1. Introduction

JAMP

Journal of Applied Mathematics and Physics

2327-4352

Scientific Research Publishing

10.4236/jamp.2022.106128

JAMP-117688

Articles

Physics&Mathematics

Modified Atomic Orbital Theory of the O+ Ion Originating from 2D0 and 4S0 Metastable States

Malick

Sow

¹^*Fatou

Ndoye

¹Babou

Diop

¹Alassane

Traoré

¹Abdou

Diouf

¹Boubacar

Sow

Department of Physics, Atoms Lasers Laboratory, Faculty of Sciences and Technologies, University Cheikh Anta Diop, Dakar, Senegal

06062022

10061873188612, April 20224, June 2022 7, June 2022

2014

This work is licensed under the Creative Commons Attribution-NonCommercial International License (CC BY-NC).http://creativecommons.org/licenses/by-nc/4.0/

We report in this paper energy positions of the 2D⁰_2s²2p²( ¹D) nd( ² F); 2D⁰_2s²2p²( ¹D) nd( ² D); 2D⁰_2s²2p²( ¹D) nd( ² P); 2D⁰_2s²2p²( ¹D) ns( ² D); 2D⁰_2s²2p³( ³D) np( ² P); 2D⁰_2s²2p³( ³D⁰) np( ² F), and 4S⁰_2s²2p³( ⁵S⁰) np( ⁴ P) Rydberg series in the photoionization spectra originating from 2D ⁰ and 4S ⁰ metastable states of O ⁺ ion. Calculations are performed up to n = 20 using the Modified Orbital Atomic Theory (MAOT) [1]. The present results are compared to the experimental data of Aguilar et al. [2] which are the only available values. The accurate data presented in this work may be a useful guideline for future experimental and other theoretical studies.

Semiempirical Calculations Modified Orbital Atomic Theory Electron Correlation Calculations Atoms and Ions Rydberg Series Quantum Defect

1. Introduction

Ion escape is of particular interest for studying the evolution of the atmosphere on geological time scales. From this study, we suggested that O⁺ ions observed in the plasma mantle and cusp have enough energy and velocity to escape the magnetosphere and are lost into the solar wind or in the distant magnetotail. Thus, this study aims to investigate Rydberg series of O⁺ ions for which photoabsorption from low-lying metastable states of open-shell ions has been shown to be important in laboratory and astrophysical plasmas [3] [4] [5] . Quantitative and qualitative measurements of photoionization of ions provide precision data on ionic structure, and guidance to the development of theoretical approaches of multielectron interactions. Thus, Aguilar et al. [2] performed the first experiment on the Absolute photoionization of O⁺ by using the merged-beam technique above the first ionization threshold at the Advanced Light Source (ALS).

For comparison with high-resolution measurements, state-of the-art-theoretical methods are required using highly correlated wave functions including relativistic effects [6] . Among the ab initio methods applied in the photoionization studies of atoms and ions are the Hartree-Fock multi-configurationnal (MCDF) method [7] , the Quantum Defect Theory [8] , the R-matrix approach [9] widely used for international collaborations such as the Opacity Project [4] , the Multi-Configuration Dirac-Hartree-Fock method (MCDHF) [10] , in the form of the grasp2k relativistic atomic structure package [11] . The MCDHF has been used to compute with high precision the 2³P_l–2³P₀ separation energy, including relativistic contributions to electron-electron correlations and radiative corrections.

In the present paper, we intend to provide accurate data on the photoionization of O⁺ ion from 2D⁰ and 4S⁰ metastable states that may be useful guideline for the physical atomic community. In addition, we aim to demonstrate the possibilities to use the Modified Atomic Orbital Theory semi-empirical procedure [1] [12] [13] to reproduce excellently experimental data from merged beam facilities. For this purpose, we report calculations of energy resonances of the 2D⁰_2s²2p²(¹D)nd(²F); 2D⁰_2s²2p²(¹D)nd(²D); 2D⁰_2s²2p²(¹D)nd(²P); 2D⁰_2s²2p²(¹D)ns(²D); 2D⁰_2s²2p³(³D)np(²P); 2D⁰_2s²2p³(³D⁰)np(²F), and 4S⁰_2s²2p³(⁵S⁰)np(⁴P) Rydberg series in the photoionization spectra originating from 2D⁰ and 4S⁰ metastable states of O⁺ ion. Calculations are performed using the MAOT method [14] [15] and analysis of the data tabulated is achieved via the MAOT procedure along with the quantum defect theory.

Section 2 presents the theoretical procedure adopted in this work with a brief description of the MAOT formalism and the analytical expressions used in the calculations. In Section 3, we present and discuss the results obtained along with comparison with the only available experimental data of Aguilar et al. [2] . In Section 4, we summarize our study and draw conclusions.

2. Theory2.1. Brief Description of the MAOT Formalism

In the framework of Modified Atomic Orbital Theory (MAOT), total energy of (νℓ)-given orbital is expressed in the form [13] [16] .

E ( υ l ) = − [ Z − σ ( l ) ] 2 υ 2 (1)

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

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

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

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

In the photoionisation study, energy resonances are generally measured relatively to the E∞ converging limit of a given (^2S⁺¹L_J) nl-Rydberg series. For these states, the general expression of the energy resonances is given by the formula of Sakho presented previously [17] (in Rydberg units):

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 μ ( 2 S + 1 L J ) × ( n − m ) × ( n − q ) ∑ k 1 f k ( n , m , q , s ) } 2 (3)

In this equation m and q (m < q) denote the principal quantum numbers of the (^2S⁺¹L_J) nl-Rydberg series of the considered atomic system used in the empirical determination of the σ_i(^2S⁺¹L_J)-screening constants, s represents the spin of the nl-electron (s = 1/2), E_∞ is the energy value of the series limit generally determined from the NIST atomic database, E_n denotes the corresponding energy resonance, and Z represents the nuclear charge of the considered element. The only problem that one may 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 term

is determined iteratively by imposing general Equation (3) to give accurate data with a constant quantum defect (δ) values along all the considered series. The value of μ is fixed to 1 and 2 during the iteration. The quantum defect (δ) is calculated from the standard formula below

E n = E ∞ − R Z c o r e 2 ( n − δ ) 2 ⇒ δ = n − Z c o r e R E ∞ − E n (4)

In this equation, R is the Rydberg constant, E_∞ denotes the converging limit, Z_core represents the electric charge of the core ion, and (δ) means the quantum defect.

Z_core is directly obtained by the photoionization process from an atomic X^p⁺ system:

X p + + h ν → X ( p + 1 ) + + e − . We find then Z_core = p + 1.

In addition, 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 (5)

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 (6)

2.2. Energy Positions of the 2D0_2s22p2(1D)nd(2F); 2D0_2s22p2(1D)nd(2D); 2D0_2s22p2(1D)nd(2P); 2D0_2s22p2(1D)ns(2D); 2D0_2s22p3(3D)np(2P); 2D0_2s22p3(3D0)np(2F), Rydberg Series from 2D0 Metastable State

Using Equation (3) we find

● For 2D⁰_2s²2p²(¹D)nd(²F) levels

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

Using the experimental data of ALS [9] , we obtain (in eV) E₅ = 32.044 ± 0.20 (m = 5) and E₆ = 32.753 ± 0.20 (q = 6) respectively for the 2D⁰_2s²2p²(¹D)5d(²F) and 2D⁰_2s²2p²(¹D)6d(²F) levels. From NIST [18] , we find E_∞ = 34.311 eV. Using these data, Equation (7) gives σ₁ = 6.022617664 ± 0.333296261 and σ₂ = −0.317908474 ± 0.012159751

● For 2D⁰_2s²2p²(¹D)nd(²D) levels:

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n + σ 2 × ( n − m ) × ( n − q ) × [ 1 ( n + m − s ) ( n + q − s ) 2 + 1 ( n − s ) 3 ] } 2 (8)

For the 2D⁰_2s²2p²(¹D)5d(²D) and 2D⁰_2s²2p²(¹D)6d(²D) levels, we find using the experimental data of ALS et al. [2] , E₅ = 32.083 ± 0.20 (m = 5) and E₆ = 32. 775 ± 0.20 (q = 6). From NIST [18] , we find E_∞ = 34.311 eV. Equation (8) provides then σ₁ = 6.020774511 ± 0.335007227 and σ₂ = −0.220533344 ± 0.012205725

● For 2D⁰_2s²2p²(¹D)nd(²P) levels

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n + σ 2 × ( n − m ) × ( n − q ) × [ 1 ( n + q − s ) 3 + 1 ( n + q + s − m ) 4 + + 1 ( n + q − m + 3 s ) 5 ] } 2 (9)

For the 2D⁰_2s²2p²(¹D)5d(²P) and 2D⁰_2s²2p²(¹D)6d(²P) levels the experimental energy positions ALS et al., [2] are, E₅ = 32.107 ± 0.20 (m = 5) and E₆ = 32.786 ± 0.20 (q = 6). From NIST [18] , we find E_∞ = 34.311 eV. In that case, we find using Equation (9) σ₁ = 6.009528595 ± 0.472635519 and σ₂ = −0.109667938 ± 0.020784332

● For 2D⁰_2s²2p²(¹D)ns(²D) levels

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n + σ 2 × ( n − m ) × ( n − q ) × [ 1 ( n + q − m + s ) + 1 ( n − s ) 2 ( n + 2 m − q ) 3 ] } 2 (10)

From ALS of Aguilar et al. [2] , we obtain for the 2D⁰_2s²2p²(¹D)6s(²D) and 2D⁰_2s²2p²(¹D)7s(²D) E₆= 32.267 ± 0.20 (m = 6) and E₇ = 32. 881 ± 0.20 (q = 7). From NIST [18] , we find E_∞ = 34.311 eV. We find then using Equation (10) σ₁ = 6.067877676 ± 0.430724379 and σ₂ = −2.360746016 ± 0.19010132

● For 2D⁰_2s²2p³(³D)np(²P) levels

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n + σ 2 × ( n − m ) × ( n − q ) × [ 1 ( n + q − s ) 2 ( n + s − m ) 2 + 1 ( n + s − m ) 3 + 1 ( n + q + s − m ) 4 ] } 2 (11)

From ALS et al. [2] , we obtain for the 2D⁰_2s²2p³(³D)3p(²P) and 2D⁰_2s²2p³(³D)4p(²P) E₃= 37.604 ± 0.20 (m = 3) and E₄ = 42.149 ± 0.20 (q = 4). From NIST [18] , we find E_∞ = 46.707 eV. We find then using Equation (11) σ₁= 6.100874328 ± 0.092439223 and σ₂ = −1.664259453 ± 0.207487744

● For 2D⁰_2s²2p³(³D⁰)np(²F) levels

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n + σ 2 × ( n − m ) × ( n − q ) × [ 1 ( n + q − s ) 2 + 1 ( n + s − m ) 2 − 1 ( n + s − m ) 3 ] } 2 (12)

From ALS et al. [2] , we obtain for the 2D⁰_2s²2p³(³D⁰)3p(²F) and 2D⁰_2s²2p³(³D⁰)4p(²F) E₃= 38.686 ± 0.20 (m = 3) and E₄ = 42.409 ± 0.20 (q = 4). From NIST [18] , we find E_∞ = 46.620 eV. We find then using Equation (12) σ₁ = 5.971438364 ± 0.124812413 and σ₂ = −0.787027868 ± 0.028785514

2.3. Energy Positions of the 4S0_2s22p3(5S0)np(4P) Rydberg Series from 4S0 Metastable State

Using Equation (3), energy positions of the 4S⁰_2s²2p³(⁵S⁰)np(⁴P) prominent Rydberg series from 4S⁰ metastable state of O⁺ are given by (in Rydberg units)

E n = E ∞ − 1 n 2 { Z − σ 1 − σ 2 n + σ 2 × ( n − m ) × ( n − q ) × [ 1 ( n + s − m ) ( n + q − m ) + 1 ( n + + q − s ) 3 ] } 2 (13)

From ALS et al. [2] , we obtain for the 4S⁰_2s²2p³(⁵S⁰)4p(⁴P) and 4S⁰_2s²2p³(⁵S⁰)5p(⁴P) E₄ = 38.216 ± 0.20 (m = 4) and E₅ = 39.919 ± 0.20 (q = 5). From NIST [18] , we find E_∞ = 42.586 eV. We find then using Equation (13) σ₁ = 5.99920618 ± 0.207827476 and σ₂ = −1.0645904 ± 0.429419911.

3. Results and Discussions

The results obtained in the present paper are listed in Tables 1-7 and compared with the Advaced Light Source experimental data of Aguilar et al. [2] .

In Table 1, we quote the present MAOT results for energy resonances (E) and quantum defect (δ) of the 2D⁰2s²2p²(¹D)nd(²F) Rydberg series relatively to the 2D⁰_metastable state of O⁺ ion. The current energy positions are calculated from Equations (5) along with Z = 8, m = 5, and q = 6, σ₁ = 6.022617664 ± 0.333296261 and σ₂ = −0.317908474 ± 0.012159751. All these screening constant are evaluated using the Advaced Light Source (ALS) experimental results of Aguilar et al. [2] , and take from NIST [18] the E_∞ energy limits which is 34.311 eV. Then our results are converted into eV for direct comparison by using the infinite Rydberg

Table 1 Energy resonances (E) and quantum defect (δ) of the 2D02s22p2(1D)nd(2F) Rydberg series observed in the photoionization spectra originating from the 2D0 metastable states of O+. The present results (MAOT) are compared to the Advanced Light Source (ALS) of Aguilar et al. [<xref ref-type="bibr" rid="scirp.117688-ref2">2</xref>] . The results are expressed in eV. The energy uncertainties in the present calculations are indicated into parenthesis

	E (eV)	δ
n	MAOT	ALS	\|ΔE\|	MAOT	ALS
5	32.044(200)	32.044	0.000	0.097	0.097
6	32.753(200)	32.753	0.000	0.084	0.084
7	33.175(150)	33.179	0.005	0.060	0.057
8	33.444(120)	33.448	0.004	0.059	0.046
9	33.629(110)	33.632	0.003	0.057	0.028
10	33.760(100)	33.763	0.003	−0.005	−0.008
11	33.857(87)	33.860	0.003	−0.005	−0.020
12	33.930(70)	33.933	0.003	−0.005	−0.045
13	33.989(55)	33.991	0.002	−0.005	−0.100
14	34.034(41)	34..036	0.002	−0.004	−0.142
15	34.070(30)	34.072	0.002	−0.004	−0.181
16	34.100(20)	34.102	0.002	−0.004	−0.249
17 18 19 20 ⁞	34.126(11) 34.148(08) 34.167(05) 34.185(04)	34.127	0.001	−0.004 −0.003 −0.004 −0.004	−0.034
∞^a	34.311

^aNIST atomic database [18] . |ΔE|: energy differences relative to the experimental data.

(1 Ry = 0.5 a.u = 13.605698 eV). It is seen that the data obtained compared very well to the experimental data of Aguilar et al. [2] .

Up to n = 17, the maximum energy differences relative to the experimental data is less than 0.006 eV. In addition, Quantum defects are positive up to n = 9 because the effective charges of these self-ionizing states are less than the charge of the O⁺ ion, and if not, they are negative from n = 10 up to n = 20. We also note that up to n = 9, the quantum defect of these resonant states decreases on average by 0.015 when the principal quantum number increases. For n = 10 up to n = 20, Quantum defects are almost constant. The agreements between the MAOT results and experimental data are seen to be very good and the quantum defect agrees well with the analysis condition of Equation (6). This allows us to expect our results on the resonance energies for this Rydberg series up to n = 20 to be accurate.

In Table 2, we compare the present MAOT energy resonances (E) and quantum defect (δ) of the 2D⁰2s²2p²(¹D)nd(²D) Rydberg series relatively to the 2D⁰_ metastable state of O⁺ ion to experimental data [2] . All our energy values are

Table 2 Energy resonances (E) and quantum defect (δ) of the 2D02s22p2(1D)nd(2D) Rydberg series observed in the photoionization spectra originating from the 2D0 metastable states of O+. The present results (MAOT) are compared to the Advanced Light Source (ALS) of Aguilar et al. [<xref ref-type="bibr" rid="scirp.117688-ref2">2</xref>] . The results are expressed in eV. The energy uncertainties in the present calculations are indicated into parenthesis

	E (eV)	δ
n	MAOT	ALS	\|ΔE\|	MAOT	ALS
5	32.083(200)	32.083	0.000	0.054	0.054
6	32.775(200)	32.775	0.000	0.042	0.042
7	33.195(194)			0.061	0.057
8	33.464(178)			0.048	0.046
9	33.649(161)			0.022	0.028
10	33.780(146)			−0.011	−0.008
11	33.877(130)			−0.012	−0.020
12	33.950(112)			−0.010	−0.045
13	34.009(99)			−0.012	−0.100
14	34.074(83)			−0.014	−0.142
15	34.090(72)			−0.011	−0.181
16	34.108(59)			−0.011	−0.249
17 18 19 20 ⁞	34.141(48) 34.168(44) 34.187(39) 34.200(37)			−0.012 −0.012 −0.012 −0.012	−0.334
∞^a	34.311

^aNIST atomic database [18] . |ΔE|: energy differences relative to the experimental data.

obtained empirically using Equation (8) and converted into (eV) for direct comparison. Here again, the agreements are seen to be very good. Along the series, the quantum defect agrees well with the analysis condition of Equation (6). We also note that, the present quantum defect is almost constant for n ≥ 10.

In Table 3, we show a comparison of the energy resonances (E) and quantum defect (δ) of the 2D⁰2s²2p²(¹D)nd(²P) Rydberg states relatively to the 2D⁰_metastable state of O⁺ ion. The current energy positions are calculated from equations (9) along with Z = 8, m = 5, and q = 6, σ₁ = 6.009528595 ± 0.472635519 and σ₂ = −0.109667938 ± 0.020784332. The agreements between the MOAT results and experimental data are seen to be very good. Along all the series investigated, we note that, the quantum defect agrees well with the MAOT analysis condition of Equation (6). This may expect our results up to n = 20 to be accurate.

In Table 4, we list the present energy resonances (E) and quantum defect (δ) for the 2D⁰2s²2p²(¹D)ns (²D) Rydberg states relatively to the 2D⁰_metastable state of O⁺ ion compared to the experimental data [2] . The current energy positions are calculated from Equations (10) along with Z = 8, m = 6, and q = 7, E_∞ = 34.311

Table 3 Energy resonances (E) and quantum defect (δ) of the 2D02s22p2(1D)nd(2P) Rydberg series observed in the photoionization spectra originating from the 2D0 metastable states of O+. The present results (MAOT) are compared to the Advanced Light Source (ALS) of Aguilar et al. [<xref ref-type="bibr" rid="scirp.117688-ref2">2</xref>] . The results are expressed in eV. The energy uncertainties in the present calculations are indicated into parenthesis

	E (eV)	δ
n	MAOT	ALS	\|ΔE\|	MAOT	ALS
5	32.107(200)	32.107	0.000	0.028	0.028
6	32.786(200)	32.786	0.000	0.021	0.021
7	33.205(180)			0.054	0.057
8	33.475(163)			0.053	0.046
9	33.659(148)			0.050	0.028
10	33.791(133)			−0.020	−0.008
11	33.888(118)			−0.019	−0.020
12	33.960(101)			−0.018	−0.045
13	34.021(84)			−0.018	−0.100
14	34.084(71)			−0.018	−0.142
15	34.100(60)			−0.017	−0.181
16 17 18 19 20 ⁞	34.118(48) 34.152(39) 34.177(33) 34.196(22) 34.208(11)			−0.017 −0.017 −0.017 −0.019 −0.019	−0.249 −0.334
∞^a	34.311

^aNIST atomic database [18] . |ΔE|: energy differences relative to the experimental data.

Table 4 Energy resonances (E) and quantum defect (δ) of the 2D02s22p2(1D)ns (2D) Rydberg series observed in the photoionization spectra originating from the 2D0 metastable states of O+. The present results (MAOT) are compared to the Advanced Light Source (ALS) of Aguilar et al. [<xref ref-type="bibr" rid="scirp.117688-ref2">2</xref>] . The results are expressed in eV. The energy uncertainties in the present calculations are indicated into parenthesis

	E (eV)	δ
n	MAOT	ALS	\|ΔE\|	MAOT	ALS
6	32.267(200)	32.267	0.000	0.836	0.836
7	32.881(200)	32.881	0.000	0.825	0.825
8	33.247(165)	33.251	0.004	0.824	0.825
9	33.496(151)	33.498	0.002	0.824	0.804
10	33.678(130)			0.824
11	33.810(91)			0.824
10	33.908(62)			0.823
11	33.976(43)			0.823
12	34.040(36)			0.823
13	34.099(28)			0.823
14	34.131(21)			0.823
15	34.163(17)			0.824
16	34.175(14)			0.824
17 18 19 20 ⁞	34.182(9) 34.197(7) 34.217(5) 34.231(2)			0.824 0.824 0.824 0.824
∞^a	34.311

^aNIST atomic database [18] . |ΔE|: energy differences relative to the experimental data.

eV; σ₁ = 6.067877676 ± 0.430724379 and σ₂ = −2.360746016 ± 0.19010132. Comparison shows that the maximum energy deviation is at 0.004 up to n =9. This indicates the very good accuracy between the results. For n ≥ 10 it should be underlined that, since the MAOT formalism reproduces excellently the experimental measurements [2] , the present results quoted in Table 4 for the 2D⁰2s²2p²(¹D)ns (²D) levels may be a very good representative of the nonexistent experimental data.

In Table 5, we compare the present MAOT energy resonances (E) and quantum defect (δ) of the 2D⁰2s²2p³(³D)np(²P) Rydberg series relatively to the 2D⁰_ metastable state of O⁺ ion to experimental data [2] . Our current energy positions are calculated from Equations (11) with Z = 8 along with m = 3, and q = 4, E_∞ = 46.707 eV, σ₁ = 6.100874328 ± 0.092439223 and σ₂ = −1.664259453 ± 0.207487744. We note the present quantum defect is almost constant and positive along the series with the effective charge Z^* ≥ Z_core. Thus up to n = 9, the agreements are seen to be very good with the maximum energy difference relative to the experimental data is at 0.004 eV.

Table 5 Energy resonances (E) and quantum defect (δ) of the 2D02s22p3(3D)np(2P) Rydberg series observed in the photoionization spectra originating from the 2D0 metastable states of O+. The present results (MAOT) are compared to the Advanced Light Source (ALS) of Aguilar et al. [<xref ref-type="bibr" rid="scirp.117688-ref2">2</xref>] . The results are expressed in eV. The energy uncertainties in the present calculations are indicated into parenthesis

	E (eV)	δ
n	MAOT	ALS	\|ΔE\|	MAOT	ALS
3	37.604(200)	37.604	0.000	0.551	0.551
4	42.149(200)	42.149	0.000	0.534	0.534
5	43.981(146)	43.977	0.004	0.520	0.511
6	44.886(102)	44.882	0.004	0.520	0.496
7	45.405(73)	45.402	0.003	0.520	0.470
8	45.660(61)	45.658	0.002	0.520	0.696
9	45.944(49)	45.943	0.001	0.519	0.397
10	46.085(42)			0.519
11	46.178(28)			0.519
12	46.250(20)			0.518
13	46.311(17)			0.518
14	46.378(12)			0.518
15	46.422(10)			0.518
16	46.460(9)			0.517
17 18 19 20 ⁞	46.496(8) 46.521(7) 46.544(6) 46.565(5)			0.519 0.519 0.519 0.519
∞^a	46.707

^aNIST atomic database [18] . |ΔE|: energy differences relative to the experimental data.

In Table 6, we compare the present MAOT energy resonances (E) and quantum defect (δ) of the 2D⁰2s²2p³(³D⁰)np(²F) Rydberg series relatively to the 2D⁰_ metastable state of O⁺ ion to experimental data [2] . All our energy values are obtained empirically using Equation (12) and converted into (eV) for direct comparison. In a few series where discrepancies are observed, the maximum energy difference relative to the experimental data is at 0.003 eV. The agreements are seen to be very good and this may expect our results for n > 6 to be accurate.

In Table 7, we quote the present MAOT results for energy resonances (E) and quantum defect (δ) of the 4S⁰2s²2p³(⁵S⁰)np(⁴P) Rydberg series relatively to the 4S⁰_metastable state of O⁺ ion. The current energy positions are calculated from Equations (13) along with Z = 8, m = 4, and q = 5, σ₁ = 5.99920618 ± 0.207827476 and σ₂ = −1.0645904 ± 0.429419911. All these screening constant are evaluated using the Advaced Light Source (ALS) experimental results of Aguilar et al. [2] , and take from NIST [18] the E_∞ energy limits which is 42.586 eV. Then our results

Table 6 Energy resonances (E) and quantum defect (δ) of the 2D02s22p3(3D0)np(2F) Rydberg series observed in the photoionization spectra originating from the 2D0 metastable states of O+. The present results (MAOT) are compared to the Advanced Light Source (ALS) of Aguilar et al. [<xref ref-type="bibr" rid="scirp.117688-ref2">2</xref>] . The results are expressed in eV. The energy uncertainties in the present calculations are indicated into parenthesis

	E (eV)	δ
n	MAOT	ALS	\|ΔE\|	MAOT	ALS
3	38.686(200)	38.686	0.000	0.391	0.391
4	42.409(200)	42.409	0.000	0.430	0.430
5	44.075(174)	44.072	0.003	0.429	0.430
6	44.920(140)	44.919	0.001	0.429	0.438
7	45.101(111)			0.428
8	45.511(76)			0.428
9	45.825(65)			0.428
10	46.000(49)			0.428
11	46.117(37)			0.427
12	46.241(25)			0.427
13	46.300(20)			0.427
14	46.338(18)			0.426
15	46.392(13)			0.426
16	46.430(8)			0.426
17 18 19 20 ⁞	46.466(6) 46.500(5) 46.528(4) 46.551(2)			0.428 0.428 0.428 0.428
∞^a	46.620

^aNIST atomic database [18] . |ΔE|: energy differences relative to the experimental data.

Table 7 Energy resonances (E) and quantum defect (δ) of the 4S02s22p3(5S0)np(4P) Rydberg series observed in the photoionization spectra originating from the 4S0 metastable states of O+. The present results (MAOT) are compared to the Advanced Light Source (ALS) of Aguilar et al. [<xref ref-type="bibr" rid="scirp.117688-ref2">2</xref>] . The results are expressed in eV. The energy uncertainties in the present calculations are indicated into parenthesis

	E (eV)	δ
n	MAOT	ALS	\|ΔE\|	MAOT	ALS
4	38.216(200)	38.216	0.000	0.477	0.477
5	39.919(200)	39.919	0.000	0.495	0.495
6	40.776(190)	40.773	0.003	0.540	0.543
7	41.209(168)	41.206	0.003	0.539	0.753
8	41.676(147)	41.674	0.002	0.538	0.336
9	41.879(122)	41.878	0.001	0.538	0.321

10	42.022(110)	42.021	0.001	0.537	0.309
11	42.122(99)	42.121	0.001	0.537	0.346
12	42.168(81)	42.167	0.001	0.537	0.795
13	42.196(66)			0.536
14	42.225(53)			0.536
15	42.251(40)			0.536
16	42.273(29)			0.538
17	42.294(21)			0.538
18 19 20 ⁞	42.317(17) 42.329(14) 42.340(12)			0.538 0.538 0.537
∞^a	42.586

^aNIST atomic database [18] . |ΔE|: energy differences relative to the experimental data.

are converted into eV for direct comparison by using the infinite Rydberg (1 Ry = 0.5 a.u = 13.605698 eV). It is seen that the data obtained compared very well to the experimental data of Aguilar et al. [2] . Up to n = 12, the maximum energy differences relative to the experimental data is less than 0.004 eV. For n = 12, our MAOT value at 42.168 eV agrees well with the experimental data of Aguilar at 42.167 eV. This may expect our results for n > 12 to be accurate.

This indicates the excellent agreements between the present calculations and the experimental measurements for energy positions.

4. Summary and Conclusion

In this paper, energy resonances of the 2D⁰_2s²2p²(¹D)nd(²F); 2D⁰_2s²2p²(¹D)nd(²D); 2D⁰_2s²2p²(¹D)nd(²P); 2D⁰_2s²2p²(¹D)ns(²D); 2D⁰_2s²2p³(³D)np(²P); 2D⁰_2s²2p³(³D⁰)np(²F), and 4S⁰_2s²2p³(⁵S⁰)np(⁴P) Rydberg series in the photoionization spectra originating from 2D⁰ and 4S⁰ metastable state of O⁺ ions are reported using the Modified Orbital Atomic Theory (MAOT). It has been demonstrated the simplicity of the MAOT semi-empirical procedure to calculate accurate values energies without having to resort to either a complex mathematical formalism nor to a specific calculation code via a tedious computer program. This work may be of interest for future experimental and theoretical studies in the photoabsorption spectrum of O⁺. The simplicity of the presented procedure, allows to obtain very accurate values of the resonance energies up to highly excited Rydberg states (n ≥ 20).

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.

Conflicts of Interest

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

Cite this paper

Sow, M., Ndoye, F., Diop, B., Traoré, A., Diouf, A. and Sow, B. (2022) Modified Atomic Orbital Theory of the O⁺ Ion Originating from 2D⁰ and 4S⁰ Metastable States. Journal of Applied Mathematics and Physics, 10, 1873-1886. https://doi.org/10.4236/jamp.2022.106128

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