jamp Journal of Applied Mathematics and Physics 2327-4379 2327-4352 Scientific Research Publishing 10.4236/jamp.2026.146111 jamp-151977 Article Physics Mathematics Calculation of the Resonance Energies, Quantum Defect and Effective Nuclear Charge of Highly Excited Rydberg Series in the Photoionization Spectra of Se II and Rb II Ions Using the Modified Atomic Orbital Theory Faye Abdou 1 Sow Malick 1 2 Kebe Moustapha 1 Ndiaye Papa Mamadou 1 Dia Oumar Baba 1 Ndiaye Cheikh 1 Diouf Cheikh Tidjane 1 Ngom Astou 1 Niasse Oumar Absatou 1 2 Department of Physics, Faculty of Sciences and Technologies, University Cheikh Anta Diop of Dakar, Dakar, Senegal Institute of Applied Nuclear Technology, University Cheikh Anta Diop of Dakar, Dakar, Senegal

The authors declare no conflicts of interest regarding the publication of this paper.

 11 06 2026 06 2026 14 06 2273 2297 30 04 2026 19 06 2026 22 06 2026 © 2026 by the authors and Scientific Research Publishing Inc. 2026 This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( https://creativecommons.org/licenses/by/4.0/ ). https://doi.org/10.4236/jamp.2026.146111

In this paper, we reported the energy positions, quantum defect and effective nuclear charge of the

4

s

2

4

p

2

(

3

P

2

)nd

,

4

s

2

4

p

2

(

1

D

2

)ns

,

4

s

2

4

p

2

(

2

D

2

)ns(

2

D

2

)

,

4

s

2

4

p

2

(

1

D

2

)nd(

2

P

1/2

o

)

and

4

s

2

4

p

2

(

2

D

2

)ns(

2

S

)

Rydberg series of Se+ ions and

4

s

2

4

p

2

(

2

P

1/2

o

)nd(

1

P

1

o

)

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.

 Rydberg Series Resonance Energy Quantum Defect Photoionization Effective Charge and Modified Atomic Orbital Theory 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 4 s 2 4 p 2 ( 3 P 2 ) n d , 4 s 2 4 p 2 ( 1 D 2 ) n s , 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 D 2 ) , 4 s 2 4 p 2 ( 1 D 2 ) n d ( 2 P 1 / 2 o ) and 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 S ) Rydberg series originating from the 4 s 2 4 p 3 ( 2 P 1 2 , 3 2 o ) and 4 s 2 4 p 3 ( 2 D 3 2 , 5 2 o ) states of Se+ and 4 s 2 4 p 2 ( 2 P 1 / 2 o ) n d ( 1 P 1 o ) Rydberg series of Rb+ converging to the ( 3 d 10 4 s 2 4 p 5 2 P ∘ 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 ( ν ℓ ) -given orbital is expressed in the form [13]-[15]

E = − [ Z − σ ( ℓ ) ] 2 ν 2

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

E = − ∑ i = 1 M [ Z − σ i ( ℓ ) ] 2 ν i 2

With respect to the usual spectroscopic notation ( N ℓ , n ℓ ′ ) 2 S + 1 L π , this equation becomes

E = − ∑ i = 1 M [ Z − σ i ( 2 S + 1 L π ) ] 2 ν i 2 .

In the photoionization of atoms and ions, energy resonances are generally measured relatively to the E ∞ converging limit of a given ( 2 S + 1 L J ) n l -Rydberg serie. For these states, the general expression of the energy resonance E n is given

E n = E ∞ − 1 n 2 { Z − σ 1 ( 2 S + 1 L J ) − σ 2 ( 2 S + 1 L J ) × 1 n − σ 2 α ( 2 S + 1 L J )     × ( n − m ) × ( n − q ) ∑ k 1 f k ( n , m , q , s ) } 2

In this equation, m andq ( m < p ) denote the principal quantum numbers of the ( 2 S + 1 L J ) n l -Rydberg serie of the considered atomic system used in the empirical determination of the σ i ( 2 S + 1 L J ) -screening constants s represents the spin

of the n l -electron (s= ½), E ∞ 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 ∑ k 1 f k ( n , m , q , s ) -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 ∞ − R Z c o r e 2 ( n − δ ) 2

In the equation, R, E ∞ , Z c o r e and δ are the Rydberg constant, the converging limit, the electric charge of the core ion and the quantum defect, respectively.

Z c o r e is directly obtained by the photoionization process from an atomic Xp+ system:

X p + + h ν → X ( p + 1 ) + + e −

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

δ = n − Z c o r e × R E ∞ − E n 2.2. Resonances Energy of the Transitions States of S <sub>e</sub> <sup>+</sup> Ions

In the framework of the Modified Atomic Orbital Theory (MOAT) formalism the energy position of the 4 s 2 4 p 2 ( 3 P 2 ) n d , 4 s 2 4 p 2 ( 1 D 2 ) n s , 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 D 2 ) , 4 s 2 4 p 2 ( 1 D 2 ) n d ( 2 P 1 / 2 o ) and 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 S ) Rydberg series originating from the 4 s 2 4 p 3 ( 2 P 1 2 , 3 2 o ) and 4 s 2 4 p 3 ( 2 D 3 2 , 5 2 o ) states of Se+ and 4 s 2 4 p 2 ( 2 P 1 / 2 o ) n d ( 1 P 1 o ) Rydberg series of Rb+ converging to the ( 3 d 10 4 s 2 4 p 5 2 P ∘ 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 4 s 2 4 p 3 ( 2 P 1 2 , 3 2 o ) and 4 s 2 4 p 3 ( 2 D 3 2 , 5 2 o ) and of the Rydberg series of the Rb+ series.

For the 4 s 2 4 p 3 ( 2 D 3 / 2 o ) → 4 s 2 4 p 2 ( 3 P 2 ) n d transition

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n − σ 2 2 ( n − m ) ( n − q ) [ 1 ( n + m + q + s ) 5     + 1 ( n − m + q + 4 s ) 6 + 1 ( n + m − q + s ) 7 ] } 2

Using the experimental data of the Advanced Light Sources (ALS) measurements of Esteves et al. [5], we obtain for the state 4 s 2 4 p 2 ( 3 P 2 ) 11 d and 4 s 2 4 p 2 ( 3 P 2 ) 12 d 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): σ 1 = 32.005995 and σ 2 = − 0.169044 .

For the 4 s 2 4 p 3 ( 2 D 5 / 2 o ) → 4 s 2 4 p 2 ( 3 P 2 ) n d transition

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n − σ 2 2 ( n − m ) ( n − q ) [ 1 ( n + 4 m + q − 2 s ) 2 ( n + q + m − 2 s )     + 1 ( n + m − q + s ) 4 + 1 ( n + m − q + s ) 5 ] } 2

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 4 s 2 4 p 3 ( 2 D 5 / 2 o ) → 4 s 2 4 p 2 ( 3 P 2 ) 11 d and 4 s 2 4 p 3 ( 2 D 5 / 2 o ) → 4 s 2 4 p 2 ( 3 P 2 ) 12 d . From ALS, we find E∞ = 19.973 eV. Thus, Equation (9) gives us: σ 1 = 32.003467 and σ 2 = − 0.043651 .

For the 4 s 2 4 p 3 ( 2 D 3 2 o ) → 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 D 2 ) transition

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n − σ 2 2 ( n − m ) ( n − q ) [ 1 ( n + 7 q + m + s ) 3     + 1 ( n + 6 m − 2 q − s ) 4 + 1 ( n + m − q − s ) 5 + 1 ( n + m − q − 2 s ) 6 ] } 2

For the 4 s 2 4 p 3 ( 2 D 3 2 o ) → 4 s 2 4 p 2 ( 2 D 2 ) 11 s ( 2 D 2 ) and 4 s 2 4 p 3 ( 2 D 3 2 o ) → 4 s 2 4 p 2 ( 2 D 2 ) 12 s ( 2 D 2 ) 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) σ 1 = 32.046604 and σ 2 = − 1.390788 .

For the 4 s 2 4 p 3 ( 2 P 1 / 2 ) → 4 s 2 4 p 2 ( 1 D 2 ) n s transition

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n − σ 2 2 ( n − m ) ( n − q ) [ 1 ( n − 2 s ) 2 ( n + q + m + s ) 2       + 1 ( n − m + q + 6 s ) 4 + 1 ( n + 2 m − q + 9 s ) 5 ] } 2

Using the experimental data of the Advanced Ligth Sources (ALS) measurements of Esteves et al. [5], we obtain for the state 4 s 2 4 p 3 ( 2 P 1 / 2 ) → 4 s 2 4 p 2 ( 1 D 2 ) 7 s and 4 s 2 4 p 3 ( 2 P 1 / 2 ) → 4 s 2 4 p 2 ( 1 D 2 ) 8 s 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): σ 1 = 32.040633 and σ 2 = − 1.360699 .

For the 4 s 2 4 p 3 ( 2 P 3 / 2 ) → 4 s 2 4 p 2 ( 1 D 2 ) n s transition

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n − σ 2 2 ( n − m ) ( n − q ) [ 1 ( n − q + m + 6 s ) 3     + 1 ( n + m − q + 4 s ) 4 + 1 ( n + m − q + 2 s ) 5 ] } 2

From Advanced Ligth Source (ALS) [5], we obtain for the 4 s 2 4 p 3 ( 2 P 3 / 2 ) → 4 s 2 4 p 2 ( 1 D 2 ) 7 s and 4 s 2 4 p 3 ( 2 P 3 / 2 ) → 4 s 2 4 p 2 ( 1 D 2 ) 8 s , 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 σ 1 = 32.04338 and σ 2 = − 1.409165 .

For the 4 s 2 4 p 3 ( 2 P 1 / 2 o ) → 4 s 2 4 p 2 ( 1 D 2 ) n d ( 2 P 1 / 2 o ) transition

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n − σ 2 2 × ( n − m ) × ( n − q ) [ 1 ( n + 4 q + m + 2 s ) 3     + 1 ( n + 4 m − q + 6 s ) 4 + 1 ( n + m + 3 q − s ) 5 ] } 2

Using the experimental data of the Advanced Light Source measurements of Esteves et al. [5], we obtain for the state 4 s 2 4 p 3 ( 2 P 1 / 2 o ) → 4 s 2 4 p 2 ( 1 D 2 ) 6 d ( 2 P 1 / 2 o ) and 4 s 2 4 p 3 ( 2 P 1 / 2 o ) → 4 s 2 4 p 2 ( 1 D 2 ) 7 n d ( 2 P 1 / 2 o ) 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): σ 1 = 32.022401 and σ 2 = − 0.495149 .

For the 4 s 2 4 p 3 ( 2 P 1 / 2 o ) → 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 S ) transtion

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n − σ 2 2 ( n − m ) ( n − q ) [ 1 ( n + m + q − s ) 5     + 1 ( n + m − q − s ) 6 ] } 2

For the 4 s 2 4 p 3 ( 2 P 1 / 2 o ) → 4 s 2 4 p 2 ( 2 D 2 ) 6 s ( 2 S ) and 4 s 2 4 p 3 ( 2 P 1 / 2 o ) → 4 s 2 4 p 2 ( 2 D 2 ) 7 s ( 2 S ) 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) σ 1 = 32.020399 and σ 2 = − 0.320232 .

For the 4 s 2 4 p 2 ( 2 P 1 / 2 o ) n d ( 1 P 1 o ) → 3 d 10 4 s 2 4 p 5 ( 2 P 1 / 2 o ) transition,

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n − σ 2 2 ( n − m ) ( n − q ) [ 1 ( n + m + q − s ) 5     + 1 ( n + m − q − s ) 6 ] } 2

For the 4 s 2 4 p 2 ( 2 P 1 / 2 o ) 8 d ( 1 P 1 o ) and 4 s 2 4 p 2 ( 2 P 1 / 2 o ) 9 d ( 1 P 1 o ) 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) σ 1 = 35.071807 and σ 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 δ is in the form:

Z * = Z c o r e ( 1 − δ n )

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

{ Z * ≥ Z c o r e     if   δ ≥ 0 Z * ≥ Z c o r e     if   δ < 0 lim n → ∞ Z * = Z c o r e

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

Se + + h ν → Se 2 + + e − Rb + + h ν → Rb 2 + + e −

In Table 1 we present the MAOT resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge (Z*) of the 4 s 2 4 p 2 ( 3 P 2 ) n d Rydberg series converging to the ( 3 P 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 P 3 / 2 o ) 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 4 s 2 4 p 2 ( 3 P 2 ) n d Rydberg series converging to the ( 3 P 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 P 5 / 2 o ) metastable state.

Table 1.Resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge Z* of the 4 s 2 4 p 2 ( 3 P 2 ) n d Rydberg series converging to the ( 3 P 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 P 3 / 2 o ) 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).

4 s 2 4 p 3 ( 2 P 3 / 2 o ) → 4 s 2 4 p 2 ( 3 P 2 ) n d
n MAOT ALS SCUNC Δ E
E δ Z * E δ Z * E δ Z *
11 19.595 0.0513 2.0094 19.595 0.051 2.0093 19.595 0.051 2.0094 0.0000
12 19.668 0.0484 2.0081 19.668 0.051 2.0085 19.668 0.052 2.0087 0.0000
13 19.7247 0.0454 2.0070 19.724 0.051 2.0079 19.724 0.052 2.0080 0.0007
14 19.7696 0.0424 2.0061 19.769 0.035 2.0073 19.770 0.052 2.0073 0.0004
15 19.8060 0.0395 2.0053 19.805 0.020 2.0068 19.807 0.052 2.0069 0.0010
16 19.8354 0.0365 2.0046 19.835 0.051 2.0064 19.835 0.052 2.0064 0.0004
17 19.8599 0.0335 2.0039 19.860 0.051 2.0060 19.860 0.052 2.0060 0.0001
18 19.8805 0.0305 2.0034 19.880 0.051 2.0057 19.880 0.051 2.0057 0.0005
19 19.8978 0.0275 2.0029 19.897 0.051 2.0054 19.897 0.051 2.0054 0.0008
20 19.9126 0.0245 2.0025 19.912 0.051 2.0051 19.912 0.051 2.0051 0.0006
21 19.9253 0.0216 2.0021 19.925 0.051 2.0049 19.925 0.051 2.0048 0.0003
22 19.9364 0.0186 2.0017 19.936 0.051 2.0046 19.936 0.051 2.0046 0.0004
23 19.946 0.0156 2.0014 19.946 0.051 2.0044 19.946 0.051 2.0044 0.0000
24 19.9544 0.0126 2.0010 19.954 0.051 2.0042 19.954 0.051 2.0042 0.0004
25 19.9619 0.0096 2.0008 19.962 0.051 2.0011 19.962 0.051 2.0042 0.0001
26 19.9685 0.0066 2.0005 19.968 0.051 2.0029
27 19.9743 0.0036 2.0003 19.974 0.051 2.0037
28 19.9796 0.0006 2.0000 19.979 0.051 2.0036
29 19.9843 −0.0024 1.9998 19.984 0.051 2.0033
30 19.9886 −0.0054 1.9996 19.988 0.051 2.0032
31 19.9924 −0.0084 1.9995 19.992 0.051 2.0031
32 19.9959 −0.0114 1.9993 19.996 0.051 2.0030
33 19.9991 −0.0144 1.9991 19.999 0.051 2.0029
34 20.002 −0.0174 1.9990 20.002 0.051 2.0028
35 20.0046 −0.0204 1.9988 20.004 0.051 2.0028
36 20.0071 −0.0234 1.9987 20.007 0.051 2.0027
37 20.0093 −0.0264 1.9986 20.009 0.051 2.0026
38 20.0114 −0.0294 1.9985 20.011 0.051 2.0025
39 20.0133 −0.0324 1.9983 20.013 0.051 2.0025
40 20.015 −0.0354 1.9982 20.015 0.051 2.0025
41 20.0167 −0.0384 1.9981
42 20.0182 −0.0414 1.9980
43 20.0196 −0.0444 1.9979
44 20.0209 −0.0474 1.9978
45 20.0222 −0.0504 1.9978
46 20.0233 −0.0534 1.9977
47 20.0244 −0.0564 1.9976
48 20.0254 −0.0594 1.9975
49 20.0264 −0.0624 1.9975
50 20.0273 −0.0654 1.9974
51 20.0281 −0.0684 1.9973
52 20.0289 −0.0714 1.9973
53 20.0297 −0.0745 1.9972
54 20.0304 −0.0775 1.9971
55 20.0281 −0.0805 1.9971
∞ 20.0490 20.0490 20.0490

Table 2.Resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge Z∗ of the 4 s 2 4 p 2 ( 3 P 2 ) n d Rydberg series converging to the ( 3 P 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 D o 5 / 2 ) 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).

4 s 2 4 p 3 ( 2 P 5 / 2 o ) → 4 s 2 4 p 2 ( 3 P 2 ) n d
n MAOT ALS SCUNC ΔE
E δ Z* E δ Z* E δ Z*
11 19.523 0.0028 2.0005 19.523 0.003 2.0005 19.523 0.003 2.0005 0.000
12 19.595 0.001 2.0002 19.595 0.003 2.0005 19.595 0.003 2.0005 0.000
13 19.651 −0.0007 1.9999 19.651 0.003 2.0005 19.651 0.003 2.0004 0.000
14 19.6954 −0.0024 1.9997 19.697 −0.040 1.9943 19.695 0.003 2.0004 0.001
15 19.7313 −0.0041 1.9995 19.733 −0.050 1.9933 19.731 0.003 2.0004 0.001
16 19.7606 −0.0057 1.9993 19.764 −0.110 1.9863 19.760 0.003 2.0003 0.003
17 19.7849 −0.0074 1.9991 19.789 −0.200 1.9767 19.785 0.003 2.0003 0.004
18 19.8052 −0.0091 1.999 19.811 −0.350 1.9618 19.805 0.003 2.0003 0.005
19 19.8224 −0.0108 1.9989 19.827 −0.320 1.9669 19.822 0.003 2.0003 0.004
20 19.8371 −0.0125 1.9988 19.842 −0.300 1.9704 19.837 0.003 2.0003 0.005
21 19.8498 −0.0142 1.9986 19.855 −0.400 1.9626 19.850 0.003 2.0003 0.005
22 19.8607 −0.0159 1.9986 19.855 −0.400 1.9643 19.861 0.003 2.0002 0.005
23 19.8703 −0.0176 1.9985 19.865 −0.450 1.9616 19.870 0.003 2.0002 0.005
24 19.8787 −0.0192 1.9984 19.878 0.003 2.0002
25 19.8861 −0.0209 1.9983 19.886 0.003 2.0002
26 19.8926 −0.0226 1.9983 19.892 0.003 2.0002
27 19.8985 −0.0243 1.9982 19.898 0.003 2.0002
28 19.9037 −0.026 1.9981 19.904 0.003 2.0002
29 19.9084 −0.0277 1.9981 19.908 0.003 2.0002
30 19.9126 −0.0293 1.998 19.913 0.003 2.0002
31 19.9165 −0.031 1.998 19.916 0.003 2.0002
32 19.9200 −0.0327 1.998 19.920 0.003 2.0002
33 19.9231 −0.0344 1.9979 19.923 0.003 2.0002
34 19.926 −0.036 1.9979 19.926 0.003 2.0002
35 19.9287 −0.0377 1.9978 19.929 0.003 2.0002
36 19.9311 −0.0394 1.9978 19.931 0.003 2.0001
37 19.9333 −0.041 1.9978 19.933 0.003 2.0001
38 19.9354 −0.0427 1.9978 19.935 0.003 2.0001
39 19.9373 −0.0444 1.9977 19.937 0.003 2.0001
40 19.9391 −0.046 1.9977 19.939 0.003 2.0001
41 19.9407 −0.0477 1.9977
42 19.9422 −0.0494 1.9977
43 19.9436 −0.051 1.9976
44 19.945 −0.0527 1.9976
45 19.9462 −0.0544 1.9976
46 19.9473 −0.056 1.9976
47 19.9484 −0.0577 1.9975
48 19.9494 −0.0594 1.9975
49 19.9504 −0.061 1.9975
50 19.9513 −0.0627 1.9975
51 19.9521 −0.0643 1.9975
52 19.9529 −0.066 1.9975
53 19.9537 −0.0677 1.9975
54 19.9544 −0.0693 1.9974
55 19.9551 −0.071 1.9974
∞ 19.9730 19.973 19.973

Table 3.Resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge Z∗ of the 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 D 2 ) Rydberg series converging to the ( 2 D 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 P 3 / 2 o ) 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).

4 s 2 4 p 3 ( 2 P 3 / 2 o ) → 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 D 2 )
n MAOT ALS SCUNC Δ E
E δ Z * E δ Z * E δ Z *
7 19.89 0.495 2.1521 19.890 0.495 2.1522 19.890 0.496 2.1525 0.000
8 20.214 0.479 2.1272 20.214 0.480 2.1276 20.211 0.487 2.1303 0.000
9 20.4298 0.46 2.1077 20.428 0.470 2.1102 20.425 0.485 2.1143 0.001
10 20.5805 0.44 2.0921 20.576 0.480 2.1008 20.575 0.484 2.1019 0.004
11 20.6898 0.42 2.0794 20.685 0.470 2.0893 20.684 0.483 2.0920 0.004
12 20.7716 0.399 2.0688 20.770 0.420 2.0725 20.766 0.482 2.0838 0.001
13 20.8344 0.378 2.0599 20.830 0.460 2.0734 20.829 0.481 2.0770 0.004
14 20.8836 0.356 2.0522 20.883 0.380 2.0558 20.878 0.481 2.0712 0.000
15 20.923 0.334 2.0456 20.922 0.360 2.0492 20.918 0.481 2.0663 0.001
16 20.9549 0.312 2.0398 20.955 0.300 2.0382 20.950 0.481 2.0620 0.000
17 20.9811 0.289 2.0346 20.985 0.100 2.0118 20.977 0.481 2.0582 0.004
18 21.0029 0.266 2.03 21.008 0 2.0000 20.999 0.481 2.0549 0.005
19 21.0213 0.243 2.0259 21.025 0 2.0000 21.017 0.481 2.0519 0.004
20 21.0369 0.22 2.0222 21.040 0 2.0000 21.033 0.481 2.0492 0.003
21 21.0503 0.196 2.0188 21.053 0 2.0000 21.047 0.481 2.0469 0.002
22 21.0618 0.172 2.0158 21.063 0 2.0000 21.058 0.481 2.0447 0.001
23 21.0718 0.148 2.013 21.073 0 2.0000 21.069 0.481 2.0427 0.001
24 21.0805 0.124 2.0104 21.078 0.481 2.0409
25 21.0882 0.1 2.008 21.085 0.481 2.0393
26 21.095 0.075 2.0058 21.092 0.481 2.0377
27 21.1011 0.05 2.0037 21.099 0.482 2.0363
28 21.1065 0.026 2.0018 21.104 0.482 2.0350
29 21.1113 0.001 2.0001 21.109 0.482 2.0338
30 21.1156 −0.024 1.9984 21.114 0.482 2.0327
31 21.1195 −0.049 1.9968 21.118 0.483 2.0316
32 21.1231 −0.074 1.9954 21.121 0.483 2.0306
33 21.1263 −0.099 1.994 21.125 0.483 2.0297
34 21.1293 −0.125 1.9927 21.128 0.484 2.0289
35 21.132 −0.15 1.9915 21.130 0.484 2.0280
36 21.1344 −0.176 1.9903 21.133 0.484 2.0273
37 21.1367 −0.201 1.9892 21.135 0.484 2.0265
38 21.1388 −0.227 1.9881 21.137 0.485 2.0258
39 21.1407 −0.252 1.9871 21.139 0.485 2.0252
40 21.1425 −0.278 1.9862 21.141 0.485 2.0246
41 21.1441 −0.304 1.9853
42 21.1456 −0.33 1.9844
43 21.147 −0.356 1.9836
44 21.1484 −0.382 1.9828
45 21.1496 −0.408 1.982
46 21.1508 −0.434 1.9813
47 21.1518 −0.46 1.9806
48 21.1529 −0.486 1.98
49 21.1538 −0.512 1.9793
50 21.1547 −0.538 1.9787
51 21.1555 −0.564 1.9781
52 21.1563 −0.591 1.9775
53 21.1571 −0.617 1.977
54 21.1578 −0.643 1.9765
55 21.1584 −0.67 1.9759
∞ 21.1760 21.176 21.176

Table 4.Resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge Z* of the 4 s 2 4 p 2 ( 1 D 2 ) n s Rydberg series converging to the ( 1 D 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 P 1 / 2 o ) 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).

4 s 2 4 p 3 ( 2 P 1 / 2 o ) → 4 s 2 4 p 2 ( 1 D 2 ) n s
n MAOT ALS SCUNC Δ E
E δ Z * E δ Z * E δ Z *
7 18.667 0.4997 2.1538 18.667 0.500 2.1538 18.667 0.500 2.1538 0.0000
8 18.991 0.4863 2.1295 18.991 0.488 2.1294 18.988 0.496 2.1323 0.0000
9 19.2067 0.4721 2.1107 19.207 0.470 2.1102 19.203 0.494 2.1162 0.0003
10 19.3574 0.4571 2.0958 19.357 0.460 2.0964 19.353 0.492 2.1036 0.0004
11 19.4668 0.4415 2.0836 19.467 0.440 2.0833 19.462 0.491 2.0935 0.0003
12 19.5488 0.4252 2.0735 19.549 0.420 2.0725 19.544 0.490 2.0852 0.0002
13 19.6117 0.4085 2.0649 19.613 0.380 2.0602 19.607 0.489 2.0782 0.0013
14 19.6611 0.3912 2.0575 19.663 0.355 2.0520 19.657 0.489 2.0724 0.0019
15 19.7006 0.3736 2.0511 19.702 0.340 2.0464 19.697 0.488 2.0673 0.0014
16 19.7326 0.3557 2.0455 19.729 0.488 2.0629
17 19.759 0.3374 2.0405 19.755 0.488 2.0591
18 19.7809 0.3189 2.0361 19.778 0.488 2.0557
19 19.7994 0.3001 2.0321 19.796 0.488 2.0527
20 19.815 0.2812 2.0285 19.812 0.488 2.0500
21 19.8285 0.262 2.0253 19.826 0.488 2.0476
22 19.84 0.2427 2.0223 19.837 0.488 2.0454
23 19.8501 0.2233 2.0196 19.848 0.488 2.0434
24 19.8589 0.2037 2.0171 19.857 0.488 2.0415
25 19.8666 0.184 2.0148 19.864 0.488 2.0398
26 19.8735 0.1643 2.0127 19.871 0.489 2.0383
27 19.8795 0.1444 2.0108 19.878 0.489 2.0369
28 19.885 0.1245 2.0089 19.883 0.489 2.0355
29 19.8898 0.1044 2.0072 19.888 0.489 2.0343
30 19.8942 0.0844 2.0056 19.893 0.489 2.0332
31 19.8981 0.0642 2.0042 19.897 0.490 2.0321
32 19.9017 0.0441 2.0028 19.900 0.490 2.0311
33 19.905 0.0238 2.0014 19.904 0.490 2.0302
34 19.9079 0.0036 2.0002 19.907 0.491 2.0293
35 19.9106 −0.0168 1.999 19.909 0.491 2.0284
36 19.9131 −0.0371 1.9979 19.912 0.491 2.0277
37 19.9154 −0.0575 1.9969 19.914 0.491 2.0269
38 19.9175 −0.0779 1.9959 19.916 0.492 2.0262
39 19.9194 −0.0983 1.995 19.918 0.492 2.0256
40 19.9212 −0.1188 1.9941 19.920 0.492 2.0249
41 19.9228 −0.1392 1.9932
42 19.9244 −0.1597 1.9924
43 19.9258 −0.1803 1.9917
44 19.9271 −0.2008 1.9909
45 19.9284 −0.2213 1.9902
46 19.9295 −0.2419 1.9895
47 19.9306 −0.2625 1.9889
48 19.9317 −0.2831 1.9883
49 19.9326 −0.3037 1.9877
50 19.9335 −0.3243 1.9871
51 19.9344 −0.3449 1.9866
52 19.9352 −0.3655 1.986
53 19.9359 −0.3862 1.9855
54 19.9366 −0.4068 1.985
55 19.9373 −0.4275 1.9846
… ∞ 19.955 19.955 19.955

Table 5.Resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge Z* of the 4 s 2 4 p 2 ( 1 D 2 ) n s Rydberg series converging to the ( 1 D 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 P 3 / 2 o ) 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).

4 s 2 4 p 3 ( 2 P 3 / 2 ) → 4 s 2 4 p 2 ( 1 D 2 ) n s
n MAOT ALS SCUNC Δ E
E δ Z * E δ Z * E δ Z *
7 18.5600 0.5123 2.1579 18.560 0.512 2.1578 18.560 0.512 2.1578 0.000
8 18.8860 0.498 2.1328 18.886 0.500 2.1333 18.883 0.509 2.1358 0.000
9 19.101 0.4919 2.1156 19.102 0.490 2.1152 19.099 0.506 2.1192 0.001
10 19.251 0.4902 2.1031 19.250 0.500 2.1052 19.249 0.504 2.1062 0.001
11 19.360 0.4905 2.0933 19.359 0.500 2.0952 19.359 0.503 2.0959 0.001
12 19.442 0.4915 2.0854 19.442 0.500 2.0870 19.441 0.502 2.0873 0.000
13 19.505 0.4925 2.0788 19.505 0.500 2.0800 19.505 0.501 2.0802 0.000
14 19.555 0.493 2.073 19.554 0.500 2.0741 19.554 0.501 2.0742 0.001
15 19.594 0.4928 2.0679 19.594 0.500 2.0690 19.594 0.500 2.0690 0.000
16 19.627 0.4916 2.0634 19.627 0.500 2.0652 19.626 0.500 2.0645 0.000
17 19.6534 0.4895 2.0593 19.653 0.500 2.0610 19.653 0.500 2.0606 0.000
18 19.676 0.4865 2.0556 19.675 0.500 2.0571 19.675 0.500 2.0571 0.001
19 19.6943 0.4825 2.0521 19.694 0.500 2.0541 19.694 0.500 2.0540 0.001
20 19.7102 0.4775 2.0489 19.710 0.500 2.0513
21 19.7239 0.4717 2.046 19.724 0.500 2.0488
22 19.7356 0.4651 2.0432 19.735 0.500 2.0465
23 19.7459 0.4577 2.0406 19.745 0.500 2.0444
24 19.7549 0.4495 2.0382 19.754 0.500 2.0426
25 19.7628 0.4406 2.0359 19.762 0.500 2.0408
26 19.7698 0.4311 2.0337 19.769 0.500 2.0393
27 19.776 0.421 2.0317 19.775 0.501 2.0378
28 19.7815 0.4102 2.0297 19.781 0.501 2.0364
29 19.7865 0.399 2.0279 19.786 0.501 2.0352
30 19.7909 0.3872 2.0262 19.790 0.501 2.0340
31 19.795 0.375 2.0245 19.794 0.502 2.0329
32 19.7986 0.3623 2.0229 19.798 0.502 2.0319
33 19.802 0.3493 2.0214 19.801 0.502 2.0309
34 19.805 0.3358 2.0199 19.804 0.502 2.0300
35 19.8077 0.322 2.0186 19.807 0.503 2.0291
36 19.8103 0.3078 2.0172 19.810 0.503 2.0283
37 19.8126 0.2933 2.016 19.812 0.503 2.0276
38 19.8148 0.2784 2.0148 19.814 0.504 2.0269
39 19.8167 0.2633 2.0136 19.816 0.504 2.0262
40 19.8186 0.248 2.0125 19.818 0.504 2.0255
41 19.8203 0.2323 2.0114
42 19.8218 0.2164 2.0104
43 19.8233 0.2003 2.0094
44 19.8247 0.184 2.0084
45 19.8259 0.1675 2.0075
46 19.8271 0.1507 2.0066
47 19.8282 0.1338 2.0057
48 19.8293 0.1167 2.0049
49 19.8302 0.0995 2.0041
50 19.8312 0.082 2.0033
51 19.832 0.0645 2.0025
52 19.8328 0.0467 2.0018
53 19.8336 0.0289 2.0011
54 19.8343 0.0109 2.0004
55 19.835 -0.0072 1.9997
∞ 19.8530 19.853 19.853

Table 6.Resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge Z* of the 4 s 2 4 p 2 ( 1 D 2 ) n d ( 2 P 1 / 2 o ) Rydberg series converging to the ( 1 D 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 P 1 / 2 o ) 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).

4 s 2 4 p 3 ( 2 P 1 / 2 o ) → 4 s 2 4 p 2 ( 1 D 2 ) n d ( 2 P 1 / 2 o )
n MAOT ALS SCUNC |Δ E |
E δ Z * E δ Z * E δ Z *
6 18.351 0.1751 2.0601 18.351 0.175 2.0600 18.351 0.175 2.0602 0.000
7 18.790 0.1652 2.0483 18.790 0.165 2.0483 18.785 0.179 2.0524 0.000
8 19.071 0.155 2.0395 19.071 0.152 2.0395 19.066 0.176 2.0451 0.000
9 19.261 0.1446 2.0327 19.262 0.140 2.0316 19.256 0.174 2.0394 0.001
10 19.396 0.1341 2.0272 19.395 0.140 2.0284 19.392 0.172 2.0350 0.001
11 19.495 0.1236 2.0227 19.494 0.130 2.0239 19.491 0.171 2.0315 0.001
12 19.570 0.113 2.019 19.571 0.100 2.0168 19.566 0.169 2.0286 0.001
13 19.628 0.1025 2.0159 19.633 0 2.000 19.624 0.168 2.0263 0.005
14 19.674 0.092 2.0132 19.683 −0.150 1.9788 19.671 0.168 2.0242 0.008
15 19.710 0.0814 2.0109 19.718 −0.150 1.9802 19.708 0.167 2.0225 0.007
16 19.740 0.071 2.0089 19.738 0.166 2.0210
17 19.765 0.0605 2.0071 19.763 0.166 2.0197
18 19.786 0.0501 2.0056 19.784 0.165 2.0186
19 19.8036 0.0397 2.0042 19.802 0.165 2.0175
20 19.8185 0.0293 2.0029 19.817 0.165 2.0166
21 19.8314 0.019 2.0018 19.830 0.165 2.0158
22 19.8425 0.0086 2.0008 19.841 0.164 2.0151
23 19.8521 −0.0016 1.9999 19.851 0.164 2.0144
24 19.8606 −0.0119 1.999 19.859 0.164 2.0138
25 19.8681 −0.0221 1.9982 19.867 0.164 2.0132
26 19.8747 −0.0324 1.9975 19.873 0.164 2.0127
27 19.8806 −0.0425 1.9969 19.879 0.164 2.0122
28 19.8858 −0.0527 1.9962 19.885 0.164 2.0118
29 19.8906 −0.0629 1.9957 19.890 0.164 2.0113
30 19.8948 −0.073 1.9951 19.894 0.163 2.0110
31 19.8987 −0.0832 1.9946 19.898 0.163 2.0106
32 19.9022 −0.0933 1.9942 19.901 0.163 2.0103
33 19.9053 −0.1034 1.9938 19.905 0.163 2.0100
34 19.9082 −0.1135 1.9933 19.907 0.163 2.0097
35 19.9109 −0.1236 1.993 19.910 0.163 2.0094
36 19.9133 −0.1337 1.9926 19.913 0.163 2.0091
37 19.9156 −0.1438 1.9923 19.915 0.163 2.0089
38 19.9176 −0.1538 1.9919 19.917 0.163 2.0086
39 19.9195 −0.1639 1.9916 19.919 0.163 2.0084
40 19.9213 −0.174 1.9913 19.921 0.164 2.0082
41 19.9229 −0.1841 1.9911
42 19.9244 −0.1941 1.9908
43 19.9258 −0.2042 1.9905
44 19.9272 −0.2143 1.9903
45 19.9284 −0.2244 1.9901
46 19.9295 −0.2345 1.9899
47 19.9306 −0.2445 1.9896
48 19.9316 −0.2546 1.9894
49 19.9326 −0.2647 1.9893
50 19.9335 −0.2748 1.9891
51 19.9343 −0.2849 1.9889
52 19.9351 −0.295 1.9887
53 19.9358 −0.3052 1.9886
54 19.9366 −0.3153 1.9884
55 19.9372 −0.3254 1.9882
∞ 19.9550 19.9550 19.9550

Table 7.Resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge Z* of the 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 S ) Rydberg series converging to the ( 2 D 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 P 1 / 2 o ) metastable state. The present 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).

4 s 2 4 p 3 ( 2 P 1 / 2 o ) → 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 S )
n MAOT ALS SCUNC |Δ E |
E δ Z * E δ Z * E δ Z *
6 18.393 0.0971 2.0329 18.393 0.097 2.0329 18.393 0.097 2.0330 0.000
7 18.816 0.0874 2.0253 18.816 0.087 2.0252 18.812 0.099 2.0287 0.000
8 19.0879 0.0774 2.0196 19.077 0.125 2.0317 19.083 0.098 2.0247 0.010
9 19.2729 0.0675 2.0151 19.264 0.125 2.0282 19.269 0.096 2.0216 0.008
10 19.4044 0.0574 2.0115 19.396 0.135 2.0274 19.400 0.095 2.0192 0.008
11 19.5013 0.0473 2.0086 19.495 0.100 2.0230 19.497 0.094 2.0173 0.006
12 19.5747 0.0371 2.0062 19.571 0.025 2.0168 19.571 0.093 2.0157 0.003
13 19.6316 0.027 2.0042 19.632 −0.100 2.0038 19.628 0.093 2.0144 0.000
14 19.6766 0.0168 2.0024 19.681 −0.150 1.9858 19.674 0.092 2.0133 0.004
15 19.7129 0.0066 2.0009 19.718 −0.240 1.9858 19.710 0.092 2.0123 0.005
16 19.7425 −0.0037 1.9995 19.749 0.097 1.9802 19.740 0.092 2.0115 0.006
17 19.767 −0.0139 1.9984 19.765 0.091 2.0108
18 19.7875 −0.0242 1.9973 19.785 0.091 2.0102
19 19.8048 −0.0344 1.9964 19.803 0.091 2.0096
20 19.8195 −0.0447 1.9955 19.818 0.091 2.0091
21 19.8322 −0.055 1.9948 19.831 0.091 2.0087
22 19.8432 −0.0653 1.9941 19.842 0.090 2.0083
23 19.8528 −0.0756 1.9935 19.851 0.090 2.0079
24 19.8612 −0.0859 1.9929 19.860 0.090 2.0075
25 19.8686 −0.0962 1.9923 19.867 0.090 2.0072
26 19.8751 −0.1065 1.9918 19.874 0.090 2.0069
27 19.881 −0.1168 1.9914 19.880 0.090 2.0067
28 19.8862 −0.1271 1.991 19.885 0.090 2.0064
29 19.8909 −0.1374 1.9906 19.890 0.090 2.0062
30 19.8951 −0.1477 1.9902 19.894 0.090 2.0060
31 19.8989 −0.158 1.9899 19.898 0.090 2.0058
32 19.9024 −0.1683 1.9895 19.902 0.090 2.0056
33 19.9056 −0.1786 1.9892 19.905 0.090 2.0055
34 19.9084 −0.189 1.9889 19.908 0.090 2.0053
35 19.9111 −0.1993 1.9887 19.910 0.090 2.0051
36 19.9135 −0.2096 1.9884 19.913 0.090 2.0050
37 19.9157 −0.2199 1.9882 19.915 0.090 2.0049
38 19.9178 −0.2302 1.988 19.917 0.090 2.0047
39 19.9197 −0.2406 1.9877 19.919 0.090 2.0046
40 19.9214 −0.2509 1.9875 19.921 0.090 2.0045
41 19.923 −0.2612 1.9873
42 19.9245 −0.2715 1.9872
43 19.9259 −0.2819 1.987
44 19.9273 −0.2922 1.9868
45 19.9285 −0.3025 1.9866
46 19.9296 −0.3129 1.9865
47 19.9307 −0.3232 1.9863
48 19.9317 −0.3335 1.9862
49 19.9326 −0.3438 1.9861
50 19.9335 −0.3542 1.9859
51 19.9344 −0.3645 1.9858
52 19.9352 −0.3748 1.9857
53 19.9359 −0.3852 1.9856
54 19.9366 −0.3955 1.9855
55 19.9373 −0.4058 1.9854
∞ 19.9550 19.955 19.955

Table 8.Resonance Energy (En, eV), of the 4 s 2 4 p 2 ( 2 P 1 / 2 o ) n d ( 1 P 1 o ) Rydberg series of Rb+ converging to the ( 3 d 10 4 s 2 4 p 5 2 P ∘ 1 / 2 ) serie limits in Rb2+. The present Modified Atomic Orbital Theory (MOAT) calculations are compared to the SCUNC calculations of Sakho et al. [10], the synchrotron radiation (SR) and dual laser plasma (DPL) measurements of Kilbane et al. [17] along with their theoretical results from Hartree-Fock with exchange plus relativistic corrections (HXR) [17] and with the Dirac R-matrix calculations of McLaughlin and Babb [18]. |ΔE| denotes the energy difference between the MAOT calculations and the SR measurement. The resultats are expression in electron-volt (eV). (1 Ryd = 0.5 au = 13.605698 eV).

4 s 2 4 p 2 ( 2 P 1 / 2 o ) n d ( 1 P 1 o )
Resonance Energie E in eV
n MOAT SCUNC R-matrix HXR SR-DLP ΔE
8 27.3 27.3000 27.3008 27.3099 27.30 0.0000
9 27.5 27.4949 27.4805 27.4595 27.50 0.0000
10 27.6404 27.6330 27.6223 27.5900 27.63 0.010
11 27.7426 27.7343 27.7263 27.6814 27.74 0.002
12 27.8194 27.8108 27.8046 27.7519 27.82 0.0006
13 27.8786 27.8700 27.8652 27.8072 27.89 0.011
14 27.925 27.9168 27.9129 27.8515 27.95 0.025
15 27.9622 27.9544 27.9512
16 27.9924 27.9851 27.9824
17 28.0173 28.0104
18 28.0381 28.0316
19 28.0556 28.0495
20 28.0704 28.0648
21 28.0831 28.0778
22 28.0941 28.0892
23 28.1037 28.0990
24 28.1121 28.1077
25 28.1194 28.1153
26 28.1259 28.1221
27 28.1317 28.1281
28 28.1369 28.1334
29 28.1415 28.1383
30 28.1457 28.1426
31 28.1495 28.1465
32 28.1529 28.1501
33 28.156 28.1533
34 28.1588 28.1563
35 28.1614 28.1590
36 28.1638 28.1615
37 28.1659 28.1638
38 28.1679 28.1659
39 28.1698 28.1678
40 28.1715 28.1696
41 28.1731
42 28.1746
43 28.1759
44 28.1772
45 28.1784
∞ 28.204 28.204

Table 9.Quantum defect (δ) and effective nuclear charge Z* of the 4 s 2 4 p 2 ( 2 P 1 / 2 o ) n d ( 1 P 1 o ) Rydberg series of Rb+ converging to the ( 3 d 10 4 s 2 4 p 5 2 P ∘ 1 / 2 ) serie limits in Rb2+. The present Modified Atomic Orbital Theory (MOAT) calculations are compared to the SCUNC calculations of Sakho et al. [10] the synchrotron radiation (SR) and dual laser plasma (DPL) measurements of Kilbane et al. [17] along with their theoretical results from Hartree-Fock with exchange plus relativistic corrections (HXR) [17] and with the Dirac R-matrix calculations of McLaughlin and Babb [18].

Quantum Defects Effective charge
n MOAT SCUNC R-matrix HXR SR-DLP Z *
8 0.241 0.2411 0.2376 0.1977 0.1982 2.0621
9 0.2077 0.2391 0.3265 0.2077 0.3922 2.0472
10 0.1736 0.2375 0.3257 0.2628 0.5853 2.0353
11 0.1389 0.2362 0.3253 0.1699 0.7952 2.0256
12 0.1039 0.2351 0.3246 0.0951 1.0283 2.0175
13 0.0685 0.2342 0.3243 -0.1651 1.2887 2.0106
14 0.0329 0.2334 0.3241 -0.6377 1.5846 2.0047
15 −0.0029 0.2327 0.3239 1.9996
16 −0.0389 0.2321 0.3239 1.9951
17 −0.0751 0.2316 1.9912
18 −0.1113 0.2312 1.9877
19 −0.1477 0.2308 1.9846
20 −0.1841 0.2304 1.9818
21 −0.2206 0.2301 1.9792
22 −0.2572 0.2298 1.9769
23 −0.2938 0.2295 1.9748
24 −0.3305 0.2293 1.9728
25 −0.3673 0.2290 1.971
26 −0.404 0.2288 1.9694
27 −0.4408 0.2286 1.9679
28 −0.4777 0.2285 1.9665
29 −0.5145 0.2283 1.9651
30 −0.5514 0.2281 1.9639
31 −0.5883 0.2280 1.9628
32 −0.6252 0.2279 1.9617
33 −0.6622 0.2277 1.9607
34 −0.6991 0.2276 1.9597
35 −0.7361 0.2275 1.9588
36 −0.7731 0.2274 1.958
37 −0.8101 0.2273 1.9571
38 −0.8471 0.2272 1.9564
39 −0.8842 0.2271 1.9557
40 −0.9212 0.2271 1.955
41 −0.9583 1.9543
42 −0.9953 1.9537
43 −1.0324 1.9531
44 −1.0695 1.9525
45 −1.1066 1.952

Results presented in this table are calculated via Equation (9). 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 = 23, 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]. The comparison shows very good agreement. |ΔE| energy differences from experimental data are less than 0.005 eV. For n = 24 up to n = 40, our results are one only compared with SCUNC theorical calculation of Sakho [11] and comparison is considered very satisfactory. Thus, our value of 19.9391 eV for n = 40 is in good agreement with Sakho’s calculated result of 19.9391 eV. This allows us to expect the present results on the resonance energies for this Rydberg series up to n = 55 to be accurate.

In Table 3 we present the MAOT, resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge (Z*) of the 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 D 2 ) Rydberg series converging to the ( 2 D 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 P 3 / 2 o ) metastable state. Results presented in this table are calculated via Equation (10). The quantum defect and effective nuclear charge Z* agree well with the MAOT analysis condition of Equation (6). We also note that the quantum defect decreases as the principal quantum number increases. For n = 7 up to n = 23, Our results are compared with of the experimental data of ALS and those of Sakho (2022) [11] obtained theoretically. The comparison shows very good agreement. The absolute deviations |ΔE| between MOAT and ALS [5] are small (less than 0.005 eV). For n = 24 up to 40, our results are one only compared with SCUNC theorical calculation of Sakho [11] and comparison is considered very satisfactory. Thus, our value of 21.1425 eV for n = 40 is in good agreement with Sakho’s calculated result of 21.141 eV. This allows us to expect the present results on the resonance energies for this Rydberg series up to n = 55 to be accurate.

In Table 4 we present the MAOT, resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge (Z*) of the 4 s 2 4 p 2 ( 1 D 2 ) n s Rydberg series converging to the ( 1 D 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 P 1 / 2 o ) metastable state. Results presented in this table are calculated via Equation (11). 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 = 7 up to n = 15, Our results are compared with of the experimental data of ALS and those of Sakho (2022) obtained theoretically. The comparison shows very good agreement. The absolute deviations |ΔE| between MOAT and ALS are small, less than 0.001 eV. Thus, our value of 19.7006 eV for n = 15 is in good agreement with Sakho’s calculated result of 19.697 eV and those of the experimental data of ALS [5] of 19.702 eV. For n = 16 up to 40, our results are one only compared with SCUNC théorical calculation of Sakho [11] and comparison is considered very satisfactory. Thus, our value of 19.9213 eV for n = 40 is in good agreement with Sakho’s calculated result of 19.920 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 5resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge (Z*) of the 4 s 2 4 p 2 ( 1 D 2 ) n s Rydberg series converging to the ( 1 D 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 P 3 / 2 o ) metastable state. Results presented in this table are calculated via Equation (12). 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 = 23, 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. The comparison shows very good agreement. |ΔE| energy differences from experimental data are less than 0.001 eV. Thus, our value of 19.6943 eV for n = 15 is in good agreement with Sakho’s calculated result of 19.694 eV and those of the experimental data of ALS [5] of 19.694 eV. For n = 20 up to n = 40, our results are one only compared with SCUNC theorical calculation of Sakho and comparison is considered very satisfactory. Thus, our value of 19.8186 eV for n = 40 is in good agreement with Sakho’s calculated result of 19.818 eV. This allows us to expect the present results on the resonance energies for this Rydberg series up to n = 55 to be accurate.

In Table 6 we present the MAOT, resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge (Z*) of the 4 s 2 4 p 2 ( 1 D 2 ) n d ( 2 P 1 / 2 o ) Rydberg series converging to the ( 1 D 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 P 1 / 2 o ) metastable state. Results presented in this table are calculated via Equation (13). 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 = 6 up to n = 15, Our results are compared with of the experimental data of ALS and those of Sakho (2022) [11] obtained theoretically. The comparison shows very good agreement. |ΔE| energy differences from experimental data are less than 0.008 eV. Thus, our value of 19.710 eV for n = 15 is in good agreement with Sakho’s calculated result of 19.708 eV and those of the experimental data of ALS [5] of 19.718 eV. For n = 16 up to n = 40, our results are one only compared with SCUNC theorical calculation of Sakho and comparison is considered very satisfactory. Thus, our value of 19.9213 eV for n = 40 is in good agreement with Sakho’s calculated result of 19.921 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 7resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge (Z*) of the 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 S ) Rydberg series converging to the ( 2 D 2 ) serie limits in Se2+ threshold origin Se+ 4 s 2 4 p 3 ( 2 P 1 / 2 o ) metastable state. Results presented in this table are calculated via Equation (14). 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. Forn = 6 up to n = 16, 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]. The comparison shows very good agreement. |ΔE| energy differences from experimental data are less than 0.008 eV. Thus, our value of 19.7425 eV for n = 16 is in good agreement with Sakho’s calculated result of 19.740 eV and those of the experimental data of ALS of 19.749 eV. For n = 17 up to n = 40, our results are one only compared with SCUNC theorical calculation of Sakho and comparison is considered very satisfactory. Thus, our value of 19.9214 eV for n = 40 is in good agreement with Sakho’s calculated result of 19.921 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 8 andTable 9 resonance Energy (En, eV), quantum defect (δ) and effective nuclear charge (Z*) of the 4 s 2 4 p 2 ( 2 P 1 / 2 o ) n d ( 1 P 1 o ) Rydberg series of Rb+ converging to the ( 3 d 10 4 s 2 4 p 5 2 P ∘ 1 / 2 ) serie limits in Rb2+. Results presented in this table are calculated via Equation (15). 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 = 8 up to n = 14, the present Modified Atomic Orbital Theory (MOAT) calculations are compared to the SCUNC calculations of Sakho et al. [10], the synchrotron radiation (SR) and dual laser plasma (DPL) measurements of Kilbane et al. [17] along with their theoretical results from Hartree-Fock with exchange plus relativistic corrections (HXR) [17] and with the Dirac R-matrix calculations of McLaughlin and Babb [18]. The comparison shows very good agreement. |ΔE| energy differences from experimental data are less than 0.01 eV. For n = 15 up to n = 40, our results are one only compared with SCUNC theorical calculation of Sakho and comparison is considered very satisfactory. This allows us to expect the present results on the resonance energies for this Rydberg series up to n = 45 to be accurate.

These strong agreements are justified by the fact that, within the MAOT formalism, σi all relativistic and electron-electron correlation effects are implicitly accounted for through the adjustment parameters which are evaluated using experimental data. For all investigated Rydberg series, the slight discrepancies between the present calculations and experimental values may be explained by the inherent simplicity of the MAOT formalism, which does not explicitly incorporate relativistic corrections. The reliability of this extrapolation procedure relies on the asymptotic convergence of the quantum defect, an approach widely validated for the prediction of highly excited Rydberg states [15][16][19].

 4. Conclusion

In this paper, energy resonances, quantum defects and of the: effective nuclear charge of the 4 s 2 4 p 2 ( 3 P 2 ) n d , 4 s 2 4 p 2 ( 1 D 2 ) n s , 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 D 2 ) , 4 s 2 4 p 2 ( 1 D 2 ) n d ( 2 P 1 / 2 o ) and 4 s 2 4 p 2 ( 2 D 2 ) n s ( 2 S ) Rydberg series originating from the 4 s 2 4 p 3 ( 2 P 1 2 , 3 2 o ) and 4 s 2 4 p 3 ( 2 D 3 2 , 5 2 o ) states of Se+ and 4 s 2 4 p 2 ( 2 P 1 / 2 o ) n d ( 1 P 1 o ) Rydberg series of Rb+ converging to the ( 3 d 10 4 s 2 4 p 5 2 P 1 / 2 o ) serie limits in Rb2+. Calculations are performed using the Modified Orbital Atomic Atomic Theory (MOAT) semi-empirical procedure for high

lying states. Very good agreements with available experimental and theoretical literature data are found. Current calculations provide reference data for the diagnosis and modelling of astrophysical and laboratory plasmas for understanding the chemical evolution of the Se element. The simplicity of the presented procedure, allows to obtain very accurate values of the resonance energies up to highly excited Rydberg states (n = 55).

 Acknowledgements

The authors are grateful to the Orsay Institute of Molecular Sciences (OIMS), Paris, France and the Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy.

 Credit Author Statement

Abdou Faye: Conceptualization; Methodology, Software, Formal analysis, validation, Data curation, Writing-Original draft preparation; Writing-Reviewing and Editing, Validation.

Malick Sow: Conceptualization; Methodology, Formal analysis, validation, Data curation, Writing-Original draft preparation; Writing-Reviewing and Editing, Validation.

Moustapha Kébé: Reviewing; Formal analysis, Validation.

Papa Mamadou Ndiaye; Reviewing; Formal analysis, Validation.

Omar Baba Dia: Reviewing; Formal analysis, Validation.

Cheikh Ndiaye; Reviewing; Formal analysis, Validation.

Cheikh Tidiane Diouf: Reviewing; Formal analysis, Validation.

Oumar Absatou Niass: Reviewing; Formal analysis, Validation.

 References Sterling, N.C. and Dinerstein, H.L. (2008) The Abundances of Light Neutron-Capture Elements in Planetary Nebulae. II. S-Process Enrichments and Interpretation. The Astrophysical Journal Supplement Series, 174, 158-201. https://doi.org/10.1086/520845 10.1086/520845https://doi.org/10.1086/520845 Sterling, N.C. Dinerstein, H.L. 2008 The Abundances of Light Neutron-Capture Elements in Planetary Nebulae II. S-Process Enrichments and Interpretation. The Astrophysical Journal Supplement Series 174 10.1086/520845 Karakas, A.I., van Raai, M.A., Lugaro, M., Sterling, N.C. and Dinerstein, H.L. (2008) Nucleosynthesis Predictions for Intermediate-Mass Asymptotic Giant Branch Stars: Comparison to Observations of Type I Planetary Nebulae. The Astrophysical Journal, 690, 1130-1144. https://doi.org/10.1088/0004-637x/690/2/1130 10.1088/0004-637x/690/2/1130https://doi.org/10.1088/0004-637x/690/2/1130 Karakas, A.I. Raai, M.A. Lugaro, M. Sterling, N.C. Dinerstein, H.L. 2008 Nucleosynthesis Predictions for Intermediate-Mass Asymptotic Giant Branch Stars: Comparison to Observations of Type I Planetary Nebulae The Astrophysical Journal 690 10.1088/0004-637x/690/2/1130 Langanke, K. and Wiescher, M. (2001) Nuclear Reactions and Stellar Processes. Reports on Progress in Physics, 64, 1657-1701. https://doi.org/10.1088/0034-4885/64/12/202 10.1088/0034-4885/64/12/202https://doi.org/10.1088/0034-4885/64/12/202 Langanke, K. Wiescher, M. 2001 Nuclear Reactions and Stellar Processes Reports on Progress in Physics 64 10.1088/0034-4885/64/12/202 Macaluso, D.A., Bogolub, K., Johnson, A., et al. (2017) Absolute Single Photoionization Cross-Section Measurements of Rb 2+ Ions: Experiment and Theory. JournalofPhysicsB: Atomic, MolecularandOpticalPhysics, 50, Article 119501. https://doi.org/10.1088/1361-6455/aa6d1b 10.1088/1361-6455/aa6d1bhttps://doi.org/10.1088/1361-6455/aa6d1b Macaluso, D.A. Bogolub, K. Johnson, A. Atomic, M 2017 Absolute Single Photoionization Cross-Section Measurements of Rb2+ Ions: Experiment and Theory Journal of Physics B: Atomic 50 119501 10.1088/1361-6455/aa6d1b Esteves, D.A., Bilodeau, R.C., Sterling, N.C., Phaneuf, R.A., et al. (2011) Absolute High-Resolution Se + Photoionization Cross-Section Measurements with Rydberg-Series Analysis. PhysicalReviewA, 84, Article 013406. https://doi.org/10.1103/physreva.84.013406 10.1103/physreva.84.013406https://doi.org/10.1103/physreva.84.013406 Esteves, D.A. Bilodeau, R.C. Sterling, N.C. Phaneuf, R.A. 2011 Absolute High-Resolution Se+ Photoionization Cross-Section Measurements with Rydberg-Series Analysis Physical Review A 84 013406 10.1103/physreva.84.013406 Esteves, D.A., Aguilar, A., Bilodeau, R.C., Phaneuf, R.A., Kilcoyne, A.L.D., Red, E.C., et al. (2012) Absolute Single Photoionization Cross-Section Measurements of Se 3+and Se 5+. JournalofPhysicsB: Atomic, MolecularandOpticalPhysics, 45, Article 115201. https://doi.org/10.1088/0953-4075/45/11/115201 10.1088/0953-4075/45/11/115201https://doi.org/10.1088/0953-4075/45/11/115201 Esteves, D.A. Aguilar, A. Bilodeau, R.C. Phaneuf, R.A. Kilcoyne, A.L.D. Red, E.C. Atomic, M 2012 Absolute Single Photoionization Cross-Section Measurements of Se3+and Se5+ Journal of Physics B: Atomic 45 115201 10.1088/0953-4075/45/11/115201 Macaluso, D.A., Aguilar, A., Kilcoyne, A.L.D., et al. (2015) Absolute Single-Photoionization Cross-Sections of Se 2+: Experiment and Theory. PhysicalReviewA, 92, Article 063424. https://doi.org/10.1103/physreva.92.063424 10.1103/physreva.92.063424https://doi.org/10.1103/physreva.92.063424 Macaluso, D.A. Aguilar, A. Kilcoyne, A.L.D. 2015 Absolute Single-Photoionization Cross-Sections of Se2+: Experiment and Theory Physical Review A 92 063424 10.1103/physreva.92.063424 McLaughlin, B.M. and Ballance, C.P. (2012) Photoionization Cross Sections for the Trans-Iron Element Se + from 18 to 31 eV. JournalofPhysicsB: Atomic, MolecularandOpticalPhysics, 45, Article 095202. https://doi.org/10.1088/0953-4075/45/9/095202 10.1088/0953-4075/45/9/095202https://doi.org/10.1088/0953-4075/45/9/095202 McLaughlin, B.M. Ballance, C.P. Atomic, M 2012 Photoionization Cross Sections for the Trans-Iron Element Se+ from 18 to 31 eV Journal of Physics B: Atomic 45 095202 10.1088/0953-4075/45/9/095202 Sakho, I. and Gning, M.T. (2023) Photoionization Studies of Se 2+ and S 4+ Ions Using the Screening Constant Per Unit Nuclear Charge Method. Sakho, I. Gning, M.T. 2023 Photoionization Studies of Se2+ and S4+ Ions Using the Screening Constant Per Unit Nuclear Charge Method Sakho, I. (2023) High Lying Precise Resonance Energies from Photoionization Studies of Se 3+ and Rb + Ions Using the Screening Constant per Unit Nuclear Charge Formalism. Atoms, 11, Article 26. https://doi.org/10.3390/atoms11020026 10.3390/atoms11020026https://doi.org/10.3390/atoms11020026 Sakho, I. 2023 High Lying Precise Resonance Energies from Photoionization Studies of Se3+ and Rb+ Ions Using the Screening Constant per Unit Nuclear Charge Formalism Atoms 11 26 10.3390/atoms11020026 Sakho, I. (2022) Screening Constant per Unit Nuclear Charge Studies of Rydberg Resonances Due to the 4p→Nd and Ns Transitions in Se + Ions. Atoms, 10, Article 107. https://doi.org/10.3390/atoms10040107 10.3390/atoms10040107https://doi.org/10.3390/atoms10040107 Sakho, I. 2022 Screening Constant per Unit Nuclear Charge Studies of Rydberg Resonances Due to the 4p→Nd and Ns Transitions in Se+ Ions Atoms 10 107 10.3390/atoms10040107 Macaluso, D.A., Bogolub, K., Johnson, A., Aguilar, A., Kilcoyne, A.L.D., Bilodeau, R.C., et al. (2016) Absolute Single Photoionization Cross-Section Measurements of Rb 2+ Ions: Experiment and Theory. Journal of Physics B: Atomic, Molecular and Optical Physics, 49, Article 235002. https://doi.org/10.1088/0953-4075/49/23/235002 10.1088/0953-4075/49/23/235002https://doi.org/10.1088/0953-4075/49/23/235002 Macaluso, D.A. Bogolub, K. Johnson, A. Aguilar, A. Kilcoyne, A.L.D. Bilodeau, R.C. Atomic, M 2016 Absolute Single Photoionization Cross-Section Measurements of Rb2+ Ions: Experiment and Theory Journal of Physics B: Atomic 49 235002 10.1088/0953-4075/49/23/235002 Sakho, I. (2013) Photoabsorption of H-and He via the Modified Atomic Orbital Theory: Application to the 1 P-Rydberg State. Chinese Journal of Physics, 51, Article 209. Sakho, I. 2013 Photoabsorption of H-and He via the Modified Atomic Orbital Theory: Application to the 1 P-Rydberg State Chinese Journal of Physics 51 209 Sow, M., Sakho, I., Sow, B., Diouf, A., Gning, Y., et al. (2020) Modified Atomic Orbital Calculations of Energy of the (2 s21S) Ground-State, the (2 p21D); (3 d21G) and (4 f21I) Doubly Excited States of Helium Isoelectronic Sequence from H − to Ca 18+. Journal of Applied Mathematics and Physics, 8, 85-99. https://doi.org/10.4236/jamp.2020.81007 10.4236/jamp.2020.81007https://doi.org/10.4236/jamp.2020.81007 Sow, M. Sakho, I. Sow, B. Diouf, A. Gning, Y. 2020 Modified Atomic Orbital Calculations of Energy of the (2s2 1S) Ground-State, the (2p2 1D); (3d2 1G) and (4f2 1I) Doubly Excited States of Helium Isoelectronic Sequence from H− to Ca18+ Journal of Applied Mathematics and Physics 8 10.4236/jamp.2020.81007 Faye, A., Sow, M., Kebe, M., Astou, T.N., et al. (2024) Étude de Photoionisation des Singlets [1s 22p(2p3/2) Ns 1p, 1s22p (2p) Nd 1p, 1s22p (2p1/2) Np 1p, 1s22p (2p3/2) Np 1d] et des Triplets [1s22p (2po) Ns 3p0, 1, 2] Série de Rydberg Pour Les Ions Niv via La Théorie Des Orbitales Atomiques Modifiée. International Journal, 12, 1013-1023. Faye, A. Sow, M. Kebe, M. Astou, T.N. 2024 Étude de Photoionisation des Singlets [1s22p(2p3/2) Ns 1p, 1s22p (2p) Nd 1p, 1s22p (2p1/2) Np 1p, 1s22p (2p3/2) Np 1d] et des Triplets [1s22p (2po) Ns 3p0, 1, 2] Série de Rydberg Pour Les Ions Niv via La Théorie Des Orbitales Atomiques Modifiée International Journal 12 Sow, M., Ndoye, F., Traoré, A., Diouf, A., Sow, B., Gning, Y. and Diagne, P.A.L. (2021) Photoionization Study of the 2s 22p 2( 1D)ns( 2D), 2s 22p 2( 1D)nd( 2P), 2s 22p 2( 1D)nd( 2S), 2s 22p 2( 1S)nd 2D and 2s 22p 3( 3P)np( 2D) Rydberg Series of O + Ions via the Modified Atomic Orbital Theory. Journal of Modern Physics, 12, 1375-1386. https://doi.org/10.4236/jmp.2021.1210086 10.4236/jmp.2021.1210086https://doi.org/10.4236/jmp.2021.1210086 Sow, M. Ndoye, F. Diouf, A. Sow, B. Gning, Y. Diagne, P.A.L. 2021 Photoionization Study of the 2s22p2(1D)ns(2D), 2s22p2(1D)nd(2P), 2s22p2(1D)nd(2S), 2s22p2(1S)nd 2D and 2s22p3(3P)np(2D) Rydberg Series of O+ Ions via the Modified Atomic Orbital Theory Journal of Modern Physics 12 10.4236/jmp.2021.1210086 Kilbane, D., Folkmann, F., Bizau, J.M., Banahan, C., Scully, S., Kjeldsen, H., et al. (2007) Absolute Photoionization Cross-Section Measurements of the Kr I Isoelectronic Sequence. Physical Review A, 75, Article 032711. https://doi.org/10.1103/physreva.75.032711 10.1103/physreva.75.032711https://doi.org/10.1103/physreva.75.032711 Kilbane, D. Folkmann, F. Bizau, J.M. Banahan, C. Scully, S. Kjeldsen, H. 2007 Absolute Photoionization Cross-Section Measurements of the Kr I Isoelectronic Sequence Physical Review A 75 032711 10.1103/physreva.75.032711 McLaughlin, B.M. and Babb, J.F. (2019) Single Photoionization of the Kr-Like Rb II Ion in the Photon Energy Range 22-46.0 eV. Monthly Notices of the RoyalAstronomical Society, 486, 245-250. https://doi.org/10.1093/mnras/stz790 10.1093/mnras/stz790https://doi.org/10.1093/mnras/stz790 McLaughlin, B.M. Babb, J.F. 2019 Single Photoionization of the Kr-Like Rb II Ion in the Photon Energy Range 22-46 0 eV. Monthly Notices of the Royal Astronomical Society 486 10.1093/mnras/stz790 Dia, O.B., Sow, M., Kebe, M., Ndiaye, P.M., et al. (2025) Photoionization Study of the 4s 24p 4( 3P1)nd, 4s 24p 4( 1D2)nd and 4s 24p 4( 1S0)nd Rydberg Series of Rb 2+ via the Modified Atomic Orbital Theory. Journal of Applied Mathematics and Physics, 13, 2775-2790. https://doi.org/10.4236/jamp.2025.138158 10.4236/jamp.2025.138158https://doi.org/10.4236/jamp.2025.138158 Dia, O.B. Sow, M. Kebe, M. Ndiaye, P.M. 2025 Photoionization Study of the 4s24p4(3P1)nd, 4s24p4(1D2)nd and 4s24p4(1S0)nd Rydberg Series of Rb2+ via the Modified Atomic Orbital Theory Journal of Applied Mathematics and Physics 13 10.4236/jamp.2025.138158 Moore, C.E. (2022) Atomic Energy Levels. https://nvlpubs.nist.gov/nistpubs/Legacy/circ/nbscircular467v2.pdf Moore, C.E. 2022 Atomic Energy Levels Cowan, R.D. (2022) The Theory of Atomic Structure and Spectra. https://www.ucpress.edu/book/9780520038219/the-theory-of-atomic-structure-and-spectra Cowan, R.D. 2022 The Theory of Atomic Structure and Spectra