(N\ell ,n{\ell }^{\prime }\right)}^{2S+1}{L}^{\pi }$ , this equation becomes

$E=-{\displaystyle \sum _{i=1}^M\frac{{\left[Z-{\sigma }_i\left({}^{2S+1}{L}^{\pi }\right)\right]}^2}{{\nu }_i{}^2}.}$ (3)

In the photoionization of atoms and ions, energy resonances are generally measured relatively to the $E_{\infty }$ converging limit of a given $\left({}^{2S+1}{L}_J\right)nl$ -Rydberg serie. For these states, the general expression of the energy resonance $E_n$ is given

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1\left({}^{2S+1}{L}_J\right)-{\sigma }_2\left({}^{2S+1}{L}_J\right)\times \frac{1}{n}-{\sigma }_2^{\alpha }\left({}^{2S+1}{L}_J\right)\\ \times \left(n-m\right)\times {\left(n-q\right){\displaystyle \sum _k\frac{1}{f_k\left(n,m,q,s\right)}}\}}^2\end{array}$ (4)

In this equation, m and q ($m<p$ ) denote the principal quantum numbers of the $\left({}^{2S+1}{L}_J\right)nl$ -Rydberg serie of the considered atomic system used in the empirical determination of the ${\sigma }_i\left({}^{2S+1}{L}_J\right)$ -screening constants s represents the spin

of the $nl$ -electron (s = ½), $E_{\infty }$ is the energy value of the serie limit generally determined from NIST atomic database, $E_n$ denote correspond energy resonance and Z represents the nuclear charge of the considered element. The only

problem that one lay face by using the MAOT formalism is linked to the determination of the ${\displaystyle \sum _k\frac{1}{f_k\left(n,m,q,s\right)}}$ -term. The correct expression of this ter mis determined iteratively by imposing general Equation (4) to give accurate data with a

constant quantum defect or decreasing throughout the series considered. The value of α is generally fixed to 1 and or 2 during the iteration. The standard quantum defect expansion is given as follows:

$E_n=E_{\infty }-\frac{R{Z}_{core}^2}{{\left(n-\delta \right)}^2}$ (5)

In the equation, R, $E_{\infty }$ , $Z_{core}$ and δ are the Rydberg constant, the converging limit, the electric charge of the core ion and the quantum defect, respectively.

$Z_{core}$ is directly obtained by the photoionization process from an atomic X^p+ system:

$X^{p+}+h\nu \to X^{\left(p+1\right)+}+{\text{e}}^{-}$ . We find then $Z_{core}=p+1$ .(6)

From Equation (5) we obtain the expression for the quantum defect δ:

$\delta =n-Z_{core}\times \sqrt{\frac{R}{E_{\infty }-E_n}}$ (7)

2.2. Resonances Energy of the Transitions States of S_e⁺ Ions

In the framework of the Modified Atomic Orbital Theory (MOAT) formalism the energy position of the$4s^{2}4p^2\left({}^3{P}_2\right)nd$ , $4s^{2}4p^2\left({}^1{D}_2\right)ns$ , $4s^{2}4p^2\left({}^2{D}_2\right)ns\left({}^2{D}_2\right)$ , $4s^{2}4p^2\left({}^1{D}_2\right)nd\left({}^2{P}_{1/2}^o\right)$ and $4s^{2}4p^2\left({}^2{D}_2\right)ns\left({}^{2}S\right)$ Rydberg series originating from the $4s^{2}4p^3\left({}^2{P}_{\frac{1}{2},\frac{3}{2}}^o\right)$ and $4s^{2}4p^3\left({}^2{D}_{\frac{3}{2},\frac{5}{2}}^o\right)$ states of Se⁺ and $4s^{2}4p^2\left({}^2{P}_{1/2}^o\right)nd\left({}^1{P}_1^o\right)$ Rydberg series of Rb⁺ converging to the ($3d^{10}4s^{2}4p^{5}2P_{\circ 1/2}$ ) serie limits in Rb²⁺. Using the general Equation (4), we obtain the expressions of the resonant energies of the Rydberg series of the Se⁺ ions originating from the $4s^{2}4p^3\left({}^2{P}_{\frac{1}{2},\frac{3}{2}}^o\right)$ and $4s^{2}4p^3\left({}^2{D}_{\frac{3}{2},\frac{5}{2}}^o\right)$ and of the Rydberg series of the Rb⁺ series.

• For the $4s^{2}4p^3\left({}^2{D}_{3/2}^o\right)\to 4s^{2}4p^2\left({}^3{P}_2\right)nd$ transition

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1-\frac{{\sigma }_2}{n}-{\sigma }_2^2\left(n-m\right)\left(n-q\right)[\frac{1}{{\left(n+m+q+s\right)}^5}\\ +\frac{1}{{\left(n-m+q+4s\right)}^6}+{\frac{1}{{\left(n+m-q+s\right)}^7}]\}}^2\end{array}$ (8)

Using the experimental data of the Advanced Light Sources (ALS) measurements of Esteves et al. [5], we obtain for the state $4s^{2}4p^2\left({}^3{P}_2\right)11d$ and $4s^{2}4p^2\left({}^3{P}_2\right)12d$ respectively E₁₁ = 19.595 eV (m = 11) and E₁₂ = 19.668 eV (q = 12). Energy limit is given by ALS data, we find E_∞ = 20.049 eV. Using these data, We then get using Equation (8): ${\sigma }_1=32.005995$ and ${\sigma }_2=-0.169044$ .

• For the $4s^{2}4p^3\left({}^2{D}_{5/2}^o\right)\to 4s^{2}4p^2\left({}^3{P}_2\right)nd$ transition

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1-\frac{{\sigma }_2}{n}-{\sigma }_2^2\left(n-m\right)\left(n-q\right)[\frac{1}{{\left(n+4m+q-2s\right)}^2\left(n+q+m-2s\right)}\\ +\frac{1}{{\left(n+m-q+s\right)}^4}+{\frac{1}{{\left(n+m-q+s\right)}^5}]\}}^2\end{array}$ (9)

The screening constants in (9) are determined from the transition energies measured by Esteves et al. [5]. Using the experimental data of the Advanced Ligth Sources (ALS) measured by Esteves, we obtain E₁₁ = 19.523 eV and E₁₂ = 19.595 eV respectively for $4s^{2}4p^3\left({}^2{D}_{5/2}^o\right)\to 4s^{2}4p^2\left({}^3{P}_2\right)11d$ and $4s^{2}4p^3\left({}^2{D}_{5/2}^o\right)\to 4s^{2}4p^2\left({}^3{P}_2\right)12d$ . From ALS, we find E_∞ = 19.973 eV. Thus, Equation (9) gives us: ${\sigma }_1=32.003467$ and ${\sigma }_2=-0.043651$ .

• For the $4s^{2}4p^3\left({}^2{D}_{\frac{3}{2}}^o\right)\to 4s^{2}4p^2\left({}^2{D}_2\right)ns\left({}^2{D}_2\right)$ transition

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1-\frac{{\sigma }_2}{n}-{\sigma }_2^2\left(n-m\right)\left(n-q\right)[\frac{1}{{\left(n+7q+m+s\right)}^3}\\ +\frac{1}{{\left(n+6m-2q-s\right)}^4}+\frac{1}{{\left(n+m-q-s\right)}^5}+{\frac{1}{{\left(n+m-q-2s\right)}^6}]\}}^2\end{array}$ (10)

For the $4s^{2}4p^3\left({}^2{D}_{\frac{3}{2}}^o\right)\to 4s^{2}4p^2\left({}^2{D}_2\right)11s\left({}^2{D}_2\right)$ and $4s^{2}4p^3\left({}^2{D}_{\frac{3}{2}}^o\right)\to 4s^{2}4p^2\left({}^2{D}_2\right)12s\left({}^2{D}_2\right)$ levels the experimental data of the Advanced Light Sources by Esteves et al. [5] are E₁₁ = 19.890 eV (m = 11) and E₁₂ = 20.214 eV (q = 12) in eV. ALS [5] gives us the limit energy, E_∞ = 21. 176 eV. In that case, we find using Equation (10) ${\sigma }_1=32.046604$ and ${\sigma }_2=-1.390788$ .

• For the $4s^{2}4p^3\left({}^2{P}_{1/2}\right)\to 4s^{2}4p^2\left({}^1{D}_2\right)ns$ transition

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1-\frac{{\sigma }_2}{n}-{\sigma }_2^2\left(n-m\right)\left(n-q\right)[\frac{1}{{\left(n-2s\right)}^2{\left(n+q+m+s\right)}^2}\\ +\frac{1}{{\left(n-m+q+6s\right)}^4}+{\frac{1}{{\left(n+2m-q+9s\right)}^5}]\}}^2\end{array}$ (11)

Using the experimental data of the Advanced Ligth Sources (ALS) measurements of Esteves et al. [5], we obtain for the state $4s^{2}4p^3\left({}^2{P}_{1/2}\right)\to 4s^{2}4p^2\left({}^1{D}_2\right)7s$ and $4s^{2}4p^3\left({}^2{P}_{1/2}\right)\to 4s^{2}4p^2\left({}^1{D}_2\right)8s$ respectively E₇ = 18.667 eV (m = 7) and E₈ = 18.991 eV (q = 8). Energy limit is given by ALS data, we find, E_∞ = 19.955 eV. Using these data, We then get using Equation (11): ${\sigma }_1=32.040633$ and ${\sigma }_2=-1.360699$ .

• For the $4s^{2}4p^3\left({}^2{P}_{3/2}\right)\to 4s^{2}4p^2\left({}^1{D}_2\right)ns$ transition

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1-\frac{{\sigma }_2}{n}-{\sigma }_2^2\left(n-m\right)\left(n-q\right)[\frac{1}{{\left(n-q+m+6s\right)}^3}\\ +\frac{1}{{\left(n+m-q+4s\right)}^4}+{\frac{1}{{\left(n+m-q+2s\right)}^5}]\}}^2\end{array}$ (12)

From Advanced Ligth Source (ALS) [5], we obtain for the $4s^{2}4p^3\left({}^2{P}_{3/2}\right)\to 4s^{2}4p^2\left({}^1{D}_2\right)7s$ and $4s^{2}4p^3\left({}^2{P}_{3/2}\right)\to 4s^{2}4p^2\left({}^1{D}_2\right)8s$ , E₇ = 18.560 eV (m = 7) and E₈ = 18.886 eV (q = 8). We find, E_∞ = 19.853 eV. We find then using Equation (12), we obtain ${\sigma }_1=32.04338$ and ${\sigma }_2=-1.409165$ .

• For the $4s^{2}4p^3\left({}^2{P}_{1/2}^o\right)\to 4s^{2}4p^2\left({}^1{D}_2\right)nd\left({}^2{P}_{1/2}^o\right)$ transition

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1-\frac{{\sigma }_2}{n}-{\sigma }_2^2\times \left(n-m\right)\times \left(n-q\right)[\frac{1}{{\left(n+4q+m+2s\right)}^3}\\ +\frac{1}{{\left(n+4m-q+6s\right)}^4}+{\frac{1}{{\left(n+m+3q-s\right)}^5}]\}}^2\end{array}$ (13)

Using the experimental data of the Advanced Light Source measurements of Esteves et al. [5], we obtain for the state $4s^{2}4p^3\left({}^2{P}_{1/2}^o\right)\to 4s^{2}4p^2\left({}^1{D}_2\right)6d\left({}^2{P}_{1/2}^o\right)$ and $4s^{2}4p^3\left({}^2{P}_{1/2}^o\right)\to 4s^{2}4p^2\left({}^1{D}_2\right)7nd\left({}^2{P}_{1/2}^o\right)$ respectively E₆ = 18.351 eV (m = 6) and E₇ = 18.790 eV (q = 7). Energy limit is given by ALS [5] data, we find E_∞ = 19.955 eV. Using these data, We then get using Equation (13): ${\sigma }_1=32.022401$ and ${\sigma }_2=-0.495149$ .

• For the $4s^{2}4p^3\left({}^2{P}_{1/2}^o\right)\to 4s^{2}4p^2\left({}^2{D}_2\right)ns\left({}^{2}S\right)$ transtion

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1-\frac{{\sigma }_2}{n}-{\sigma }_2^2\left(n-m\right)\left(n-q\right)[\frac{1}{{\left(n+m+q-s\right)}^5}\\ +{\frac{1}{{\left(n+m-q-s\right)}^6}]\}}^2\end{array}$ (14)

For the $4s^{2}4p^3\left({}^2{P}_{1/2}^o\right)\to 4s^{2}4p^2\left({}^2{D}_2\right)6s\left({}^{2}S\right)$ and $4s^{2}4p^3\left({}^2{P}_{1/2}^o\right)\to 4s^{2}4p^2\left({}^2{D}_2\right)7s\left({}^{2}S\right)$ levels the experimental data of the Advanced Ligth Sources by Esteves et al. [5] are E₆ = 18.393 eV (m = 6) and E₇ = 18.816 eV (q = 7) in eV. ALS [5] gives us the limit energy, E_∞ = 19.955 eV. In that case, we find using Equation (14) ${\sigma }_1=32.020399$ and ${\sigma }_2=-0.320232$ .

• For the $4s^{2}4p^2\left({}^2{P}_{1/2}^o\right)nd\left({}^1{P}_1^o\right)\to 3d^{10}4s^{2}4p^5\left({}^2{P}_{1/2}^o\right)$ transition,

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1-\frac{{\sigma }_2}{n}-{\sigma }_2^2\left(n-m\right)\left(n-q\right)[\frac{1}{{\left(n+m+q-s\right)}^5}\\ +{\frac{1}{{\left(n+m-q-s\right)}^6}]\}}^2\end{array}$ (15)

For the $4s^{2}4p^2\left({}^2{P}_{1/2}^o\right)8d\left({}^1{P}_1^o\right)$ and $4s^{2}4p^2\left({}^2{P}_{1/2}^o\right)9d\left({}^1{P}_1^o\right)$ levels the experimental data of the synchrotron radiation (SR) and dual laser plasma (DPL) [17] are E₈ = 27.39 (m = 8) and E₉ = 27.50 (q = 9) in eV. SR-DPL [17] gives us the limit energy, E_∞ = 28.204 eV. In that case, we find using Equation (15) ${\sigma }_1=35.071807$ and ${\sigma }_2=-1.071424$ .

3. Resultants and Discussions

The calculated resonance energies (in eV) for the Se⁺ Rydberg series are presented in Tables 1-7. These results, obtained via the MOAT formalism, are compared with SCUNC calculations (Sakho et al. 2022) and ALS experimental data from Esteves et al. [5] for the Se II ion. Similarly, our results for the Rb⁺ Rydberg series are summarized in Table 8 and Table 9. These are compared with theoretical data from Sakho et al. using the SCUNC method, Hartree-Fock calculations with exchange and relativistic corrections (HXR), and Dirac R-matrix results from McLaughlin and Babb. Comparisons are also made with Synchrotron Radiation (SR) and Dual Laser Plasma (DLP) experimental data, which, to our knowledge, represent the only available benchmarks. For all series studied, the MOAT resonance energies demonstrate high stability, enabling calculations up to n = 55 including both quantum defects and effective charge.

Theoretical and measured energy positions can be analyzed by calculating the Z effective charge in relationship with the quantum defect (δ). The relationship between $Z^{*}$ and $\delta$ is in the form:

$Z^{*}=\frac{Z_{core}}{\left(1-\frac{\delta }{n}\right)}$ (16)

According to this equation, each Rydberg series must satisfy the following conditions:

$\{\begin{array}{l}{Z}^{*}\ge Z_{core}\text{if}\delta \ge 0\\ Z^{*}\ge Z_{core}\text{if}\delta <0\\ \underset{n\to \infty }{\lim }{Z}^{*}=Z_{core}\end{array}$ (17)

For the photoionisation of the Se⁺ and Rb⁺ ion considered in this work Equation (6) give.

${\text{Se}}^{+}+h\nu \to {\text{Se}}^{2+}+{\text{e}}^{-}$ $Z_{core}=2$ (18)

${\text{Rb}}^{+}+h\nu \to {\text{Rb}}^{2+}+{\text{e}}^{-}$ $Z_{core}=2$ (19)

In Table 1 we present the MAOT resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge (Z*) of the $4s^{2}4p^2\left({}^3{P}_2\right)nd$ Rydberg series converging to the (${}^3{P}_2$ ) serie limits in Se²⁺ threshold origin Se⁺ $4s^{2}4p^3\left({}^2{P}_{3/2}^o\right)$ metastable state. Results presented in this table are calculated via Equation (8). The quantum defect and effective nuclear charge Z* agree well with the MAOT analysis condition of Equation (17). We also note that the quantum defect decreases as the principal quantum number increases. For n = 11 up to n = 25 Our results are compared with of the experimental data of ALS [5] and those of Sakho (2022) [11] obtained theoretically. The comparison shows very good agreement. The absolute deviations |ΔE| between MOAT and ALS are small (on the order of 0.0001 eV). For n = 26 up to 40, our results are one only compared with SCUNC theorical calculation of Sakho and comparison is considered very satisfactory. Thus, our value of 20.015 eV for n = 40 is in good agreement with Sakho’s calculated result of 20.015 eV. This allows us to expect the present results on the resonance energies for this Rydberg series up to n = 55 to be accurate.

We present in Table 2 resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge (Z*) of the $4s^{2}4p^2\left({}^3{P}_2\right)nd$ Rydberg series converging to the (${}^3{P}_2$ ) serie limits in Se²⁺ threshold origin Se⁺ $4s^{2}4p^3\left({}^2{P}_{5/2}^o\right)$ metastable state.

Table 1. Resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge Z* of the $4s^{2}4p^2\left({}^3{P}_2\right)nd$ Rydberg series converging to the (${}^3{P}_2$ ) serie limits in Se²⁺ threshold origin Se⁺ $4s^{2}4p^3\left({}^2{P}_{3/2}^o\right)$ metastable state. The present Modified Atomic Orbital Theory (MOAT) calculations are compared to the SCUNC calculations of Sakho et al. [11] and the ALS experimental measurements Esteves et al. [5]. |ΔE| denotes the energy difference between the MAOT calculations and the ALS measurement. The resultats are expression in electron-volt (eV). (1 Ryd = 0.5 au = 13.605698 eV).

Table 2. Resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge Z ∗ of the $4s^{2}4p^2\left({}^3{P}_2\right)nd$ Rydberg series converging to the $\left({}^3{P}_2\right)$ serie limits in Se²⁺ threshold origin Se⁺ $4s^{2}4p^3\left({}^2{D}^o{}_{5/2}\right)$ metastable state. The present Modified Atomic Orbital Theory (MOAT) calculations are compared to the SCUNC calculations of Sakho et al. [11] and the ALS experimental measurements Esteves et al. [5]. |ΔE| denotes the energy difference between the MAOT calculations and the ALS measurement. The resultats are expression in electron-volt (eV). (1 Ryd = 0.5 au = 13.605698 eV).

Table 3. Resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge Z ∗ of the $4s^{2}4p^2\left({}^2{D}_2\right)ns\left({}^2{D}_2\right)$ Rydberg series converging to the (${}^2{D}_2$ ) serie limits in Se²⁺ threshold origin Se⁺ $4s^{2}4p^3\left({}^2{P}_{3/2}^o\right)$ metastable state. The present Modified Atomic Orbital Theory (MOAT) calculations are compared to the SCUNC calculations of Sakho et al. [11] and the ALS experimental measurements Esteves et al. [5]. |ΔE| denotes the energy difference between the MAOT calculations and the ALS measurement. The resultats are expression in electron-volt (eV). (1 Ryd = 0.5 au = 13.605698 eV).